Dark Energy Survey Year 1 Results: The Impact of Galaxy

DES 2017-0727

FERMILAB-PUB-17-291-E

MNRAS 000, 000–000 (0000) Preprint 7 August 2017 Compiled using MNRAS L

X style ﬁle v3.0

Neighbours on Weak Lensing Cosmology with IM3SHAPE

S. Samuroff

, S. L. Bridle

, J. Zuntz

, M. A. Troxel

3,4

, D. Gruen

5,6

†, R. P. Rollins

, G. M. Bernstein

T. F. Eiﬂer

, E. M. Huff

, T. Kacprzak

, E. Krause

, N. MacCrann

3,4

, F. B. Abdalla

10,11

, S. Allam

J. Annis

, K. Bechtol

, A. Benoit-L

evy

14,10,15

, E. Bertin

14,15

, D. Brooks

, E. Buckley-Geer

, A.

Carnero Rosell

16,17

, M. Carrasco Kind

18,19

, J. Carretero

, M. Crocce

, C. B. D’Andrea

, L. N. da

Costa

16,17

, C. Davis

, S. Desai

, P. Doel

, A. Fausti Neto

, B. Flaugher

, P. Fosalba

, J. Frieman

12,23

J. Garc

ıa-Bellido

, D. W. Gerdes

25,26

, R. A. Gruendl

18,19

, J. Gschwend

16,17

, G. Gutierrez

K. Honscheid

3,4

, D. J. James

27,28

, M. Jarvis

, T. Jeltema

, D. Kirk

, K. Kuehn

, S. Kuhlmann

T. S. Li

, M. Lima

32,16

, M. A. G. Maia

16,17

, M. March

, J. L. Marshall

, P. Martini

3,34

, P. Melchior

F. Menanteau

18,19

, R. Miquel

36,20

, B. Nord

, R. L. C. Ogando

16,17

, A. A. Plazas

, A. Roodman

5,6

E. Sanchez

, V. Scarpine

, R. Schindler

, M. Schubnell

, I. Sevilla-Noarbe

, E. Sheldon

, M. Smith

M. Soares-Santos

, F. Sobreira

40,16

, E. Suchyta

, G. Tarle

, D. Thomas

, D. L. Tucker

(DES Collaboration)

Author afﬁliations are listed at the end of the paper.

7 August 2017

ABSTRACT

We use a suite of simulated images based on Year 1 of the Dark Energy Survey to explore the

impact of galaxy neighbours on shape measurement and shear cosmology. The HOOPOE im-

age simulations include realistic blending, galaxy positions, and spatial variations in depth

and PSF properties. Using the IM3SHAPE maximum-likelihood shape measurement code,

we identify four mechanisms by which neighbours can have a non-negligible inﬂuence on

shear estimation. These effects, if ignored, would contribute a net multiplicative bias of

m ∼ 0.03 − 0.09 in the DES Y1 IM3SHAPE catalogue, though the precise impact will be

dependent on both the measurement code and the selection cuts applied. This can be reduced

to percentage level or less by removing objects with close neighbours, at a cost to the effective

number density of galaxies n

eﬀ

of 30%. We use the cosmological inference pipeline of DES

Y1 to explore the cosmological implications of neighbour bias and show that omitting blend-

ing from the calibration simulation for DES Y1 would bias the inferred clustering amplitude

≡ σ

(Ω

/0.3)

0.5

by 2σ towards low values. Finally, we use the HOOPOE simulations to

test the effect of neighbour-induced spatial correlations in the multiplicative bias. We ﬁnd the

impact on the recovered S

of ignoring such correlations to be subdominant to statistical error

at the current level of precision.

Key words: cosmological parameters - cosmology: observations - gravitational lensing: weak

- galaxies: statistics

1 INTRODUCTION

A standard and well tested prediction of General Relativity is that

a concentration of mass will distort the spacetime around it, and

thus produce a curious phenomenon called gravitational lensing.

[email protected]

† NASA Einstein Fellow

The most obvious manifestation is about massive galaxy clusters,

where background galaxies can be elongated into cresent-shaped

arcs. So-called strong lensing of galaxies was ﬁrst observed in the

late 1980s and has been conﬁrmed many times since. A subtler, but

from a cosmologist’s perspective more powerful, consequence of

gravitational lensing is that background ﬂuctuations in the density

of dark matter will induce coherent distortions to photons’ paths.

This effect is known as cosmic shear, and it was ﬁrst detected

 0000 The Authors

arXiv:1708.01534v1 [astro-ph.CO] 4 Aug 2017

2 S. Samuroff, S. L. Bridle, J. Zuntz et al

by four groups at around the same time close to two decades ago

(Bacon et al. 2000; Van Waerbeke et al. 2000; Kaiser et al. 2000;

Wittman et al. 2000). Cosmic shear has the potential to be the sin-

gle most powerful probe in the toolbox of modern cosmology. The

spatial correlations due to lensing are a direct imprint of the large

scale mass distribution of the Universe. Thus it allows one to study

the total mass of the Universe and the growth of structure within

it (Maoli et al. 2001; Jarvis et al. 2006; Massey et al. 2007; Kil-

binger et al. 2013; Heymans et al. 2013; Abbott et al. 2016; Jee

et al. 2016; Hildebrandt et al. 2017; K

ohlinger et al. 2017), or to

map out the spatial distribution of dark matter on the sky (eg Kaiser

1994; Van Waerbeke et al. 2013; Chang et al. 2015). As a probe of

both structure and geometry, cosmic shear is also attractive as a

method for shedding light on the as yet poorly understood compo-

nent of the Universe known as dark energy (Albrecht et al. 2006;

Weinberg et al. 2013). Alternatively, lensing will allow us to place

ever more stringent tests of our theories of gravity (Simpson et al.

2013; Harnois-D

eraps et al. 2015; Brouwer et al. 2017). Is also the-

oretically very clean, responding directly to the power spectrum of

dark matter, which is affected by baryonic physics only on small

scales, and avoids recourse to poorly-understood phenomenologi-

cal rules. Indeed galaxy number density enters only at second order

as a weighting of the observed shear due to the fact that one can

only sample the shear ﬁeld where there are real galaxies (Schmidt

et al. 2009).

Though well modelled theoretically, cosmic shear is techni-

cally highly challenging to measure; as with all these probes it is

not without its own sources of systematic error. It also cannot be

reiterated too many times that the shear component of even the

most distant galaxy’s shape is subdominant to noise by an order of

magnitude. Indeed, the ambitions of the current generation of cos-

mology surveys will require sub-percent level uncertainties (both

systematic and statistical) on what is already a tiny cosmological

ellipticity component g ∼ 0.01.

It was realised early on how signiﬁcant the task of translat-

ing photometric galaxy images into unbiased shear measurements

would be. In response came a series of blind shear measurement

challenges, designed to review, test and compare the best methods

available. The ﬁrst of these, called STEP1 (Heymans et al. 2006)

grew out of a discussion at the 225th IAU Symposium in 2004.

The exercise was based around a set of simple SKYMAKER sim-

ulations (Bertin & Fouqu

e 2010), which were designed to mimic

ground based observations but with analytic galaxies and PSFs and

constant shear. The algorithms at this point represented a ﬁrst wave

of shear measurement codes and included several moments-based

algorithms (Kaiser et al. 1995; Kuijken 1999; Rhodes et al. 2001),

some early forward modelling methods (Bridle et al. 2002), as well

as a technique called shapelets, which models a light proﬁle as a set

of 2D basis functions (Bernstein & Jarvis 2002; Refregier & Bacon

2003).

The simulations and the codes themselves steadily grew in

complexity. STEP2 was followed by series of GREAT challenges

(Massey et al. 2007; Bridle et al. 2009; Kitching et al. 2010; Man-

delbaum et al. 2014), which focused on different aspects of shape

measurement bias and have been essential in quantifying a num-

ber of signiﬁcant effects. In recent years the drive to ﬁnd ever

more accurate ways to measure shear has intensiﬁed, with many

novel approaches being suggested. For example Fenech Conti et al.

(2017) use a form of self-calibration, which repeats the shape mea-

surement on a test image based on the best-ﬁtting model for each

galaxy. A related approach, named metacalibration, involves de-

riving corrections to the galaxy shape measurements directly from

the data, using modiﬁed copies of the image with additional shear

(Huff & Mandelbaum 2017; Sheldon & Huff 2017). More advanced

moments-based approaches include the BFD method (Bernstein &

Armstrong 2014), which derives a prior on the ensemble elliptic-

ity distribution using deeper ﬁelds, and SNAPG (Herbonnet et al.

2017), a similar approach which builds ensemble shear estimates

using shear nulling.

This paper is intended as a companion study to Zuntz et al.

(2017) (Z17), where we present two shear catalogues derived from

DES Y1 dataset. It is also presented alongside a raft of other pa-

pers, which use both catalogues and show them to be consistent in

a number of different scientiﬁc contexts (Troxel et al. 2017; Prat

et al. 2017; Chang et al. 2017; DES Collaboration et al. 2017) Con-

taining 22 million and 35 million galaxies respectively, these cat-

alogues are the product of two independent maximum likelihood

codes. The ﬁrst, called IM3SHAPE, implements simultaneous ﬁts

using multiple models and we calibrate externally using simula-

tions. The second implements a Gaussian model ﬁtting algorithm,

supplemented by shear response corrections using METACALIBRA-

TION. Whereas in Z17 we focus on the catalogues themselves, pre-

senting a raft of calibration tests and a broad overview of the value-

added data products, here we use the same resources to explore

a narrower topic: the impact of image plane neighbours on shear

measurement. Speciﬁcally we use the image simulations described

in Z17, from which the Y1 IM3SHAPE calibration is derived, to ex-

plore the mechanisms for neighbour bias, and then propagate the re-

sults to mock shear two-point data to investigate the consequences

for weak lensing cosmology. The results presented in this paper will

be somewhat dependent on the choice of measurement algorithm,

selection cuts and the conﬁguration of the object detection code.

Unlike previous studies on this subject, however, we make use of a

highly realistic simulation and measurement pipeline. Our choices

on each of aspects are realistic, if not unique, for a leading-edge

cosmology analysis.

It is worth remarking, however, that the tests described in this

paper make use of IM3SHAPE only, and should not be assumed to

apply generically to its sister Y1 METACALIBRATION catalogue.

A complementary set of tests using METACALIBRATION are pre-

sented in §4.5 of Z17.

This paper is structured as follows. In Section 2 we brieﬂy re-

view the formalism of lensing, and the observables discussed in this

work. In Section 3 we present a series of numerical calculations us-

ing a toy model to characterise neighbour bias. Section 4 decribes

the simulated DES Y1 datasets, generated using our HOOPOE sim-

ulator. We test the earlier predictions under more typical observing

conditions in Section 5, and extend them into a quantitative set of

results using the more extensive Y1 HOOPOE dataset. Section 6

then presents a numerical analysis designed to test the cosmolog-

ical implications of neighbour bias of the nature and magnitude

found in our simulations. We conclude in Section 7.

2 THE SHEAR MEASUREMENT PROBLEM

The problem of shape measurement is far more intricate than it

might ﬁrst appear. Any cosmological analysis based on cosmic

shear is reliant on a series of technical choices, which can have

a non-trivial impact on measurement biases, precision and cosmo-

logical sensitivity. Speciﬁcally we must choose (a) how to parame-

terize each galaxy’s shape, and which measurement method to use

to estimate it, (b) what selection criteria are needed to obtain data

of sufﬁciently high quality for cosmology and (c) how biased is the

MNRAS 000, 000–000 (0000)

Cosmic Shear & Galaxy Neighbours 3

measurement and what correction is needed? These choices should

be made on a case-by-case basis, since the optimal solutions are

dependent on a number of survey-speciﬁc factors. We discuss each

brieﬂy in turn below.

2.1 Shape Measurement with IM3SHAPE

The shape measurements upon which the following analyses are

based make use of the maximum likelihood model ﬁtting code

IM3SHAPE

(Zuntz et al. 2013). It is a well tested and understood

algorithm, which has since been used in a range of lensing studies

(Abbott et al. 2016; Whittaker et al. 2015; Kacprzak et al. 2016;

Clampitt et al. 2017). It was also one of two codes used to produce

shear catalogues in the Science Veriﬁcation (SV) stage and Year

1 of the Dark Energy Survey. We refer the reader to Jarvis et al.

(2016) (hereafter J16) and Z17 for the most recent modiﬁcations to

the code.

We use the deﬁnition of the ﬂux signal-to-noise ratio of Z17,

J16 and Mandelbaum et al. (2015):

S/N ≡

pix

i=1

/σ

pix

i=1

/σ

. (1)

The indices i = (1, 2...N

pix

) run over all pixels in a stack of im-

age cutouts at the location of a galaxy detection. The model pre-

diction and observed ﬂux in pixel i are denoted f

and f

re-

spectively and σ

is the RMS noise. This signal-to-noise measure

is maximised when the differences between the model and the im-

age pixel ﬂuxes are small. Note that if the best-ﬁtting model f

is identical for two different postage stamps, S/N will favour the

image with the greater total ﬂux.

A useful size measure, referred to as R

is deﬁned as

the measured Full Width at Half Maximum (FWHM) of the galaxy

after PSF convolution, normalised to the PSF FWHM. Real galaxy

images are are not perfectly symmetric (i.e. size is not independent

of azimuthal angle about a galaxy’s centroid), and single-number

size estimates are obtained by circularising (azimuthally averag-

ing) the galaxy proﬁle and computing the weighted quadrupole mo-

ments of the resulting image. For each galaxy we take the mean

measured size across exposures.

2.2 Shear Measurement Bias

There are many ways bias can enter an ensemble shear estimate

based on a population of galaxies. Although the list is not exhaus-

tive, a handful of mechanisms are particularly prevalent, and have

been extensively discussed in the literature.

• Noise Bias: On addition of pixel noise to an image, the best-

ﬁtting parameters of a galaxy model will not scale linearly with

the noise variance. This is as an estimator bias as much as a mea-

surement bias, and results in an asymmetric, skewed likelihood sur-

face (Hirata & Seljak 2003; Refregier et al. 2012; Kacprzak et al.

2012; Miller et al. 2013). Any code which uses the point statis-

tics of the distribution (either mean or maximum likelihood) as a

single-number estimates of the ellipticity results in a bias. This is

https://bitbucket.org/joezuntz/im3shape-git

true even in the idealised case where the galaxy we are ﬁtting can

be perfectly decribed by our analytic light proﬁle. The bias is sen-

sitive to the noise levels and also the size and ﬂux of the galaxy,

and thus is speciﬁc to the survey and galaxy sample in question.

For likelihood-based estimates one solution would be to impose

a prior on the ellipticity distribution and propagate the full poste-

rior. However, the results can become dependent on the accuracy of

that prior, and such codes require cautious testing using simulations

(Bernstein & Armstrong 2014; Simon & Schneider 2016)

• Model Bias: In reality galaxies are not analytic light proﬁles

with clear symmetries. For the purposes of model-ﬁtting, however,

we are constrained to use models with a ﬁnite set of parameters.

A model which does not allow sufﬁcient ﬂexibility to capture the

range of morphological features seen in the images will produce

biased shape measurements (Lewis 2009; Voigt & Bridle 2010;

Kacprzak et al. 2014).

• Selection Bias: Even if we were to devise an ideal shape mea-

surement algorithm, capable of perfectly reconstructing the his-

togram of ellipticities in a certain population of galaxies, our at-

tempts to estimate the cosmological shear could still be biased. If

a measurement step prefers rounder objects or those with a partic-

ular orientation, the result would be a net alignment that could be

mistaken as having cosmological origin. In practice selection bias

commonly arises from imperfect correction of PSF asymmetries

(eg Kaiser et al. 2000; Bernstein & Jarvis 2002), and the fact that

many detection algorithms fail less frequently on rounder galaxies

(Hirata & Seljak 2004). It is such effects that make post facto qual-

ity cuts on quantities such as signal-to-noise or size (both of which

correlate with ellipticity) particularly delicate.

• Neighbour bias: In practice, galaxies in photometric surveys

like DES are not ideal isolated objects. Rather, they are extracted

from a crowded image plane using imperfect deblending algo-

rithms. The term “neighbour bias” refers to any biases in the re-

covered shear arising from the interaction between galaxies in the

image plane. This can include both the direct impact on the per-

galaxy shapes (e.g. Hoekstra et al. 2017) and changes in the selec-

tion function (e.g. Hartlap et al. 2011). Neighbour bias is the subject

of relatively few previous studies, and is the focus of this paper.

3 TOY MODEL PREDICTIONS

To develop a picture of how image plane neighbours affect shear

estimates with IM3SHAPE, we build a simpliﬁed toy model. Using

GALSIM

we generate a 48 × 48 pixel postage stamp containing a

single exponential disc proﬁle convolved with a tiny spherically

symmetric PSF (though we conﬁrm that our results are insensi-

tive to the exact size of the PSF). We can then apply a small shear

along one coordinate axis prior to convolution and use IM3SHAPE

to ﬁt the resulting image. In the absence of noise or model bias

the maximum of the likelihood of the measured parameters coin-

cides exactly with the input values. The basic setup then has four

adjustable parameters: the ﬂux and size of the galaxy plus two el-

lipticity components, denoted f

, r

, g

and g

. Unless otherwise

stated we ﬁx these to the median values measured from the DES Y1

IM3SHAPE catalogue. We do not model miscentering error between

the true galaxy centroid and the stamp centre.

It is worth noting that neither this basic model nor the more

https://github.com/GalSim-developers/GalSim

MNRAS 000, 000–000 (0000)

4 S. Samuroff, S. L. Bridle, J. Zuntz et al

complex simulations that follow attempt to model spatial correla-

tions in shear. Even at different redshifts, a real neighbour-central

pair share some portion of their line of sight. These spatial cor-

relations will amplify the impact of blending, and are worthy of

future investigation. This is, however, likely a second-order effect

of neighbours, and we postpone such study to a future date.

3.1 Single-Galaxy Effects

To explore the interaction in single neighbour-galaxy instances we

introduce a second galaxy into the postage stamp, convolved with

the same nominal PSF. This adds four more model parameters:

neighbour size r

, ﬂux f

, radial distance from the stamp centre

and azimuthal rotation angle relative to the x coordinate axis

θ. At this stage the neighbour has zero ellipticity.

We show this setup at three neighbour positions in Fig. 1. Un-

der zero shear, the system has perfect rotational symmetry, and the

measured ellipticity magnitude ˜g(θ|g

= 0) is independent of θ

As a ﬁrst exercise, we generate a circular central galaxy with a cir-

cular Gaussian neighbour, which is gradually shifted outwards from

the stamp centre. Following the usual convention for galaxy-galaxy

lensing, tangential shear is deﬁned such that it is negative when

the major axis of the measured shape is oriented radially towards

the neighbour. The measured two-component ellipticity shown by

the solid and dot-dashed lines in Fig. 2. The decline in the mea-

sured tangential shear to zero at small separations is understand-

able, as there is no reason to expect drawing one circular proﬁle

directly atop another should induce spurious non-zero ellipticity. In

the regime of a few pixels, however, strong blending can increase

the ﬂux of the best-ﬁtting model.

Next, we repeat the calculation, now applying a moderate

cross-component shear to the neighbour (g

= 0.1). The result

is shown by the blue lines in Fig. 3. Unsurprisingly the measured

tangential shear is unaffected by a true shear along an orthogonal

axis. In cases where the objects share a large portion of their half-

light radii, we are ﬁtting a strongly blended pair with a single pro-

ﬁle, and the neighbour/central distinction becomes difﬁcult to de-

ﬁne. The best-ﬁtting ellipticity recovered from the blended image

is not a pure measurement of either galaxy’s shape; rather it is a

linear combination of the two. We repeat the zero-offset measure-

ment using a range of neighbour ﬂuxes and ﬁnd that the best-ﬁtting

follows roughly as a ﬂux-weighted sum over the two galaxies

˜g

≈ (f

i,c

+ f

i,n

)/(f

+ f

3.2 Ensemble Biases

While useful for understanding what follows, the impact of neigh-

bours on individual galaxy instances is not particularly informative

about the impact on cosmic shear measurements. Even signiﬁcant

bias in the per-object shapes could average away over many galax-

ies with no residual impact on the recovered shear. More impor-

tant is the collective response to neighbours. To explore this we

build on the toy model concept. To estimate the ensemble effect,

we measure a neighbour-central image at 70 positions on a ring

of neighbour angles. Again, under zero shear g

= 0 the mea-

sured shape is constant in magnitude, and simply oscillates about

0 with peaks of amplitude |˜g(θ|0, d

)|. This sinusoidal variation

is shown by the dotted lines in Fig. 3b at two values of d

(7 and

Unless otherwise stated we ﬁx the other model parameters to their ﬁdu-

cial values.

8 pixels). By averaging over a (large) number of neighbours one is

effectively marginalising over θ, which results in an unbiased mea-

surement of the shear h˜g(θ|g

= 0, d

= g

= 0. A non-

zero shear g

6= 0, however breaks the symmetry of the system. A

galaxy sheared along one axis will not respond to a neighbour in the

same way irrespective of θ, which can result in a net bias. To show

this we ﬁx g

= −0.05 and proceed as before. The solid lines in

Fig. 3b show the periodicity in the measured shear at two d

. The

mean value averaged over θ is shifted incrementally away from the

input shear, shown by the horizontal dot-dashed line. Speciﬁcally

we should note that the peaks below g

at π/2 and 3π/2 radi-

ans are deeper and narrower than those above it. The cartoon in

Fig. 4 shows how this arises. The purple lines are iso-light con-

tours in a strongly sheared S

ersic disc proﬁle (g

= −0.3). Clearly

rotating the neighbour from position A to C carries it from the rel-

atively ﬂat low wings of the central galaxy’s light proﬁle closer to

the core. Perturbing an object about C by a small angle results in

a much greater change in the local gradient, 5f

(x, y) than doing

the same about A. All other parameters ﬁxed, an incremental shift

along the blue tangent vector will have a larger impact at θ = 0

than at π/2, resulting in asymmetry in the width of the positive and

negative peaks in Fig. 4. The depth of the peak can be explained

qualititatively by similar arguments. At C a neighbour of given ﬂux

is closer to the centre of the light distribution and thus has a greater

ﬂux overlap with the central galaxy than at A. Naturally, then, one

might expect neighbour A to have less impact than C. Returning to

Fig. 3, we can see that the two effects are in competition. Depend-

ing on the exact neighbour conﬁguration, the simultaneous narrow-

ing and deepening the negative peaks can result in a bias in the

neighbour-averaged ellipticity towards large or small values.

The level of this effect will clearly correlate with the magni-

tude of the shear, and so induce a multiplicative bias. To illustrate

this point the above exercise is repeated with a range of different

input shears. The results for our ﬁducial setup are shown in in Fig.

5. Each point on these axes corresponds to a ring of neighbour po-

sitions for a given input shear. The equivalent measurements with-

out the neighbour are indistinguishable from the x axis. At small

shears, the neighbour induced bias ˜g − g

is well aproximated

as a linear in g

. We leave exploration of the possible nonlinear

response at large ellipticities for future investigations. Though the

above numerical exercise demonstrates that it is possible for signif-

icant multiplicative bias to arise as a result of neighbours, it does

not make a clear prediction of the magnitude or even the sign. In-

deed, our toy model is effectively marginalised over θ, but there is

nothing to guarantee that ﬁxing the other neighbour parameters to

the median measured values is representative of the real level of

neighbour bias in a survey like DES. Motivated by this observa-

tion we add a ﬁnal layer of complexity to the model, as follows.

A single neighbour-central realisation is created as before, deﬁned

by a unique set of model parameters. Now, however, the values

of those parameters p = (d

, f

, r

, f

, r

) are drawn randomly

from the DES data. As these quantities will, in reality, be corre-

lated we sample from the 5-dimensional joint distribution rather

than each 1D histogram individually. We then ﬁt the model at 70

neighbour angles and two input shears g

= ±0.05 (a total of 140

measurements), and estimate the multiplicative bias as a two-point

ﬁnite-difference derivative:

m + 1 =

h˜g(θ|g

− h˜g(θ|g

−

− g

−

. (2)

This process is repeated to create 1.33M unique toy model reali-

MNRAS 000, 000–000 (0000)

Cosmic Shear & Galaxy Neighbours 5

Figure 1. Postage stamp snapshots of the basic two-object toy model described in Section 3. The overlain ellipse shows the maximum likelihood ﬁt to the

image. The panels show three neighbour positions in the range θ = [0, π/2] rad. The best ﬁt ellipticity and half light radius are shown above each image. In

all cases the input values are e = (0, 0), r = 0.5 arcseconds.

0 5 10 15 20 25 30

Neighbour Distance d

/ pixels

0.4

0.3

0.2

0.1

0.0

Tangential or Cross Shear

(round neighbour)

2,n

=0.1)

2,n

=0.1)

Figure 2. Tangential shear measured using the numerical toy model de-

scribed in Section 3.1 as a function of radial neighbour distance. The solid

purple line shows the shape component aligned with the central-neighbour

separation vector and the dot-dashed line is measured along axes rotated

through 45

◦

. Note that the latter is smaller than 10

−6

at all points on this

scale. The dashed and dotted black lines show the same ellipticity compo-

nents when the neighbour is sheared in the e

direction by g

= 0.1.

sations. Binning by neighbour distance we can then make a rough

prediction for the level of neighbour-induced bias and the angular

scales over which it should act. The result is shown in Fig. 6, where

full results using all model realisations are indicated by the dashed

blue line. The majority of cases yield a negative bias, particularly

at low neighbour separation (referring back to Fig. 4, the broad-

ening of the peak around position A dominates over the increased

ﬂux overlap at C). In the real data, of course, we apply a quality

based selection and

uberseg object masking (J16), both of which

are neglected here. We can, however, test the impact of selecting

on ﬁtted quantities that respond to neighbour bias. Imposing a ﬂat

prior on the centroid offset ∆r

= (x

+ y

)

(i.e. discarding

randomly generated model realisations where the galaxy centroid

is displaced from the stamp centre by more than a ﬁxed number of

pixels) changes the shape of this curve signiﬁcantly, as illustrated

by the thick purple line.

We can understand the difference between the results with and

without the centroid cut as a form of selection bias, whereby the cut

preferentially removes toy model realisations in which the neigh-

bour is bright relative to the central galaxy. At any given d

we are

left with a relative overrepresentation of galaxies with f

 1.

Figure 3. Best-ﬁt galaxy ellipticity as a function of neighbour position angle

at ﬁxed neighbour distance d

from the toy model described in the text.

The two panels (left, right) show the same central-neighbour system (g

−0.05), but with different d

(7 and 8 pixels) and biases m (shown atop

each panel). The solid line in each case is the recovered galaxy shape at each

θ, and the integrated mean along this range is shown by the horizontal dot-

dashed line. The dotted lines show the zero-shear shape (ie. the ellipticity

that would be measured if the input shear were zero), but shifted downwards

such that the mean is at −0.05. Finally, to illustrate the (a)symmetry of the

system we show the solid line ﬂipped about y = g

and shifted by π/2

radians as a dashed curve.

Faint neighbours, which in reality tend to be compact high redshift

objects, have little impact when they sit on the outskirts of the cen-

tral proﬁle (A in the cartoon picture in Fig. 4; the regime which

produces negative m). The same faint galaxy has a stronger im-

pact if it is rotated to a position closer to the centre of the central’s

ﬂux proﬁle. Thus one might expect a selection on ∆r

to make

the mean m in a particular bin less negative (or even positive) by

preferentially removing brighter galaxies.

4 HOOPOE IMAGE SIMULATIONS

In this section we provide a brief overview of the simulation

pipeline. The process is the same as that described in §4 of Z17,

and we refer the reader to that work for more detail. The end point

of the pipeline is a cloned set of survey images with many of the

observable characteristics of a chosen set of parent images, but for

which we know the input noise properties and galaxy population

pefectly. The simulated images inherit the pixel masking, PSF vari-

MNRAS 000, 000–000 (0000)

6 S. Samuroff, S. L. Bridle, J. Zuntz et al

Figure 4. Cartoon diagram of a neighbour-central system. The purple con-

tours show the lines of constant ﬂux in a S

ersic disc proﬁle with extreme

negative ellipticity (g

= −0.3). The blue crosses labelled A, B and C are

points on a ring of equal distance from the centre of the proﬁle. The blue

arrows show the local unit vector along a tangent to the ring.

Figure 5. Measured shear minus input shear plotted as a function of input

shear. The purple points show the recovered ˜g

from averaging over ring

of 70 neighbour positions. The dark blue lines show the linear relation ˜g −

= mg

at m = (−0.4, −0.45, −0.5). The dotted line shows what

would be measured using the same central proﬁle in the absence of the

neighbour, and is near indistinguishable from the x axis line on all points

within this range of g

ation and noise maps measured from the progenitor data. Each sim-

ulated galaxy is then inserted into a subset of overlapping exposures

and into the coadd at the position of a real detection in the DES Y1

data. Object detection is rerun on the new coadd images and galaxy

cutouts and new segmentation masks are extracted and stored in the

MEDS format decribed by J16. The mock survey footprint is shown

in Fig. 7. In the lower panels we show an example of a simulated

coadd (left) and the spatial variation in PSF orientation within the

same image (right).

4.1 Parent Data

We use reduced images from Year One of the Dark Energy Survey

(DES Y1; Diehl et al. 2014) as input to the simulations discussed

in this paper. The Dark Energy Survey is undertaking a ﬁve year

0 5 10 15 20 25 30 35 40 45

Neighbour Distance d

/ pixels

1.0

0.8

0.6

0.4

0.2

0.0

Multiplicative Bias m

No r

Cut

r

< 4 pixels

Figure 6. Multiplicative bias estimated using the Monte Carlo toy model

described in the text. For each neighbour realisation, deﬁned by a particular

distance, ﬂux and size we compute the average of the measured ellipticity

components over 70 rotations on a ring of neighbour angles. To estimate the

bias we perform this averaging twice at two non-zero shears, g

and g

−

and compute the ﬁnite-difference deriviative using equation 2. The dashed

thin blue line shows the result of using all measurements, while the bold

purple line has a cut based on the offset between the centroid position of the

best-ﬁtting model and the stamp centre.

programme with the ultimate aim of observing ∼ 5000 square de-

grees of the southern sky to ∼ 24th magnitude in ﬁve optical bands,

grizY, covering 0.40−1.06 microns. The dataset is recorded using a

570 megapixel camera called DECam (Flaugher et al. 2015), which

has a pixel size of 0.26 arcseconds. In full it will consist of ∼ 10

interwoven sets of exposures in the g, r, i, z and Y bands.

The Y1 data were collected between August 2013 and Febru-

ary 2014, and cover a substantially larger footprint than the prelim-

inary Science Veriﬁcation (SV) stage at 1500 square degrees, albeit

to a reduced depth. Details of the reduction and processing are pre-

sented in Z17. Our HOOPOE simulations use a selection of the total

3000 0.75 × 0.75 degree coadded patches known as “tiles”.

4.2 Input Galaxy Selection

For populating the mock survey images a sample of real galaxy pro-

ﬁles from the HST COSMOS ﬁeld, imaged at signiﬁcantly lower

noise and higher resolution than DES by the Hubble Space Tele-

scope Advanced Camera for Surveys (HST ACS) (Scoville et al.

2007). The COSMOS catalogue extends signiﬁcantly deeper than

the Y1 detection limit of M

r,lim

= 24.1, extending to roughly 27.9

mag in the SDSS r-band. A main sample for our DES Y1 simula-

tions is deﬁned by imposing a cut at < 24.1 mag.

Since the DES images do not cut off abruptly at 24th magni-

tude, in reality they contain a tail of fainter galaxies that contribute

ﬂux are not identiﬁable above the pixel noise. To assess the impact

of these objects on shape measurements in Y1, we simulate a pop-

ulation of sub-detection galaxies in addition to the main sample.

In brief we use the full histogram of COSMOS magnitudes to esti-

mate the number of faint galaxies within a given tile. The required

proﬁles are selected randomly from the faint end of the COSMOS

distribution. Each undetected galaxy is paired with a detection, and

inserted at a random location within the overlapping bounds of the

MNRAS 000, 000–000 (0000)

Cosmic Shear & Galaxy Neighbours 7



350



50



40



30



2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0

/arcmin

2



320



330



340



350



(a) (b)

Figure 7. Top: The projected footprint of the simulated survey, visualised using the SKYMAPPER package

. The colour indicates the local raw number density

in HEALPIX cells of nside = 1024. The axes shown are right ascension and declination in units of degrees. The full simulation comprises 1824 0.73 × 0.73

degree tiles drawn randomly from the DES Y1 area. The solid blue line indicates the bounds of the planned area to be covered by the complete Y5 dataset.

Bottom: A random tile (DES0246-4123) selected from the HOOPOE area. The left panel (a) shows a square subregion of approximately 4 × 4 arcminutes.

The right hand panel (b) shows a PSF whisker plot covering the full 0.73 × 0.73 tile. The length and orientation of each line represents the magnitude and

position angle of the spin-2 PSF ellipticity at that position. Only galaxies which pass IM3SHAPE quality cuts are shown. The white patches show the spatial

masking inherited from the GOLD catalogue, and correspond to the positions of bright stars in the parent data.

https://github.com/pmelchior/skymapper

same (subset of) single-exposure images. A more detailed descrip-

tion of this process can be found in Z17.

If these galaxies were present in the data they would enter

the background ﬂux calculation, and thus the subtraction applied

would change due to their presence. Since the simulation pipeline

produces images effectively in a post-background subtraction state

this is not captured. To test this we rerun the SEXTRACTOR back-

ground calculation on a handful of tiles drawn with and without

the faint galaxies. The impact was found to be well approximated

as a uniform shift in the background correction. A ﬂux correction

equal to the pixel-averaged ﬂux of the sub-detection galaxies over

each image plane is, then, applied to postage stamps prior to shape

measurement.

In reality the overdensity of sub-threshold galaxies will be

coupled to the density of detectable objects, which is clearly not

the case in our simulations. To gauge the impact of this we per-

form the following test. Each tile is divided into a 6 × 6 grid, and

the mean multiplicative bias is calculated in each sub-patch. We

bin sub-patches according to the ratio f

faint

≡ N

faint

det

, or the

total number of faint galaxies relative to the number of detectable

ones. The impact is signiﬁcant, but not leading order; excluding

patches outside the range 0.9 < f

faint

< 1.1 induces a shift of

∆m ∼ −0.005.

An independent noise realisation is generated for each expo-

sure using the weight map from the parent data. We simulate the

noise in each pixel by drawing from a Gaussian of corresponding

width. The coaddition process is not rerun, but rather we compute

an independent noise ﬁeld by drawing the ﬂux in each pixel from a

MNRAS 000, 000–000 (0000)

8 S. Samuroff, S. L. Bridle, J. Zuntz et al

Figure 8. An example of an object in the main DES Y1 calibration simula-

tion and the neighbour-free resimulation. The upper panels show the coadd

cutout in the original simulated images (left, labelled HOOPOE) and in the

neighbour-subtracted version (right, labelled WAXWING). The lower panels

are the segmentation masks for the same galaxy. A number of neighbours,

both masked (upper left and centre left) and unmasked (lower right) are

visible within the stamp bounds.

zero-centred Gaussian of width determined by the measured vari-

ance in that pixel.

4.3 Neighbour-Free Resimulations

For the purpose of untangling the impact of image plane neighbours

we use the simulated HOOPOE images to create a new spin-off

dataset. In a subset of a little over 500 tiles we store the (convolved)

input proﬁle for each object and the noise-only cutout, taken from

the same position in the image plane prior to objects being drawn.

By adding together these two components we can generate a suite

of spin-off MEDS ﬁles, which are equivalent to the results of a

simpler neighbour-free simulation (eg Miller et al. 2013, J16). The

pixel noise realisation, COSMOS selection and input shears, how-

ever, are identical to the progenitor HOOPOE simulations.

We will call this process “resimulating”, and the basic concept

is illustrated in Fig. 8. The 506-tile set of neighbour-free data are

named the WAXWING resimulations. Finally the (now empty) seg-

mentation masks corresponding to the subtracted neighbours are

also removed. In subsequent IM3SHAPE runs on these data we ig-

nore the SEXTRACTOR ﬂags obtained from the main simulations.

5 QUANTIFYING NEIGHBOUR BIAS WITH HOOPOE

Equipped with qualitative predictions from Section 3, we now turn

to the question of neighbour bias in the more complete simulations

described in Section 4. The mock survey was designed to capture

as much of the complexity of shape measurements on real photo-

metric data as possible. We refer to Section 4 of this paper for a

short overview and to §5 of Z17 for a more detailed discussion of

the simulation pipeline and validation tests. The simulated galaxy

catalogue used in the following is identical to the one used to cali-

brate the DES Y1 IM3SHAPE catalogue, including quality cuts and

selection masks.

5.1 Single-Galaxy Effects

The most straightforward way to assess the impact of neighbours

on individual shape measurements in our simulations is to rotate

the measured shapes into a frame deﬁned by the central-neighbour

separation vector. Whereas in the earlier toy model we had only one

neighbour per galaxy, we now have a crowded image plane contain-

ing many objects simultaneously. For simplicity, in the earlier case

we included no masking. For HOOPOE we wish to mimic the pro-

cess of shape measurement on real data as closely as possible. We

generate new segmentation maps by running SEXTRACTOR on the

simulated images, and incorporate them into our shape measure-

ments using the

uberseg algorithm (J16). Each simulated galaxy is

allocated a nearest neighbour using a k-d tree matching algorithm

constructed on the coadd pixel grid using every galaxy simulated at

r-band magnitude M

< 24.1. The quantities d

and θ are now

redeﬁned slightly as nearest-neighbour distance and angle. We de-

ﬁne the tangential shear of a galaxy relative to its nearest neighbour

using the standard convention,

= − [e

cos(θ) + e

sin(θ)] , (3)

and the cross shear

= − [e

cos(θ) − e

sin(θ)] . (4)

Note that negative values of e

imply a net tangential alignment

of the measured shapes towards neighbours. By analogy, we de-

ﬁne e

1,n

and e

2,n

, which are the measured ellipticity components,

rotated into a reference frame deﬁned by the major axis of the

neighbour. Non-zero e

i,n

would indicate leakage of the neighbour’s

shape into the measurement, which might conceivably be induced

by inadequate deblending of very close neighbours or by extensive

non-circular masking. We ﬁrst divide the main simulated catalogue

into bins according to d

, and measure the tangential shear about

nearest neighbours in each bin. The result is shown by the pur-

ple curve in Fig. 9. Note that the statistical uncertainty is within

the width of the line in all bins. The results here show qualitative

agreement with the numerical predictions in Fig. 3. As we found

earlier, the exact shape of this curve is sensitive to the properties

of both the neighbour and the central galaxy. Despite small dif-

ferences, the range of variation is comfortably within the scale of

the postage stamp for the bulk of galaxies in DES Y1. Repeating

the measurement, rotated into the plane of the neighbour shape re-

sults in the dotted and dot-dash lines in this ﬁgure. As noted above,

there are not necessarily reliable ellipticity measurements for each

neighbour, so we instead use the sheared input ellipticities. Both

components of e

i,n

are seen to be negligible over all scales.

5.2 Neighbour Ensemble Biases

To explore the more practical question of how neighbours impact

shear estimates we divide the catalogue into bins according to

neighbour distance. Within each d

bin, the galaxies are further

split into twelve bins of input shear, which are ﬁtted to estimate the

multiplicative and additive bias. We show the result as the purple

MNRAS 000, 000–000 (0000)

Cosmic Shear & Galaxy Neighbours 9

0 5 10 15 20 25 30

Nearest Neighbour Distance d

/ pixels

−0.35

−0.30

−0.25

−0.20

−0.15

−0.10

−0.05

0.00

0.05

Tangential or Cross Shear

Tangential Shear e

Cross Shear e

Tangential Shear e

+,n

Cross Shear e

×,n

−0.004

−0.002

0.000

0.002

Figure 9. Tangential shear around image plane neighbours in the full

HOOPOE simulation. The purple solid line shows the mean component of

the measured galaxy shapes radial to the nearest image plane neighbour.

Dashed blue shows the component rotated by 45

◦

, which we have no rea-

son to expect should be non-zero. The dotted and dot-dash lines show the

measured ellipticity components when rotated into a coordinate frame de-

ﬁned by the major axis of the neighbour. The inset shows the same range

in d

(the x-axis tick markers are the same), but with a magniﬁed vertical

axis.

points in Fig. 10, which can be compared with the earlier numerical

model prediction in Fig. 6. The horizontal band on these axes shows

the 1σ mean m measured using all galaxies in the HOOPOE cat-

alogue, and sits at m ∼ −0.12. We note a steeper decline than in

the bold line (without the centroid cut), more akin to the case with

the centroid cut (∆r

< 1 arcsec). This is not surprising given

that the quality selection implemented by IM3SHAPE includes ex-

actly this cut. We do not report a local peak at ∼ 11 pixels, which

we saw before in Fig. 6. We suggested previously that effect was

the result of positive m in galaxies where the nearest neighbour is

relatively faint and at middling distance. It is likely that many of

these objects manifest themselves as large changes in other quan-

tities to which IM3SHAPE’s INFO FLAG (see Z17) is sensitive such

as ellipticity magnitude and ﬁt likelihood, or are ﬂagged by the

SEXTRACTOR deblending cuts.

When divided into broad bins according to the r-band magni-

tude of the nearest neighbour M

r,neigh

(the coloured stripes in Fig.

10) we ﬁnd the surviving objects show relatively weak dependence

on neighbour brightness, except at the neighbour distances, where

bright neighbours have a slightly stronger (negative) impact than

faint ones.

We measure the additive bias components in the same bins,

but ﬁnd no systematic variation with d

above noise.

Finally we show the analogous measurement in bins of galaxy

magnitude in Fig. 11. The steep inﬂation of |m| at the faint end

of this plot has been seen elsewhere (e.g. Zuntz et al. 2017; Fenech

Conti et al. 2017), and is easily understandable as the result of noise

bias. We ﬁnd that splitting by neighbour magnitude does not reveal

any obvious trend here.

5.3 Untangling the Knot of Neighbour Bias

A central plank of this analysis rests on a comparison of the main

HOOPOE simulations with the neighbour-free WAXWING resimu-

0 5 10 15 20 25 30

Neighbour Distance d

/ pixels

−0.8

−0.6

−0.4

−0.2

0.0

Multiplicative Bias m

r,neigh

= 21.5

r,neigh

= 23.0

r,neigh

= 23.6

r,neigh

= 24.0

all galaxies

Figure 10. Multiplicative bias as a function of separation from the nearest

image plane neighbour. The purple points show the bias calculated in bins

of neighbour distance using the main HOOPOE simulated shape catalogue.

The coloured bands show the same dataset divided into four equal-number

bins according to the r-band magnitude of the neighbour. As shown in the

legend, the median values in the four bins are 21.5, 23.0, 23.5 and 24.0.

The mean bias and its uncertainty across all distance bins is indicated by

the horizontal band.

20.5 21.0 21.5 22.0 22.5 23.0 23.5 24.0

r-band Magnitude

−0.8

−0.6

−0.4

−0.2

0.0

Multiplicative Bias m

r,neigh

= 21.5

r,neigh

= 23.0

r,neigh

= 23.6

r,neigh

= 24.0

all galaxies

Figure 11. Multiplicative bias as a function of r-band magnitude. As in Fig.

10 the four coloured bands represent equal number bins of neighbour mag-

nitude. Purple points show the full catalogue, with no magnitude binning.

The mean bias and its uncertainty are shown by the purple horizontal band.

lations described in Section 4.3. The simplest comparison would

be between multiplicative bias values, calculated using all galax-

ies in each catalogue after cuts. These values are shown by the two

upper-most lines (purple) in Fig. 12. The difference is an indicator

of the net impact of neighbours through any mechanism, which we

ﬁnd to be ∆m ∼ −0.05.

To untangle the various contributions to this shift, we con-

struct a matched catalogue. Galaxies in the overlap between

HOOPOE and WAXWING (12M galaxies over 183 square degrees)

are matched by ID; quality cuts are calculated for each set of mea-

surments (see Appendix E from Z17). Geometric masking from the

DES Y1 GOLD catalogue (Drlica-Wagner et al. 2017) and SEX-

MNRAS 000, 000–000 (0000)

10 S. Samuroff, S. L. Bridle, J. Zuntz et al

Figure 12. Graphical illustration of the measured multiplicative bias in the various scenarios considered in this paper. The upper two lines show the mean m in

the main DES Y1 HOOPOE simulations and a spin-off neighbour-free resimulation named WAXWING, as described in Section 4.3. The middle section (green)

shows results using only galaxies which appear in both the HOOPOE and WAXWING simulations. The matching process alone does not imply any quality-

based selection function. The ﬁnal three lines in red are from a similar matching between a smaller rerun of the simulation with and without sub-detection

limit galaxies. See the text for details about each of these cases.

TRACTOR deblending ﬂags are included for HOOPOE. Since the

latter are irrelevant to WAXWING, we omit them from quality ﬂags

on that dataset. For conciseness we will refer to the two measure-

ments as “matched HOOPOE” and “matched WAXWING”, and their

cuts as “HOOPOE cuts” and “WAXWING cuts”. Since the images

are identical in all respects, but for the presence of neighbours, the

statistical noise on the change in measured quantities should be

smaller than the face-value uncertainties.

The appropriate cuts are ﬁrst applied to each catalogue, then

the results are divided into equal number signal-to-noise bins and

ﬁtted for the multiplicative bias in each. The result is shown by

the points in the upper left-hand part of Fig. 13. The equivalent in

bins of PSF-normalised size is shown on the right. The difference

between the blue and the purple points gives an indication of the

total effect of all neighbour-induced effects on m, indicated by the

solid purple line in the lower panel. The generic shift attributed to

“neighbour bias” is in reality a collection of distinct effects. By

comparing the matched catalogues we identify four main mecha-

nisms: direct contamination, selection bias, S/N bin shifting and

neighbour dilution. Each of these components that we describe is

shown by one of the lines in Fig. 13. For a visual summary of the

various tests designed to isolate them see Fig. 12.

5.3.1 Direct Flux Contamination

The most intuitive form of neighbour bias arises from the fact

that, even after masking, neighbours contribute some ﬂux to the

cutout image of a galaxy. To gauge its impact we take the com-

mon sample of galaxies, which pass cuts in both datasets. The com-

parison is complicated somewhat by binning in measured S/N or

/Rp; for this test, we divide both sets of galaxies using the

WAXWING-derived quantities. The resulting m measured using the

HOOPOE galaxies is unrealistic in the sense that we are binning

measurements made in the presence of neighbours by quantities de-

rived from neighbour-subtracted images. This exercise does, how-

ever, isolate the impact of the neighbour ﬂux on the measured el-

lipticity. The result is shown by the purple dotted and purple dot-

dashed lines in the upper and lower panels of Fig. 13. The effect

scales signiﬁcantly with signal-to-noise and size. Faint small galax-

ies are affected strongly by neighbour light, while larger brighter

ones are relatively immune.

MNRAS 000, 000–000 (0000)

Cosmic Shear & Galaxy Neighbours 11

−0.25

−0.20

−0.15

−0.10

−0.05

0.00

Multiplicative Bias m

HOOPOE, matched per galaxy

HOOPOE, matched per galaxy (diluted)

HOOPOE, matched per bin

WAXWING, matched per galaxy

HOOPOE, full selection

WAXWING, full selection

1.0 1.2 1.4 1.6 1.8 2.0 2.2

Signal-to-Noise log(S/N)

−0.08

−0.06

−0.04

−0.02

0.00

0.02

∆m

Direct Neighbour Bias

Selection Bias

Neighbour Dilution

Bin Shifting

Total Neighbour Bias

−0.15

−0.10

−0.05

0.00

Multiplicative Bias m

1.2 1.3 1.4 1.5 1.6 1.7

−0.08

−0.06

−0.04

−0.02

0.00

0.02

0.04

0.06

∆m

Figure 13. Top half of each panel Multiplicative bias as a function of signal-to-noise and size. The purple circles show the measured bias using the main

HOOPOE simulation, and the blue diamonds are the resimulated neighbour-free version. The lines show permutations of the same measurements to highlight

the neighbour-induced effects causing the two to differ. The dot-dashed blue and dashed purple lines show the impact of applying the HOOPOE selection mask

to WAXWING and vice versa. The impact of bin shifting is shown by the purple dotted line, which is calculated from the same matched galaxies, using the

HOOPOE shape measurements for the bias and WAXWING size and S/N for binning. The pink curve is the same as the dashed purple, but with a fraction of

heavily blended galaxies added back with randomised shear (see Section 5.3.4). Bottom half of each panel The change in bias due to the effects described

above. The green (dashed) line shows the impact of selection effects (the difference between the blue diamonds and the dashed line in the top panel). The direct

neighbour bias due to light contamination is shown by the purple dash-dotted line (purple dotted minus blue dash-dot top). The impact of shifting between

bins is shown by the blue dotted (dashed minus purple dotted, top). The pink dot-dot-dashed line illustrates the impact of adding back randomised shears, as

described. Finally the solid line represents the total neighbour bias, which includes all these effects (circles minus diamonds, top).

5.3.2 Neighbour-Induced Selection Bias

To gauge the neighbour-induced selection effect, we take the

WAXWING catalogue but now impose, in addition to its own qual-

ity cuts, the selection function derived from the with-neighbour

HOOPOE dataset. The double masking removes an additional 0.5M

galaxies, which survive cuts in WAXWING but would be cut from

the HOOPOE catalogue. The resulting change in m is shown by the

dot-dash blue lines in the upper panels of Fig. 13 (dashed green in

the lower). The multiplicative bias arising from this cut is less than

one percent in all but the faintest and smallest galaxies, where it

can reach up to m ∼ −0.02.

5.3.3 Bin Shifting

The above two tests encapsulate the impact on the measured ellip-

ticities, and the selection ﬂags from neighbour ﬂux. An additional

subtlety arises from the fact that the measured quantities used to

bin galaxies (S/N and R

) are themselves affected by the

presence of neighbours. To test this we recalculate m using the

same galaxy selection as in Section 5.3.1 (i.e. passing both sets of

cuts), but now binned by the appropriate measured S/N. For clar-

ity, the bin edges are unchanged, deﬁned to contain equal numbers

of WAXWING galaxies. The result is shown by the dashed lines in

Fig. 13. The difference compared with the case using ﬁxed binning

is purely the result of galaxies moving between bins. This shift-

ing contributes multiplicative bias if one bins galaxies by observed

quantities such as S/N, as we do in order to calibrate IM3SHAPE’s

shear estimates. The amplitude of this is illustrated by the blue dot-

ted line in the lower panels. Such neighbour-induced shifting is

noticable if we plot out the S/N of objects in HOOPOE against the

S/N of the same objects in WAXWING. Objects which are strongly

shifted in S/N are more likely to scatter upwards than downwards.

A similar skew can be seen in the R

plane; when galaxies are

scattered in size they tend to be thrown further and more often up-

wards than downwards. Small galaxies (which we know already are

more strongly affected by noise bias) are shifted strongly upwards

by the presence of neighbour ﬂux in the HOOPOE images. The re-

sult is a net negative m added to the upper R

bins, and a

simultaneous positive shift in the lowest bins from which galaxies

are lost. In the case of galaxy size we see the effects of bin scatter

and direct neighbour bias almost negate each other, although the

degree of cancellation is likely dependent on the speciﬁcs of the

measurement code and the dataset.

5.3.4 Neighbour Dilution

A ﬁnal point can be gleaned from Fig. 13: that applying the

WAXWING cuts to HOOPOE induces a shift in m. Naively one

might expect the HOOPOE selection function, which includes

neighbours, to remove the same galaxies as the WAXWING selec-

tion, plus some extra strongly blended galaxies. It is true that a

sizeable number of galaxies are cut in the presence of neighbours,

but would otherwise not be. There is also, however, a smaller popu-

lation that survive cuts because they have image plane neighbours.

We can see this clearly from the fact that the purple points and

the dashed purple lines Fig. 13 are non-identical. We identify three

separate (but partially overlapping) galaxy selections in this ﬁgure:

(a) galaxies passing both sets of cuts, (b) galaxies passing cuts in

the absence of neighbours, but cut by the HOOPOE selection and (c)

galaxies which pass cuts in the presence of neighbours, but cut by

the WAXWING selection. We ﬁnd that populations (b) and (c) have

much smaller mean neighbour separation than the full population

(the histograms of d

show a sharp peak at under 10 pixels). In

MNRAS 000, 000–000 (0000)

12 S. Samuroff, S. L. Bridle, J. Zuntz et al

contrast, both the full catalogue and population (a) objects a much

broader distribution (

∼ 24 pixels).

Based on the toy model predictions in Section 3 we set out a

working proposal: that population (c), objects cut out only when

neighbours are removed, are extreme blends dominated by a super-

posed neighbour. We will assume these objects are boosted con-

siderably in size, S/N or both, such that what would otherwise

be a small faint galaxy is now sufﬁciently bright to pass quality

cuts. In these cases the measured shape of a simulated galaxy might

be expected to be only weakly linked with the input ellipticity. To

approximate this effect we take population (a) HOOPOE galaxies,

subject to both sets of cuts, and add back some of the population

by more than 20%. The true shears associated with these

galaxies are now randomised to eliminate any correlation with the

measured ellipticity. The result is shown as a pink dot-dot-dashed

line in Fig. 13. We can see that this effect, which we call neigh-

bour dilution, to good approximation accounts for the residual dif-

ference between the population (a) and (c) samples. Particularly

in the upper size bins of the right hand panel the differences are

not eliminated entirely. This is thought to be the result of resid-

ual (albeit weakened) covariance between the measured shapes of

strongly blended objects and the input shears. Clearly the scenario

in which a neighbour totally overrides the original shape of a galaxy

is extreme, and there will be an indeterminate number of moderate

blends which are boosted sufﬁciently to survive cuts but which re-

tain some correlation with their original unblended shapes. Such

cases are, however, extremely difﬁcult to model accurately with the

resources available for this investigation.

5.4 Isolating the Impact of Sub-detection Galaxies

A handful of previous studies have attempted to quantify the impact

of galaxies below, or close to, a survey’s limiting magnitude. For

example, Hoekstra et al. (2015) and Hoekstra et al. (2017) suggest

they can induce a non-trivial multiplicative bias, which is depen-

dent on the exact detection limit. They recommend using a shear

calibration sample at least by 1.5 magnitudes deeper than the sur-

vey in question (which ours does). Their ﬁndings, however, make

exclusive use of the moments-based KSB algorithm (see Kaiser

et al. 1995); such techniques are known to probe a galaxy’s elliptic-

ity at larger radii than other methods, which could in principle make

them more sensitive to nearby faint galaxies. It is thus a worthwhile

exercise to to gauge their impact in our case with IM3SHAPE.

5.4.1 Impact on Multiplicative Bias

We ﬁrst compare our HOOPOE simulations with the neighbour-free

WAXWING resimulations. Since WAXWING postage stamps con-

sist of only a single proﬁle added to Gaussian pixel noise, they are

unaffected by neighbours of any sort (faint or otherwise). We have

seen that the impact of neighbours is strongly localised, with the

excess m converging within a nearest neighbour distance d

of a

dozen pixels or so. Thus selecting galaxies that are well separated

from their nearest visible neighbour will isolate the impact of the

undetected ones.

A further cut is thus imposed on d

< 20 pixels. Relative to

the case with quality cuts only, the global multiplicative bias now

shifts from m ∼ −0.119 to m = −0.064 ± 0.006 (the ﬁrst and

second lines in green on Fig. 12). This measurement is in mild ten-

sion with the value measured from WAXWING (again under its own

0 10 20 30 40 50 60 70 80

Nearest Subdetection Galaxy d

/ pixels

With Faint, Own Cuts

Only Without Faint

Only With Faint

Figure 14. Histogram of radial distances between galaxies in our measured

shape catalogues (the full HOOPOE simulations) and the nearest object be-

low the DES detection limit. The dotted line includes all objects prior to

quality cuts, while the solid line shows the impact of applying IM3SHAPE’s

INFO FLAG cuts (see J16). The dashed blue line shows the population of

galaxies which survive cuts only in the presence of the faint galaxies.

cuts, with the selection on d

). This difference, which amounts to

a negative shift in m of ∼ 0.01 is, we suggest, the net effect of the

sub-detection galaxies. From these numbers alone we cannot tell if

this is a result of selection effects, ﬂux contamination, bin shifting

or some combination thereof.

Interestingly we ﬁnd that imposing both the HOOPOE and

WAXWING selection functions in addition to the cut on d

brings

m into agreement to well within the level of statistical precison

(compare the ﬁnal and penultimate lines in green in Fig. 12). That

is, when restricted to a subset of galaxies that pass quality cuts in

both simulations the ﬂux contributed by the faint objects has little

impact. Their main impact is rather that they allow a population

of marginal faint galaxies which would otherwise be ﬂagged and

removed by quality cuts to pass into the ﬁnal HOOPOE catalogue.

To test this idea further we rerun a subset of 100 random tiles

from the simulated footprint, without the ﬁnal step of adding sub-

detection galaxies. To minimise the statistical noise in this com-

parison we enforce the same COSMOS proﬁles, shears and ro-

tations as well as the per-pixel noise realisation as before. SEX-

TRACTOR source detection is applied and the blending ﬂags are

propagated into the postprocessing cuts.

The raw m values calculated from the rerun and the main

HOOPOE simulations, matched to the same tiles, are shown by the

upper two red lines on Fig. 12. The downward shift of ∼ 0.01 is

consistent with the previous result based on the main simulation.

This comparison should encapsulate the full effect of the faint ob-

jects (since there are no other differences between these datasets).

For each galaxy we next measure the distance to the nearest

faint object d

, the distribution of which is shown under vari-

ous selections in Fig. 14. Like in the comparison in Section 5.3.4,

there is a population of galaxies that survive cuts only in the sim-

ulation with the sub-detection objects, and these galaxies tend to

be ones with extremely close faint neighbours. Interestingly the

inverse population surviving only when they are removed do not

preferentially have small d

. This is intuitively understandable: a

faint object might boost its neighbour’s apparent size or S/N if it

MNRAS 000, 000–000 (0000)

Cosmic Shear & Galaxy Neighbours 13

were centred within a few pixels. Otherwise it would act as a source

of background noise, which would reduce the quality of the ﬁt.

Finally we ﬁnd that if we apply both selection functions to

the with-faint galaxies, the measured biases become roughly con-

sistent. These ﬁndings, combined with the observations in the pre-

vious section lead us to an interesting conclusion: the major effect

of the faint galaxies in the DES Y1 IM3SHAPE catalogue is to al-

low a population of small faint galaxies to pass quality cuts, where

otherwise they would have been removed. This is analogous to the

neighbour dilution effect described above, but is subdominant to

the inﬂuence of visible neighbours.

5.4.2 Impact on Background Flux Subtraction

As a test of the robustness of this result we recompute our

IM3SHAPE ﬁts on the faint-free images, with and without the cor-

rection for the shift in the background ﬂux that would have been

applied if the sub-detection galaxies had been drawn. The mean

per-tile correction is ∆f ∼ 0.05, against typical noise ﬂuctua-

tions σ

∼ 6.5. Matching galaxies and examining the histograms

of ∆S/R and ∆R

reveals weak downwards scatter in both

quantities (i.e. the ﬂux subtraction alone makes galaxies appear

smaller and fainter). The magnitude of the shift is, however, tiny,

peaking at ∼ −0.1 and −0.005 respectively. This is logical given

the deﬁnition in equation 1. If the change is small enough such

that the best-ﬁtting model is stable, then an incremental reduction

in ﬂux will reduce the signal-to-noise of the measurement. Looking

at the best-ﬁt shapes, we ﬁnd a small shift towards high ellipticities,

which can likewise be understood as a numerical effect; imposing

a ﬂat positive ﬁeld of zero ellipticity will dilute the measured shear,

producing a bias towards round |e|. The reverse logic applies with

the ﬂux correction, and subtracting a ﬂat value from all pixels will

make galaxies appear slightly more elliptical. In practice we ﬁnd a

sharp peak at ∆e ∼ 0.001.

5.5 Suppressing Neighbour Bias

There is no universal deﬁnition for the shape-weighted effective

number density commonly used as proxy for cosmological con-

straining power in a shear catalogue. One which is particularly use-

ful in the context of weak lensing, and which has been adpoted in

DES Y1 is the prescription of Chang et al. (2013), which is de-

signed to account for shape noise and ﬁtting error (see equation

7.5 in Zuntz et al. 2017). A second useful deﬁnition is set out by

Heymans et al. (2012) in terms of the (see also Z17). We com-

pute a neighbour distance d

for every object in the real data,

which allows us to cut on this quantity. Removing any galaxy with

a neighbour detected within a radius of d

= 20 pixels reduces

the effective number density of sources using either deﬁnition to

about 70% of its initial value, from n

H13

eﬀ

= 5.48 to n

H13

eﬀ

= 3.68

arcmin

−2

using Heymans et al. (2012)’s deﬁnition. Using the pre-

scription of Chang et al. (2013), the equivalent density drops from

C13

eﬀ

= 3.18 prior to cuts and n

C13

eﬀ

= 2.18 arcmin

−2

afterwards.

This cut is stringent, as we have shown that beyond ∼ 12 pixels

the multiplicative bias becomes insensitive to further selection on

. There are, however, a number of limitations in our analysis,

including the fact that d

is deﬁned using the true input positions,

and indeed that we are using only the detected positions in DES to

draw our simulated M

< 24.1 galaxies. We thus judge that a level

of conservatism is appropriate here. Relaxing the cut to d

> 14

pixels leaves n

eﬀ

at 84% of its full value.

6 COSMOLOGICAL IMPLICATIONS

As we have shown in the previous sections, if ignored completely

image plane neighbours can induce negative calibration biases in

IM3SHAPE of a few per cent or more. The earlier part of the inves-

tigation focused on when and how neighbour bias can arise, ﬁrst

in the context of single-galaxies and then on ensemble shear esti-

mates. We now turn to a more pressing question from the general

cosmologist’s perspective: how far should I be concerned about

these effects in practice? We present a set of numerical forecasts us-

ing the MULTINEST nested sampling algorithm (Feroz et al. 2013)

to sample trial cosmologies. Each of the likelihood analyses pre-

sented in this paper has been repeated using a Markov Chain Monte

Carlo sampler (EMCEE). Although we see the same small shift in

contour size noted by DES Collaboration et al. (2017) (see their

Appendix A), which diminishes as the length of the MCMC chains

increases, we ﬁnd our conclusions are robust to the choice of sam-

pler. Our basic methodology here follows previous numerical fore-

casts (e.g. Joachimi & Bridle 2010; Krause & Eiﬂer 2016; Krause

et al. 2017a). We construct mock DES Y1 cosmic shear measure-

ments using a matter power spectum derived from the Boltzmann

code CAMB

with late-time modiﬁcations from HALOFIT. The

cosmic shear likelihood surface is sampled at trial cosmologies us-

ing COSMOSIS

. The ﬁnal data used for the likelihood calcula-

tion have the form of real-space ξ

correlations. For the photo-

metric redshift distributions we use the measured estimates in four

tomographic bins, obtained from runs of the BPZ code on the Y1

IM3SHAPE catalogue, as described by Hoyle et al. (2017). Since

this analysis was completed before the details of the photometric

redshift calculation for DES Y1 had been ﬁnalised, these distribu-

tions differ marginally from (but are qualitatively the same as) the

ﬁnal version used in Troxel et al. (2017) and DES Collaboration

et al. (2017). In all chains which follow we maginalise over two

nuisance parameters (an amplitude and a power-law in redshift) for

intrinsic alignments, photo-z bias and shear calibration bias. In to-

tal this gives 10 extra free parameters in addition to six for cosmol-

ogy (Ω

, Ω

, A

, n

, h, Ω

), which are also allowed to vary.

Apart from the difference in redshift distributions remarked upon

above, our analysis choices match the DES Y1 cosmic shear anal-

ysis of Troxel et al. (2017). We refer the reader to that paper for

details of the priors and scale cuts, and their derivation. Finally,

the following adopts shear-shear covariance matrices derived from

the analytic halo model calculations of Krause et al. (2017b). We

assume a ﬁducial ΛCDM cosmology σ

= 0.83, Ω

= 0.295,

Ω

= 0.047, n

= 0.97, h = 0.688, τ = 0.08, with non-zero

comoving neutrino density Ω

= 0.00062.

6.1 Mean Multiplicative Bias

We ﬁrst seek to quantify the bias that would be present in a

cosmic shear analysis in a survey like DES, if we were to use

a simple postage stamp simulation of the sort presented in J16

and Miller et al. (2013). To this end we use the neighbour-free

WAXWING dataset to construct an alternative shear calibration. In

Z17 we compared three methods for shear calibration using the

HOOPOE simulations and found our results to be robust to the

differences. We now use the ﬁducial (grid-based) scheme to de-

rive an alternative set of bias corrections from WAXWING. These

http://camb.info

https://bitbucket.org/joezuntz/cosmosis

MNRAS 000, 000–000 (0000)

14 S. Samuroff, S. L. Bridle, J. Zuntz et al

Angular Scale ✓ / arcseconds

4

2

⇠

(✓ ) / ⇥10

3

hmmi

h

-0.1

0.0

0.1

(4, 1) (3, 1) (2, 1)

10 100

(1, 1)

waxwing Calibration

Spatial Correlations

-0.1

0.0

0.1

(4, 2) (3, 2)

10 100

(2, 2)

-0.1

0.0

0.1

(1, 1)

-0.1

0.0

0.1

(4, 3)

10 100

(3, 3)

-0.1

0.0

0.1

(2, 2) (1, 2)

1 10 100

θ / arcmin

-0.2

-0.1

0.0

0.1

(4, 4)

-0.1

0.0

0.1

(3, 3) (2, 3) (1, 3)

10 100

θ / arcmin

-0.2

-0.1

0.0

0.1

(4, 4)

10 100

(3, 4)

10 100

(2, 4)

10 100

(1, 4)

obs

(θ)/ξ

(θ) − 1

obs

−

(θ)/ξ

−

(θ) − 1

Figure 15. Top: The observed two point correlation of multiplicative bias, as measured from the main HOOPOE simulation set presented in this paper. Sub-

patches are used to compute m in spatial patches of dimension 0.15 × 0.15 degrees and the correlation function calculated as described in the text. The

dashed vertical line shows the diagonal scale of the sub-patches, below which we do not attempt to directly measure spatial correlations. The shaded blue

bands show the minimum and maximum scales used in the DES Y1 cosmic shear analysis of Troxel et al. (2017). Bottom: Residuals between the mock two

point shear-shear data used in this paper, before and after different forms of bias have been applied. The upper-left and lower-right triangles show the ξ

and ξ

−

correlations respectively, calculated using the redshift distributions of Hoyle et al. (2017). The dotted black lines, which are ﬂat across all scales but

vary between panels, show the result of calibrating our Y1 shear measurements with a simple postage stamp simulation without image plane neighbours. The

dashed lines illustrate the impact of ignoring scale-dependent selection effects, which are not captured by our simulation-based calibration. The shaded blue

regions of each panel show the excluded scales for each particular tomographic bin pairing.

are then applied to the same galaxies in the matched HOOPOE sim-

ulation, and residual biases are measured in four DES-like tomo-

graphic bins. The process is very similar to the diagnostic tests in

§5 of Z17, and so we defer to that work for details of the redshift

bin assignment of simulated galaxies.

Using the neighbour-free simulation we under-correct the

measurement bias by several percent in each bin. The remeasured

residual bias after calibration provides an estimate for the level of

systematic that would be present were we to calibrate DES Y1 us-

ing the simpler WAXWING simulations. In the four tomographic

MNRAS 000, 000–000 (0000)

Cosmic Shear & Galaxy Neighbours 15

Figure 16. Expected cosmology constraints from DES Y1 cosmic shear

only. The purple (solid) contours show the results of calibrating using a sim-

ulation which fully encapsulates all biases in the data, leaving no residual

m in the ﬁnal catalogue. In blue (dash-dotted) we show the result of cal-

ibrating with an insufﬁciently realistic simulation, which leaves a residual

bias between −0.03 and −0.08 in each of the redshift bins. For reference

we mark the input cosmology with a black cross.

bins used in DES Y1 we ﬁnd (∆m

(1)

, ∆m

(2)

, ∆m

(3)

, ∆m

(4)

) =

(−0.037, −0.044, −0.064, −0.073), and apply these biases to our

mock data. The resulting shift in the shear two-point correlations is

shown by the black dotted lines in the lower panel of Fig. 15. Since

the calibration scheme does not explicitly include neighbour dis-

tances, but rather orders galaxies into cells of S/N and R

this test does not include any scale dependent neighbour effects.

The calibration effectively marginalises out d

, and the residual

biases are an average over the survey. For the moment we will as-

sume this mean shift in m is sufﬁcient, and return to the question

of scale dependence in the following section.

Our predicted cosmology constraints with weak lensing alone

in DES Y1 are shown in Fig. 16. In purple we show the results of

the ﬁducial analysis, in which the shear calibration fully captures

all neighbour effects and leaves no residual multiplicative bias. The

blue (solid) contours then show the impact of residual neighbour

biases per bin at the level described. As we can see, even when

marginalising over m

with an (erroneously) zero-centred Gaussian

prior of width σ

= 0.035, our cosmology constraints are shifted

enough to place the input cosmology outside the 1σ conﬁdence

bounds. We reiterate here that this calculation highlights the bias

that would arise were we to naively apply a calibration of the sort

used in DES SV based on neighbour-free simulations to the Y1

data. Since we are conﬁdent that the HOOPOE code captures the

effects of image plane neighbours correctly (at least to ﬁrst order)

this is a hypothetical scenario only and not a prediction of actual

bias in DES Y1.

6.2 Scale Dependence

It is not trivial that including an mean neighbour-induced compo-

nent to m over the entire survey will be sufﬁcient to mitigate all

forms of neighbour bias. The local mean m on a patch of sky is sen-

sitive to spatial ﬂuctuations in source density, which could induce

scale dependent bias on arcminute scales. Clearly, when correlat-

Figure 17. The same as Fig. 16, but now showing the impact of residual

scale dependent selection bias. The two sets of conﬁdence contours repre-

sent different assumptions about the small scale extrapolation of the ξ

correlation, as outlined in the Section 6.2. In green (dashed) we show a

mildly optimistic case, using the linear ﬁt shown in Fig. 15. The pink dot-

ted contours show a (strongly pessimistic) power law extrapolation. The

dark blue solid line makes identical assumtions to the pink, but incorporates

small-scale information, to a minimum separation of θ

min

= 0.5 arcmin-

utes in ξ

(θ) and θ

−

min

= 4.2 arcminutes in ξ

−

(θ). As in Fig. 16 the input

cosmology is shown by a black cross.

ing galaxy pairs on small scales one can expect a larger fraction in

which the objects come from a similar image plane environment,

and more often than not that enviroment will be densely populated.

Thus the true multiplicative bias should become more negative on

small scales.

Two subtly different effects emerge from this thought experi-

ment. First, the multiplicative bias of galaxies will be spatially cor-

related i.e. a correlation involving two galaxy populations





is not just the product of the means ¯m

¯m

. Second, in the small

θ bins one is selecting galaxies with close partners with which to

correlate, and thus oversampling the dense parts of the image. To

gauge the level of these effects, we divide each simulated coadd tile

into a grid of 25 square sub-patches with dimension 0.15 × 0.15

degrees. We ﬁt for m using the galaxies in each sub-patch and as-

sign the resulting value to these objects. While this only allows a

noisy measurement of m, it should capture the spatial variations

in number density to the level of a few percent. We next measure

the two-point correlation function of multiplicative bias values as-

signed in this way using TREECORR

, excluding galaxy pairs at

angular separation smaller than the scale of the sub-patches. We re-

fer to this bias-bias autocorrelation as ξ

, which we show as a

function of angular scale in the upper panel of Fig. 15. analogously

one could use the sub-patches to construct correlations between m

and galaxy number density ξ

or density with density ξ

. The

statistical noise on these correlations is signiﬁcantly lower than that

on the individual sub-patches by virtue of the large simulation foot-

print. Note that in Fig. 15 we subtract ¯m

, measured from all galax-

ies in the simulation, from the measured ξ

. If there were no θ

dependence the correlation hm

i should simply average to the

square of the global mean in all scale bins. As we can see from

http://rmjarvis.github.io/TreeCorr

MNRAS 000, 000–000 (0000)

16 S. Samuroff, S. L. Bridle, J. Zuntz et al

the circular points in this ﬁgure, scales larger than the diagonal size

the sub-patches (shown by the vertical dashed line) exhibit non-

negligible excess ξ

. One obvious question is whether this could

be the result of ﬁnite binning error, which scatters galaxies in the

same sub-patch into different θ bins. To verify this is not the case

we repeat the measurement as before, but halve the parameter con-

trolling binning error tolerance (“bin slop”) and obtain the same

results.

To extend this measurement down to scales below the sub-

patch size we must make some assumptions about the functional

form of the mm correlation. We ﬁt a power law, ∆ξ

(θ) ≡

(θ) − ¯m

∆ξ

(θ) = βθ

−α

, (5)

which is shown by the dotted purple line in this ﬁgure. This pro-

vides a qualitiatively good ﬁt to the measured points, but as we can

see implies a rather dramatic inﬂation on small scales.

In the limited range over which we have a nonzero measured

correlation, however, a linear function of θ (truncated at θ = 27

arcminutes) also provides a reasonable by-eye ﬁt. The small-scale

extrapolation in this case is signiﬁcantly milder. The hmδ

i and

hδ

i measurements are linear with θ to good approximation, and

so we use linear ﬁts to extrapolate them below the patch size.

Assuming the bias per tomographic bin can be written as the

sum of a redshift dependent contribution (i.e. a scale invariant mean

in each bin), and a scale dependent term, one can write the corre-

lation per bin as m

= ¯m

¯m

+ ∆ξ

(θ). A more complete

derivation of this expression can be found in Appendix A. The ﬁrst

part can be extracted from the DES Y1 calibration, and we can ﬁt

for ∆ξ

(θ) as described above. A set of modiﬁed ξ

are thus

computed. These appear in the lower panels of Fig. 15 as dashed

lines. As we can see, the scale cuts of Troxel et al. (2017) (ex-

cluded scales are shaded in blue) are sufﬁciently stringent to re-

move almost all of the visible scale dependence. Though reassur-

ing for the immediate prospects of DES Y1, this will not trivially

be true for all future (or indeed ongoing) lensing surveys. It is thus

important that the effects we identify here are properly understood

at a level beyond the resources of the current paper. These biased

data are then passed through our likelihood pipeline to gauge the

cosmological impact, which is shown in Fig. 17. In the linear case

(dashed green) there is no discernable bias in the σ

Ω

pair; even

the much harsher power-law extrapolation (pink dotted) induces

only an incremental shift along the degeneracy direction. In both

cases the input cosmology still sits comfortably within the 1σ con-

ﬁdence contour.

Finally we test the impact of relaxing the stringency of our

scale cuts. The minimum scales used for ξ

and ξ

−

are shifted

downwards to 0.5 and 4.2 arcminutes respectively, irrespective of

bin pair, which are the cut-off values used in ﬁducial cosmic shear

analysis of Hildebrandt et al. (2017). This increases the size of our

datavectors considerably. Incorporating smaller angular scales will

clearly improve the constraining power of the data to an extent. Pri-

marily the effect is to shorten the lensing degeneracy ellipse, cutting

out much of the peripheral curvature, but it also reduces the width

in the S

direction. These scales, however, contain biased informa-

tion, which induces tension between the small and large angular

scales. With the strongest (power law) scale dependence consid-

ered, the input cosmology is displaced marginally along the degen-

eracy curve, though it remains comfortably within the 1σ conﬁ-

dence bound.

7 CONCLUSIONS

The Dark Energy Survey is the current state of the art in cosmo-

logical weak lensing. Multi-band imaging down to 24th magnitude

across 1500 square degrees of the southern sky has yielded hith-

erto unparalleled late-time constaints on the basic parameters of

the Universe (see Troxel et al. 2017 and DES Collaboration et al.

2017).

In this paper we have used one of two DES Y1 shear cata-

logues, and large-area simulations based upon them, to quantify

the impact of image plane neighbours on both ensemble shear mea-

surements, and on the inferred cosmological parameters.

In order to properly mitigate the inﬂuence of galaxy neigh-

bours, and thus avoid drawing ﬂawed conclusions about cosmology

from the data, it is important to ﬁrst understand the mechanisms

by which they enter the shape measurement. Using a simple toy

model of the galaxy-neighbour system we have shown that shear

bias can arise even when the distribution of neighbours is isotropic

(i.e. there is no preferred direction). This is the result of a small

difference in the impact of the same neighbour, when it is placed

on or away from the axis of the shear. We have furthermore shown

that the resulting multiplicative shear bias m can be either positive

or negative, depending on the model parameters. With slight mod-

iﬁcations to the toy model, whereby we Monte Carlo sample input

parameters from the joint distribution of the equivalent properties

measured in DES Y1, we have shown that a mild negative m is

dominant when marginalising over a realistic ensemble of neigh-

bours. This was seen to be strongly dependent on the distance of

the neighbour, and to be mitigated but not eliminated by basic cuts

on the centroid position of the best-ﬁtting model.

Using the DES Y1 HOOPOE simulations, which were

also used to derive shear calibration corrections for the Y1

IM3SHAPE catalogue of Z17, we have presented a detailed study of

the ensemble effects of galaxy neighbours. In this analysis we have

identiﬁed four mechanisms for neighbour bias, which we call ﬂux

contamination, selection effects, bin shifting and neighbour dilua-

tion. All can be understood in intuitive terms, resulting from close-

by or moderately close neighbours. Our results from the full sim-

ulation are consistent with the toy model calculation. Though we

have shown strong dependence on distance to the nearest neigh-

bour (and thus on number density) we found only weak sensitiv-

ity to neighbour brightness, when averaged across broad bins of

magnitude. In addition to this, cuts on the DES Y1 catalogue suf-

ﬁcient to null the impact of the detectable neighbours would result

in a degradation of over 20% in source number density. We can-

not recommend such measures for a code like IM3SHAPE, in part

because the data contains correlations between shear and number

density. Unless the link is preserved in the calibration simulations,

such selection could conceivably induce additional bias towards

low shear

Our investigation also assessed the impact of the faintest

galaxies, which are not reliably detected but nonetheless contribute

ﬂux to the survey images. Via two different routes, ﬁrst using a

spin-off neighbour-free resimulation, and also using a subset of im-

ages simulated again with the sub-detection galaxies missing, our

ﬁndings suggested a net contribution to the multiplicative bias bud-

get of m ∼ −0.01.

Unlike most earlier works on shear measurement, we have

Although the sister catalogue to Y1 IM3SHAPE uses a form of internal

calibration, which should allow one to correct for the additional selection

bias.

MNRAS 000, 000–000 (0000)

Cosmic Shear & Galaxy Neighbours 17

propagated these ﬁndings to the most meaningful metric for cos-

mic shear: bias on the inferred cosmological parameters. The study

we have presented here uses the DES Y1 cosmology pipeline, as

well as real non Gaussian shear covariance matrices and photo-

metric redshift distributions to implement MCMC forecasts. In the

ﬁrst case considered, the data included a (different) multiplica-

tive bias in each redshift bin, designed to approximate the resid-

ual m that would arise were we to calibrate DES Y1 with a simple

neighbour-free simulation. Even marginalising over m with a prior

of N (0, 0.035) this scenario was demonstrated to result in a shift in

the favoured cosmology towards low clustering amplitude of more

than 1σ.

Finally, we have explored a second possible source of mea-

surement bias arising from the link between number density and

neighbour bias. This enters two-point measurements as an addi-

tional correlation between the multiplicative bias in galaxy pairs

at small angular separation. In the ﬁnal section we have measured

such a correlation from the HOOPOE mock images. With the most

pessimistic small-scale extrapolation, this was found to result in

a shift in the best-ﬁtting cosmology of under 1σ in the negative

direction, which is not remedied by marginalising over m. A

less dramatic, though still considerable, increase in the correlation

strength on small scales was demonstrated to result in no discern-

able cosmological bias.

Both of these effects are of primary concern for the next gen-

eration of cosmological surveys. By the end of their lifetime KiDS,

DES and HSC are set to offer lensing-based cosmological con-

straints comparable to the CMB. The ﬁrst, dominant, effect can be

remedied relatively easily by calibrating our shear measurements

with sufﬁciently complex image simulations. Indeed, the most re-

cent shear constraints of Hildebrandt et al. (2017), K

ohlinger et al.

(2017) and Troxel et al. (2017) have done just that. Unfortunately,

the correct treatment of scale dependent bias is not as clear, though

it should be captured at some level by the per-galaxy responses

upon which METACALIBRATION relies. Though further statements

about the likely small scale dependence of the mm correlation are

beyond the scope of the present study, understanding this intricate

topic will be crucial for future surveys if we are to fully exploit the

constraining power of the data. The massive simulation efforts of

LSST and Euclid, combined with advancement in neighbour miti-

gation using techniques such as multi-object ﬁtting will be invalu-

able in this task. With the enhanced understanding these will pro-

vide and the exquisite data of the next generation surveys, the com-

ing decade will be an exciting time for cosmology.

8 ACKNOWLEDGEMENTS

We thank Nicolas Tessore, Catherine Heymans and Rachel Man-

delbaum for various insights that contributed to this work. We are

also indebted to the many DES “eyeballers” for lending their holi-

day time to help us understand and validate our simulations. The

HOOPOE simulations were generated using the National Energy

Research Scientiﬁc Computing Center (NERSC) facility, which is

maintained by the U.S. Department of Energy. The likelihood cal-

culations were performed using NERSC and the Fornax computing

cluster, which was funded by the European Research Council. SLB

acknowledges support from the European Research Council in the

form of a Consolidator Grant with number 681431. Support for DG

was provided by NASA through Einstein Postdoctoral Fellowship

grant number PF5-160138 awarded by the Chandra X-ray Center,

which is operated by the Smithsonian Astrophysical Observatory

for NASA under contract NAS8-03060.

Funding for the DES Projects has been provided by the

U.S. Department of Energy, the U.S. National Science Founda-

tion, the Ministry of Science and Education of Spain, the Sci-

ence and Technology Facilities Council of the United Kingdom, the

Higher Education Funding Council for England, the National Cen-

ter for Supercomputing Applications at the University of Illinois at

Urbana-Champaign, the Kavli Institute of Cosmological Physics at

the University of Chicago, the Center for Cosmology and Astro-

Particle Physics at the Ohio State University, the Mitchell Institute

for Fundamental Physics and Astronomy at Texas A&M Univer-

sity, Financiadora de Estudos e Projetos, Fundac¸

ao Carlos Chagas

Filho de Amparo

a Pesquisa do Estado do Rio de Janeiro, Con-

selho Nacional de Desenvolvimento Cient

ıﬁco e Tecnol

ogico and

the Minist

erio da Ci

encia, Tecnologia e Inovac¸

ao, the Deutsche

Forschungsgemeinschaft and the Collaborating Institutions in the

Dark Energy Survey.

The Collaborating Institutions are Argonne National Labora-

tory, the University of California at Santa Cruz, the University of

Cambridge, Centro de Investigaciones Energ

eticas, Medioambien-

tales y Tecnol

ogicas-Madrid, the University of Chicago, Univer-

sity College London, the DES-Brazil Consortium, the University

of Edinburgh, the Eidgen

ossische Technische Hochschule (ETH)

urich, Fermi National Accelerator Laboratory, the University of

Illinois at Urbana-Champaign, the Institut de Ci

encies de l’Espai

(IEEC/CSIC), the Institut de F

ısica d’Altes Energies, Lawrence

Berkeley National Laboratory, the Ludwig-Maximilians Univer-

sit

at M

unchen and the associated Excellence Cluster Universe, the

University of Michigan, the National Optical Astronomy Observa-

tory, the University of Nottingham, The Ohio State University, the

University of Pennsylvania, the University of Portsmouth, SLAC

National Accelerator Laboratory, Stanford University, the Univer-

sity of Sussex, Texas A&M University, and the OzDES Member-

ship Consortium.

The DES data management system is supported by the Na-

tional Science Foundation under Grant Numbers AST-1138766

and AST-1536171. The DES participants from Spanish institu-

tions are partially supported by MINECO under grants AYA2015-

71825, ESP2015-88861, FPA2015-68048, SEV-2012-0234, SEV-

2016-0597, and MDM-2015-0509, some of which include ERDF

funds from the European Union. IFAE is partially funded by the

CERCA program of the Generalitat de Catalunya. Research leading

to these results has received funding from the European Research

Council under the European Union’s Seventh Framework Pro-

gram (FP7/2007-2013) including ERC grant agreements 240672,

291329, and 306478. We acknowledge support from the Australian

Research Council Centre of Excellence for All-sky Astrophysics

(CAASTRO), through project number CE110001020.

This manuscript has been authored by Fermi Research Al-

liance, LLC under Contract No. DE-AC02-07CH11359 with the

U.S. Department of Energy, Ofﬁce of Science, Ofﬁce of High En-

ergy Physics. The United States Government retains and the pub-

lisher, by accepting the article for publication, acknowledges that

the United States Government retains a non-exclusive, paid-up, ir-

revocable, world-wide license to publish or reproduce the published

form of this manuscript, or allow others to do so, for United States

Government purposes.

Based in part on observations at Cerro Tololo Inter-American

Observatory, National Optical Astronomy Observatory, which is

operated by the Association of Universities for Research in As-

MNRAS 000, 000–000 (0000)

18 S. Samuroff, S. L. Bridle, J. Zuntz et al

tronomy (AURA) under a cooperative agreement with the National

Science Foundation.

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APPENDIX A: DERIVATION OF A TWO-POINT

MODIFIER FOR SCALE DEPENDENT BIAS

In the following we set out a brief derivation of the analytic modiﬁ-

cations to account for scale-dependent neighbour effects the shear-

shear two-point correlations used in the earlier section. We do not

claim that this is a precise calculation of the sort that could be used

to derive a robust calibration. Rather it is an order of magnitude

estimate to allow us to assess the approximate size of the cosmo-

logical bias these effects could induce in the data.

First, with complete generality it is possible to write the i com-

ponent of the measured shear at angular position θ as

obs

(θ) = [1 + m

(θ)] γ

(θ), (A1)

where γ

is the underlying true shear, which is sensitive to cos-

mology only. Extending this to the level of a two-point correlation

between two populations α and β this implies:

obs,αβ

(θ) ≡

obs,α

(θ

)γ

obs,β

(θ

+ θ)

[1 + m

(θ

)][1 + m

(θ

+ θ)]˜γ

(θ

)˜γ

(θ

+ θ)

(A2)

Note that the observed shear used in a particular bin correlation

is now weighted by the overdensity of galaxies in the image, in

addition to the calibration bias, such that

˜γ

(θ) ≡



1 + δ

(θ)



× γ

(θ). (A3)

Expanding each of the terms one ﬁnds:

MNRAS 000, 000–000 (0000)

Cosmic Shear & Galaxy Neighbours 19

obs,αβ

(θ) =

(θ

)γ

(θ

+ θ)

(θ

)γ

(θ

)γ

(θ

+ θ)

(θ

+ θ)γ

(θ

)γ

(θ

+ θ)

(θ

)γ

(θ

)γ

(θ

+ θ)

(θ

+ θ)γ

(θ

)γ

(θ

+ θ)

(θ

+ θ)γ

(θ

)γ

(θ

+ θ)

(θ

)δ

(θ

+ θ)γ

(θ

)γ

(θ

+ θ)

(A4)

The terms contributing to the measured two-point shear correla-

tion, then, is sensitive to both spatial correlations between the m

in different galaxies and to the correlations with the source den-

sity. Note that we’ve chosen to neglect a higher-order (six-point)

term. In reality there will also be a connection between galaxy den-

sity and shear, but we will follow the normal convention and as-

sume the contribution is small enough to be neglected. In simple

terms, an excess in the hmmi term above the product of the mean

m values indpendently could arise because galaxy pairs separated

on small scales tend to come from similar image plane environ-

ments. In contrast the density weighted correlations hδ

mi would

be zero, but for a simple observation; selecting a random galaxy

with a suitable correlation pair at a distance θ is not the same as un-

conditionally selecting a random galaxy. In the small scale bins we

will over-sample the dense regions, where m tends to be larger (see

Section 6.2). The angular brackets here indicate averaging over all

galaxy pairs separated by θ. If we can assume the bias is indepen-

dent of the underlying cosmology the above expression simpliﬁes

signiﬁcantly:

obs,αβ

(θ) = (1 + ¯m

+ ¯m

(θ

+ θ)

(θ

+ θ)

(θ

)δ

(θ

+ θ)

(θ

)δ

(θ

+ θ)

) × ξ

αβ

(θ|p), (A5)

with ξ

αβ

being the true correlation function of cosmological shears

hγ

i, which is contingent on the underlying cosmological param-

eters p. It can be shown that

(θ) ≡



(θ

)γ

(θ

+ θ)





(θ

)γ

(θ

+ θ)





(θ

)γ

(θ

+ θ)





(θ

)γ

(θ

+ θ)



= ξ

(θ) + ξ

(θ), (A6)

and so one can use equation A5 to construct the observed ξ

cor-

relation functions

obs,αβ

(θ) =



1 + ¯m

+ ¯m

(θ

+ θ)

(θ

+ θ)

(θ

)δ

(θ

+ θ)

(θ

)δ

(θ

+ θ)



αβ

(θ|p). (A7)

The i subscript has been discarded here under the assumption that

and m

are approximately equal for a given set of galaxies.

Next, let’s say imagine that we have a measured datavector.

Our measurements are biased, but we’ll assume it is possible to

devise a correction that recovers the true cosmological signal pre-

cisely. Our observed datavector is then just,

obs,αβ

(θ) = Υ

tr,αβ

αβ

(θ|p), (A8)

which follows trivially from equation A7. Since we do not trivially

know Υ

tr,αβ

ab initio (that’s why we need simulations!) we can

only construct a best-estimate approximation. By applying a cor-

rection factor to the raw measurements we construct a best-estimate

datavector:

BE,αβ

(θ) =

BE,αβ

obs,αβ

(θ) =

tr,αβ

BE,αβ

αβ

(θ|p). (A9)

Of course, if our best correction is perfect then the ratio goes to

unity, and we recover the underlying cosmology. Since we apply

corrections to the single-galaxy shears we will assume Υ

BE,αβ

in-

cludes the hδ

i term, but neglects the correlations involving m.

We then can write:

BE,αβ



1 + ¯m

+ ¯m

¯m

(θ

)δ

(θ

+ θ)



(A10)

We can measure the mean bias in each bin that would be obtained

from the calibration directly. As we show in Z17, using the full

DES Y1 HOOPOE catalogues, these biases are ∼ −0.08 to −0.20.

Finally, assume that although m clearly varies between

redhshift bins, the strength of the correlation does not That is, the

bias-bias term is the product of the mean ms (which varies between

z bins) plus a scale dependent shift (which doesn’t). One then has:

(θ

+ θ)

= ¯m

¯m

+ ∆ξ

(θ). (A11)

The additive part can be measured directly from the simulation us-

ing sub-patches, as described earlier. The density-density correla-

tion can be obtained in the same way. This, then, leaves only the

m × δ

cross-correlation. This should vanish in the case of zero

correlation, but it also seems reasonable to assume that the magni-

tude should be proportional to the mean bias ¯m

in a particular bin.

This allows the scale dependent (non-tomographic) cross correla-

tion measured from HOOPOE to be rescaled appropriately for each

bin pair:

(θ

+ θ)



¯m



(θ), (A12)

where ¯m is the global multiplicative bias and ξ

(θ) ≡ hmδ

each measured using all simulated galaxies. Using the above equa-

tions, with our ﬁducial calibration and three measured correlations,

one can derive a scale dependent modiﬁcation to shear-shear two

point correlation data using equation A8.

AFFILIATIONS

Jodrell Bank Centre for Astrophysics, School of Physics and

Astronomy, University of Manchester, Oxford Road, Manchester,

M13 9PL, UK

Institute for Astronomy, University of Edinburgh, Edinburgh

EH9 3HJ, UK

Center for Cosmology and Astro-Particle Physics, The Ohio

State University, Columbus, OH 43210, USA

Department of Physics, The Ohio State University, Columbus,

MNRAS 000, 000–000 (0000)

20 S. Samuroff, S. L. Bridle, J. Zuntz et al

OH 43210, USA

Kavli Institute for Particle Astrophysics & Cosmology, P. O. Box

2450, Stanford University, Stanford, CA 94305, USA

SLAC National Accelerator Laboratory, Menlo Park, CA 94025,

USA

Department of Physics and Astronomy, University of Pennsylva-

nia, Philadelphia, PA 19104, USA

Jet Propulsion Laboratory, California Institute of Technology,

4800 Oak Grove Dr., Pasadena, CA 91109, USA

Department of Physics, ETH Z

urich, Wolfgang-Pauli-Strasse 16,

CH-8093 Z

urich, Switzerland

Department of Physics & Astronomy, University College

London, Gower Street, London, WC1E 6BT, UK

Department of Physics and Electronics, Rhodes University, PO

Box 94, Grahamstown, 6140, South Africa

Fermi National Accelerator Laboratory, P. O. Box 500, Batavia,

IL 60510, USA

LSST, 933 North Cherry Avenue, Tucson, AZ 85721, USA

CNRS, UMR 7095, Institut d’Astrophysique de Paris, F-75014,

Paris, France

Sorbonne Universit

es, UPMC Univ Paris 06, UMR 7095,

Institut d’Astrophysique de Paris, F-75014, Paris, France

Laborat

orio Interinstitucional de e-Astronomia - LIneA, Rua

Gal. Jos

e Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil

Observat

orio Nacional, Rua Gal. Jos

e Cristino 77, Rio de

Janeiro, RJ - 20921-400, Brazil

Department of Astronomy, University of Illinois, 1002 W.

Green Street, Urbana, IL 61801, USA

National Center for Supercomputing Applications, 1205 West

Clark St., Urbana, IL 61801, USA

Institut de F

ısica d’Altes Energies (IFAE), The Barcelona Insti-

tute of Science and Technology, Campus UAB, 08193 Bellaterra

(Barcelona) Spain

Institute of Space Sciences, IEEC-CSIC, Campus UAB, Carrer

de Can Magrans, s/n, 08193 Barcelona, Spain

Department of Physics, IIT Hyderabad, Kandi, Telangana

502285, India

Kavli Institute for Cosmological Physics, University of Chicago,

Chicago, IL 60637, USA

Instituto de Fisica Teorica UAM/CSIC, Universidad Autonoma

de Madrid, 28049 Madrid, Spain

Department of Astronomy, University of Michigan, Ann Arbor,

MI 48109, USA

Department of Physics, University of Michigan, Ann Arbor, MI

48109, USA

Astronomy Department, University of Washington, Box

351580, Seattle, WA 98195, USA

Cerro Tololo Inter-American Observatory, National Optical

Astronomy Observatory, Casilla 603, La Serena, Chile

Santa Cruz Institute for Particle Physics, Santa Cruz, CA 95064,

USA

Australian Astronomical Observatory, North Ryde, NSW 2113,

Australia

Argonne National Laboratory, 9700 South Cass Avenue,

Lemont, IL 60439, USA

Departamento de F

ısica Matem

atica, Instituto de F

ısica, Univer-

sidade de S

ao Paulo, CP 66318, S

ao Paulo, SP, 05314-970, Brazil

Station, TX 77843, USA

Department of Astronomy, The Ohio State University, Colum-

bus, OH 43210, USA

Department of Astrophysical Sciences, Princeton University,

Peyton Hall, Princeton, NJ 08544, USA

Instituci

o Catalana de Recerca i Estudis Avanc¸ats, E-08010

Barcelona, Spain

Centro de Investigaciones Energ

eticas, Medioambientales y

Tecnol

ogicas (CIEMAT), Madrid, Spain

Brookhaven National Laboratory, Bldg 510, Upton, NY 11973,

USA

School of Physics and Astronomy, University of Southampton,

Southampton, SO17 1BJ, UK

Instituto de F

ısica Gleb Wataghin, Universidade Estadual de

Campinas, 13083-859, Campinas, SP, Brazil

Computer Science and Mathematics Division, Oak Ridge

National Laboratory, Oak Ridge, TN 37831

Institute of Cosmology & Gravitation, University of Portsmouth,

Portsmouth, PO1 3FX, UK

MNRAS 000, 000–000 (0000)