One of the papers

MNRAS 430, 197–208 (2013) doi:10.1093/mnras/sts571

Optimizing cosmic shear surveys to measure modifications to gravityon cosmic scales

Donnacha Kirk,1‹ Istvan Laszlo,2 Sarah Bridle1 and Rachel Bean2

1Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK2Department of Astronomy, Cornell University, Ithaca, NY 14853, USA

Accepted 2012 December 6. Received 2012 November 11; in original form 2011 October 25

ABSTRACTWe consider how upcoming photometric large-scale structure surveys can be optimized tomeasure the properties of dark energy (DE) and possible cosmic-scale modifications to Gen-eral Relativity in light of realistic astrophysical and instrumental systematic uncertainties. Inparticular, we include flexible descriptions of intrinsic alignments (IAs), galaxy bias and pho-tometric redshift uncertainties in a Fisher Matrix analysis of shear, position and position–shearcorrelations, including complementary cosmological constraints from the cosmic microwavebackground. We study the impact of survey trade-offs in depth versus breadth, and redshiftquality. We parametrize the results in terms of the Dark Energy Task Force figure of merit,and deviations from General Relativity through an analogous modified gravity (MG) figure ofmerit. We find that IAs weaken the dependence of figure of merit on area and that, for a fixedobserving time, halving the area of a Stage IV reduces the figure of merit by 20 per cent whenIAs are not included and by only 10 per cent when IAs are included. While reducing photo-metric redshift scatter improves constraining power, the dependence is shallow. The variationin constraining power is stronger once IAs are included and is slightly more pronounced forMG constraints than for DE. The inclusion of IAs and galaxy position information reduces therequired prior on photometric redshift accuracy by an order of magnitude for both the fiducialStage III and IV surveys, equivalent to a factor of 100 reduction in the number of spectroscopicgalaxies required to calibrate the photometric sample.

Key words: equation of state – gravitational lensing: weak – cosmological parameters – darkenergy – large-scale structure of Universe.

1 IN T RO D U C T I O N

The dawn of ‘precision cosmology’ was heralded by the results ofsurveys which, for the first time, produced data of sufficient quantityand quality that our cosmological probes could begin to accuratelymeasure some of the fundamental properties of the Universe. Theoutcome has been the � cold dark matter (�CDM) concordancecosmology, a description of the Universe compatible with the jointconstraints from many sources of cosmological information. Thenext decades will see a step-change in our ability to measure cos-mological parameters as new surveys produce orders of magnitudemore data than has previously been available. This will allow us totest the standard cosmological model as never before.

The standard �CDM model describes a Universe made up of∼75 per cent dark energy (DE), a smooth, negative pressure fluid,∼20 per cent dark matter, collisionless massive particles which

� E-mail: [email protected]

interact solely via gravity, and just ∼5 per cent baryons makingup the potentially visible mass of the Universe (Dunkley et al.2009). The standard model also assumes gravity to be described byEinstein’s General Relativity (GR).

A Universe governed by GR and populated by standard gravitat-ing matter cannot explain the observed acceleration. The DE fluidwas proposed to solve this paradox, reviving the idea of a cosmo-logical constant which Einstein had originally included in his GRfield equations and subsequently discarded. Much effort has beendevoted to characterizing the nature of DE – particularly attempts todiscriminate between a pure cosmological constant and a dynamicscalar potential with time-varying equation of state w(a) (Albrechtet al. 2006). In all its forms, DE poses problems of fine-tuning forwhich we have, as yet, no physical motivation.

Rather than invoke DE, an alternative explanation for cosmicacceleration has been proposed – that our standard theory of gravityis incomplete and that a correct theory of gravity would explaincosmic acceleration at late times and large scales in a universepopulated by matter with positive pressure. There are a large number

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198 D. Kirk et al.

of theoretically motivated modified theories of gravity, see Jain &Khoury (2010) for a review. Testing the theory of gravity on cosmicscales will be one of the most interesting opportunities afforded byupcoming survey data.

Weak gravitational lensing (WL) is a particularly useful probe ofgravity because it is sensitive to φ + ψ , the sum of the metric poten-tials. Through the modified growth of structure and geometry via thelensing integral cosmic shear can constrain both the ratio of metricpotentials and modifications to the Poisson equation. In contrast,probes such as galaxy redshift surveys and galaxy peculiar veloc-ities depend only on the Newtonian potential, ψ . WL constraintsalso have the benefit that they probe the dark matter distribution di-rectly, avoiding the impact of galaxy biasing. Joint constraints fromcombining multiple probes can help break degeneracies betweenfree parameters, producing the tightest constraints.

Cosmic shear is the name given to WL in random patches ofthe sky and was first detected a decade ago (Bacon, Refregier &Ellis 2000; Kaiser, Wilson & Luppino 2000; van Waerbeke et al.2000; Wittman et al. 2000). The latest constraints on cosmologycome from the Hubble Space Telescope COSMOS survey (Masseyet al. 2007; Schrabback et al. 2010), and the Canada-France-HawaiiTelescope Legacy Survey (Fu et al. 2008). In addition, the 100square degree survey (Benjamin et al. 2007) combines data fromseveral smaller surveys (Hoekstra et al. 2002; Le Fevre et al. 2004;Hetterscheidt et al. 2007).

Cosmic shear has been identified as the method with the mostpotential to uncover the nature of DE (Albrecht et al. 2006; Peacocket al. 2006) and therefore a number of surveys are planned with amajor cosmic shear component. Upcoming ‘Stage III’ projects in-clude Kilo-degree Survey on the Very Large Telescope (VLT) Sur-vey Telescope (VST), the Panoramic Survey Telescope and RapidResponse System project, the Subaru Measurement of Images andRedshifts survey using HyperSUPRIMECam, the Dark Energy Sur-vey (DES) on the Blanco Telescope. More ambitious ‘Stage IV’imaging projects are the Large Synoptic Survey Telescope ground-based project and in space the proposed European Space Agencymission Euclid and the NASA proposed Wide-Field Infrared SurveyTelescope.

Greater accuracy demands better treatment of systematic effects.One of the main systematics in cosmic shear studies is the con-tamination by the intrinsic alignment (IA) of galaxy shapes. Thissystematic manifests in two main ways: in the first, physically closegalaxies have preferentially aligned ellipticities, this is known as theintrinsic–intrinsic (II) correlation. In the second, galaxies close onthe sky separated radially are anticorrelated as the foreground grav-itational potential shapes the closer galaxy while also gravitation-ally lensing the background galaxy, this is called the gravitational–intrinsic (GI) correlation. The former adds to the cosmic shear signalwe wish to measure, the latter subtracts from it. In an earlier pa-per (Laszlo et al. 2012), herein ‘LBKB’, we established how, fora fixed survey specification, the inclusion of a realistic IA modeland the inherent uncertainties in the model significantly degradethe constraining power of a cosmic shear survey for DE and mod-ified gravity (MG) cosmological models. This work deepened theconclusions of Bernstein (2009) and Joachimi & Bridle (2010), andexpanded them into the MG domain.

What is encouraging, however, is that the use of position–position correlations, nn, and particularly position–shear, nε, cross-correlations can go some way towards mitigating the impact of IAs.These data are already collected by a standard WL survey. In thepresence of IAs, constraints from εε + nε + nn are roughly twiceas strong as those from εε alone. Until now, the discussion of op-

timal survey parameters has generally assumed that εε correlationsalone will be included, without IAs. In this paper, we investigate theimpact of survey strategy and design on different combinations ofprobes, with and without IAs. This is a continuation of the cosmicshear survey optimization work begun by, among others, Amara &Refregier (2007), Bridle & King (2007), Joachimi & Bridle (2010),Ma, Hu & Huterer (2006), Huterer et al. (2006) and Huterer et al.(2006).

In this paper, we expand on the work on LBKB to fully under-stand how the observing strategy of a given survey is central tothe type of data and the quality of results that survey will produce.Different survey geometries with the same instrument will producedifferent statistical errors due to a different balance between surveycharacteristics, the most important for cosmic shear being surveyarea galaxy number density on the sky and median redshift of thegalaxy distribution. Other properties such as required redshift ac-curacy are important guidelines for instrument designers and thosepreparing analysis pipelines and follow-up or calibration studies(Amara & Refregier 2007). While more area and increased numberdensity will always be desirable, it is important to keep firm goalsin mind when producing desiderata for future surveys which willalways be limited by technology and finite observing time. In partic-ular, learning about DE and MG may benefit from different surveystrategies and call for the prioritization of different properties.

The paper is organized as follows: in Section 2, we summarizeour standard cosmology, fiducial surveys and models for deviationsfrom GR, IAs and bias parametrization, as well as our figures ofmerit (FoMs) for DE and MG. In Section 3, we investigate thesensitivity of DE and MG constraints to changes in survey spec-ification. In Section 3.1, we present the effect of varying surveyarea and the impact of finite survey time on overall strategy andthe ability to constrain MG. The importance of photometric redshiftaccuracy is addressed in Sections 3.2 and 3.3. Conclusions are madein Section 4.

2 C O S M O L O G I C A L S E T-U P

The paper deals with the impact of various aspects of survey strat-egy and redshift quality on the power of cosmic shear and galaxyposition information to constrain DE and deviations from GR. InSection 2.1, we summarize our cosmic shear formalism, Section 2.2extends it to include IAs and Section 2.3 adds galaxy position auto-and cross-correlations and introduces a coherent biasing formalismfor IAs and galaxy bias. Section 2.4 reviews the MG parametriza-tion we use and how it enters our angular power spectrum inte-grals. In Section 2.5, we summarize some basic cosmology we usethroughout the paper and Section 2.6 describes the fiducial surveyparameters we use in the following sections.

2.1 Cosmic shear

Cosmic shear is the shape-distortion induced in the image of adistant galaxy due to the bending of its light by gravity as it passesmassive structure in the Universe. If we assume that GR holds, thenwe can define the cosmic shear angular power spectrum under theLimber approximation as

CGGij (l) =

∫dχ

χ2Wi(χ )Wj (χ )Pδδ(k, z), (1)




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Optimizing WGL surveys for modified gravity 199

where Pδδ(k, z) is the three-dimensional matter power spectrum,χ is the comoving distance in units of h−1 Mpc and W(χ ) is thelensing efficiency function

Wi(χ ) = 4πG

c2ρ(z)a2(z)χ

∫dχ ′ni(χ

′)χ ′ − χ

χ ′ . (2)

2.2 Intrinsic alignments

The IA of galaxy ellipticities is a prime contaminant to the measuredcosmic shear signal. A naive approach to cosmic shear assumes thatgalaxy’s intrinsic ellipticities are randomly distributed on the sky so,when we average over observed ellipticity in a small patch, intrinsicellipticity cancels and we are left with the induced shear.

Unfortunately, this assumption is invalid because galaxy elliptic-ities are aligned due to two effects arising from the same physical IAorigin. Physically close galaxies tend to align with the local gravi-tational tidal field and so are positively correlated with each other,this is the II alignment. Background galaxies can have their lightlensed by foreground gravitational fields which align the intrinsicellipticity of foreground galaxies. This induces an anticorrelationand is the GI alignment.

We follow the procedure of LBKB and implement IAs usingan updated version of the corrected LA model of Hirata & Seljak(2004), incorporating their correction of erratum 2010 and assum-ing that all IA physics occurs at a high-z epoch of galaxy formationhence the IA signal depends only on the linear matter power spec-trum rather than the subsequent non-linear evolution. The rest ofthe LA model enters as a factor of

bI = −C1ρm(z = 0), (3)

where ρm(z = 0) is the matter density today and C1 = 5 ×10−14(h2 M� Mpc−3)−1 is the amplitude of the IA term, normal-ized to redshift zero (Bridle & King 2007). This factor appears inthe IA projected angular power spectra, linearly in the GI correla-tion and squared for the II. The window function associated withIAs is the galaxy redshift distribution, n(z), see Table 1 for the fullangular power spectra equations.

In Section 2.3, we expand our notation to allow nuisance pa-rameters to parametrize our knowledge of IAs, galaxy bias andtheir cross-correlations. In this more general case, bI given in equa-tion (3) becomes the fiducial value of a variable IA term whichappears quadratically in the II power spectrum and linearly in theGI and gI power spectra.

The total observed lensing signal is then the sum of the cosmicshear and the IA terms

Cεεl = CGG

l + CIIl + CGI

l . (4)

2.3 Galaxy position data

A cosmic shear survey contains galaxy position information (an-gular position on the sky and redshift) as well as measurementsof galaxy shear. Joachimi & Bridle (2010) provide a formalismfor including this additional information in cosmological parameterconstraints and show how this extra information can serve to par-tially mitigate the impact of IAs. This approach follows from thework of Zhang (2010) and Hu & Jain (2004).

The extra observables we use are galaxy position–position powerspectra and the position–shear cross-spectra, defined analogouslyto the shear–shear power spectrum:

Cnnl = C

ggl (5)

Cnεl = C

gIl + C

gGl . (6)

We now assign nuisance parameters to each component Cl in aself-consistent way as explained in LBKB. The four bias functionare bg, bI, rg and rI. bg and bI model bias from galaxy position andIAs amplitudes, and their cross-correlations are modelled by rg andrI. This unified approach was first introduced by Bernstein (2009).Table 1 summarizes the full power spectra equations, consistentwith the definitions in LBKB eqns. 37, 38, 43 and 44.

We let each bias parameter vary in amplitude and as a functionof scale and redshift using an Nk × Nz grid of free parametersinterpolated over k-, z-space, i.e. each nuisance factor, bX = AxQX(k,z) is the product of a variable constant amplitude parameter, AX, anda variable grid, QX(k, z) in k and z. Throughout this paper, thegrid size is set to the fiducial value of Nk = Nz = 5, which meansmarginalization over 104 nuisance parameters when the full εε +nε + nn probe combination is considered in the presence of IAs.The nuisance parameters multiply linearly into the angular powerspectra integrands and each power spectrum depends on a subset ofthe bias parameters as follows:

CGGl : − , C

gGl : bgrg

CIIl : bI bI , C

gIl : bgbI rgrI

CGIl : bI rI , C

ggl : bgbg.

(7)

Table 1. Summary of the projected angular power spectra considered in this work.

Correlation 2D PS

Shear CGGij (l) = ∫ dχ

χ2 Wi (χ )Wj (χ )[Q(z) 1+R(z)

2

]2Pδδ(k, z)

Intrinsic-shear CGIij (l) = ∫ dχ

χ2 Wi (χ )nj (χ )Q(z)Q(zf )R(zf )(1 + R(z)2 )bI (k, z)rI (k, z)

√Pδδ (k,zf )Pδδ (k,z)

D(zf )

Intrinsic CIIij (l) = ∫ dχ

χ2 ni (χ )nj (χ )b2I (k, z)Q2(zf )R2(zf )Pδδ(k, z)

Galaxy clustering Cggij (l) = ∫ dχ

χ2 ni (χ )nj (χ )b2g(k, z)Pδδ(k, z)

Clustering-shear CgGij (l) = ∫ dχ

χ2 ni (χ )Wj (χ )Q(z) 1+R(z)2 bg(k, z)rg(k, z)Pδδ(k, z)

Clustering-intrinsic CgIij (l) = ∫ dχ

χ2 ni (χ )nj (χ )Q(zf )R(zf )bg(k, z)bI (k, z)rg(k, z)rI (k, z)√

Pδδ (k,zf )Pδδ (k,z)D(zf )

Galaxy ellipticity (observable) Cεεij = CGG

ij + CIIij + CGI

ij

Galaxy number density (observable) Cnnij = C

ggij

Number density–ellipticity (observable) Cnεij = C

gIij + C

gGij




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200 D. Kirk et al.

As mentioned in the previous section, the fiducial value of bI is givenby equation (3). bg has fiducial value 1 while rg and rI vary aroundthe fiducial value 0.9 to avoid more than perfect cross-correlation.

We have ignored the effect of lensing magnification and followedLBKB and Joachimi & Bridle (2010) in applying a cut on multipolel to any redshift bin combination ij which includes galaxy positioninformation (i.e. nn or nε) according to max(i) = kmax

lin (z(i)med)χ (z(i)

med).This aims to account for uncertainties in the galaxy bias model atsmall scales. For nε bin pairs, the cut is made on the galaxy, i.e. n,bin. For position–position, nn, pairs there is a choice of bin on whichto apply the cut. We follow Joachimi & Bridle (2010) in makingthe optimistic choice and cutting on the higher redshift bin. Shear–shear, εε, pairs do not depend on galaxy bias and are therefore usedup to the full range in l.

2.4 Modified gravity

There are a large number of modifications or extensions of Einstein’sgeneral theory of relativity on cosmic scales which come under thegeneral heading of MG theories. These can be motivated by thepresence of extra dimensions, as in Dvali, Gabadadze & Porrati(2000), or extra degrees of freedom compared to the GR actionequation, as in f(R), TeVeS, etc. (Dvali, Gabadadze & Porrati 2000;Carroll et al. 2006; Skordis 2009; Jain & Khoury 2010). As inLBKB, we do not assume a particular modified theory of gravitybut rather concentrate on ‘trigger parameters’ whose deviation fromtheir GR values would indicate the presence of some physics beyondthat in the standard GR picture.

In the conformal Newtonian gauge the metric for a flat FRWspace–time is written

ds2 = −a(τ )2 [1 + 2ψ(x, t)] dτ 2 + a(τ )2 [1 − 2φ(x, t)] dx2, (8)

where ψ and φ are the scalar potentials which describe perturba-tions to the time- and space-parts of the metric, respectively. Weparametrize deviations from GR through two parameters, Q and R.One alters the way the Newtonian potential responds to mass viathe Poisson equation,

k2ψ(x, t) = −4πGQρa2δ (9)

and the other modifies the ratio of the metric potentials

ψ(x, t) = Rφ(x, t). (10)

There are a variety of similar parametrizations in the literature, weare following the formalism of Bean & Tangmatitham (2010) whichwas also used in Laszlo et al. (2012).

Q and R are assumed to be scale independent and vary withredshift as

Q = (Q0 − 1)as (11)

R = (R0 − 1)as, (12)

where a is the scale factor and Q0, R0 are the free parameters wevary for MG, and s = 3. We are interested in MG theories whichexplain the accelerating expansion of the Universe observed at latetimes. s = 3 allows any modification to ‘turn-on’ and late times,and avoids violating early Universe constraints from the cosmicmicrowave background (CMB) and big bang nucleosynthesis.

The MG parameters enter the projected angular power spectrain different combinations, as shown in Table 1, due to their differ-ent dependencies on the mater density field. As well as modifyingthe angular power spectra, Q and R enter the angular power spec-trum integrals via their response to the metric potentials through

the linear growth function. The potential for weak lensing to con-strain deviations from GR has previously been noted in Bean &Tangmatitham (2010), Laszlo & Bean (2008) and Beynon, Bacon& Koyama (2010).

The full projected angular power spectra, including IAs, MG andbias parameters, is summarized in Table 1. Note also that deviationsfrom GR will also manifest in the growth function which producesthe matter power spectra in Table 1. We include modifications togrowth via a ratio of MG/GR power spectra calculated using themodified version of CAMB used for LBKB. For ease, LBKB alsoprovide a fitting function to compute this ratio over a range of Q0,R0 and s values.

2.5 General cosmology

Throughout this paper, we assume a flat �CDM cosmologi-cal model with fiducial parameter values equal to the WilkinsonMicrowave Anisotropy Probe 7 best-fitting values: �m = 0.262,�b = 0.0445, w0 = −1, wa = 0, σ 8 = 0.802, h = 0.714, ns =0.969, �ν = 0. Where �m,�b and �ν are the dimensionless matter,baryon and neutrino densities, respectively, w0 and wa are the DEequation of state parameters (Albrecht et al. 2006), σ 8 is the normal-ization of the linear matter power spectrum, h is the dimensionlessHubble parameter today and ns is the power law of the primor-dial power spectrum. The linear matter power spectrum is givenby the fitting formula of Eisenstein & Hu (1998), with non-linearcorrections from Smith et al. (2003).

When we treat MG, we assume that the background expansion ofthe Universe is consistent with our fiducial DE model and deviationsdue to non-GR physics enter through the growth of structure frominitial perturbations. Unless otherwise specified all plots includeCMB priors from a Planck-type survey as described in LBKB. Wedivide our survey galaxy redshift distributions into redshift sliceswith equal galaxy number density to allow redshift tomographywhere auto- and cross-correlations of these redshift bins are consid-ered, allowing us to measure redshift evolution of the observables.

We calculate constraints on cosmological parameters using theFisher Matrix formalism,

Fμν =Nd∑m,n

Nmaxl∑l

∂Dm(l)

∂pμ

Cov−1mn(l)

∂Dn(l)

∂pν

, (13)

where m, n label tomographic redshift bins, Dm(l) is the data vector,in our case some combination of the angular power spectra Cεε

l , Cnεl

and Cnnl . Covmn(l) is the covariance matrix, defined as in Joachimi

& Bridle (2010), pμ are the parameters varied and Nd is the numberof independent combinations of tomographic bins. Unless other-wise stated the Fisher matrix varies the cosmological parameters:p = {�m, w0, wa, h, σ8,�b, ns}.

Using the Fisher matrix formalism, the lower limit on the min-imum variance bound on the error of a parameter pμ marginal-ized over all other parameters of interest, is given by σ (pμ) =√

(F−1)μμ. We quote results in terms of the DETF FoM for DE

FoMDE = 1√det

[(F−1

GR )w0,wa

] . (14)

Here (F−1GR )w0,wa is the 2 × 2 submatrix of the inverted Fisher

matrix for cosmological and nuisance parameters excluding theMG parameters are excluded (these are fixed at their GR values).The FoM is proportional to the inverse of the area of the constraintcontour in w0−wa space.




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By analogy, we define a MG FoM as

FoMMG = 1√det

[(F−1)Q0,Q0(1+R0/2)

] , (15)

where we have inverted the full Fisher Matrix including the MGparameters and DE parameters.

2.6 General survey parameters

LBKB investigated the degeneracy between IAs and MG parametersfor a fixed survey specification. It was found that the inclusion ofa realistic IA model reduced the FoM (DE or MG) for a typicalStage IV cosmic shear survey by ∼70 per cent. The effect could besubstantially mitigated by the inclusion of galaxy-position data andgalaxy-shear cross-correlations. The constraining power of εε +nε + nn, including IAs, is roughly double that of εε, including IAs,alone. We showed that our results were robust to different numbersof free nuisance parameters, accounting for flexibility in the galaxyclustering and IA bias models. The results were broadly similar forattempts to constrain DE equation of state parameters and deviationsfrom GR.

In this paper, we vary certain survey parameters around a fiducialcosmic shear survey set-up, corresponding to Stage IV project. Thissurvey has an area = 20 000 deg2, number density of galaxies pro-jected on to the sky, ng = 35 arcmin−2 and galaxy intrinsic ellipticitydispersion, σγ = 0.35, a Gaussian photometric redshift scatter ofwidth δz = 0.05 which is related to the (redshift-dependent) rmsphotometric dispersion via σ z(z) = (1 + z)δz, with a fraction ofcatastrophic outliers, fcat = 0. We analyse the results using cos-mic shear tomography with 10 tomographic redshift bins of equalnumber density.

The redshift distribution of galaxies n(z) is assumed to be givenby a Smail-type distribution (Smail, Ellis & Fitchett 1994)

n(z) ∝ zαexp

(− z

z0

)β

, (16)

with α = 2, β = 1.5 and z0 = zm/1.412, where the median redshift,zm = 0.9.

For comparison purposes, some results are also presented for theStage III DES, which saw first light in 2012. Table 2 summarizesthe specifications for both surveys.

Table 2. Fiducial survey parameter values for the DES-like (Stage III) and Euclid-like (Stage IV) surveys.Note that zmedian is related to z0 in equation (16) asz0 = zmedian/

√2.

Fiducial survey paramters

Parameter Stage III Stage IV

Area 5000 deg2 20 000 deg2

ng 10 35σγ 0.23 0.35Nz 5 10δz 0.07 0.05fcat 0 0�z 1 1zmedian 0.8 0.9α 2 2β 1.5 1.5

3 SU RV E Y SP E C I F I C AT I O N S A N D F O R E C A S TC O N S T R A I N T S

In this section, we analyse the sensitivity of the prospective DE andMG FoMs on variations in survey specification, principally focusingon their relationship to the survey area and depth, and the precisionof the photometric redshift measurements when uncertainties inhow IAs are included in the modelling.

Throughout, we quantify constraints using the DE and MG FoMsin (14) and (15) normalized relative to a ‘baseline’ FoM for theshear-only autocorrelations ‘GG’, with IAs excluded, using the fidu-cial survey described in Table 2.

3.1 Survey area

Figs 1 and 2 present the key results showing how the constrainingpower of a survey, for DE equation of state and MG parameters,varies with the survey area. We consider two alternative approaches,first a simple option where the increased area derives from an in-crease in survey time, and secondly a more realistic option whensurvey time is fixed, and increased survey area is achieved by areduction in survey depth/limiting flux.

Let us first consider the case where all survey parameters arekept fixed except for the survey area. We focus on the quantitativeimplications for the Stage IV survey, the results for a Stage IIIsurvey, presented in Fig. 2 are qualitatively similar.

When IAs are excluded from the analysis the improvement inFoM with increasing area can be roughly described by a power-law dependence FoM∝Areax with x = 0.4 and 0.7 for CMB+shearcorrelations alone for the DE and MG FoMs. The MG FoM isslightly more sensitive to changes in area than the DE FoM. TheDE index is lower than that reported in Amara & Refregier (2007)only because of the inclusion of, survey area independent, CMBpriors.

As discussed in LBKB, including IAs in the analysis deterioratesboth the DE and MG FoMs, with the MG FoM more negatively im-pacted. For the fiducial Stage IV survey, the FoMs are reduced by 60and 65 per cent relative to when IAs are excluded, consistent withLBKB. When IAs and their associated uncertainties are marginal-ized over, the power-law scaling is weaker, with x = 0.35 for DE, sothat the relative degradation of the FoM becomes more pronouncedas survey area increases. This reflects that the IAs are astrophysi-cal systematics that are not significantly removed by increasing thesurvey area.

Because of the uncertainties in the galaxy bias, through the grid-ded bias and cross-correlation coefficients, bg and rg, the cosmo-logical information from the galaxy position data is massively sup-pressed. For the fiducial full survey area FoMs for CMB+nn are 22and 1 per cent of the CMB+pure lensing signal and for the scenarioswith and without IAs, respectively.

If one adds in galaxy position autocorrelations to shear autocor-relations, ‘nn + εε’, we find a 12 and 36 per cent improvement inthe DE and with and without IAs, respectively, and 19 and 40 percent for MG FoMs. While the cross-correlations, ‘nn + nε + εε’,give only a small improvement when IAs are excluded, they providean 34 per cent improvement on ‘nn + εε’ when uncertainties in theIAs are included. This results from the nε correlations breaking thedegeneracies between the cosmological and IA amplitude nuisanceparameters.

In reality most surveys have a fixed observing time. This in-evitably leads to a trade off between total survey area and surveydepth, which impacts both number density of lensed galaxies, ng,




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202 D. Kirk et al.

Figure 1. Figures showing the variation with the survey area of a relative Stage IV survey FoM for dark energy (DE) parameters w0, wa (blue lines), andmodified gravity (MG) parameters Q0, Q0(1 + R0)/2 (red lines), as described in (14) and (15), relative to a ‘baseline’ FoM. The baseline includes shearautocorrelations ‘GG’ alone, excluding IAs, over a 20 000 square degree survey. All FoMs contain priors from a Planck-like CMB survey. The figures showrelative FoMs for an optimistic scenario, in which uncertainties in the IA model have been excluded (top panels) and a conservative scenario, in whichuncertainties in the IA model are marginalized over using an Nk = Nz = 5 gridded bias model (lower panels). The baseline model (relative FoM = 1) is shownas a black cross in the top panels. We consider the impact on the relative FoM of increasing survey area, by increasing survey time for fixed limiting magnitude(left-hand panels) and fixing the survey time to trade-off survey area and depth (right-hand panels). Four data combinations are considered in each panel,shear–shear correlations ‘εε’ (dotted lines), galaxy–galaxy positions ‘nn’ alone (dot–dashed), ‘εε+nn’ (dashed) and when shear–position cross-correlations‘nε’ are included (full lines). For the galaxy position correlations a Nk = Nz = 5 gridded bias model is used throughout.

and median redshift achieved, zm. To implement the fixed surveytime option, we use a fitting formula for the number of galaxies perunit redshift per square arcminute,

n(z) = �0 × 3z2

2z30

exp

[−

(z

z0

)3/2]

(17)

(e.g. Baugh & Efstathiou 1993) and we interpolate the numbersfrom table 1 of Blake & Bridle (2005) using

z0 = 0.055(rlim − 24) + 0.39

�0 = 35400

602

( rlim

24

)19, (18)

where rlim is the limiting apparent magnitude in the SDSS r filterand �0 is an overall surface density in units of deg−2. We use (18)to scale the number density of galaxies with area relative to thefiducial values. We estimate the change in the limiting magnitude,�mag, with area in the usual way as

�mag = −1.25 log10

(A

Afid

). (19)

As one might expect, the benefits of going to larger survey areaare now less strong because of the lost depth, with the shear-onlypower-law dependence weakening to x = 0.3 for DE and 0.35 forMG. The baseline result for cosmic shear analyses is that it is stillbetter to go to larger survey area despite the lost depth. Fig. 4




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Figure 2. Same as in Fig. 1 but for a Stage III large-scale structure survey with fiducial survey area of 5000 square degrees.

of Amara & Refregier (2007) suggests that, in the absence of aCMB prior and IA modelling uncertainties, the FoM is roughlyproportional to the square root of the area for fixed observing time.We find that with the inclusion of CMB priors, we obtain a similarresult although the dependence on area is slightly weakened as inthe constant depth case. When the galaxy position information isadded the same story holds, but the dependence on area is slightlystronger.

When IAs are added the improvement with increased area issignificantly reduced, particularly in the case of εε-only whichreduces to a power law of x ∼ 0.15 for DE. A survey with anarea of 5000 deg2 gathers ∼80 per cent of the DE information ofa ∼20 000 deg2 survey. This is because IAs dominate over cosmicshear at very low redshift, and by marginalizing out the IA nuisanceparameters, we effectively remove the information at low redshift.Therefore, we expect any cosmological FoM to eventually reducein the (unphysical) limit of very large area and small depth.

The galaxy clustering alone constraints are insensitive to the IAsand therefore increase as usual. When using the full εε + nε +nn data set there is more benefit to increased area even with the

trade-off of shallower depth, although in this case a 5000 squaredegree survey still gathers ∼70 per cent of the information in an∼20 000 deg2 survey. Note that the exact details of these conclusionswill depend on the fiducial IA model and the number of IA and biasnuisance parameters marginalized over. Without galaxy position–shear cross-correlations the constraints bear out the same trend inrelative FoM with area, just with an overall lower magnitude.

We repeated the exercise using the default survey parametersfor a Stage III survey in Fig. 2. We vary the survey area for fixedsurvey time, and cover the same range as in Fig. 1. The surveydepth for each survey time is correspondingly shallower, so wehave 10 galaxies per square arcminute for the 5000 square degreesurvey area. The left-hand plots in Fig. 2 tell a similar story tothose in Fig. 1. When a constant survey time is considered thereis a more pronounced tailing off of improvement with area above∼5–10 000 deg2 than was the case for a Stage IV survey. As thearea is increased above 5000 deg2 the number density of galaxiesfalls from its fiducial value of 10 arcmin−2. For larger areas thenumber density becomes very low and shot noise rapidly increasesat small scales. The loss of information from non-linear scales has




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204 D. Kirk et al.

a strong impact on the constraining power of the survey, limitingthe improvement with increasing area. The fact that nn alone doesnot display this behaviour, displaying the same behaviour for fixeddepth or survey time, supports our analysis – the galaxy positioninformation is already subject to a stringent cut on l, removing thenon-linear information and protecting nn from the impact of shotnoise.

3.2 Photometric redshifts

Accurate redshift information is essential for weak lensing tomog-raphy. As survey size increases the number of redshifts becomesof the order of hundreds of millions to billions. The expense ofacquiring spectroscopic redshifts for this number of galaxies is pro-hibitive. As a result the next generations of WL surveys will rely onphotometric redshifts.

In this paper, we follow the photometric redshift model of Amara& Refregier (2007), including the statistical dispersion of measuredphotometric redshifts and catastrophic failures in redshift measure-ment. The statistical dispersion is modelled assuming that measuredphotometric redshifts zphot are described by a Gaussian probabilitydistribution centred on the true redshift, zt. The width of the Gaus-sian is the photometric redshift error which we assume scales as(1 + z) and is controlled by a parameter δz,

Pzphot (z) = 1√(2π)δz(1 + z)

. (20)

In addition to this uncertainty in the photometric probability dis-tribution function (PDF) galaxy redshifts can be entirely misidenti-fied if, for example, the wavelength range examined is insufficientto identify important features in the spectrum. A misidentified red-shift can be said to be assigned a redshift offset by an amount �z

from the true redshift. Affected galaxies are known as catastrophicoutliers.

We construct a PDF to describe these catastrophic outliers, Pzcat ,and combine it with the scatter PDF, writing the full probabilitydistribution

P (zphot|zt ) = (1 − fcat)Pzphot (zt ) + (fcat)Pzcat (zt ), (21)

where fcat is the fraction of our measured galaxies which suffercatastrophic redshift estimation failures.

The distribution of Pzcat is bimodal, reflecting the fact that galaxiescan either be misidentified as being higher or lower than their trueredshifts,⟨Pz±

cat

⟩= z±

cat = zt ± �z, (22)

where Pzcat = Pz−cat

+ Pz+cat

. The value of �z will depend on thesurvey filters, photometric reconstruction technique and spectralproperties of the survey galaxies. We assume a fiducial value of�z = 1 and that the uncertainty on the PDF of the population ofcatastrophic outliers is the same as for the slightly scattered galaxies

σ(Pz±

cat

)= δz

(1 + z±

cat

). (23)

Fig. 3 shows the variation in FoMs with δz and Nz, the numberof tomographic redshift bins, for different combinations of observ-ables, with and without IAs. There is a parallel between varying δz

and Nz, lower redshift error focuses the redshift distribution per bininto a tighter shape, allowing us to learn from changes with redshiftover shorter intervals. Somewhat equivalently, increasing the num-ber of tomographic bins gives us, in principle, the opportunity todetect changes with redshift at higher resolution.

Increasing δz decreases the constraining power of a survey be-cause the loss of redshift precision ‘smears out’ the redshift dis-tribution of our sample. In particular, galaxies leak between tomo-graphic redshift bins meaning that we can extract less informationfrom tomography. The impact of increased δz on constraining DE isrelatively weak. For εε alone the DE FoM drops by only 1/4 fromδz = 0 (spectroscopic, i.e. perfect redshift precision) to δz = 0.2, al-ready a much higher value than the fiducial redshift uncertainties oftypical Stage III (δz = 0.07) and Stage IV (δz = 0.05) experiments.MG FoMs display very similar behaviour.

The reasons for this behaviour are relatively well understood.The lensing integral (equation 1) modulates the 3D matter powerspectrum by the lensing weight functions Wi(χ ) of the redshiftbins being correlated. These lensing weight functions are broadin redshift and act as kernels smoothing the redshift informa-tion. This effect limits the usefulness of increasingly accurate red-shift information and explains why the trend with δz is relativelyshallow.

The inclusion of galaxy position information produces slightlysteeper trends with increased δz because their angular power spectraintegrals are modulated by the redshift distribution ni(χ ) of thebin(s) in question. This is a narrower function than the lensingweight function, providing more scope for improvement in accuracywith better redshift information.

The slightly steeper curves for the MG FoMs may come fromthe larger number of parameters that need to be constrained in thismodel, which can be done with new types of information such asincreased redshift resolution. Broadly the results for MG are verysimilar to those for DE.

All of the data combinations with varying number of tomographicredshift bins, Nz, show the characteristic plateaux behaviour as Nz

increases. Without IAs, increasing the number of tomographic binsbecomes much less effective after Nz ∼ 2–3. After this number εε iseffectively flat, while εε + nn and εε + nε + nn flatten out slowly,gaining little benefit after Nz ∼ 10–12. This behaviour is expected asthe lensing kernel and photo-z scatter smear out redshift informationand limit the ability of ever-finer tomography to improvement ourknowledge of redshift behaviour.

The inclusion of IA terms for the same probes shows the expecteddecrease in constraining power as we have to marginalize over IAbias terms. As we have seen previously, the MG FoMs take moreof a hit due to the IA effects than their DE counterparts.

The decrease in FoM with increasing δz is significantly morepronounced once IAs are included, with εε including IAs losing67 per cent of its constraining power for DE over the interval probed.Qualitatively, we might hope to understand this from the fact thatthe cosmic shear and each IA term, II and GI, have a different red-shift dependence. Improved redshift knowledge allows us to betterdiscriminate between the cosmic shear signal and IA contamination.This produces stronger constraints on the IA nuisance parametersand consequently tighter constraints on DE and MG.

Similarly, the presence of IAs for increasing Nz makes the plateaubehaviour less pronounced and pushes the levelling off of the FoMback towards Nz ∼ 6–10 for all probes. The use of smaller slicesin redshift space gains information about the z-distribution of theIA signals and aids our constraining power. The effect is curtailedby the same fundamental limit of redshift accuracy in the projectedangular power spectrum. Again the hit on FoM due to IAs is morepronounced for MG than DE but the qualitative behaviour of FoMwith Nz in the presence of IAs is the same for both.

The effect of changing the fraction of catastrophic outlier red-shifts in our survey sample is shown in Fig. 4. As we would expect




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Figure 3. Figures showing the variation with photometric redshift error (δz, left-hand panels) and number of tomographic bins (Nz, right-hand panels) of aStage IV survey FoM for dark energy (DE) parameters w0, wa (blue lines) and modified gravity (MG) parameters Q0, Q0(1 + R0)/2 (red lines), as describedin (14) and (15), relative to a ‘baseline’ FoM (black cross). All FoMs contain priors from a Planck-like CMB survey. The figures show relative FoMs for anoptimistic scenario, in which uncertainties in the IA model have been excluded (top panels) and a conservative scenario, in which uncertainties in the IA modelare marginalized over using an Nk = Nz = 5 gridded bias model (lower panels). Three data combinations are considered in each panel, shear–shear correlations‘εε’ (dotted lines), ‘εε+nn’ (dashed) and when shear–position cross-correlations ‘nε’ are included (full lines). For the galaxy position correlations an Nk =Nz = 5 gridded bias model is used throughout.

increasing the fraction of completely misestimated redshifts makesfor poorer constraints on both MG and DE, for similar reasons tothe impact of δz.

All the lines in Fig. 4 are relatively flat, suggesting that theprojected nature of the angular power spectra and the effect of thelensing weight function continue to limit the impact of redshiftinformation on overall constraints. No probe combination losesmore than 30 per cent of its DE or MG FoM as fcat increases from0 to 0.2 which is more than twice the typical requirement value ofprojected Stage IV surveys.

The general pattern that MG FoMs decrease slightly morestrongly than their DE equivalents with the addition of IAs is re-peated here. One interesting feature is that the MG constraints for εε

without IAs and all the probe combinations with IAs decrease moresteeply with increasing fcat than their DE counterparts. The differ-

ence is most pronounced for εε with IAs where the MG constraintsfalls by twice as much over the fcat range as the DE FoM.

3.3 Redshift priors

As well as describing the change in FoMs with the size of thephotometric uncertainty, it is important to the quantify the effect ofour degree of knowledge of that uncertainty on our FoMs. In ourcase, we follow the standard procedure of inserting this degree ofknowledge as a prior on the parameter δz.

To test this assumption, we extend our photometric redshift model– breaking the redshift range 0–3 into 30 bins in z of width 0.1,following Ma et al. (2006). Note that these bins are used only toparametrize our photo-z model, we continue to use the standard 10bins for shear tomography. We allow δz to vary independently in




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206 D. Kirk et al.

Figure 4. Figures showing the variation with fraction of catastrophic red-shift outliers (fcat) of a Stage IV survey FoM for dark energy (DE) parametersw0, wa (blue lines) and modified gravity (MG) parameters Q0, Q0(1 + R0)/2(red lines), as described in (14) and (15), relative to a ‘baseline’ FoM (blackcross). The figures show relative FoMs for an optimistic scenario, in whichuncertainties in the IA model have been excluded (left-hand panel), and aconservative scenario, in which uncertainties in the IA model are marginal-ized over using an Nk = Nz = 5 gridded bias model (right-hand panel).Notation is the same as in Fig. 3.

each of these z-bins and introduce a new parameter zbias, the amountby which the photometric distribution of redshifts is offset from thetrue redshift. zbias is allowed to vary independently in each of our30 z-bins. Each δz has the standard fiducial value of 0.05 and eachzbias has fiducial value zero. We have enlarged our Fisher Matrix by60 new free parameters, all of which are marginalized over in theresults given in Fig. 5.

We assume that the Gaussian prior assigned to each of the 60 freeparameters is the same and we allow the prior to vary from a widthof 0.0001 to 10. Clearly the constraining power a given surveyachieves decreases as we reduce our amount of prior knowledgeabout the redshift distribution.

Apart from the expected decrease with wider priors we see twodistinct regimes – relatively flat FoMs below prior 0.001 and above1 with a transition regime in between. It is reassuring to note thatwe cannot achieve arbitrarily large FoMs with tighter redshift priors– this is connected to the previously noted ‘smearing’ of redshiftinformation in the shear angular power spectrum. At large valuesof the prior width, we reach a ‘self-calibration’ regime in whichthe information in the survey is sufficient to constrain the unknownparameters (Huterer et al. 2006).

When IAs are ignored, the decrease in constraining power ismore pronounced. Mostly, the lines start higher for narrow priorsand converge to a similar low FoM for wide priors. It is probablethat this behaviour is an artefact of there being fewer nuisanceparameters when IAs are ignored. This produces strong constraintsin the narrow prior regime, while for wide priors redshift uncertaintybecomes the dominant source of error, and probe combinationswith/without IAs are more similar in terms of constraining power.

What is also clear is that the DE FoM falls off much more steeplyover the prior range than the MG FoM. This is true for all probecombinations, with and without IAs but is most spectacular for thelines which include IAs, here there is actually a cross-over, withthe MG FoM ending higher than the DE FoM at prior = 10. Forεε, including IAs, we observe a factor of 10 decrease in DE FoMbut a factor of 3 decrease in MG FoM. This trend is repeated whenall data are used (εε + nε + nn). We expect the DE parameterwa to be particularly sensitive to incomplete redshift information

Figure 5. Figure showing the variation with the priors on the bias onmedian redshift, zbias, of 30 redshift bins spanning the survey redshift rangedue to the effect of photometric redshift uncertainties and the photometricredshift statistical dispersion, δz. All other survey parameters are fixed attheir fiducial values. Results are for a Stage IV survey FoM for dark energy(DE) parameters w0, wa (blue lines) and modified gravity (MG) parametersQ0, Q0(1 + R0)/2 (red lines), as described in (14) and (15), relative toa ‘baseline’ FoM (black cross). As shown in the key in the figure, shear–shear correlations, ‘εε’, and combined shear and galaxy position informationincluding cross-correlations ‘εε+nε+nn’ are shown when uncertainties inthe IA model have been included and excluded.

because it describes the time evolution of the DE equation of state.By contrast, the MG parameters are more robust to poor redshiftknowledge because they are constant with redshift and the redshiftevolution of the MG functions is fixed by our choice of redshiftpower law s.

We can relate the required prior on redshift parameters to thenumber of spectroscopic-quality galaxy redshifts needed to calibrateour photometric sample as given by Ma et al. (2006):

Nspec = 2δ2z (1 + z)2

�2σz

. (24)

For εε, ignoring IAs, a prior of 1 × 10−3 is sufficient to recover90 per cent of the peak FoM for either DE or MG. With a photometricscatter of σ z = δz(1 + z) = 0.05(1 + z) and a median redshift ofzm = 0.9/

√2, this translates into 1.34 × 104 spectroscopic galaxies

per redshift bin. Including all galaxy position information εε + nε +nn and IAs reduces the required prior to ∼1 × 10−2, or of the orderof 100 spectroscopic galaxies per bin. This represents a factor of100 reduction in required numbers compared to the standard result.However, note that there are other factors which must be taken intoconsideration when determining the number of galaxies per redshiftbin, especially cosmic variance (e.g. see the discussion in Ishak &Hirata 2005; Van Waerbeke et al. 2006; Bordoloi, Lilly & Amara2010; Cunha et al. 2012).




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Figure 6. Same as in Fig. 5, but for a Stage III survey specification.

Fig. 6 shows similar results for the fiducial Stage III survey.Qualitatively, the results are similar to the Euclid survey. In this case,with a photometric scatter of σ z = 0.07(1 + z) and a median redshiftof zm = 0.8/

√2, εε without IAs requires a prior of ∼1 × 10−2

which translates into around 100 galaxies per bin. Again εε + nε +nn with IAs requires a factor of 100 fewer galaxies, a substantialsaving. Interestingly, while the MG constraints produce the samesaving in spectroscopic galaxies required when IAs and all probesare included, the absolute values are a factor of 2 tighter in priorterms (i.e. a factor of 4 more spectroscopic galaxies are required).

4 C O N C L U S I O N S

Previous work (Joachimi & Bridle 2010) has shown how constraintsfrom cosmic shear are significantly degraded by the inclusion ofIA systematics, using a comprehensive grid of k- and z-dependentnuisance parameters. The impact of IAs can be mitigated by theinclusion of galaxy position correlations and shear–position cross-correlations using only the photometric information gathered as partof the shear survey.

In a related paper, LBKB, we confirmed and extended theseresults showing that using the full εε + nε + nn information withup to 36 nuisance parameters can regain the constraining power ofa naive εε analysis which ignores IAs and that this is true whenconstraining deviations from GR as well as DE. In this paper, weconfirm that these results are robust across a wide range of surveyparameters, encompassing fiducial Stage III and Stage IV cosmicshear surveys.

Increasing survey area improves both DE and MG FoMs, as oneexpects, whether we ignore or include IAs. Previous work, whichdid not include the CMB, found a linear variation of FoM with sur-vey area even when IAs were included. We have shown, however,that if one includes priors from upcoming CMB experiments, thisdependence is weakened to vary roughly as a power law with an ex-

ponent of 0.4. The inclusion of IAs slightly weakens the relationshipbetween FoM and survey area.

More realistically, survey time is a constant so increased areamust be traded off against depth. We find the dependence on areasignificantly flatter when IAs are included. This is to be expectedbecause IAs dominate over the cosmic shear signal at low redshift,and we effectively marginalize out information from this regime. Onincreasing the area, and thus decreasing the depth to lower redshift,an increasing proportion of the signal is lost by the marginalization.The effect is more pronounced for a Stage III survey than a Stage IVsurvey, which is expected because the depth is very shallow indeedif it has to cover half of the sky.

The quality of photometric redshift information is crucial to thesuccess of future cosmic shear missions. We show that the inclu-sion of IAs and the full use of galaxy position information requiresmore accurate redshift information than a naive cosmic shear-onlyanalysis. GG-only retains 97 per cent of its constraining power forperfect redshift knowledge with a photometric redshift scatter ofδz = 0.05, the fiducial value for a Stage IV survey. By contrast, thecombined εε + nε + nn including IAs requires δz = 0.015 to retainthe same proportion. Similarly, the characteristic plateau behaviour,where WL constraining power does not improve over a tomographicresolution of ∼4−5, bins is pushed back on the inclusion of IAsand galaxy position information to ∼10−12 redshift bins. This isunderstandable as the standard cosmic shear redshift dependence,modulated by a broad lensing kernel for each redshift bin, has beenjoined by galaxy position correlations which depend on the nar-rower redshift distribution function and the attempt to constrain IAssignals, each with their own redshift dependence. Taken togetherthese place increased importance on accurate redshift knowledge.

We have presented results showing the effect of varying our de-gree of prior knowledge of an extended parametrization of redshiftinformation, allowing 60 free parameters to be shared between red-shift dispersion and redshift bias. Around �zbias = �δz = 0.1, theprobes enter a self-calibration regime and the FoMs flatten off. TheDE FoMs fall off much more steeply and relatively lower than theirMG counterparts, highlighting the sensitivity of the DE equation ofstate, particularly wa, to incomplete redshift information. By con-trast, the MG parameters have a fixed redshift dependence in thiswork.

This work assumed a relatively simple model for deviations fromGR. We vary two MG parameters, each with the same fixed red-shift dependence. In the future, we hope to extend these results totake into account an MG gravity parametrization which could varymore generally as a function of scale and redshift. In contrast, ourfiducial parametrization of galaxy bias and IA nuisance parametersis robust, containing over 100 nuisance parameters. With a greaterunderstanding of the physics of galaxy formation mechanism re-sponsible for IAs it may be possible to motivate a more targeted,simpler parametrization.

AC K N OW L E D G M E N T S

We thank the Aspen Center for Physics for support and for host-ing two coincident workshops on ‘Wide-Fast-Deep Surveys: NewAstrophysics Frontier’ and ‘Testing General Relativity in the Cos-mos’ in 2009 where this work was conceived. We would also liketo thank the Kavli Royal Society International Centre for hostingthe ‘Testing general relativity with cosmology’ workshop in 2011that supported fruitful discussion and collaboration.

We thank Filipe Abdalla, Adam Amara, David Bacon, Scott Do-delson, Ole Host, Martin Kilbinger, Andrew Jaffe, Bhuvnesh Jain,




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208 D. Kirk et al.

Benjamin Joachimi, Ofer Lahav, Rachel Mandelbaum, Anais Ras-sat, Shaun Thomas, Alexandre Refregier and Jochen Weller forhelpful discussions.

We are very grateful to Josh Frieman for spotting an error inequation 19 of the manuscript, and to Barney Rowe for discussingwith us the correction.

RB’s and IL’s research is supported by NSF CAREER grantAST0844825, NSF grant PHY0968820, NASA Astrophysics The-ory Program grants NNX08AH27G and NNX11AI95G and by Re-search Corporation. SLB thanks the Royal Society for support inthe form of a University Research Fellowship and acknowledgessupport from European Research Council in the form of a StartingGrant with number 240672.

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