arXiv:1112.4752v1 [astro-ph.CO] 20 Dec 2011 5Nvme 2018 November 15

ita ta.(00 n a areee l 20) For (2000). (2000); al. and al. et Waerbeke et (1992) van Kaiser gravita- and Kaiser (2000); (2000) of al. by al. et measure et Bacon proposed correla- Wittman a by The first observed as was signal. first shapes lensing shear galaxy cosmic tional of the the inform function cosmological to tion in the access contained recover no tion to themselves. have statistically smaller galaxies observers, galaxies significantly the as ex- is galaxies of we, lensed be which on ellipticity course 1%, to intrinsic Of induced order is the of shear potential than is cosmic interest true typical of its The if ploited. challenges technical whatever expansion. or accelerated state, apparent us of the allows equation causing it energy way is dark this the In Universe. constrain an the to history in expansion function structure the a of about lensing us growth as tells The distribution thus and tool. matter redshift, of the cosmological the probes powerful has directly a effect images be galaxy to distant of potential lensing gravitational The INTRODUCTION 1 o.Nt .Ato.Soc. Astron. R. Not. Mon. oncaKirk Donnacha Shear Model Cosmic Alignment for Intrinsic Choice of Impact Cosmological The c 4 3 2 1 aoaor I,URCACR-ai ,If,SD-A,Ser SEDI-SAP, Irfu, 7, CEA-CNRS-Paris UMR AIM, Laboratoire ATO ws eea nttt fTcnlg nLuan ( Lausanne in Technology of Institute Federal Colleg Swiss Lond Imperial LASTRO, Laboratory, College Blackett University Group, Astronomy, & Physics of Department 09RAS 2009 ekgaiainllnigpssanme ftough of number a poses lensing gravitational Weak aayiae ow uttetapplto of population a treat must we so images galaxy 1 , 000 2 ni Rassat Anais , –2(09 rne 5Nvme 08(NL (MN 2018 November 15 Printed (2009) 1–12 , etmdlo A,floigtebifnt fHrt ejk(2010) Seljak & bias cosmological Hirata parameters the of show state We note of (2011). brief equation al. ergy the et of following Laszlo version in IAs, latest en interpretation the of dark summarise on model We (IAs) ment lensing. alignments intrinsic gravitational galaxy weak of from effect the consider We ABSTRACT efidthat find We otoefo oua rvosipeetto n xli h differ the explain s and observed words: implementation compare Key previous we popular appendix, a an from In those recover th state. to and between bias of difference cosmological equation remove but large to energy model, sufficient the IA is wor despite flexibility latest previous grid that the in the find used used We universe as the version. parameters if experiments earlier bias happen III galaxy would and Stage what IA future gate of for grid account flexible into t a taken Therefore implementation. be con IA must This earlier IAs approximation. the for matrix larger Fisher times the several under unity than greater fUies aais evolution galaxies: – Universe of w 3 omlg:osrain rvttoa esn ag-cl stru large-scale – lensing gravitational – observations : 0 , 4 and l Host Ole , ,Pic osr od odn W B,UK 2BZ, SW7 London, Road, Consort Prince e, w un- a a- d r ohctsrpial isd ya bouevleo just of value absolute an by biased, catastrophically both are n oe tet odn CE6T UK 6BT, WC1E London, Street, Gower on, PL,Osraor eSuen,C-20 esi,Switze Versoix, CH-1290, Sauverny, de Observatoire EPFL), iedAtohsqe E aly -19 i /Yet,Fr Yvette, s/ Gif F-91191 Saclay, CEA d’Astrophysique, vice rvttoa-nrni G)creain hc a eof the term. be as II (known the can than potential which magnitude same greater correlation) the by (GI) lensed galaxGravitational-Intrinsic by are background shaped which and ies potential galaxies neg- gravitational foreground additional particular an a between cor- noted (II) correlation (2004) Intrinsic-Intrinsic ative aligne Seljak the & preferentially as Hirata be (known relation). ex- other to we each galaxies so with form close they physically which pect in potentials gravitational scale al. et Crittenden 2001; 2000). Metzler al. & et assump- Croft Catelan this 2001; 2000; unjus that al. is cos- out et distribution (Heavens ellipticity the pointed intrinsic recover random soon of was to tion average it averaged would ellipticity cer- However be intrinsic a zero. the could on because ellipticities sky shear, observed mic of case acr the patch random was tain this is If ellipticities sky. galaxy the of distribution trinsic (2008). Jain al. & et Hoekstra Munshi and (2001); (2003), Schneider Refregier & (2006); Bartelmann see reviews 1 aa Bridle Sarah , fe h lgmn fglx litcte rmlinear from ellipticities galaxy of alignment the After large the with align to expected be may galaxies fact In in- the that assumes shear cosmic to approach naive A w 0 and w A T a E tl l v2.2) file style X htwudocri A eeignored. were IAs if occur would that 1 eei odutthat doubt no is here n eod euse We beyond. and erpwrspectra power hear rsswt bias a with trasts w Amodels, IA two e h ieralign- linear the ryconstraints ergy h orc dark correct the easmdthe assumed we ntedr en- dark the on ,adinvesti- and k, ences. n further and rland cture tified ance oss to d - 2 Donnacha Kirk, Anais Rassat, Ole Host, Sarah Bridle response to the gravitational potential was proposed as an The result of this evolution in the treatment of IAs effect by Catelan et al. (2001) the study was put on a firm is that much useful theoretical and observational work has analytic footing by the introduction of the Linear Alignment been conducted using an incorrect implementation of the LA (LA) model by Hirata & Seljak (2004), hereafter HS04. This model, using often unjustified treatments of the small scale approach, in which the orientation of galaxies responds lin- seeding of galaxy IAs. In this work our aim is to present early to the large-scale gravitational potential in which they basic results for the most up to date implementation of the form, has become the standard model when including the LA model for IAs. We present angular power spectra for effects of Instrinsic Alignments (IA). components of the shear-shear (ǫǫ), position-position (nn) Correlated shears of galaxies have been observed at low and position-shear (nǫ) observables as well as the reduction redshift, where they can be attributed to intrinsic align- in constraining power in measuring caused by ments, by Brown et al. 2002, and have been measured in a robust treatment of IAs and the biasing of cosmological galaxies selected to be physically close in Mandelbaum et al. parameters that results from ignoring IAs or treating them (2006); Hirata et al. (2007); Okumura et al. (2009); using an old model. Many of these results reproduce previ- Brainerd et al. (2009) and Mandelbaum et al. (2006). Num- ous work in the literature which was conducted using an old ber density - shear correlations are easier to observe since implementation of the LA model. Specifically we reproduce the random galaxy ellipticity only enters the calculation Fig. 1 from Joachimi & Bridle (2009) for the new implemen- once, and this has been constrained in Mandelbaum et al. tation of the LA model which we hope will act as a reference (2006); Hirata et al. (2007); Okumura & Jing (2009); for those wishing to apply it in the future. Furthermore we Mandelbaum et al. (2009) and Joachimi et al. (2011). investigate the ability of a flexible parameterisation of in- Hirata et al. (2007) noted that the scale dependence trinsic alignments to compensate for using the old LA model of the signal they measured was better matched to theory compared to the latest version we summarise here. if the non-linear matter power spectrum was used in the This paper is organised as follows: In section 2 we de- LA instead of the linear power spectrum implied in HS04. scribe the most up to date implementation of the LA model Bridle & King (2007) then used the non-linear matter power of IAs, detailing our formalism for shear-shear (ǫǫ), position- spectrum in their cosmological forecast calculations. This position (nn) and position-shear (nǫ) correlations. Section approach has been called the Non-Linear Alignment (NLA) 3 presents the bias on cosmological parameter estimation ansatz. Schneider & Bridle (2010) attempted a more moti- caused by mistreatment of IAs for the latest model, the old vated solution for assigning power to the IA model at small model and an intermediate model; we then conclude in sec- scales through the use of the halo model of galaxy cluster- tion 4. In an appendix we summarise the history of the LA ing. By design their halo model reproduced the LA model model and present the major differences between the latest at large scales. implementation and the previous widely used version. The II contribution to intrinsic alignments can be taken into account in cosmological analyses by removing galaxies with small physical separation from the analysis 2 THE LINEAR ALIGNMENT MODEL e.g. King & Schneider (2002, 2003); Heymans & Heavens In this section we summarise the latest linear alignment (2003); Takada & White (2004). This was attempted in a model interpretation and its contribution to (i) shear- real analysis of COSMOS data in Schrabback et al. (2009) shear correlations (ii) position-position correlation and (iii) by removing the autocorrelation tomographic bin. An exten- position-shear cross correlation. sion of this method for GI was suggested in King (2005) and developed to a sophisticated level in Joachimi & Schneider (2008, 2009). Alternatively a model may be assumed for all 2.1 Shear-shear correlations, ǫǫ the intrinsic alignment contributions, and the free parame- ters can be marginalised over, as demonstrated at the Fisher The Linear Alignment (LA) model for galaxy intrinsic ellip- matrix level in Bridle & King (2007); Bernstein (2009); ticities was suggested by Catelan et al. (2001) and applied Joachimi & Bridle (2009); Laszlo et al. (2011); Kirk et al. quantitatively by Hirata & Seljak (2004), see section 4 below (2011) and demonstrated on real data in Kirk et al. (2010). for a more complete history. It assumes that, under gravita- The redshift evolution of the IA contributions in the LA tional collapse, forms into triaxial halos whose model was found to be incorrect due to a mistake in HS04 major axis is aligned with the maximum curvature of the in the conversion between the primordial potential and the large scale gravitational potential. Elliptical galaxies are ex- matter power spectrum. This was corrected in a new version pected to trace the ellipticity of their parent halos, meaning of HS04, issued as Hirata & Seljak (2010), hereafter HS10. that the ellipticity distribution of a population of elliptical In addition there is an ambiguity in HS04 as to which cos- galaxies will be linearly related to the curvature of the grav- mological epoch is responsible for the “imprinting” of galaxy itational potential, φ, via 2 2 intrinsic alignments. Most starkly the question is, are IAs ǫ+ = C(∂y ∂x)φ (1) frozen in at some redshift of formation or do they evolve − ǫ× = 2C∂ ∂ φ, (2) with the growth of structure, particularly nonlinear clus- y x tering on small scales? Blazek et al. (2011) note this as an where C is an unknown constant, fixed for our formalism by issue in the HS04 approach and estimate its effect on the GI a normalisation given below. amplitude as of order 20%. Laszlo et al. (2011), Kirk et al. In Weak Gravitational Lensing (WGL) we are inter- (2011) implement one physically motivated solution to this ested in IAs as systematic effects which contaminate a mea- question (as well as including the redshift evolution correc- sured cosmic shear signal, specifically the shear-shear power tion of Hirata & Seljak 2010) which we adopt in this work. spectrum. The LA model motivates the amplitude of two

c 2009 RAS, MNRAS 000, 1–12 The Cosmological Impact of Intrinsic Alignment Model Choice 3

−14 2 −3 −1 separate terms which affect our ability to accurately measure use a fiducial value of 5 10 (h M⊙Mpc ) , following × cosmic shear. The first, Intrinsic-Intrinsic (II), term arises Bridle & King (2007) who match to the power spectra in when galaxies are physically close. These galaxies form in Hirata & Seljak (2004) which are based on SuperCOSMOS the same gravitational potential, hence their intrinsic ellip- data from Brown et al. (2002). ticities are preferentially aligned. This produces a spurious The II term is related to the linear matter power spec- lin positive correlation which adds to the measured cosmic shear trum, Pδδ , because we assume that galaxy intrinsic ellip- signal. ticity is imprinted at the (early) epoch of galaxy formation The second, Gravitational-Intrinsic (GI), term becomes and subsequently “frozen in”. As such the power spectrum important when treating galaxies which are close on the sky of its distribution does not undergo the nonlinear evolution but separated in redshift. In this case foreground galaxies that the distribution of matter clustering does at late times. lin will align to a foreground gravitational potential which will The factor of Pδδ (k,χ = 0)Pδδ(k,χ)/D(z) in the GI term itself contribute to the gravitational lensing of the back- reflects the dependence on both the intrinsic ellipticity dis- p ground galaxies. This produces an anti-correlation in ob- tribution and the full, late time, mass distribution via grav- served ellipticities and subtracts from the expected cosmic itational lensing. The exact relation between intrinsic ellip- shear signal. ticity, the epoch of galaxy formation and the evolution of In terms of projected angular power spectra we can galaxy clustering is a vexed and disputed one. A more de- ǫǫ write the measured ellipticity power spectrum, Cl ,as a sum tailed discussion of the history of the treatment of this re- of the gravitational lensing shear power spectrum and two lationship can be found in the appendix below. We believe IA terms that our approach best reflects the physical understanding

ǫǫ GG II GI of the LA model as proposed by Hirata & Seljak (2004) who C (l)= C (l)+ C (l)+ C (l). (3) related the intrinsic ellipticity distribution to the Newtonian GG The cosmic shear power spectrum, Cij (l), under the potential at the time of galaxy formation. Limber approximation, is written as the integrated product of the matter power spectrum, P (k,χ), and the lensing δδ 2.2 Position-position correlations, nn weight function, Wi(χ) (Hu 1999),

χhor Any cosmic shear survey provides not only a catalogue of GG dχ C (l)= Wi(χ)Wj (χ)Pδδ(k,χ), (4) measured galaxy ellipticities but also a record of galaxy po- ij χ2 Z0 sitions through their location on the sky and an estimate of l where Pδδ(k,χ) Pδδ( ,χ), χ is the comoving distance their redshift. Analogously to the cosmic shear power spec- ≡ fK (χ) along the line of sight,fK (χ) is the comoving angular diame- trum we may define the galaxy position density power spec- nn ter distance, i, j denote a tomographic redshift bin pair and trum, C (l), the Fourier transform of the galaxy position two-point correlation function. It is conceptually useful to

2 χhor ′ divide the contributions up as follows, 3 H0 Ωm χ ′ ′ χ χ Wi(χ)= dχ ni(χ ) − , (5) nn gg mm gm 2 c2 a χ′ C (l)= C (l)+ C (l)+ C (l). (10) Zχ with H0 the Hubble parameter today, Ωm the dimensionless Here we have separated the contribution from galaxy clus- matter energy density, c the speed of light, a the dimension- tering itself, gg, from the contribution due to lensing mag- ′ less scale factor and ni(χ ) is the galaxy redshift distribution nification, mm, and the cross-correlation of the two, gm. for a particular bin i. The lensing contribution manifests because a galaxy may Similarly, the projected angular power spectra for the increase (or decrease) in size as a result of the image distor- IA terms are (Hirata & Seljak 2004) tion. The application of Liouville’s theorem tells us that the

χhor process of WGL conserves surface brightness density hence II dχ a larger (smaller) galaxy will appear brighter (dimmer) due Cij (l) = 2 ni(χ)nj (χ)PII (k,χ) (6) 0 χ to the lensing effect. For a magnitude limited survey this Z χhor GI dχ can affect the statistics of galaxy clustering as otherwise too Cij (l) = Wi(χ)nj (χ)PGI (k,χ), (7) χ2 faint galaxies are promoted into the survey by magnifica- Z0 where, unsurprisingly, the II correlation depends only on the tion and vice-versa. As the nn correlation is effectively just galaxy redshift distribution, n(χ), while GI depends on the a counting of galaxies and their relative positions there is no product of the galaxy redshift distribution and the lensing ellipticity contribution and we can ignore IAs. weight function. The equation for the pure galaxy clustering term is rel- atively straightforward in the Limber approximation The IA “power spectra”, PII (k,χ) and PGI (k,χ), are χhor unknown functions of scale and cosmic epoch. In the case of gg dχ 2 C (l)= ni(χ)nj (χ)b (k,z)P (k,χ), (11) the LA model they become ij χ2 g δδ Z0 2 lin PII (k,χ) = ( C1ρ(χ = 0)) Pδδ (k,χ = 0) (8) where the window function is the galaxy redshift distribu- − tion as we would expect and we have introduced the galaxy lin Pδδ(k,χ) PGI (k,χ) = C1ρ(χ = 0) P (k,χ = 0) (9) bias term, bg, to reflect our understanding that galaxies are − δδ D(z) q p a biased tracer of the underlying matter distribution. As l where Pδδ is the matter power spectrum, ρ(χ = 0) is the before, Pδδ(k,χ) Pδδ( ,χ). In general bg is an un- ≡ fK (χ) matter density today and D(z) is the linear growth factor as known function of scale and cosmic epoch. The details of a function of redshift z, where it is implicitly meant that z = the galaxy bias formalism we assume in this paper, along z(χ). C1 is the normalisation of the IA contribution and we with other bias terms, are explained in section 3 below.

c 2009 RAS, MNRAS 000, 1–12 4 Donnacha Kirk, Anais Rassat, Ole Host, Sarah Bridle

The magnification power spectrum and galaxy Observables clustering-magnification cross-term can be written ǫǫ shear-shear mm GG nn position-position Cij (l) = 4(αi 1)(αj 1)Cij (l) (12) − − nǫ position-shear gm gG Cij (l) = 2(αj 1)Cij (l), (13) − Fields where αi is defined as the slope of the luminosity function, G gravitational lensing evaluated at the median redshift of the bin i, and CgG(l) I intrinsic alignment is defined in section 2.3. In this approach we are following g galaxy clustering m cosmic magnification Joachimi & Bridle (2009). We treat each of the αi terms as a free parameter in the model, and take fiducial parameters from eqns. 32, 33 and Table 2 of Joachimi & Bridle (2009) Table 1. Summary of observables considered in this paper and (see Appendix A of that paper for more details). different fields which contribute to each observable.

survey specification: 20,000 deg2 with a galaxy number den- 2.3 Position-shear correlations, nǫ sity of 35 arcmin−2 and a redshift distribution given by the As well as ǫǫ and nn correlations themselves, we can cross- Smail-type n(z), correlate the two fields to form the galaxy position-shear, nǫ, β α z cross-correlation functions as first proposed by Hu & Jain n(z)= z exp , (19) − z0 (2004) in the context of dark energy. This is often referred to   ! as galaxy-galaxy lensing, in which the mass of a foreground with α = 2, β = 1.5, z0 = 0.9/√2, divided into 10 tomo- galaxy distorts the shape of a background galaxy. Here we graphic bins with equal number density out to redshift 3 consider the general cross-correlation which includes contri- and normalised so that n(z)dz = 1. A Gaussian photo- butions from larger dark matter structures, from magnifi- metric redshift error of σz = 0.05(1 + z) is assumed with no R cation and also from intrinsic alignments if the correlated catastrophic outliers in redshift. We constrain the set of cos- galaxies are physically close. We can write the contributions mological parameters Ωm,w0,wa,h,σ8, Ωb, ns which take { } to the full nǫ term as the values Ωm = 0.25, w0 = 1, wa = 0, h = 0.7, σ8 = 0.8, − nǫ gG gI mG mI Ω = 0.05, n = 1. All results assume a flat, ΛCDM cosmol- C (l)= C (l)+ C (l)+ C (l)+ C (l) (14) b s ogy. Linear matter power spectrum is calculated using the (Joachimi & Bridle 2009). Eisenstein & Hu (1998) fitting function and nonlinear cor- The individual expressions for the terms in the Cnǫ(l) rections are applied where appropriate using the Smith et al. expansion contain combinations of quantities already con- (2003) formalism. sidered Fig. 1 is presented and formatted in the same way as χhor Fig. 3 in Joachimi & Bridle (2009). The purpose is to supply gG dχ Cij (l) = ni(χ)Wj (χ) a new reference plot using the most up to date implementa- χ2 Z0 tion of the LA model for IAs. All lines which do not include b (k,z)r (k,z)P (k,χ) (15) g g δδ an “I” term will be identical in both versions of the plot. χhor gI dχ However, as Joachimi & Bridle (2009) use the original HS04 C (l) = ni(χ)nj (χ) ij χ2 Z0 implementation, from before the publication of the erratum bg(k,z)rg(k,z)bI (k,z)rI (k,z)PGI (k,χ) (16) incorporated into Hirata & Seljak (2010), and present lines mG GG for the NLA ansatz, there are differences in any lines con- C (l) = 2(αi 1)C (l) (17) ij − ij mI GI taining an I contribution. The NLA version of HS04 is the C (l) = 2(αi 1)C (l), (18) ij − ij most widely used in the literature to date; we refer to it where we have introduced the additional free functions rg subsequently as HS04NL. Note that that we are employing and rI which appear due to the possible stochastic re- the NLA prescription in HS04NL even though this was not lation between the galaxy and dark matter distributions explicitly discussed in the original Hirata & Seljak (2004) (Dekel & Lahav 1999) as discussed in section 3 below. As paper. See the appendix for a more detailed explanation of l the evolution of the LA model and the differences between before Pδδ(k,χ) Pδδ( f χ ,χ). We note that PGI (k,χ) ≡ K ( ) various implementations. appears in the CgI (l) term because it describes how the ij The upper triangle of Fig. 1 shows the contributions to dark matter distribution relates to the intrinsic alignments, the shear-shear correlation function in black for every other and we assume the relation between the galaxy distribution tomographic bin pairing. As usual the lensing contribution and the dark matter distribution can be accounted for by is the largest on all angular scales, but is most dominant the product b r . g g in the auto-correlations at high redshift. The II intrinsic alignment term is most important in the auto-correlations 2.4 Summary of Observables and Fields and negligible for widely spaced bins. The GI contribution is largest for separated bins (shown as IG for compactness - see The observables considered in this paper and the different caption). The intrinsic alignment contributions are typically fields which contribute to each observable are summarised one to ten per cent of the lensing signal. in Table 1. In Fig. 1 we plot the angular power spectra of The contributions to the position-position correlation all considered components for a fiducial stage-IV lensing sur- function are shown in pink in the upper triangle and as vey. For our fiducial model in this paper we use the following expected, the auto-correlations are dominated by the in-

c 2009 RAS, MNRAS 000, 1–12 The Cosmological Impact of Intrinsic Alignment Model Choice 5

Figure 1. Fiducial power spectra for all correlations considered. The upper right panels depict the contributions to the ǫǫ (in black) and nn (in magenta) correlations. The lower left panels show the contributions to correlations between number density fluctuations and ij ellipticity. Since we only show correlations Cαβ (l) with i 6 j, we make in this plot a distinction between nǫ (in red; number density contribution in the foreground, e.g. gG) and ǫn (in blue; number density contribution in the background, e.g. Gg) correlations. In each sub-panel a different tomographic redshift bin correlation is shown. For concision only odd bins are displayed. In the upper right panels the usual cosmic shear signal (GG) is shown as black solid lines; the intrinsic alignment GI term is shown by the black dashed lines; the intrinsic alignment II term is shown by the dotted black lines; the usual galaxy clustering signal (gg) is shown by the magenta solid lines; the cross correlation between galaxy clustering and lensing magnification (gm) is shown by the magenta dashed lines; the lensing magnification correlation functions (mm) are shown by the magenta dotted lines. In the lower left panels the solid blue lines show the correlation between lensing shear and galaxy clustering (Gg); the blue dashed lines show the correlation between lensing shear and lensing magnification (gm); the blue dot-dashed lines show the correlation between intrinsic alignment and galaxy clustering (Ig or equivalently gI); the red solid lines show the correlation between galaxy clustering and lensing shear (gG), which is equivalent to the blue solid lines with redshift bin indices i and j reversed; similarly the red dashed lines show the correlation between lensing magnification and lensing shear (mG), for cases where the magnification occurs at lower redshift than the shear (i

3 εε, no felxibility in IA model εε+nε+nn, no felxibility in IA model Latest IA model in data; latest IA model in fit 8 8 Latest IA model in data; no IAs in fit 2 HS10NL in data; no IAs in fit 6 6 HS04NL in data; no IAs in fit 4 4 a a

1 w w 2 2

0 0 0 −2 −2 −3 −2 −1 0 −3 −2 −1 0 a −1 w w w 0 0 εε, marginalised over IA grid εε + nε+nn, marginalised over IA grid −2 8 8 Latest IA model in data; 6 6 latest IA model in fit −3 Latest IA model in data; 4 4 HS04NL in fit a a w w −4 To Red Ellipse 2 2 0 0

−5 −2 −2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −3 −2 −1 0 −3 −2 −1 0 w w w 0 0 0

Figure 2. 95% confidence limits on the dark energy parame- Figure 3. 95% confidence limits on the dark energy parameters ters w0 and wa for the IA implementation from this work [blue w0 and wa for the IA implementation from this work [blue con- contour] and the biased constraints if it is assumed that IAs do tours] and the biased constraints if the old HS04NL implementa- not exist but in fact the new implementation [black contour], the tion is assumed [red contours]. The cosmological bias is given by HS10NL implementation [green contour] or the HS04NL imple- the offset between the displaced contour and the fiducial values of mentation [off plot, direction indicated by red arrow] are true. The (−1, 0) [black dotted lines]. Results are presented for an observ- cosmological bias is given by the offset between each displaced able data vector of shear-shear (ǫǫ) correlations alone [left-hand contour and the fiducial values of (−1, 0) [black cross]. Results panels] and the full combination of shear-shear,shear-position and are presented for an observable data vector of shear-shear (ǫǫ) position-position (ǫǫ + nǫ + nn) correlations [right-hand panels]. correlations only. It is assumed that we enjoy perfect knowledge Each probe combination is shown for the case of perfect knowl- of the IA contribution for each implementation. edge of IAs and galaxy bias [top panels] and for the case where our ignorance is accounted for by marginalisation over the nuisance parameter grid with Nk = Nz = 7 [bottom panels].

8σFM for the HS10NL implementation and 30σFM for ∼ ∼ the HS04NL model. So far we have not taken into account any uncertainty take the form of this grid, in the IA model, but assumed they are zero in the param- bI (k,z) = AbI QbI (k.z) (23) eter fitting, whichever model we employ to describe them b (k,z) = A Q (k.z) (24) in the true model. In reality, we are aware that our knowl- g bg bg edge of IAs from simulations and observations is still de- rI (k,z) = ArI QrI (k.z) (25) veloping and relatively uncertain. The case is similar for rg(k,z) = Arg Qrg (k.z). (26) galaxy bias, introduced in section 2.2, and employed be- low. To parameterise our ignorance of both effects and their The “b” terms can be understood as bias terms between cross-correlations we use a grid of nuisance parameters the power spectrum of a field X and the matter power spec- trum, i.e.: X = AX QX (k,z), (22) PXX (k,z) 2 = bX (k,z) (27) where AX is a constant amplitude parameter, free to vary Pδδ(k,z) about a fiducial value of 1, and QX (k,z) is a grid of Nk Nz × and “r” terms are biasing terms which arise in cross- nodes logarithmically spaced in k/z space, each of which correlations, i.e.: is allowed to vary independently around a fiducial value of 1. A final smooth grid is created by spline interpola- PXY (k,z) tion over the values of the grid nodes. For more details of = bX (k,z)rX (k,z)bY (k,z)rY (k,z). (28) Pδδ(k,z) this nuisance parameter grid see Joachimi & Bridle (2009), Laszlo et al. (2011), Kirk et al. (2011). This was inspired by The “b” and “r” terms are both functions of the clustering the marginalisation of ? which led to the grid implementa- and stochastic bias as described in Dekel & Lahav (1999). tion in Joachimi & Bridle (2009). These sets of nuisance parameters then appear, in differ- We define four sets of nuisance parameters which each ent combinations, in each angular power spectrum integral

c 2009 RAS, MNRAS 000, 1–12 8 Donnacha Kirk, Anais Rassat, Ole Host, Sarah Bridle where galaxy clustering, g, or the IA term, I, appear: fectly known. This corresponds to a factor of 2.6 reduction in DETF FoM (Albrecht et al. 2006). II dχ 2 C (l) = ni(χ)nj (χ)b (k,z)PII (k,χ) (29) ij χ2 I Z GI dχ C (l) = Wi(χ)nj (χ)bI (k,z)rI (k,z)PGI (k,χ)(30) 4 CONCLUSIONS ij χ2 Z gg dχ 2 Galaxy IAs are the most important physical systematic in C (l) = ni(χ)nj (χ)b (k,z)P (k,χ) (31) ij χ2 g δδ the study of cosmic shear. As increasing volumes of data Z become available from WGL surveys more interest is be- gG dχ C (l) = ni(χ)Wj (χ)bg(k,z)rg(k,z)P (k,χ)(32) ij χ2 δG ing paid to their correct treatment. Much of this work has Z been dominated by the LA model, originally introduced in gI dχ C (l) = ni(χ)nj (χ)bI (k,z)rI (k,z) HS04 (and the NLA extension which we refer to as HS04NL). ij χ2 Z This original implementation was subsequently corrected in bg (k,z)rg(k,z)PδI (k,χ), (33) Hirata & Seljak (2010) and detailed attention paid to the

gm mI treatment of non-linear clustering in Laszlo et al. (2011); while nuisance parameters appear in Cij (l) and Cij (l) due Kirk et al. (2011). gG GI to their dependence on Cij (l) and Cij (l) respectively. In However, much of the existing literature has been pro- general this means that, for the full suite of observables duced using the uncorrected HS04NL model. In this paper ǫǫ + nǫ + nn, we marginalise over a total of 4(1 + Nk Nz ) we have provided a brief explanation of the evolution of, and × nuisance parameters, plus 30 that parameterise the magnifi- context surrounding, the LA model for IAs, highlighting the l cation effects. As before Pδδ(k,χ) Pδδ( ,χ). When we most important differences between HS04NL and the latest ≡ fK (χ) employ the nuisance grid in this work we set Nk = Nz = 7, implementation. giving a total of 230 nuisance parameters. This follows the We have calculated the angular power spectra of the practice used in Joachimi & Bridle (2009) for the purposes cosmic shear observables, correcting the implicit mistake of of comparison (see their Table 2). We also restrict the in- Joachimi & Bridle (2009), which was based on HS04, which formation used in the galaxy field to linear scales as in we hope will act as a new reference for those interested in Rassat et al. (2008). applying IAs to their cosmic shear analysis. This current paper presents the most advanced inter- The main motivation for the study of IAs is the mea- pretation of the LA model. However, much work of the surement of unbiased constraints of key cosmological pa- work on IAs in the literature has been conducted using a rameters. We show that the new LA implementation signif- non-linear version of the original interpretation presented in icantly reduces the impact of IAs, and hence the bias, but Hirata & Seljak (2004). In Fig. 3 we quantify the bias intro- that the effect is still very significant, producing a bias at duced on the dark energy parameters from the use of this the tens of σ level when we assume perfect knowledge of IAs. HS04NL approach rather than the latest implementation in- If we did know the IA signal perfectly then we could troduced in this work. When the IA signals (whether from produce unbiased measurements by simply subtracting the the old or new models) are assumed to be known perfectly IA signal from our measured cosmic shear. In practice it is there is a very strong biasing effect which would result in a useful to parameterise our ignorance of the true IA signal very worrying systematic mis-measurement of cosmology. through a set of nuisance parameters which are marginalised In the case where ǫǫ correlations alone are considered, over to produce weaker but hopefully unbiased cosmological the bias is 25σFM of the constraining power of the cosmic estimates. We show that a robust grid of 130 nuisance pa- ∼ shear signal given the true IA contribution. When the full rameters for IAs and magnification uncertainties, allowed to set of correlations including position information, ǫǫ + nǫ + vary in scale and redshift, effectively removes the bias due nn, are considered this biasing increases to 65σFM. As to assuming an incorrect IA model. ∼ previously discussed, when the effect is this strong the exact The same effect holds when our observables are ex- position/direction of the biasing is likely to be well outside tended to include shear-shear, position-position and shear- the competence of the FM-based formalism but the fact that position correlations. The extra observables increase con- there is a very significant effect should not be doubted. straining power so that we are able to produce unbiased When we employ the robust parameterisation of both constraints on ǫǫ + nǫ + nn which recover 40% of the con- IA and galaxy bias uncertainties via the grid of nuisance straining power of the highly biased constraint from ǫǫ alone parameters described above we see a striking change in the when the wrong IA model is assumed. results. As we would expect the constraining power of either The LA model has allowed a firm foothold on the study probe combination decreases as we have marginalised over of IAs to develop over the last decade. We have detailed an 130 free parameters which represent our ignorance of the updated implementation of the most used IA model and its exact details of IAs (and another 100 for galaxy bias). What physical motivation, showing a reduction in biasing of cos- is also clear is that the biasing of our cosmological estimates mological constraints. In addition we reiterate that a robust has decreased to well within the 1σ error of the respective nuisance parameter model can control the biasing due to IAs probes. for either the old or new implementations. Together, these The constraints on w0 and wa from the full ǫǫ + nǫ + nn results should give us confidence that the IA effect is un- combination after marginalisation over 230 nuisance param- der control as we begin to analyse the first data from large eters are only reduced by a factor of two compared to the cosmic shear surveys. naive (and heavily biased if the wrong IA model is used) However our somewhat pessimistic approach carries the constraints from ǫǫ alone when it is assumed IAs are per- cost of reduced constraining power. Future work, focused

c 2009 RAS, MNRAS 000, 1–12 The Cosmological Impact of Intrinsic Alignment Model Choice 9 on accurate simulation and measurement of IAs is sure to Heavens A., Refregier A., Heymans C., 2000, Monthly No- provide more detail on the physical mechanisms responsi- tices of the Royal Astronomical Society, 319, 649 ble for the initial intrinsic ellipticity distribution and its Heymans C., Brown M., Heavens A., Meisenheimer K., evolution. This better knowledge of IAs will improve our Taylor A., Wolf C., 2004, Monthly Notices of the Royal ability to model them and reduce our dependence on brute- Astronomical Society, 347, 895 force marginalisation over nuisance parameters. As such the Heymans C., Heavens A., 2003, Monthly Notices of the marginalised constraints which we present in Fig. 3 may be Royal Astronomical Society, 339, 711 a worst-case scenario. With better knowledge of IAs, better Hirata C. M., Mandelbaum R., Ishak M., Seljak U., Nichol constraints on cosmology will be possible. R., Pimbblet K. A., Ross N. P., Wake D., 2007, Monthly Notices of the Royal Astronomical Society, 381, 1197 Hirata C. M., Mandelbaum R., Seljak U., Guzik J., Pad- manabhan N., Blake C., Brinkmann J., Bud´avari T., Con- ACKNOWLEDGEMENTS nolly A., Csabai I., Scranton R., Szalay A. S., 2004, The authors are very grateful to Lisa Voigt who was in- Monthly Notices of the Royal Astronomical Society, 353, timately involved as a contributor to the preparatory work 529 leading to this paper. The authors are thankful to Benjamin Hirata C. M., Seljak U., 2004, PRD, 70, 063526 Joachimi for useful discussions regarding Joachimi & Bridle Hirata C. M., Seljak U., 2010, arXiv:astro-ph/0406275 (2009). We also thank Rachel Bean and Istvan Laszlo for Hoekstra H., Jain B., 2008, Annual Review of Nuclear and very useful discussion and work on the best implementation Particle Science, 58, 99 of the lA model. Sarah Bridle acknowledges support from Hu W., 1999, Astrophysical Journal Letters, 522, L21 the Royal Society in the form of a University Research Fel- Hu W., Jain B., 2004, PRD, 70, 043009 lowship and the European Research Council in the form of Huterer D., Takada M., Bernstein G., Jain B., 2006, a Starting Grant with number 240672. Monthly Notices of the Royal Astronomical Society, 366, 101 J. Buergers, H. van de Hulst H. ed. 1951, Problems of cos- A This paper has been typeset from a TEX/ LTEX file prepared mical aerodynamics; proceedings of a Joint Symposium 1 by the author. on the Motion of Gaseous Masses of Cosmical Dimensions held at Paris, August 16-19, 1949. Jing Y. P., 2002, Monthly Notices of the Royal Astronom- ical Society, 335, L89 REFERENCES Joachimi B., Bridle S. L., 2009, arXiv:0911.2454 Albrecht A., Bernstein G., Cahn R., Freedman W. L., He- Joachimi B., Mandelbaum R., Abdalla F. B., Bridle S. L., witt J., Hu W., Huth J., Kamionkowski M., Kolb E. W., 2011, Astronomy and Astrophysics, 527, A26 Knox L., Mather J. C., Staggs S., Suntzeff N. B., 2006, Joachimi B., Schneider P., 2008, Astronomy and Astro- ArXiv: 0609591 physics, 488, 829 Amara A., Refregier A., 2006, arXiv:astro-ph/0610127 Joachimi B., Schneider P., 2009, Astronomy and Astro- Bacon D., Refregier A., Ellis R., 2000 physics, 507, 105 Bartelmann M., Schneider P., 2001, Phys. Rept., 340, 291 Kaiser N., 1992, Astrophysical Journal, 388, 272 Bernstein G. M., 2009, Astrophysical Journal, 695, 652 Kaiser N., Wilson G., Luppino G. A., 2000 Blazek J., McQuinn M., Seljak U., 2011, Journal of Cos- King L., Schneider P., 2002, Astronomy and Astrophysics, mology and Astroparticle Physics, 5, 10 396, 411 Brainerd T. G., Agustsson I., Madsen C. A., Edmonds King L. J., 2005, Astronomy and Astrophysics, 441, 47 J. A., 2009, arXiv:0904.3095 King L. J., Schneider P., 2003, Astronomy and Astro- Bridle S., King L., 2007, New Journal of Physics, 9, 444 physics, 398, 23 Brown M. L., Taylor A. N., Hambly N. C., Dye S., 2002, Kirk D., Bridle S., Schneider M., 2010, Monthly Notices of Monthly Notices of the Royal Astronomical Society, 333, the Royal Astronomical Society, 408, 1502 501 Kirk D., Laszlo I., Bridle S., Bean R., 2011, Catelan P., Kamionkowski M., Blandford R. D., 2001, arXiv:1109.4536 Monthly Notices of the Royal Astronomical Society, 320, Laszlo I., Bean R., Kirk D., Bridle S., 2011, L7 arXiv:1109.4535 Ciotti L., Dutta S. N., 1994, Monthly Notices of the Royal Mackey J., White M., Kamionkowski M., 2002, Monthly Astronomical Society, 270, 390 Notices of the Royal Astronomical Society, 332, 788 Crittenden R. G., Natarajan P., Pen U.-L., Theuns T., Mandelbaum R., et al., 2009, arXiv:0911.5347 2001, Astrophysical Journal, 559, 552 Mandelbaum R., Hirata C. M., Broderick T., Seljak U., Croft R. A. C., Metzler C. A., 2000, Astrophysical Journal, Brinkmann J., 2006, Monthly Notices of the Royal Astro- 545, 561 nomical Society, 370, 1008 Dekel A., Lahav O., 1999, Astrophysical Journal, 520, 24 Munshi D., Valageas P., Van Waerbeke L., Heavens A., Doroshkevich A. G., 1970, Astrofizika, 6, 581 2006 Eisenstein D. J., Hu W., 1998, Astrophysical Journal, 496, Okumura T., Jing Y. P., 2009, Astrophysical Journal Let- 605 ters, 694, L83 Heavens A., Peacock J., 1988, Monthly Notices of the Royal Okumura T., Jing Y. P., Li C., 2009, Astrophys. J., 694, Astronomical Society, 232, 339 214

c 2009 RAS, MNRAS 000, 1–12 10 Donnacha Kirk, Anais Rassat, Ole Host, Sarah Bridle

Peebles P. J. E., 1969, Astrophysical Journal, 155, 393 Croft & Metzler (2000) point out that future tomographic Rassat A., Amara A., Amendola L., Castander F. J., Kitch- measurements will also be significantly affected. ing T., Kunz M., Refregier A., Wang Y., Weller J., 2008, Hirata & Seljak (2004) (which we are calling HS04) dis- arXiv:0810.0003 cussed both models and used the term ‘linear alignment Refregier A., 2003, Ann. Rev. Astron. Astrophys., 41, 645 model’ to describe the alignment of the galaxy halo with Schaefer B. M., 2008 the tidal stretch of the local gravitational potential. Previ- Schneider M. D., Bridle S., 2010, Monthly Notices of the ous to HS04 the cosmic shear contamination was believed Royal Astronomical Society to come from physically close galaxies due to both galaxies Schrabback T., et al., 2009, arXiv:0911.0053 forming in the same large-scale structure. HS04 identified Sciama D. W., 1955, Monthly Notices of the Royal Astro- a second term due to the intrinsic alignment of one galaxy nomical Society, 115, 3 near to a large mass which would correlate with the grav- Smith R. E., Peacock J. A., Jenkins A., White S. D. M., itational alignment of a distant galaxy lensed by the same Frenk C. S., Pearce F. R., Thomas P. A., Efstathiou G., mass. The first of these two was labelled ‘intrinsic-intrinsic’ Couchman H. M. P., 2003, Monthly Notices of the Royal (II) correlation and the second ‘gravitational-intrinsic’ (GI) Astronomical Society, 341, 1311 correlation. Takada M., White M., 2004, Astrophysical Journal Letters, Following the approach of Catelan et al. (2001), HS04 601, L1 assume that galaxy intrinsic ellipticity follows the linear re- van den Bosch F. C., Abel T., Croft R. A. C., Hernquist lation

L., White S. D. M., 2002, Astrophys. J., 576, 21 I C1 2 2 γ = ∆ ∆ , 2∆x∆y [ΦP ], (34) van Waerbeke L., et al., 2000, Astron. Astrophys., 358, 30 −4πG x − y S White S. D. M., 1984, Astrophysical Journal, 286, 38 where ΦP is the Newtonian potential at the time of galaxy Wittman D. M., Tyson J. A., Kirkman D., Dell’Antonio I., formation, smoothed by some filter that cuts off fluctu- Bernstein G., 2000, Nature, 405, 143 S ations on galactic scales, G is the Newtonian gravitational constant, ∆ is a comoving derivative and C1 is a normalisa- tion constant. After relating the primordial potential to the linear APPENDIX: HISTORY OF THE LA MODEL matter power spectrum, HS04 calculate the II and GI power spectra The history of galaxy intrinsic alignment studies goes 2 back to J. Buergers, H. van de Hulst H. (1951) and Sciama HS04 C1ρ¯(z) lin PII (k,χ) = − Pδδ (k,χ) (35) (1955) who discussed that galaxies may acquire angular D¯ (z)   momentum through tidal torquing, in which anisotropic C ρ¯(z) P HS04(k,χ) = 1 P lin(k,χ) (36) shear flows distort the protogalaxy and lead to a lo- GI − D¯(z) δδ cal rotation. The theory was further developed analyti- in terms of the linear theory matter power spectrum at co- cally by Peebles (1969), Doroshkevich (1970) and White lin (1984), and first simulated by Heavens & Peacock (1988). moving distance χ corresponding to redshift z, Pδδ (k,χ). ¯ If galaxy orientations are correlated with their angular Hereρ ¯(z) is the mean density of the Universe, D(z) = momenta then we may expect neighbouring galaxies to (1 + z)D(z) is the rescaled growth factor D(z). be aligned (e.g. van den Bosch et al. 2002). Heavens et al. Note that HS04 actually present the power spectra for (2000); Croft & Metzler (2000); Catelan et al. (2001) simul- γ˜I , the density weighted intrinsic shear (and δ, γ˜I for the GI taneously pointed out that this could lead to a contami- term). This is because galaxies are not randomly positioned, nating term in weak gravitational lensing, and this was in- but are expected to form preferentially in regions of higher vestigated further in Crittenden et al. (2001); Mackey et al. matter density. This leads to higher order terms in the IA (2002); Jing (2002); Heymans et al. (2004); Hirata et al. power spectra which are usually ignored because they are (2004) and Bridle & King (2007). See Schaefer (2008) for about an order of magnitude smaller than the leading term a more recent review. in the above equations (Bridle & King 2007). These equations were used in several publications un- Ciotti & Dutta (1994) described a second ansatz to mo- 2 tivate the intrinsic alignments of elliptical galaxies, based on til the discovery of missing factors of a which arose in tidal stretching of the host halo by the surrounding large- the conversion factor between the density perturbation and scale structure. Catelan et al. (2001) showed that the corre- the primordial potential. These were corrected in an erra- lation function of the intrinsic alignments should be propor- tum Hirata & Seljak (2010), which we call HS10. In the up- tional to that of the tidal field and thus be significant even dated version, eqns. 35 and 36 become 2 on large scales. This contrasts with the correlation function C1ρ¯(z) P HS10(k,χ) = − a2 P lin(k,χ) (37) expected from tidal torque theory which has a higher order II D¯ (z) δδ dependence on the tidal field and thus decays more rapidly.   C ρ¯(z) Heavens et al. (2000); Croft & Metzler (2000) used the HS10 1 2 lin PGI (k,χ) = ¯ a Pδδ (k,χ). (38) shapes of dark matter halos in N-body simulations as a − D(z) proxy for elliptical galaxy shapes and estimated the pos- Originally HS04 related their PII and PGI to the lin sible contamination to the (then) recent cosmic shear de- linear matter power spectrum Pδδ . It was subsequently tections. They found the contamination to be a fraction of indicated by Hirata et al. (2007) that more power at the measured signal but Heavens et al. (2000) commented small scales may fit the data better and Bridle & King that low redshift surveys would be strongly affected and (2007) proposed that the non-linear matter power spec-

c 2009 RAS, MNRAS 000, 1–12 The Cosmological Impact of Intrinsic Alignment Model Choice 11 trum be substituted for the linear in what became known 0.2 and the Non-Linear Alignment (NLA) model, which was δδ II HS04 then applied in Bridle & King (2007); Schneider & Bridle II this work (2010); Mandelbaum et al. (2009); Joachimi et al. (2011); 0.15 GG Kirk et al. (2010); Laszlo et al. (2011) and Kirk et al. (2011). It is this common NLA approach, which we describe 0.1 as HS04NL, 2 HS04NL C1ρ¯(z) PII (k,χ) = − Pδδ(k,χ) (39) 0.05 D¯(z)   HS04NL C1ρ¯(z) PGI (k,χ) = Pδδ(k,χ), (40) − D¯ (z) Integrand at l=2000 without P 0 0 1 2 3 where the non-linear matter power spectrum is used in both z equations. 1.5 II HS04NL In this work we employ a more physically motivated II This Work approach based on the idea that galaxy intrinsic alignments GG are seeded at some early epoch of galaxy formation and do not evolve with redshift. This was the original intent of the at l=2000 1 LA model in Catelan et al. (2001) and HS04. This principle 1/2 is apparent from the original HS04 equations by the presence of the 1/D(z) terms which divide out the growth. Therefore lin 0.5 we relate our II term directly to Pδδ but the GI term picks up some contribution from the later (linear and nonlinear) growth of large scale structure via its dependence on cos- Integrand*P(k) 0 mic shear. This is included in the formalism by the geomet- 0 1 2 3 ric mean of the linear and nonlinear matter power spectra z in equation 9. This was proposed and discussed further in Laszlo et al. (2011); Kirk et al. (2011). Figure 4. Integrand of the GG angular power spectrum at a mul- tipole of l = 2000 (2πGρ(z)a2c−2W (χ) P (k,χ)) [blue dotted] Note that equations 39 and 40 are the same as our ex- i p δδ pressions in eqns. 8 and 9 with the prefactor in brackets sim- and the II integrand for the original Hirata & Seljak (2004) LA model (C ρ(z)an (χ) P (k,χ)) [red dotted] and the LA model plified, and there is a different treatment of the linear/non- 1 i p δδ C ρ z n χ P lin k,χ linear matter power spectra. given in this work ( 1 ( = 0) i( )q δδ ( )) [red solid], both An improvement on the contributions at non-linear shown for l = 2000. We present integrands without [upper panel] scales was developed by Schneider & Bridle (2010) by devel- and with the appropriate matter power spectrum term [lower oping a halo model of intrinsic alignments. We do not use it panel]. Each angular power spectrum is plotted for the first and tenth tomographic redshift bins of our fiducial survey, these ap- here as it does not contain full flexibility of the cosmological pear to the left and right hand of each plot respectively. Here model. the term P (k) corresponds to either the linear or non-linear mat- In this paper we discuss and illustrate the difference be- ter power spectrum depending on the term (G or I), and on the tween the models, summarised in Fig. 4. The changes can implementation used (HS04NL or This Work). be divided into two types: (i) the correct use of the a2 terms as included in Hirata & Seljak (2010) and (ii) the consistent use of the linear matter power spectrum for the II correla- tion but the square root product of the linear and nonlinear creases with redshift in the HS04NL prescription. As IAs are power spectra for the GI correlation (see Equation 9). The normalised against a C1 value measured at low redshift this upper panel of Fig. 4 includes only the first change by plot- means that the new II term is comparable to the HS04NL ting the square root of the Cl integrand without the contri- term at low redshift (if it were possible to show the II terms bution from the matter power spectrum. The terms plotted at z = 0 on the top panel of Fig. 4 they would be identical) for bins i = 1, 10 are but increasingly diverge a higher redshifts. This is why the 2 II kernel for the 10th (high-z) redshift bin is reduced more 2πGρ(z)a GG : c2 Wi(χ) (41) as we move from HS04NL to the new model than the equiv- HS04 II : C1ρ(z)ani(χ) (42) alent lines for the 1st (low-z) bin. As the shear-shear term is unaffected by our treatement of IAs this means that IAs IIThis Work : C ρ(z = 0)n (χ). (43) 1 i make a less significant contribution to the total shear signal The lower panel includes the square root of the matter power at higher redshift than in the HS04NL approach. spectrum contribution, plotting The impact of moving from the common NLA approach 2 2πGρ(z)a to our motivated treatment of nonlinear clustering and IAs GG : c2 Wi(χ) Pδδ(k,χ) (44) is to weaken the contribution of IAs relative to cosmic shear HS04NL II : C1ρ(z)ani(χ) pPδδ(k,χ) (45) because we discard all nonlinear clustering power for the IA terms. This means we lose all power due to nonlinear IIThis Work : C ρ(z = 0)n (χ) P lin(k,χ). (46) 1 i p δδ clustering in the II term and half the nonlinear power from Correct inclusion of the a2pterms produces an IA signal the GI term where the I contribution is linear but the G which is constant with redshift, rather than one which in- contribution remains dependent on the fully nonlinear mat-

c 2009 RAS, MNRAS 000, 1–12 12 Donnacha Kirk, Anais Rassat, Ole Host, Sarah Bridle

Figure 5. Projected angular power spectra which contain IA contributions. Computed for our fiducial survey and displayed for a variety of tomographic redshift bin combinations as in Fig 1. Power spectra are computed using the HS04NL approach [narrow lines] and the new approach described in this work [thick lines]. The GG (solid black narrow) power spectrum is shown for comparison.

ter power spectrum. The impact is felt at all redshifts but ing to the latest LA implementation are thick versions of the effect is strongest at low-z where nonlinear clustering is the same line-style as their HS04NL equivalents. The shear- strongest. Fig. 5 shows the projected angular power spectra shear (GG) angular power spectra are, of course, unaffected for those components which are affected by the change from by the change, they are shown in the upper triangle for com- the HS04NL model (as used in Joachimi & Bridle (2009)) parison. to the latest implementation described in this work. The format of the plot is the same as Fig. 1, lines correspond-

c 2009 RAS, MNRAS 000, 1–12