Simultaneous measurement of cosmology and intrinsic alignments using joint cosmic shear and galaxy number density correlations

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1 A&A 523, A1 (21) DOI: 1.151/4-6361/ c ESO 21 Astronomy & Astrophysics Simutaneous measurement of cosmoogy and intrinsic aignments using joint cosmic shear and gaaxy number density correations B. Joachimi 1,2 and S. L. Bride 1 1 Department of Physics and Astronomy, University Coege London, London WC1E 6BT, UK e-mai: joachimi@astro.uni-bonn.de 2 Argeander-Institut für Astronomie (AIfA), Universität Bonn, Auf dem Hüge 71, Bonn, Germany Received 12 November 29 / Accepted 9 August 21 ABSTRACT Aims. Cosmic shear is a powerfu method to constrain cosmoogy, provided that any systematic effects are under contro. The intrinsic aignment of gaaxies is expected to severey bias parameter estimates if not taken into account. We expore the potentia of a joint anaysis of tomographic gaaxy eipticity, gaaxy number density, and eipticity-number density cross-correations to simutaneousy constrain cosmoogy and sef-caibrate unknown intrinsic aignment and gaaxy bias contributions. Methods. We treat intrinsic aignments and gaaxy biasing as free functions of scae and redshift and marginaise over the resuting parameter sets. Constraints on cosmoogy are cacuated by combining the ikeihoods from a two-point correations between gaaxy eipticity and gaaxy number density. The information required for these cacuations is aready avaiabe in a standard cosmic shear data set. We incude contributions to these functions from cosmic shear, intrinsic aignments, gaaxy custering and magnification effects. Resuts. In a Fisher matrix anaysis we compare our constraints with those from cosmic shear aone in the absence of intrinsic aignments. For a potentia future arge area survey, such as Eucid, the extra information from the additiona correation functions can make up for the additiona free parameters in the intrinsic aignment and gaaxy bias terms, depending on the fexibiity in the modes. For exampe, the dark energy task force figure of merit is recovered even when more than 1 free parameters are marginaised over. We find that the redshift quaity requirements are simiar to those cacuated in the absence of intrinsic aignments. Key words. cosmoogy: theory gravitationa ensing: weak arge-scae structure of the Universe cosmoogica parameters methods: data anaysis 1. Introduction On its way to Earth the ight from distant gaaxies is continuousy defected by the matter density inhomogeneities which it passes by. This induces distortions of the shapes of the projected gaaxy images on the sky, causing modifications of these shapes of order 1%, known as weak gravitationa ensing, or cosmic shear. Hence, to measure this distortion effect on distant gaaxies, one requires statistica methods; see e.g. Bartemann & Schneider (21)and Schneider (26) for detaied reviews. Detected in 2 (Bacon et a. 2; Kaiser et a. 2; van Waerbeke et a. 2; Wittman et a. 2), cosmic shear has since rapidy evoved into a mature technique that produces increasingy stringent constraints on cosmoogica parameters (see e.g. Jarvis et a. 26; Hoekstra et a. 26; Sembooni et a. 26; Hetterscheidt et a. 27; Benjamin et a. 27; Massey et a. 27; Schrabback et a. 27; Fu et a. 28; Schrabback et a. 21). Probing both the evoution of structure and the geometry of the Universe at ow redshifts, it is compementary to observations of the cosmic microwave background (e.g. Tereno et a. 25; Das & Sperge 29) and considered as potentiay the most powerfu method to pin down the properties of dark energy (Hu 22; Abrecht et a. 26; Peacock et a. 26). Upcoming and future surveys ike Pan-STARRS 1,DES 2, LSST 3, 1 Panoramic Survey Teescope & Rapid Response System, 2 Dark Energy Survey, 3 Large Synoptic Survey Teescope, JDEM 4, and Eucid 5 wi produce deep imaging over arge fractions of the sky and thereby yied unprecedented insight into the properties of dark matter, dark energy and gravitation (e.g. Takada & Jain 24; Réfrégier et a. 24, 26; Kitching et a. 28a; Thomas et a. 29). The great statistica power of cosmic shear demands a carefu assessment of possibe systematic errors that might bias the resuts. A serious imitation may arise from a physica systematic caused by the intrinsic aignment of gaaxies. The matter structure around gaaxies can modify their intrinsic shape and their orientation. Firsty, this can resut in correations between the intrinsic shapes of gaaxies which are cose both on the sky and in redshift (intrinsic eipticity correations, or II correations). Moreover, a dark matter hao can intrinsicay aign a physicay cose gaaxy in the foreground and at the same time contribute to the ensing signa of a background object, which induces gravitationa shear-intrinsic eipticity correation (GI, Hirata & Sejak 24). Detaied investigations have been performed on the aignment between haoes (Croft & Metzer 2; Heavens et a. 2; Lee & Pen 2; Catean et a. 21; Crittenden et a. 21; Jing 22; Mackey et a. 22; Hirata & Sejak 24; Bride & Abdaa 27; Schneider & Bride 21), as we as the aignment of the spin or the shape of a gaaxy with its own 4 Joint Dark Energy Mission, fareaid=12 Artice pubished by EDP Sciences Page 1 of 21

2 A&A 523, A1 (21) dark matter hao (e.g. Pen et a. 2; van den Bosch et a. 22; Okumura et a. 29; Okumura & Jing 29; Brainerd et a. 29; see aso Schäfer 29). Intrinsic aignments have aso been investigated observationay, where non-vanishing II and GI signas have been detected in severa surveys (Brown et a. 22; Heymans et a. 24; Mandebaum et a. 26; Hirata et a. 27; Brainerd et a. 29). The resuts of both theoretica studies and observations show arge variations, but most are consistent with a contamination of the order 1% by both II and GI correations for future surveys that further divide the gaaxy sampe into redshift sices (cosmic shear tomography). Hence, the contro of intrinsic aignments in cosmic shear studies is crucia to obtain unbiased resuts on cosmoogica parameters. Accurate modes woud sove the probem, but progress is hampered due to the dependence of intrinsic aignments on the intricacies of gaaxy formation and evoution within their dark matter environment. Currenty, the eve of modes is crude, and party ony motivated phenomenoogicay (see e.g. Schneider & Bride 21, for recent progress). The II contamination can be controed reativey easiy by excuding cose pairs of gaaxies from the anaysis (King & Schneider 22, 23; Heymans & Heavens 23; Takada & White 24). Joachimi & Schneider (28, 29) introduced a nuing technique which transforms the cosmic shear data vector and discards a entries of the transformed data set that are potentiay contaminated by the GI signa. Whie this approach ony reies on the we-known redshift dependence of gravitationa ensing, King (25) projects out the GI term by making use of tempate functions. Furthermore the work by Mandebaum et a. (26) andhirata et a. (27) suggests that the intrinsic aignment may be dominated by uminous red gaaxies which coud be eiminated from the cosmic shear cataogues. A these remova techniques require exceent redshift information, and sti they can cause a significant reduction in the constraints on cosmoogy. Deep imaging surveys not ony provide information about the shape of gaaxies, but aow in addition for a measurement of gaaxy number densities, as we as cross-correations between shape and number density information. This substantia extension of the set of observabes increases the cosmoogica information to be extracted and, more importanty, enabes one to internay caibrate systematic effects (Hu & Jain 24; Bernstein 29). By adding gaaxy number density information one adds signas that are capabe of pinning down the functiona form of intrinsic aignments, but one aso introduces as another systematic, the gaaxy bias, which quantifies the ack of knowedge about how gaaxies, i.e. the visibe baryonic matter, foow the underying dark matter distribution. It is the scope of this work to eucidate the performance of a joint anaysis of gaaxy shape and number density information as regards the abiity to constrain cosmoogica parameter in presence of genera and fexibe modes of intrinsic aignments and gaaxy bias. In doing so we incorporate severa cosmoogica signas which have been considered before as promising probes of cosmoogy themseves, incuding gaaxy custering from photometric redshift surveys (Bake & Bride 25; Doney et a. 26; Zhan 26; Bake et a. 27; Padmanabhan et a. 27) gaaxy-gaaxy ensing (e.g. Schneider & Rix 1997; Guzik & Sejak 21, 22; Sejak 22; Sejak et a. 25; Yoo et a. 26; Johnston et a. 27; Cacciato et a. 29) and ensing magnification (Broadhurst et a. 1995; Zhang & Pen 25, 26; van Waerbeke 21). We foow the ansatz outined in Bernstein (29) and extend the investigation by Bride & King (27) who considered the residua information content in gaaxy shape correations after marginaising over the parameters of two oginear grid modes representing the II and GI terms. We quantify the cross-caibration properties of the joint set of observabes and determine the requirements on cosmoogica surveys to efficienty appy this joint approach. This paper is organised as foows: in Sect. 2 we give an overview on the two-point correations that form part of the gaaxy shape and number density observabes, and we derive their expicit form. Two appendices provide further detais. Section 3 demonstrates how we mode the different signas and their dependence on cosmoogy. We introduce a genera grid parametrisation for the intrinsic aignments and the gaaxy bias. Furthermore we summarise our Fisher matrix formaism and the figures of merit we empoy. In Sect. 4 we present our resuts on the dependence of the parameter constraints on the freedom in the mode of intrinsic aignments and gaaxy bias, the characteristics of the redshift distributions, and the priors on the different sets of nuisance parameters. Finay, in Sect. 5 we summarise our findings and concude. 2. Two-point correations from cosmoogica surveys Cosmoogica imaging surveys observe the anguar positions and the projected shapes of huge numbers of gaaxies over increasingy arge areas on the sky. In addition, by means of muticoour photometry, it is possibe to perform a tomographic anaysis, i.e. obtain coarse information about the ine-of-sight dimension in terms of photometric redshifts (photo-z). From the gaaxy shapes in a given region of space, one can infer the eipticity ɛ (i) (θ) = γ (i) G (θ) + γ(i) I (θ) + ɛ (i) rnd (θ), (1) where the superscript in parentheses assigns a photo-z bin i.the observed eipticity ɛ has contributions from the gravitationa shear γ G and an intrinsic shear γ I, which is caused by the aignment of a gaaxy in its surrounding gravitationa fied. Moreover, ɛ is assumed to have an uncorreated component ɛ rnd, which accounts for the purey random part of the intrinsic orientations and shapesof gaaxies.note that (1) is ony vaid if the gravitationa shear is weak, see e.g. Seitz & Schneider (1997); Bartemann & Schneider (21) and for certain definitions of eipticity. Likewise, the positions of gaaxies can be used to construct an estimate of the number density contrast n (i) (θ) = n (i) m (θ) + n (i) g (θ) + n (i) rnd (θ), (2) which is determined by the intrinsic number density contrast of gaaxies n g and the ateration of gaaxy counts due to ensing magnification n m. An uncorreated shot noise contribution is added via n rnd. In contrast to ɛ (i) (θ) the number density contrast n (i) (θ) can obviousy not be estimated from individua gaaxies. One can understand n (i) (θ) as the ensembe average over a hypothetica, Poisson-distributed random fied of which the observed gaaxy distribution is one particuar representation. The forma reation between the projected number density contrast as used in (2) and the three-dimensiona gaaxy number density fuctuations wi be provided beow, see (12). As was aready noted in Bernstein (29), (1) and(2) are symmetric in the sense that they both contain an intrinsic contribution and a term caused by gravitationa ensing effects. Under usua circumstances the correated part of the eipticity is dominated by the gravitationa shear, whereas the argest term in (2) is due to the intrinsic number density contrast. Page 2 of 21

3 B. Joachimi and S. L. Bride: Simutaneous measurement of cosmic shear and gaaxy number density correations Tabe 1. Overview on the two-point correations considered in this work. Measured correation 2D PS 3D PS shear C GG P δδ intrinsic-shear C IG P δi intrinsic C II P II gaaxy custering C gg P gg custering-magnification C gm P gδ magnification C mm P δδ custering-shear C gg P gδ custering-intrinsic C gi P gi magnification-shear C mg P δδ magnification-intrinsic C mi P δi gaaxy eipticity (observabe) C ɛɛ gaaxy number density (observabe) C nn number density-eipticity (observabe) Notes. Listed are the symbos used for the two-dimensiona projected power spectra and the underying three-dimensiona power spectra. Both eipticity and number over-density vanish if averaged over sufficienty argescaes. Thus, one considersto owest order two-point statistics of these quantities. Since a rea-space twopoint measures are reated to the power spectrum (see e.g. Kaiser 1992), we can work in terms of power spectra without oss of generaity, which is desirabe in particuar due to a simper structure of the signa covariances in Fourier space. Denoting the Fourier transform by a tide, the power spectrum ab () between redshift bins i and j can then be defined by x (i) a () x ( j) b ( ) = (2π) 2 δ (2) D ( ) ab (), (3) where δ (2) D is the two-dimensiona Dirac deta-distribution, and where denotes the anguar frequency, the Fourier variabe on the sky. The measures x a and x b can correspond to any of the set { } γ G,γ I, n g, n m. The random contributions in (1) and(2) are not correated with any of the other measures and ony yied a contribution to the noise, see Sect Inserting (1) and(2) into(3), one obtains the compete set of tomographic two-point observabes which are avaiabe from shape and number density information C ɛɛ (ij) () = ji) GG () + C(ij) IG () + C( IG () + C(ij) II () (4) C nn (ij) () = C gg (ij) () + C gm (ij) () + C gm ( ji) () + C mm() (ij) (5) nɛ () = gg () + C(ij) gi () + C(ij) mg C nɛ () + C(ij) mi (), (6) see Bernstein (29). We name signas stemming from gaaxy shape information by capita etters ( G for gravitationa shear, I for intrinsic shear) and signas reated to gaaxy number densities by sma etters ( g for intrinsic number density fuctuations, m for ensing magnification). An overview of the nomencature of the correations in (4) to(6) is provided in Tabe 1. Note that (4) and(5) are symmetric with respect to their photo-z bin arguments. Hence, if N zbin denotes the number of avaiabe photo-z bins, one has N zbin (N zbin + 1)/2 observabes for every considered anguar frequency. In contrast, one can expoit Nzbin 2 eipticity-number density cross-correation power spectra (6)per. The set of observabes in (4) is the one that cosmic shear anayses are based on. The shear correation signa (GG) is a cean probe of the underying matter power spectrum and is thus powerfu in constraining cosmoogica parameters (e.g. Hu 1999). However, shape measurements incorporate further terms stemming from correations of intrinsic eipticities (II) and shear-intrinsic cross-correations (IG, or equivaenty GI) whose contribution can be substantia, but is to date poory known (Hirata & Sejak 24). These terms exist because the shapes and orientations of gaaxies are infuenced via the tida gravitationa fieds of the matter structures in their surrounding, which firsty induce correations between neighbouring gaaxies, and secondy cause correations by determining the intrinsic shape of a foreground object and adding to the shear signa of a background gaaxy. Intrinsic gaaxy custering (gg) add a strong signa to the correations of gaaxy number densities (5), but its use to obtain cosmoogica parameter estimates is imited due by poor knowedge of the gaaxy bias (e.g. Lahav & Suto 24). Gravitationa ensing modifies the fux of objects and thus reduces or increases number counts of gaaxies above a certain imiting magnitude, depending on the form of the gaaxy uminosity function cose to the imiting magnitude. This produces magnification correations (mm) and intrinsic number density-magnification crosscorreations (gm). The gm correations occur when a foreground mass overdensity (underdensity) contains an overdensity (underdensity) of gaaxies and (de)-magnifies background objects aong the same ine of sight causing an apparent over- or underdensity of gaaxies at higher redshift. Cross-correations between gaaxy number densities and eipticities (6) contain contributions from cross terms between intrinsic custering and shear (gg), intrinsic custering and intrinsic shear (gi), magnification and shear (mg), and magnification and intrinsic shear (mi). For instance, one expects to find gi and gg signas when a mass structure eads to an overdensity in the oca gaaxy distribution and infuences the intrinsic shape of gaaxies at the same redshift or contributes to the shear of background objects. The atter is the usua gaaxy-gaaxy ensing signa. Because a foreground overdensity can in addition enhance gaaxy counts due to ensing magnification, the mg and mi signas wi aso be non-vanishing. The form of a these correations wi be further discussed in Sect. 3. A non-random terms in (1) and(2), given for a photometric redshift bin i, can be reated to a source term S, whichisa function of spatia coordinates, i.e. x (i) a (θ) = dχ w (i) (χ) S a ( f K (χ)θ,χ), (7) where we defined a weight function w that depends on the photoz bin i (for a simiar approach see Hu & Jain 24). Here, χ denotes comoving distance, and f K (χ) is the comoving anguar diameter distance. If (7) hods for two quantities x (i) a and x ( j) b, their projected power spectrum is given by the ine-of-sight integra of the three-dimensiona source power spectrum P Sa S b via Limber s equation in Fourier space (Kaiser 1992), ab () = dχ w(i) (χ) w ( j) (χ) P Sa S b f K (χ),χ. (8) By identifying weights and source terms for gravitationa and intrinsic shear, as we as intrinsic custering and magnification, we can derive Limber equations for a power spectra entering (4) (6). To compute the equivaent of (7) for the cosmic shear case, we first note that in Fourier space the shear and the convergence are reated by the simpe equation κ G () = γ G ()e 2iϕ,whereϕ is the poar ange of. As a consequence, the power spectra of Page 3 of 21

4 A&A 523, A1 (21) shear and convergence are identica. Therefore, we can equivaenty use the convergence κ (i) (θ) as the cosmic shear observabe. It is reated to the three-dimensiona matter density contrast δ via κ (i) G (θ) = dχ q (i) (χ) δ ( f K (χ)θ,χ), (9) where the weight is given by q (i) (χ) = 3H2 Ω m f K (χ) χhor 2 c 2 a(χ) χ dχ p (i) (χ ) f K(χ χ) f K (χ, (1) ) see Bartemann & Schneider (21); Schneider (26) for detais. Here a denotes the scae factor and p (i) (χ) the comoving distance probabiity distribution of those gaaxies in bin i for which shape information is avaiabe. Anaogousy to the ensing case, one can define a convergence of the intrinsic shear fied κ (i) I (θ), which is directy reated to the intrinsic shear via κ I () = γ I () e 2iϕ. This intrinsic convergence is a projection of the three-dimensiona intrinsic shear fied κ I, which can be written as κ (i) I (θ) = dχ p (i) (χ) κ I ( f K (χ)θ,χ), (11) see e.g. Hirata & Sejak (24) for the anaogous expression in terms of intrinsic shear. Here we have assumed that the intrinsic shear fied is ike the gravitationa shear fied cur-free to good approximation. This hods for instance for the inear aignment mode deveoped in Hirata & Sejak (24). Then κ I corresponds to the Fourier transform of γ E I (k)asdefinedinschneider &Bride(21). Likewise, anguar gaaxy number density fuctuations n (i) g (θ) are given by the ine-of-sight projection of three-dimensiona number density fuctuations δ g as (e.g. Hu & Jain 24) n (i) g (θ) = dχ p (i) (χ) δ g ( f K (χ)θ,χ). (12) It is important to note that p (i) (χ)isthesameasin(1)and(11), i.e. the number counts are restricted to those gaaxies with shape measurements, which require a higher signa-to-noise than the position determination. In principe, number density information coud be obtained for a arger number of gaaxies, in particuar fainter ones. But, to determine the contribution to number density correations by magnification, it is necessary to measure the sope of the uminosity function α (i) at the faint end of the used gaaxy distribution. We wi detai the exact definition and the determination of α (i) in Sect Since it is desirabe to extract the vaues of the α (i) internay from the survey, one needs to be abe to measure fuxes down to vaues sighty beow the magnitude imit of the gaaxies incuded in p (i) (χ). Moreover gaaxy number density measurements may require photometric redshifts which are of the same or better quaity than for cosmic shear studies, imiting the number of faint usabe gaaxies. Hence, we argue that the choice of identica distance probabiity distributions for both shape and number density signas is a fair assumption. We add the warning that one may have to account for seection biases, for instance if one investigates cosmic shear and magnification effects with the same gaaxy sampe (e.g. Schmidt et a. 29; Krause & Hirata 29). We write the contribution of magnification effects to the number density measurement as n (i) m (θ) = 2(α (i) 1) Page 4 of 21 = 2(α (i) 1) κ (i) G (θ). dχ q (i) (χ) δ ( f K (χ)θ,χ) (13) As severa assumptions enter this equation, in particuar concerning the treatment of the sope α (i), we provide a detaied derivation of (13) in Appendix A. Comparing the projection Eqs. (9) (13) to the genera form (7), one can derive a possibe cross- and auto-power spectra in the form of the genera Limber Eq. (8). For competeness we have given these Limber equations in Appendix B. Good modes of the three-dimensiona source power spectra in the Limber equations (see aso the right hand coumn of Tabe 1) are unknown except for the non-inear theory matter power spectrum P δδ. The distribution of gaaxies is expected to foow the distribution of dark matter, so that the gaaxy custering power spectra shoud be reated to P δδ. However, to date it is unknown how much the gaaxy custering deviates from dark matter custering, in particuar on sma scaes. This is usuay expressed in terms of the gaaxy bias b g, which is a function of both anguar scae k and redshift or ine-of-sight distance χ. Hence, one can write P gg (k,χ) = b 2 g(k,χ) P δδ (k,χ) (14) P gδ (k,χ) = b g (k,χ) r g (k,χ) P δδ (k,χ), where to describe the cross-correation between matter and gaaxy custering, we introduced a correation coefficient r g in the second equaity. The intrinsic aignment power spectra depend on the intricacies of gaaxy formation and evoution within their dark matter environment. Again, precise modes of the intrinsic aignment have to rey on baryonic physics and are currenty not avaiabe. For symmetry reasons we parametrise our ack of knowedge about the intrinsic aignment power spectra simiary to the gaaxy bias as P II (k,χ) = b 2 I (k,χ) P δδ(k,χ) (15) P δi (k,χ) = b I (k,χ) r I (k,χ) P δδ (k,χ) with the intrinsic aignment bias b I and correation coefficient r I (foowing Bernstein 29). Athough the power spectrum P gi coud in principe contain a third, independent correation coefficient, we assume that it is sufficient to write P gi (k,χ) = b I (k,χ) r I (k,χ) b g (k,χ) r g (k,χ) P δδ (k,χ), (16) i.e. we hypothesise that correations between intrinsic number density fuctuations and intrinsic aignments can entirey be traced back to the effects of the intrinsic aignment bias and the gaaxy bias. This is a strong assumption since instead of introducing a fifth competey unconstrained bias term, (16) estabishes a ink between the gaaxy bias and intrinsic aignment biases. Our equation (16)iseffectivey incuded within the ast term in cury brackets of Bernstein (29), Eq. (19), and we have effectivey set s gκ = inbernstein (29), Eq. (35). Bernstein (29)fixeshiss gκ to be a singe unknown scaar across the survey, stating that we expect this type of cross-correation to have a minima effect on cosmoogica constraints. It woud be interesting to check this by comparing resuts in which s gκ is aowed to vary with those in which s gκ =. However this is beyond the scope of this paper. We note that the unknown quantity in question is the crosscorreation between the intrinsic aignment fied and the gaaxy position fied (r gκ in the Bernstein 29, notation), which is precisey the correation measured in the observationa constraints papers (e.g. Mandebaum et a. 26; Hirata et a. 27; Mandebaum et a. 21). Our ansatz states that this is simpy

5 reated to the cross-correation between the intrinsic aignment fied and the mass, and the cross-correation between the gaaxy position fied and the mass (r gκ = r g r κ in the Bernstein 29, notation). These atter two quantities are much harder to measure because we do not know the mass fied we. Indeed gravitationa ensing and the observations we discuss in this paper are ikey the best probe of these quantities. Utimatey these data coud be used to constrain this additiona freedom. To shed further ight on this question, we now discuss a simpe toy mode to iustrate the nature of our assumption. In the simpest case the gaaxy distribution may trace the mass distribution, and the intrinsic shear fied is the gradient of the mass distribution. In a toy universe we coud dispace the position mass fied on the sky reative to the other two fieds which woud render the correation coefficient r I (or r κ in the notation of Bernstein 29) zero and aso the correation r g = (for arge dispacements), eaving a strong correation between the intrinsic shear fied and the gaaxy position fied (unity r gκ in the Bernstein 29, notation, thus s gκ = 1). Simiar more physica arguments coud be made by considering stochasticity instead of dispacements. However, we might expect the physica origin of both the gaaxy position fied and the intrinsic shear fied to both ie in the mass fied, since most of the interactions are mediated by gravity. Therefore it seems reasonabe to expect there to be no additiona interpay between the intrinsic shear fied and gaaxy position fieds that woud produce further cross-correations (non-zero s gκ in the Bernstein 29, notation). We wi not imit the vaues of the correation coefficients to the interva [ 1; +1]. It is formay possibe that r > 1 if our assumption about the statistics of the gaaxy distribution, usuay taken to be Poissonian, is incorrect (Bernstein 29). Treating the correation coefficients as competey free parameters, our choice of parametrisation in (14) and(15) is equivaent to modeing P gg and P gδ, or ikewise P II and P δi, independenty. We insert the parametrisations (14), (15), and (16) into the set of Limber equations and can this way reate a power spectra entering (4) (6) to the three-dimensiona matter power spectrum: GG () = B. Joachimi and S. L. Bride: Simutaneous measurement of cosmic shear and gaaxy number density correations ( b I f K (χ),χ r I f K (χ),χ P δδ dχ q(i) (χ) q ( j) (χ) P δδ ( ) f K (χ),χ (17) IG () = dχ p(i) (χ) q ( j) (χ) fk 2(χ) (18) b I f K (χ),χ r I f K (χ),χ P δδ f K (χ),χ II () = dχ p(i) (χ)p ( j) (χ) fk 2(χ) b 2 I f K (χ),χ P δδ f K (χ),χ (19) C gg (ij) () = dχ p(i) (χ)p ( j) ( (χ) fk 2(χ) b 2 g )P f K (χ),χ δδ f K (χ),χ (2) C gm (ij) () = 2(α ( j) 1) gg () (21) C mm() (ij) = 4(α (i) 1) (α ( j) 1) GG () (22) gg () = dχ p(i) (χ) q ( j) (χ) fk 2 (χ) b g f K (χ),χ r g f K (χ),χ P δδ f K (χ),χ (23) gi () = dχ p(i) (χ) p ( j) (χ) fk 2(χ) b g f K (χ),χ r g f K (χ),χ ) f K (χ),χ (24) mg () = 2(α(i) 1) GG () (25) mi () = 2(α(i) 1) C ( ji) IG (). (26) The matter power spectrum and the distances f K (χ), which aso enter q (i) (χ), depend on cosmoogy and can therefore be expoited to constrain cosmoogica parameters. Whie distances and P δδ are we known from theory, the probabiity distribution of gaaxies p (i) (χ) has to be measured by using additiona spectroscopic redshift information (e.g. Huterer et a. 26; Ma et a. 26; Abdaa et a. 27; Bride & King 27; Bernstein & Huterer 21) with a certain eve of uncertainty. It may aso be possibe to infer some additiona information from the cosmic shear data itsef (Newman 28; Schneider et a. 26; Zhang et a. 21). The same hods for the sopes of the uminosity function α (i), which can be determined from the survey by studying the fux of gaaxies cose to the magnitude imit. The east known quantities in the equations above are the bias terms { bg, b I, r g, r I }, for which we wi thus introduce a very genera parametrisation in Sect Modeing In this section we detai the modeing of the terms entering (17) to (26). We specify how we parametrise the uncertainty in the gaaxy redshift distributions, the sope of the uminosity function, and the bias terms. Moreover we describe our Fisher matrix approach and the way we infer the resuting errors on cosmoogica parameters Matter power spectrum and survey characteristics As the basis for our anaysis we compute matter power spectra for a spatiay fat CDM universe with fiducia parameters Ω m =.25, Ω DE =.75, and H = 1 h 1 km s 1 Mpc 1 with h 1 =.7. We incorporate a variabe dark energy mode by parametrising its equation of state, reating pressure p DE to density ρ DE,as(Chevaier & Poarski 21; Linder 23) ( z ) p DE (z) = w + w a ρ DE (z) c 2, (27) 1 + z where the ΛCDM Universe is chosen as the fiducia mode, i.e. w = 1 andw a =. The dark energy density parameter is then given by integrating Eq. (3) of Linder (23), { Ω DE (z) =Ω DE exp 3 ( (w + w a + 1) n(1 + z) w a z 1 + z )} (28) The primordia power spectrum of matter density fuctuations is assumed to be a power aw with fiducia sope n s = 1. We empoy the fiducia normaisation σ 8 =.8. The transfer function of Eisenstein & Hu (1998) is used without baryonic wigges, computing the shape parameter with a fiducia vaue of Ω b =.5. The non-inear corrections to the power spectrum are computed by means of the fit formua by Smith et a. (23). We account for the infuence of dark energy on structure growth by modifying the hao mode fitting routine of Smith et a. (23) foowing the approach of Réfrégier et a. (28). We provide a summary of this modification in Appendix D. The survey characteristics foow the rough specifications of a Stage IV experiment (Abrecht et a. 26) such as the ESA Eucid sateite mission. To compute the noise properties, we assume the maximum extragaactic sky coverage of Page 5 of 21

6 A&A 523, A1 (21) A = 2 deg 2 and a tota number density of gaaxies n = 35 arcmin 2. Shape noise is characterised by a tota dispersion of intrinsic eipticities of σ ɛ =.35. We refer to this survey as Eucid-ike in the remainder of this paper. According to Smai et a. (1994) we assume an overa number of gaaxies per unit redshift, per square arcminute { 3z 2 ( z z) β } n tot (z) = Σ exp (29) 2 z 3 with the gaaxy surface density Σ and β = 1.5. The probabiity distribution over a the gaaxies p tot (z) is proportiona to the number density n tot (z). We set z =.64, which produces a distribution with median redshift z med =.9. The distribution is cut at z max = 3 and then normaised to unity. For the tomography we define photometric bins by dividing the distribution (29) such that every bin contains the same number of gaaxies. This choice is merey for computationa convenience and to aow for an easy comparison between resuts with a different numberof bins. As defaut we wi use N zbin = 1 bins. To account for photometric redshift errors, we assume that the fraction of catastrophic faiures in the assignment of photometric redshifts is negigibe, but incude the spread of the true redshifts in the bin-wise distributions by writing the conditiona probabiity of obtaining a photometric redshift z ph given the true redshift z as p(z ph z) exp ( zph z ) 2 (1 + z)2, (3) 2σ 2 ph where σ ph denotes the photometric redshift dispersion. The redshift distribution of an individua photo-z bin p (i) (z) is then obtained by integrating (3) over the bin width and by weighting the resut by the overa redshift distribution (29), see Joachimi & Schneider (29) for detais. We use σ ph =.5 as our defaut vaue. This aso foows fiducia Mode 1 of Ma et a. (26). Since the underying redshift distributions p (i) (z) aredetermined by measurement, they are not perfecty known, but introduce further uncertainty into the anaysis. A detaied anaysis of the dependence of the joint anaysis of gaaxy shape and number density information on redshift parameters, and aso the potentia of caibrating these errors internay, wi be investigated esewhere (e.g. Zhang et a. 21). Bride & King (27) have undertaken a more detaied study of the effect of redshift errors in the case of eipticity correations ony. For the purpose of this work we assume that the vaue of σ ph is unknown, i.e. we use it as a singe, goba parameter to account for the uncertainty in the redshift distributions. We empoy a wide Gaussian prior on σ ph of 1 for reasons of numerica stabiity Gaaxy uminosity function In order to cacuate power spectra which incude the ensing magnification signa, we need to mode the sope of the cumuative gaaxy uminosity function at the magnitude imit of the gaaxy number density cataogue. In Appendix C we extend observationa resuts for the normaisation and redshift scaing of the gaaxy redshift distribution (29)by Bake & Bride (25)to provide a fitting formua for the uminosity function sope as a function of redshift and survey magnitude imit. We use the fit given by (C.2) and(c.3) with the parameters isted in Tabe C.1 to compute the sope of the uminosity function at r im = 24. The discussion in this work appies to groundbased surveys because the COMBO-17 uminosity functions are cacuated for the SDSS r fiter as observed from the ground. A space mission to a depth of r im = 24 wi have a different uminosity function sope, corresponding more cosey to a deeper ground based survey, depending on the resoution of the spacebased survey. We use resuts for r im = 24 throughout this paper for both ground and space surveys. We note that from Fig. C.1, top pane, the sope of the uminosity function is changed itte on increasing the survey depth beyond r im = 24. The fiducia sope in a photo-z bin i is defined as α (i) is the median redshift of bin i, see Appendix A. We assume α (i) is aso measured from the survey itsef, and therefore adds another source of uncertainty to the anaysis which we account for by setting α (i) to be free parameters for a i = 1,..., N zbin. Again we appy a wide Gaussian prior of 1 on every sope parameter, which does not have a measurabe infuence on our resuts, but merey ensures numerica stabiity. α(z (i) med, r im = 24), where z (i) med 3.3. Gaaxy and intrinsic aignment bias As aready outined in Sect. 2, the bias terms b X = { } b g, b I, r g, r I encoding the gaaxy bias and intrinsic aignments are the east accuratey known contributions to (17) to(26). We parametrise each of these terms on a grid in k and z, foowing Bride & King (27) whose ansatz is in turn simiar to the recommendations by the Dark Energy Task Force (Abrecht et a. 26)andBernstein (29). Every bias term is assumed to vary around a fiducia functiona form b base X as b X (k,χ) = A X Q X (k, z(χ)) b base X (k,χ), (31) where Q X (k, z(χ)) is an unknown two-dimensiona function which comprises the aforementioned grid, and where A X denotes an additiona free overa ampitude. We use N K bins in k and N Z bins in redshift for each bias term and ineary interpoate in the ogarithms of Q X, k,and1+ z, sothatq X is given by n Q X (k, z) = K i (k) Z j (z) B X ij + [ 1 K i (k) ] Z j (z) B X (i+1) j (32) +K i (k) [ 1 Z j (z) ] B X i ( j+1) + [ 1 K i (k) ][ 1 Z j (z) ] B X (i+1) ( j+1) for k i < k k i+1 and z j < z z j+1, where we defined K i (k) n(k) n(k i) n(k i+1 ) n(k i ) Z j (z) n(1 + z) n(1 + z j) n(1 + z j+1 ) n(1 + z j ) (33) The free parameters are the grid nodes Bij X.SinceQ X is a mutipicative function, one reproduces the base mode b base X if A X = 1 and if a Bij X vanish. The effect of this parametrisation on the observabe projected power spectra is iustrated in Bride & King (27) for the case of intrinsic aignments. The indices in (32) run from i =,...,N K + 1and = 1,...,N Z +1. We fix a parameters at the edge of the grid by setting the parameters with indices i, j = ; i = N K +1or j = N Z +1 to Bij X =, so that we have N K N Z free grid parameters per bias term. We pace the owest and highest grid nodes at the imits of our integration ranges, so k = h Mpc 1 and k NK +1 = h Mpc 1 in k, and in the redshift dimension z = andz NZ +1 = 19. The grid nodes, which are free to vary, are og-ineary spaced in a smaer range, respectivey. We use k 1 = 1 3 h Mpc 1 and k NK = 2 h Mpc 1, and for the redshift range z 1 = z and z NZ = 3. In the specia case of N K = 1we Page 6 of 21

7 B. Joachimi and S. L. Bride: Simutaneous measurement of cosmic shear and gaaxy number density correations Tabe 2. Overview on the tota number of nuisance parameters used for different setups. N K N Z ɛɛ nn a Notes. The abe ɛɛ corresponds to using eipticity correations ony as the observabes. Likewise, nn corresponds to using gaaxy number density correations ony, and a to using a avaiabe correations. For a entries we have assumed ten photometric redshift bins used for the tomography, N zbin = 1. position the ony free parameter in the k dimension at the centre between k 1 and k NK, and proceed ikewise for redshifts. It is important to note that whie Bride & King (27) imit the fexibe grid parametrisation to the non-inear regime of the power spectra, we attempt to cover a k ranges which substantiay contribute to the observabe power spectra. As we fix the grid vaues on the edges, the overa scaing of the bias terms is not free, so that we use the ampitude A X as a further varying parameter throughout. To a bias term parameters we add a very wide Gaussian prior of standard deviation 5 to ensure numerica stabiity. Together with the goba uncertainty on the redshift distributions, expressed in terms of σ ph, and the vaues of the sope of the gaaxy uminosity function per photo-z bin, we obtain a arge number of nuisance parameters that we determine simutaneousy with the cosmoogica parameters of interest. For ater reference, we have summarised the tota number of nuisance parameters for different setups in Tabe 2. Whie our parametrisation is fairy genera and shoud capture most of the variabiity, it is of course possibe that the bias terms depend on more parameters than k and z. For instance, it is we known that both intrinsic aignments and gaaxy bias are a function of gaaxy coour and uminosity which coud be incorporated into our approach in the future. For observationa constraints on this effect using intrinsic aignments see Mandebaum et a. (26), Hirata et a. (27) and in gaaxy biasing see McCracken et a. (28), Swanson et a. (28), Simon et a. (29), Cresswe & Perciva (29), Wang et a. (27) for recent exampes. To compute the fiducia modes for a the power spectra entering the observabes (4) to(6), we set A X = 1andaB X ij =, i.e. they are fuy determined by the base modes. We set b base g (k,χ) = 1 (34) rg base (k,χ) = 1 b base ρ(z) I (k,χ) = C 1 D(z)(1 + z) ri base (k,χ) = 1. With the choice for b base I and ri base we reproduce the non-inear modification of the inear aignment mode by Bride & King (27). Lacking soid physica motivation, it is yet in agreement with current observationa evidence (Mandebaum et a. 26; see Bride & King 27, for a comparison) and the hao mode studies by Schneider & Bride (21). It is based on the inear aignment mode (Hirata & Sejak 24) which is expected to provide a good description of intrinsic aignments on the argest scaes. We assume the gaaxy bias to be of order unity for our fiducia mode, and set rg base (k,χ) = 1. Note that most investigations of gaaxy custering consider much ess fexibiity in the gaaxy bias. In Fig. 1 we pot the fiducia anguar power spectra of a considered signas for different combinations of photo-z bins. Note that the eipticity-number density cross-correations are not symmetric under exchanging the photometric redshift bins. Hence, in this figure we treat nɛ and ɛn correations, as we as a signas contributing to them, separatey, keeping i j for a αβ (). The GG signa shows the usua behaviour of moderate increase with increasing redshift of the contributing photo-z-bins. The redshift scaing of the IG term is simiar, but peaks when the source gaaxies are at high redshift in the background (arge j), whie the gaaxies that are intrinsicay aigned are at ow redshift (sma i). For the mode used here the IG contribution can even surpass the shear signa in this case. Due to the narrow kerne containing p (i) (χ) p ( j) (χ), see (19), the II signa is strong in the auto-correations i = j, but drops off quicky as soon as the overap of the redshift distributions decreases. Due to the simiar kerne, the scaing of the gaaxy custering contribution (gg) resembes the II term, but gg constitutes a much stronger signa. Lensing magnification (mm) adds the argest fraction of the gaaxy number correations at the highest redshifts, showing a sighty stronger redshift scaing than GG. However, the mm term aways remains subdominant with respect to signas with a contribution from gaaxy custering; even for widey separated gaaxy redshift distributions, say i = 1 and j = 9, and the gm cross-term is consideraby stronger than mm. Such contributions might be a serious obstace for probing cosmoogy with the ensing magnification signa as proposed by Broadhurst et a. (1995), Zhang & Pen (25, 26), van Waerbeke (21). Yet in our approach, where the gaaxy bias is taken into account and parametrised, the magnification signa yieds a vauabe contribution to the gaaxy number correations, which heps constraining the cosmoogica mode. The signas within the number density-eipticity crosscorreations are not symmetric when swapping the photo-z bins. When the contribution by number density fuctuations stems from the foreground, the gg signa is strong, in particuar if the photo-z bins are far apart in redshift, whereas the Gg (in the notation of Fig. 1) drops off fast if i < j because the shear signa of foreground gaaxies is not correated with the custering of gaaxies at much higher redshift. The mg, Gm, and GG signas differ ony by the term incuding the sope of the uminosity function and thus have simiar ampitudes. Correations between intrinsic aignment and magnification (mi and Im) are subdominant throughout, obtaining their argest ampitudes if intrinsic aignments at ow redshifts are combined with the magnification signa from gaaxies far in the background, i.e. in the figure for Im at i = 1andj = 9. Finay, the symmetric Ig term is the argest contribution for auto-correations of number densityeipticity observabes, but decreases quicky in the cross terms, again due to the kerne p (i) (χ) p ( j) (χ) since we have assumed the photometric redshift errors are reasonaby we behaved, without catastrophic outiers. Note that we have potted the absoute vaues of the power spectra in Fig. 1 and that the correations IG, gm, Gm, mg, and gi are negative. Page 7 of 21

8 A&A 523, A1 (21) Fig. 1. Fiducia power spectra for a considered correations. The upper right panes depict the contributions to ɛɛ (in back) and nn (in magenta) correations. The ower eft panes show the contributions to correations between number density fuctuations and eipticity. Since we ony show correations αβ () with i j, we make in this pot a distinction between nɛ (in red; number density contribution in the foreground, e.g. gg) and ɛn (in bue; number density contribution in the background, e.g. Gg) correations. In each sub-pane a different tomographic redshift bin correation is shown. For carity ony odd bins are dispayed. In the upper right panes the usua cosmic shear signa (GG) is shown as a back soid ines; the intrinsic aignment GI term is shown by the back dashed ines; the intrinsic aignment II term is shown by the dotted back ine; the usua gaaxy custering signa (gg) is shown by the magenta soid ine; the cross correation between gaaxy custering and ensing magnification (gm) is shown by the magenta dashed ine; the ensing magnification correation functions (mm) are shown by the magenta dotted ine. In the ower eft panes the soid bue ine shows the correation between ensing shear and gaaxy custering (Gg); the bue dashed ine shows the correation between ensing shear and ensing magnification (gm); the bue dot-dashed ine shows the correation between intrinsic aignment and gaaxy custering (Ig or equivaenty gi); the red soid ine shows the correation between gaaxy custering and ensing shear (gg), which is equivaent to the bue soid ine with redshift bin indices i and j reversed; simiary the red dashed ine shows the correation between ensing magnification and ensing shear (mg), for cases where the magnification occurs at ower redshift than the shear (i < j); finay the dotted ine shows the correation between ensing magnification and intrinsic aignment (mi). Page 8 of 21

9 B. Joachimi and S. L. Bride: Simutaneous measurement of cosmic shear and gaaxy number density correations 3.4. Parameter constraints We determine constraints on our parameters using a Fisher matrix anaysis. To account for the errors and correations of our observabes, we compute covariances of the power spectra (4) to (6) in the Gaussian approximation, extending the resuts of Joachimi et a. (28), see aso Hu & Jain (24). If we denote the difference between estimator and its ensembe average by Δ αβ (), one can write for the covariance (ij) ΔC αβ () ΔC(k) γδ ( ) 2 π = δ (35) AΔ { C αγ (ik) () C ( j) βδ () + C (i) αδ () C ( jk) βγ ()} Cov (ijk) αβγδ (), where A is the survey size and Δ the width of the corresponding anguar frequency bin. As the Kronecker symbo δ indicates, the covariance is diagona in in the Gaussian imit, which keeps the computation and inversion of (35) tractabe. The subscripts {α, β, γ, δ} can be either ɛ or n,wherec ɛn (ij) () C nɛ ( ji) () hods. To account for the shot and shape noise contributions induced by the random terms in (1)and(2), we have defined αβ () C(ij) αβ () + N(ij) αβ, (36) the second term given by N (ij) σ 2 ɛ αβ = δ ij 2 n (i) for α = β = ɛ (37) N (ij) 1 αβ = δ ij n (i) for α = β = n N (ij) αβ = forα β. Here σ 2 ɛ denotes the tota intrinsic eipticity dispersion, and n(i) is the average gaaxy number density per steradian in photo-z bin i. Combining the observabe power spectra, we compose the tota data vector { D() = C ɛɛ (11) (),..., C (N zbinn zbin ) ɛɛ (), C nɛ (11) (),..., (38) C (N zbinn zbin ) nɛ (), C nn (11) (),..., C(N zbinn zbin ) nn () for every anguar frequency considered. The corresponding covariance, again for every, reads Cov() = Cov (ijk) ɛɛɛɛ () Cov (ijk) ɛɛnɛ () Cov (ijk) ɛɛnn () nɛnn () nnnn () Cov (ijk) nɛɛɛ () Cov (ijk) nɛnɛ () Cov (ijk) Cov (ijk) nnɛɛ () Cov (ijk) nnnɛ () Cov (ijk) } τ, (39) with the bock matrices given by (35). The number of gaaxy eipticity (ɛɛ) and number density (nn) observabes entering D() isn zbin (N zbin + 1) /2, respectivey, whie there are N 2 zbin eipticity-number density cross terms (nɛ), which are not symmetric. In the anaysis that foows we wi aso consider ɛɛ and nn correations ony. In these cases the covariance (39) isreduced accordingy to its upper eft or ower right bock. For reasons of computationa time the tota number of parameters that we can consider is imited to a few hundred. As a consequence the k dependence of the gaaxy bias can not be parametrised by more than about ten parameters per redshift grid node. This number might not provide enough freedom in b g (and r g )torepresentasufficienty genera set of functiona Tabe 3. Overview on the cuts in anguar frequency. z-bin z med g max N max Notes. Resuts are shown for the defaut set of parameters with N zbin = 1 and σ ph =.5, using (4)and(41). For each photo-z bin the median redshift z med, the maximum anguar frequency max, g and the corresponding number of usabe anguar frequency bins N max (out of the tota of 5) is given. forms, which inadvertenty may cause strong constraints on cosmoogica parameters due to the strong signa of gaaxy custering. Hence, we foow existing studies of gaaxy custering by discarding the custering contribution in the non-inear regime where the signa is argest and the form of the gaaxy bias most uncertain. Rassat et a. (28) cacuated wave vectors k max in as a function of redshift at which the three-dimensiona power spectrum has to be cut off to avoid number density fuctuations above a certain threshod, used as an indicator for non-inearity. Since we do sti have a fairy genera parametrisation of the bias terms associated with gaaxy bias, we can afford to incude the midy non-inear regime into our anaysis. Consuting Fig. 2 of Rassat et a. (28), we choose a simpe inear parametrisation of the form k max in (z).132 zhmpc 1. (4) This reation roughy coincides with the fiducia curve in the figure, producing sighty more conservative cuts at ow redshifts. We do not cut the three-dimensiona power spectrum in k-space, but instead excude projected power spectra above a threshod anguar frequency from the Fisher matrix anaysis. This maximum anguar frequency is computed via g(i) max = k max in (z(i) med ) f K ( χ(z (i) med )), (41) where we choose as a characteristic redshift of bin i the median redshift z (i) med. Hence, we obtain a cut-off for every photo-z bin. We choose that ɛɛ correations are not at a affected by this cutoff because they are not dominated by terms invoving gaaxy bias. We impose max g(i) on nɛ correations, where i is the photo-z bin from which the number density signa stems. For observabes C nn (ij) () weusethecut-offcacuated for bin j. Note that, due to the fast drop-off of the gaaxy custering signa with increasingy different median redshifts of bins i and j, the more optimistic choice of bin j over i in the atter case shoud not infuence our resuts decisivey. To compute the Fisher matrix, we use N = 5 anguar frequency bins, spaced ogarithmicay between min = 1 and max = 3, the atter vaue being a conservative maximum for future surveys. We assume that the covariance of the power spectra is independent of the cosmoogica parameters, so that Page 9 of 21