MNRAS 000, 1–13 (2018) Preprint 25 February 2020 Compiled using MNRAS LATEX style file v3.0

Statistical separation of weak gravitational lensing and intrinsicellipticities based on galaxy colour information

Tim M. Tugendhat, Robert Reischke, Björn Malte Schäfer?Astronomisches Rechen-Institut, Zentrum für Astronomie der Universität Heidelberg, Philosophenweg 12, 69120 Heidelberg, Germany

25 February 2020

ABSTRACTIntrinsic alignments of galaxies are recognised as one of the most important systematic inweak lensing surveys on small angular scales. In this paper we investigate ellipticity cor-relation functions that are measured separately on elliptical and spiral galaxies, for whichwe assume the generic alignment mechanisms based on tidal shearing and tidal torquing,respectively. Including morphological information allows to find linear combinations of mea-sured ellipticity correlation functions which suppress the gravitational lensing signal com-pletely or which show a strongly boosted gravitational lensing signal relative to intrinsicalignments. Specifically, we find that (i) intrinsic alignment spectra can be measured in amodel-independent way at a significance of Σ ' 60 with a wide-angle tomographic surveysuch as Euclid’s, (ii) the underlying intrinsic alignment model parameters can be determinedat percent-level precision, (iii) this measurement is not impeded by misclassifying galaxiesand assuming a wrong alignment model, (iv) parameter estimation from a cleaned weak lens-ing spectrum is possible with almost no bias and (v) the misclassification would not stronglyimpact parameter estimation from the boosted weak lensing spectrum.

Key words: gravitational lensing: weak – dark energy – large-scale structure of Universe.

1 INTRODUCTION

The next generation of weak lensing surveys has the goal to mea-sure correlations in the shapes of neighbouring galaxies over a widerange of angular scales and with resolution in redshift in order toinvestigate the properties of gravity through its influence on cos-mic structure formation and in order to carry out a precision deter-mination of cosmological parameters, investigate the properties ofgravity on large scales or determine details of dark energy modelsthrough their influence on cosmic structure formation. While weaklensing surveys like Euclid or LSST provide exquisite statisticalprecision due to the vast amount of data, the control of systematicsis the primary obstacle in the way of exploiting this information.

Weak cosmic shear refers to the correlated distortion of thecross-section of light bundles that reach us from distant galax-ies (Mellier sics; Bartelmann & Schneider 2001; Refregier 2003;Hoekstra & Jain 2008; Munshi et al. 2008; Kilbinger 2015). Us-ing it as a cosmological probe one primarily determines angularcorrelations in the ellipticities of galaxies, after subdividing thegalaxy sample in redshift bins. Under the assumption of intrinsi-cally uncorrelated shapes the angular ellipticity correlation func-tion measures lensing-efficiency weighted tidal shear correlationsin the cosmic large-scale structure. These correlations carry infor-mation about the structure formation history as well as the expan-sion history of the Universe (Matilla et al. 2017), and do not rely

? e-mail: bjoern.malte.schaefer@uni-heidelberg.de

on any assumption apart from the validity of a gravitational the-ory (Huterer 2002; Linder & Jenkins 2003; Amendola et al. 2008;Bernstein 2009; Huterer 2010; Carron et al. 2011; Martinelli et al.2011; Vanderveld et al. 2012).

Intrinsic alignments of galaxies mimic correlations in theshapes of neighbouring galaxies which would be naively con-tributed to gravitational lensing (for reviews, Kirk et al. 2015;Kiessling et al. 2015; Joachimi et al. 2015; Schaefer 2009; Troxel& Ishak 2015). If unaccounted for, either by modelling or by mit-igation, they would interfere with the parameter inference processand would lead to wrong conclusions about the cosmological model(Kirk et al. 2010, 2011, 2012; Laszlo et al. 2012; Capranico et al.2013; Schaefer & Merkel 2015; Krause et al. 2016; Tugendhat &Schaefer 2018). While the exact mechanisms of galaxy alignmentwith the cosmic large-scale structure are not yet clear, tidal align-ment models provide a physically motivated way to link the shapesof galaxies to the matter distribution on large-scales. In the caseof elliptical galaxies, for which the linear alignment model (alsocalled tidal alignment model) might be applicable, one assumes adistortion of the galaxy ellipsoid by tidal gravitational forces, whichact perturbatively on the galaxy and exert a shearing distortion (Hi-rata & Seljak 2004, 2010; Blazek et al. 2011, 2017). In this case, theobserved ellipticity is proportional to the tidal shear componentsperpendicular to the line of sight. In contrast, the alignment of spi-ral galaxies may be due to the quadratic alignment model, wherethe orientation of the galaxy is linked to the host halo angular mo-mentum, which in turn is generated in the early stages of structure

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2 T.M. Tugendhat, R. Reischke, B.M. Schäfer

formation by tidal torquing (Crittenden et al. 2001; Natarajan et al.2001; Mackey et al. 2002). As an orientation effect, the alignmentof spiral galaxies would not reflect the magnitude of tidal fields butwould only depend on their orientation. Applying the models inthe strictest sense would result in predicting a non-vanishing cross-correlation between the gravitational lensing effect and the lineartidal alignment of elliptical galaxies.

Our study is motivated by the fact that the distortion of galaxyellipticities due to gravitational lensing should be universal and notdepend on galaxy type. In this case one should be able to makeuse of morphological information in order to find linear combina-tions of ellipticity maps where the intrinsic alignment signal is up-or down-weighted relative to the gravitational lensing signal. Wewill investigate the usability of ellipticity spectra with weightedrelative contributions from gravitational lensing and from intrinsicalignments, with the purpose of cosmological inference from weaklensing with suppressed IA-induced biases as well as for investigat-ing alignment signals with a suppressed weak gravitational lensingeffect, that would otherwise dominate ellipticity correlations.

The fiducial cosmological model is a spatially flat ΛCDM-cosmology motivated by the Planck-results (Planck Collaborationet al. 2015), with specific parameter choices Ωm = 0.32, ns = 0.96,σ8 = 0.83 and h = 0.68, with a constant dark energy equation ofstate parameter of w = −1.0. We adopt the summation conventionand orient coordinate systems in such a way that the z-axis pointsalong the line of sight. We carry out all computations and statisticalestimates for the Euclid-missions, but all statements would be ap-plicable in a similar way to other weak lensing surveys with similarsurvey depths. After a summary of cosmology in Sect. 2 we reviewweak gravitational lensing and intrinsic alignments in Sect. 3. Weintroduce our method and demonstrate the separation technique inSect. 4 and summarise and discuss our results in Sect. 5.

2 COSMOLOGY

Under the symmetry assumption of Friedmann-Lemaître-cosmologies all fluids are characterised by their density and theirequation of state: In spatially flat cosmologies with the matterdensity parameter Ωm and the corresponding dark energy density1 − Ωm one obtains for the Hubble function H(a) = a/a theexpression,

H2(a)H2

0

=Ωm

a3 +1 −Ωm

a3(1+w) , (1)

The comoving distance χ is related to the scale factor a through

χ = −c∫ a

1

daa2H(a)

, (2)

where the Hubble distance χH = c/H0 sets the distance scale forcosmological distance measures. Small fluctuations δ in the distri-bution of dark matter grow, as long as they are in the linear regime|δ| 1, according to the growth function D+(a) (Linder & Jenkins2003),

d2

da2 D+(a) +2 − q

ad

daD+(a) −

32a2 Ωm(a)D+(a) = 0, (3)

and their statistics is characterised by the spectrum 〈δ(k)δ∗(k′)〉 =

(2π)3δD(k − k′)Pδ(k). Inflation generates a spectrum of the formPδ(k) ∝ kns T 2(k) with the transfer function T (k) (Bardeen et al.1986) which is normalised to the variance σ8 smoothed to the scale

of 8 Mpc/h,

σ28 =

∫ ∞

0

k2dk2π2 W2(8 Mpc/h × k) Pδ(k), (4)

with a Fourier-transformed spherical top-hat W(x) = 3 j1(x)/x asthe filter function. From the CDM-spectrum of the density pertur-bation the spectrum of the dimensionless Newtonian gravitationalpotential Φ can be obtained,

PΦ(k) ∝(

3Ωm

2(kχH)2

)2

Pδ(k) (5)

by applying the comoving Poisson-equation ∆Φ = 3Ωm/(2χ2H)δ for

deriving the gravitational potential Φ (in units of c2) from the den-sity δ. Because our analysis relies on the assumption of Gaussian-ity, we need to avoid nonlinearly evolving scales and will restrictour analysis to large angular and spatial scales, where the cosmicdensity field can be approximated to follow a linear evolution, con-serving the near-Gaussianity of the initial conditions. We increasethe variance of the weak lensing signal and of the intrinsic align-ment signal of elliptical galaxies on small scales because of nonlin-ear structure formation using the description of Smith et al. (2003);Casarini et al. (2011, 2012). Consequently, the cross-correlation be-tween weak lensing and elliptical galaxy shapes will likewise haveincreased variances on small scales. Shapes of spiral galaxies areset by the initial conditions of structure formation, therefore we didnot apply changes to the linear CDM-spectrum P(k).

3 WEAK GRAVITATIONAL LENSING AND INTRINSICALIGNMENTS

3.1 Gravitational tidal fields and their statistics

In our investigation, alignments of galaxies with the large-scalestructure are due to gravitational-tidal interactions with shear fields.These fields will be assumed to have Gaussian statistics. This as-sumption will not be applicable at late times and on small scales.Tidal alignment models relate correlations in the shapes of galaxiesto correlations in the tidal shear field,

Φαβ(x) =∂2Φ(x)∂xα∂xβ

, (6)

as a tensor containing the second derivatives of the Newtonian grav-itational potential. Correlations of Φαβ(x) as a function of distancer = |x − x′| will be described by the correlation function

Cαβγδ(r) ≡ 〈Φαβ(x)Φγδ(x′)〉 (7)

which Catelan & Porciani (2011) have shown to take the form

Cαβγδ(r) = (δαβδγδ + δαγδβδ + δαδδβ,γ) ζ2(r)

+ (rαrβδγδ + 5 perm.)ζ3(r) + rαrβrγ rδ ζ4(r).(8)

The fluctuation statistics of the gravitational potential PΦ(k) entersthrough the functions ζn(r),

ζn(r) = (−1)n rn−4∫

dk2π2 PΦ(k) kn+2 jn(kr), (9)

as derived by Crittenden et al. (2001). rα is the α-component of theunit vector parallel to r = x − x′.

While the tidal shear fields will directly change the shape ofan elliptical galaxy, the effect of tidal fields on a spiral galaxies isto determine its angular momentum direction and consequently theinclination of the galactic disc. In both cases, the relevant compo-nents of the tidal shear tensor are those of the traceless part. The

MNRAS 000, 1–13 (2018)

Separation of gravitational lensing and IAs 3

resulting correlation function Cαβγδ(r) of the traceless tidal shearΦαβ = Φαβ − ∆Φ/3 × δαβ, would be given by

Cαβγδ(r) = Cαβγδ(r)

−13

(δγδ (5ζ2(r) + ζ3(r)) + rγ rδ (7ζ3(r) + ζ4(r))

)δαβ

−13

(δαβ (5ζ2(r) + ζ3(r)) + rαrβ (7ζ3(r) + ζ4(r))

)δγδ

+19

(15ζ2(r) + 10ζ3(r) + ζ4(r)) δαβδγδ.

(10)

Furthermore, the apparent shapes of spiral galaxies only de-pend on the direction of the angular momentum and not its mag-nitude because their ellipticity is only an orientation effect. Conse-quently, their shape correlations can be traced back to the traceless,unit-normalised tidal shear field Φi j, which obeys the conditionsΦii = 0 and Φi jΦ ji = 1. It can be decomposed into correlationsCAB = 〈ΦAΦB〉 of the traceless tidal shear ΦA field (Natarajan et al.2001; Crittenden et al. 2001) by virtue of Wick’s theorem,

〈ΦA(x)ΦB(x) ΦC(x′)Φ′D(x′)〉 =1[

14ζ2(0)]2

(CACCBD + CADCBC

),

(11)

where A, B, C and D are containers for pairs of indices. The nor-malisation with ζ2 has the consequence that correlations of the unit-normalised tidal shear field are only due to the orientation of theeigen-systems and do not depend on the absolute magnitude of thetidal shear. The assumption of Gaussianity of the aligning large-scale structure is in fact a strong one, as alignments on filaments hasbeen demonstrated with numerical simulations (Codis et al. 2015).To what extend these alignments would, after Limber-projection,differ from those derived from a Gaussian random field, has not yetestimated. Likewise, depending on the details of halo-model basedintrinsic alignment models (Joachimi et al. 2013a,b), a separationof spiral and elliptical alignment due to their dependence with pow-ers of the tidal shear field would not be feasible.

3.2 Weak gravitational lensing

The lensing potential ψ (for a review see e.g. Bartelmann & Schnei-der 2001) is a line-of-sight projection of the gravitational potentialΦ:

ψ =

∫ χH

0dχW(χ)Φ, (12)

it thus inherits the statistical properties of the gravitational poten-tial. The changes in size and shape of a light bundle are cause bydifferential deflection and are therefore proportional to the secondderivatives of ψ perpendicular to the line sight and can be decom-posed in terms of the weak lensing convergence κ,

κ =

∫dχ W(χ) σ(0)

αβ

∂2Φ

∂xα∂xβ, (13)

which depends on the trace of the tidal shear, and the complex weaklensing shear γ,

γ = γ+ + iγ× =

∫dχ W(χ)

(σ(1)αβ + iσ(3)

αβ

) ∂2Φ

∂xα∂xβ, (14)

which reflects the traceless part of the tidal shear. σ(n)αβ are the Pauli-

matrices, and effectively only the derivatives of Φ perpendicular tothe line of sight are relevant. The weight function W(χ) is given by

W(χ) = 2D+(χ)

aG(χ)χ, (15)

with the lensing efficiency function

G(χ) =

∫χ

dχ n(χ′)dzdχ′

(1 −

χ

χ′

). (16)

with dz/dχ′ = H(χ′)/c and n(χ′) being the distribution of thesources. Because weak gravitational lensing affects both ellipti-cal and spiral galaxies alike, we scale the resulting lensing spectraCγ

i j(`) with the total number of galaxy pairs ' n2 = (ns +ne)2. In ouranalysis, we are using a fit for the nonlinear power spectrum (Smithet al. 2003) to calculate the gravitational lensing contribution.

3.3 Alignments of spiral galaxies

In the tidal torque model used in this work, the alignment of spiralgalaxies is purely due to their orientation, which in turn is relatedto the angular momentum correlation of neighbouring galaxies rel-ative to the line of sight (Croft & Metzler 2000; Crittenden et al.2001). Angular momentum correlations are mainly build up at earlytimes during structure formation and are thus due to initial corre-lations (Catelan & Theuns 1996; Theuns & Catelan 1997; Cate-lan & Theuns 1997). The correlated angular momenta result intoto correlated inclination angles of neighbouring galaxies and thusultimately into correlated ellipticities (Catelan et al. 2001). Assum-ing that the symmetry axis of the galactic disc coincides with thedirection of the angular momentum L = L/L, the ellipticity canbe written as The alignment of spiral galaxies is purely due to theorientation of their circular disks, which in turn is related to theangular momentum correlation of neighbouring galaxies relative tothe line of sight (Croft & Metzler 2000; Crittenden et al. 2001).Angular momentum correlations are mainly build up at early timesduring structure formation and are thus due to initial correlations(Catelan & Theuns 1996; Theuns & Catelan 1997; Catelan & The-uns 1997). The correlated angular momenta result into to correlatedinclination angles of neighbouring galaxies and thus ultimately intocorrelated ellipticities (Catelan et al. 2001). Assuming that the sym-metry axis of the galactic disc coincides with the direction of theangular momentum L = L/L, the ellipticity can be written as

ε =L2

x − L2y

1 + L2z

+ 2iLxLy

1 + L2z

. (17)

Angular momentum is generated by a torque exerted by the am-bient large-scale structure onto the protogalactic halo, a mecha-nism called tidal torquing (White 1984; Barnes & Efstathiou 1987;Schaefer 2009; Stewart et al. 2013). For Gaussian random fields theauto-correlation of angular momenta is given by (Lee & Pen 2001)

⟨LαLβ

⟩=

13

(1 + A

3δαβ − AΦαµΦµβ

). (18)

The free parameter A determines the strength of the coupling be-tween alignment and tidal torque. Since the correlation is deter-mined by the traceless part of the shear tensor Φαβ the resultingeffect is clearly due to orientation effects only. For a Gaussian dis-tribution p(L|Φαβ)dL and the use of eq. (17) one can express theellipticity in terms of the tidal field

ε(Φ) =A2

(ΦxαΦαx − ΦyαΦαy − 2iΦxαΦαy

). (19)

Correlations in the ellipticities can thus be traced back to the 4-point function of the shear field, which is given in eq. (11). Forkeeping a correct relative normalisation of the shape correlations,we scale the resulting angular ellipticity spectra Cs,II

i j (`) with the

MNRAS 000, 1–13 (2018)

4 T.M. Tugendhat, R. Reischke, B.M. Schäfer

squared number of spiral galaxies n2s . It is remarkable that the

shapes of spiral galaxies in the quadratic alignment model are infact sensitive to tidal shear components parallel to the line of sight,in fact, those components determine the magnitude of the align-ment effect, in contrast to the alignment of elliptical galaxies inthe linear alignment model or to gravitational lensing, which re-flect purely the tidal shear components perpendicular to the line ofsight.

Figure 1 gives a visual impression of alignments in thequadratic, angular momentum-based alignment model: From a re-alisation of a Gaussian random density field δ(x) from the CDM-spectrum P(k) we computed the traceless, unit-normalised tidalshear Φαβ, which is used to determine the variance of the distri-bution p(Lα|Φαβ). Angular momenta are drawn at random locationswhich fix the orientation of the galactic discs. The amount of cor-relation in the orientation of the galactic discs corresponds deter-mined from the realisation corresponds to the theoretically com-puted correlation function. Ellipticity correlations between spiralgalaxies are rather short-ranged, with a typical correlation lengthof about 1 Mpc/h (Schaefer & Merkel 2012), which makes them asmall-scale phenomenon at angular scales of ` ' 103 for Gpc-scalesurveys.

3.4 Alignments of elliptical galaxies

For elliptical galaxies we assume a virialised system in which starsmove randomly in a gravitational potential Φ with the velocity dis-persion σ2. In equilibrium the density profile is a solution of the ra-dially symmetric Jeans equation and scales ρ ∝ exp(−Φ/σ2). In thepresence of a tidal field induced by the ambient large-scale struc-ture, the equilibrium situation is perturbed and the galaxy finds anew equilibrium. Perturbing the Jeans equation at first order in thetidal fields ∂α∂βΦ yields the following solution for the density:

ρ ∝ exp(−

Φ(x)c2

) (1 −

12σ2

∂2Φ(x)∂xα∂xβ

∣∣∣∣∣x=x0

xαxβ). (20)

While the reaction of the halo to the tidal fields is determined by thevelocity dispersion σ2, i.e. how strongly the particles are bound inthe gravitational potential, the relationship between tidal shear fieldand ellipticity needs as well to reflect the luminous profile, whichwe absorb in the definition of a constant of proportionality D. Sincethis model gives rise to ellipticities being linear in the tidal fieldsit is commonly referred to as the linear alignment model (Hirata &Seljak 2010; Blazek et al. 2017, 2012, 2011, 2015). Assuming xand y being coordinates perpendicular to the line of sight the com-plex ellipticity is given by

ε = ε+ + ε× = D(∂2Φ

∂x2 −∂2Φ

∂y2 + 2i∂2Φ

∂x∂y

). (21)

From this equation it is clear that the linear alignment model de-pends on both the amplitude and the orientation of the tidal fields.In order to maintain the correct relative normalisation of the spec-tra, we scale the resulting Ce,II

i j (`) with the number of ellipticalgalaxies n2

e .The alignment model parameters A and D are chosen to have

values of A = 0.25 from numerical simulations of angular momen-tum generation in haloes, and D = 9.5 × 10−5c2 from CFHTLenS-data, respectively. While A is effectively a geometric, dimension-less parameter, D links the dimensionless ellipticity to the tidalshear field. For a more detailed discussion of this relationship in-cluding the scaling with mass, we refer to Piras et al. (2017) and

Tugendhat & Schaefer (2018). While there is no definitive mea-surement of ellipticity correlations of spiral galaxies, the case isnotably different for elliptical galaxies (Mandelbaum et al. 2006,2011; Singh et al. 2015).

3.5 Cross-alignments between intrinsic shapes and lensing

For spiral galaxies there exist no GI-type terms due to Wick’s theo-rem, because those terms would be proportional to a third momentof the tidal shear field. In contrast, there will be a non-vanishingcross-correlation between lensing and the intrinsic alignment ofelliptical galaxies. We would like to point out that these cross-correlations, Ce,GI

i j (`), have to be symmetrised with respect to thebin numbers: Naturally, the more distant galaxy is lensed whereasthe closer galaxy is intrinsically aligned while the inverse is notpossible. Statistical isotropy forces the covariance matrix to besymmetrical however (Tugendhat & Schaefer 2018). We scale thecross-spectra Ce,GI

i j (`) with the number ne(ne + ns) = nen of pairsinvolving at least one elliptical galaxy.

It is worth pointing out that there is a straightforward physicaldifference between the weak lensing and intrinsic alignments. Asweak lensing is an integrated effect, there will be nonzero cross-correlations between different tomographic bins, whereas intrinsicalignments will, due to their locality, only show correlations withinthe same bin. This can already be used as a method of discrimina-tion between II- and GG-spectra (Bernstein & Jain 2004; Huterer& White 2005), but will not get rid of the GI-contribution.

3.6 Shape correlations in a weak lensing survey

We carry out our investigation for a weak lensing survey similarto Euclid’s: The redshift distribution n(z)dz is assumed to have theshape,

n(z)dz ∝(

zz0

)2

exp− (

zz0

)β dz, (22)

with the choices β = 3/2 and z = 1/√

2, which generates a medianredshift of unity (Laureijs et al. 2011). In this work, we have used ayield of 40 galaxies per squared arcminute and to assume a fractionof fsky = 0.5 of the sky is observed, which is above the currentexpected specifications for Euclid.

Linking the correlations of the observable ellipticity ε to thetidal shear fields allows to express correlations in ε in terms of cor-relations in ∂2

αβΦ. A suitable Limber-projection with the redshift-distribution n(z)dz while introducing a binning allows us to com-pute angular correlation functions and in the next step, to obtaintomographic angular E-mode spectra Cs,II

i j (`), Ce,IIi j (`) and lastly

Ce,GIi j (`), which we can compare to the tomographic weak lensing

spectrum Cγi j(`) (Hu 1999; Hu & White 2001). Due to the locality

of intrinsic alignments, the two II-spectra are ∝ δi j, while the GI-spectrum or the weak lensing spectrum do not possess this property(Hirata & Seljak 2004). Central to our investigation will be the lin-ear dependence of the ellipticity with tidal shear field for gravita-tional lensing and for the intrinsic alignment of elliptical galaxies,while the shapes of spiral galaxies depend on squares of the tidalshear. For non-tomographic, 3-dimensional weak lensing surveys,the effects of intrinsic alignments are physically identical and canbe studied analogously (Merkel & Schaefer 2013).

Figure 2 shows the expected E-mode spectra for a ΛCDM-cosmology with a conventional choice of the alignment parame-ters A and D for tomography with nbin = 3 bins with Euclid, re-

MNRAS 000, 1–13 (2018)

Separation of gravitational lensing and IAs 5

Figure 1. A realisation of the cosmic matter distribution as a Gaussian random field in a cube of the side length 100 Mpc/h. The blue and green surfaces are±1σ-contours of the density field, and the orientation of the galactic discs (in red) follow from a quadratic, angular-momentum based alignment model.

sulting from a Limber-projection (Limber 1954) and subsequentFourier-transform. For a reasonably deep lensing survey such asEuclid’s, lensing-induced ellipticity correlations dominate over in-trinsic alignments. The IA contribution on large angular scales iscaused by elliptical galaxies following the tidal shearing model,whereas on small scales the contribution from spiral galaxies,which is described by the tidal torquing model, is most important.Over a wide range of angular scales the negative cross-correlationbetween gravitational lensing and intrinsic alignments shapes theellipticity spectrum. It is notable that the shapes of elliptical galax-ies and lensing measure tidal field components perpendicular tothe line of sight and are proportional to the magnitude of the tidalshears, but that in contrast spiral galaxies reflect with their shapesthe tidal field orientation including line of sight-components. Fordetails on the derivation of angular shape correlation functions andE/B-mode ellipticity spectra we refer to Capranico et al. (2013),Schaefer & Merkel (2015) and Tugendhat & Schaefer (2018).

4 SEPARATION BETWEEN WEAK LENSING ANDGALAXY ALIGNMENTS

4.1 Idea and formalism

Weak gravitational lensing and intrinsic alignments of galaxies aretidal gravitational effects of the cosmic large scale structure, but thedetails of how the observed shape of a galaxy is influenced by thetidal field ∂2

i jΦ generated by the large-scale structure depends onthe interaction mechanism: Gravitational lensing is universal andoperates on all galaxy shapes in an identical way. For the intrinsic

101 102 103

Multipole `

10−8

10−7

10−6

10−5

10−4

10−3

10−2

(2π

)−1`(`

+1)C

(`)

lensing

spirals, II

ellipticals, II

ellipticals, GI

Figure 2. Contributions to the ellipticity correlations by weak gravitationallensing Cγ

i j(`). The number of tomography bins is nbin = 3, and negativecontributions are depicted in dashed lines. The grey bands show the upperand lower limits of the six weak lensing spectra possible from 3-bin tomog-raphy; separately for linear structure formation (lower band) and nonlin-ear structure formation (upper band), the linear lensing falls off at large `.The IA contributions are: Ce,GI

i j (`) (green), Cs,IIi j (`) (blue) and Ce,II

i j (`) (red).Here, we also show all possible bin pairings, the three autocorrelations forthe II-part, as well as all six possible combinations for the GI-part. Darkercolour indicates higher bin numbers i j.

MNRAS 000, 1–13 (2018)

6 T.M. Tugendhat, R. Reischke, B.M. Schäfer

alignment contributions, we assume that elliptical galaxies changetheir shape in proportion to the tidal gravitational field according tothe tidal shearing model and that the shape of spiral galaxies is anorientation effect that depends on the squared, unit-normalised tidalfield, as stipulated by the tidal torquing model. The dependenceson the linear and squared tidal shear field have the important con-sequence that there is no cross-correlation between lensing and theintrinsic shape of spiral galaxies, 〈γε′s〉 = 0 and neither between theshapes of spiral and elliptical galaxies, 〈εsε

′e〉 = 0, if the tidal shear

field follows Gaussian statistics: In this case, the two correlationfunctions would be proportional to a third moment of a symmetricdistribution, which makes them vanish. In contrast, there will bea nonzero cross-correlation between the intrinsic shapes of ellipti-cal galaxies and weak lensing, 〈γε′e〉 , 0, which is usually dubbedGI-alignment. These model choices are consistent with Tugend-hat & Schaefer (2018), where the implications of such a physicallymotivated mixed model for the two overwhelmingly contributingmorphologies is studied. Other model choices, such as the closelyrelated Blazek et al. (2017), wouldn’t invalidate this method. Al-though correlations between the two morphologies would contami-nate the isolated gravitational lensing signal, a separation of galaxytypes using colour information would still be feasible. Using mul-tiple models for each galaxy type would make the calculation morecomplex as well, but can be done in principle. We have chosento use the aforementioned models for the two morphologies be-cause we believe them to be a physically well-motivated choice andthus being the major contributor to the IA signal. A more complexmodel, for example additional linear term dependency of spirals atsome scales, would amount to a misclassification of galaxy types inour ansatz, which we discuss in detail. Furthermore, an additionaldependency of the spiral model on the amplitude of the tidal fieldwould change the amplitude of the corresponding part of the IA sig-nal, however it would not affect the points where both IA signalsdisappear. Superficially not unlike a multi-tracer method (such asused in e.g. Leonard & Mandelbaum 2018), our formalism allowsfor a certain flexibility for which signals to boost or suppress re-spectively. For example, the lensing signal can be slightly enhancedover intrinsic alignments or eliminated completely depending onlyon the choice of a free parameter.

Starting with a tomographic observation of the ellipticity fieldin a range of redshift bins i, which contains contributions fromweak gravitational lensing and from the two alignment mechanismsand which is sampled including colour-information we define theobserved maps es,i and ee,i, where the subscripts denote spiral galax-ies and elliptical galaxies. These contain contributions from weaklensing γi in redshift bin i and their respective alignment mecha-nism εs,i and εe,i and denoting the shape noise ε,

es,i = γi + εs,i + εi, (23)

ee,i = γi + εe,i + εi, (24)

respectively. Using the same value of γi for both shapes assumesthat the change in shape due to gravitational lensing does not de-pend on the type of galaxy. In reality, the situation is more compli-cated, because an estimate of the shape of the galaxy depends on thebrightness distribution is a nonlinear process which is affected bythe tidal shear and higher derivatives of the gravitational potential.Similarly, there are dependences of the measured shape on colourbecause of variations of the telescope’s point spread function (Eret al. 2018).

Consequently, the covariance matrix Ct(`) = S t(`) + Nt(`) of

a tomographic measurement of the data vector (es,i, ee,i′ ),

Ct(`) =

(Css

i j (`) Csei j′ (`)

Cesi′ j(`) Cee

i′ j′ (`)

), (25)

is composed from the contributions of the signal S t(`)

S t(`) = n2s

(Cγ

i j(`) + Cs,IIi j (`)

)nsne

(Cγ

i j′ (`) + Ce,GIi j′ (`)

)nsne

(Cγ

i′ j(`) + Ce,GIi′ j (`)

)n2

e

(Cγ

i′ j′ (`) + 2Ce,GIi′ j′ (`) + Ce,II

i′ j′ (`)) ,

(26)

where the two intrinsic alignment autocorrelations Cs,IIi j (`) and

Ce,IIi j (`) are diagonal and therefore proportional to δi j due to the

locality of the tidal interaction process, and the noise Nt(`),

Nt(`) = σ2ε nbin

(nsδi j 00 neδi′ j′

), (27)

which depends on the number ns and ne of spiral and ellipticalgalaxies, respectively. The dispersion σ2

ε of the shape measurementis taken to be σε = 0.3 as expected from Euclid, and we set thenumber of bins to nbin = 7. The off-diagonal elements of Nt(`) arezero because a galaxy can not be spiral and elliptical at the sametime: If the ellipticity data set is split by galaxy type, a shape mea-surement would yield uncorrelated noise estimates for each galaxytype. Note that since Ct, S t and Nt are made up from four nbin×nbin-matrices, their total size is 2nbin × 2nbin. The indices i and j are thetomographic bin numbers, therefore, just as in tomographic lens-ing, increasing the number of bins increases the number of possi-ble correlations, and thus leads to a gain in statistical information.It should be noted that we assume the same ellipticity dispersionfor both galaxy types. This assumption, however, can be relaxedwithout altering the method.

Statistical inference from the covariance Ct(`) would yield ex-actly the same statistical errors on cosmological parameters as ameasurement that would not differentiate between galaxy types.The corresponding Fisher-matrix Fµν would be obtained from

Fµν = fsky

∑`

2` + 12

tr(C−1

t (`)∂µS t(`) C−1t (`)∂νS t(`)

), (28)

and the signal to noise-ratio of the non-randomness of the ellipticityfield would be established at a significance Σ of

Σ2 = fsky

∑`

2` + 12

tr(C−1

t (`)S t(`) C−1t (`)S t(`)

). (29)

The two maps es,i and ee,i can be linearly superposed in orderto up- or downweight the relative contributions of the spectra Cγ

i j(`),Ce,II

i j (`), Ce,GIi j (`) and Cs,II

i j (`): We incorporate this by rotation with amixing angle α,

e+,i = + cosα es,i + sinα ee,i, (30)

e−,i = − sinα es,i + cosα ee,i . (31)

This change of basis amounts to an orthogonal transformation ofthe data, C±i j(`) = UCt(`)UT , with the orthogonal matrix U−1 =

UT given by eq. (31). Since the statistical errors encoded in Fµν

and the signal to noise-ratio Σ are both trace-relationships whichare invariant under orthogonal transformations, it would not haveany influence on inference from a weighted measurement either. Inother words, the mapping is one-to-one and thus preserves all theinformation content by definition.

But if one now focuses on the statistical property of a sin-gle field, for instance ε+,i, it is possible to influence the relative

MNRAS 000, 1–13 (2018)

Separation of gravitational lensing and IAs 7

contribution of lensing and the intrinsic alignment terms by a suit-able choice of α: Of course this can be trivially achieved by set-ting α = 0, in which case there are only spiral galaxies in themeasurement whose shape correlations would be described byCγ

i j(`) + Cs,IIi j (`), or by choosing α = π/2, which eliminates all

galaxies apart from the elliptical ones with the the shape correla-tion Cγ

i j(`) + 2Ce,GIi j (`) + Ce,II

i j (`).Interestingly, there is a choice for α that corresponds to the

case where correlations involving weak lensing Cγi j(`) and Ce,GI

i j (`)are completely removed from the data: In fact, the correlationC++

i j (`) is given by

C++i j (`) = (ns cosα + ne sinα)2 Cγ

i j(`)

+ 2(nsne cosα sinα + n2e sin2 α) Ce,GI

i j (`)

+ n2s cos2 αCs,II

i j (`) + n2e sin2 αCe,II

i j (`) .

(32)

The prefactors of the lensing signal can now be set to zero with asuitable choice of α, eliminating weak lensing from the measure-ment. Incidentally, the prefactor cosα sinα + sin2 α will then bezero as well, which cancels the cross-correlation between lensingand the intrinsic shapes of elliptical galaxies and keeps only in-trinsic alignments. For the case ns = ne one would select α to be3π/4, and the two alignment contributions Cs,II

i j (`) and Ce,IIi j (`) en-

ter C++i j (`) with the same weight, so one effectively retains only

Cs,IIi j (`) + Ce,II

i j (`) from C++i j (`) with the choice α = 3π/4. Note that

even if ns , ne, which will be generally the case, one will findan α such that all lensing contributions vanish. On the level of theellipticity maps, the value α = 3π/4 would set in the linear combi-nation (31) cosα = −1/

√2 and sinα = +1/

√2, which cancels γ

from e+,i, e+,i = (es,i − ee,i)/√

2 = (εs,i − εe,i)/√

2. It should be notedthat eq. (32) assumes a particular alignment model, especially thatthere will be no intrinsic ellipticity correlations between spiral andelliptical galaxies and that the GI term for spiral galaxies vanishes.However, even if we consider an alignment model which allowsall possible correlations, the split can still be performed. In par-ticular, an intrinsic alignment cross-correlation between spiral andelliptical galaxies, Cse,II

i j , would appear in eq. 32 with a prefactorof (2nsne sinα cosα), whilst a GI term for the spiral galaxies Cs,GI

i j

would be weighted with 2(n2s cos2 α + nsne sinα cosα) in turn.

Peculiarly, these factors vanish always at the same choice of αthat also eliminates Cs,II

i j , e.g. α = π/2 for q = 0.5. In general, thus,it is always possible to remove all spiral alignment contributionsat once using this method – at the price of statistically diminishingthe lensing signal and boosting the relative contribution of ellipticalgalaxies’ alignment.

The aforementioned point that eliminates lensing completely– meaning choosing α to be 3π/4 for q = 0.5 – still holds as well:The spiral GI contributions vanish here alongside the elliptical GIsignal, whilst a spiral-elliptical cross-alignment would remain, al-beit with a negative sign.

As shown by Figure 3, the choice of α = π/4 would boostthe lensing signal relative to the intrinsic alignments for the casens = ne, but would acquire nonzero contributions from everyother correlation, with the peculiarity that the relative weightingof the two intrinsic alignments are identical. On the level of theellipticity maps, α = π/4 sets cosα = sinα = 1/

√2 such that

e+,i = (2γi + εs,i + εe,i)/√

2, showing the maximal enhancement ofγ, but with a non-zero contribution from all other ellipticity fields.We will investigate if other choices of α will lead to a better rel-ative suppression of intrinsic alignments in the lensing signal, butone should point out that this procedure is not general and will de-

0 14π

12π

34π π

mixing angle α

−0.2

0.0

0.2

0.4

0.6

rela

tive

contr

ibu

tion

toS

++

Cγij(`)

Ce,GIij (`)

Cs,IIij (`)

Ce,IIij (`)

Figure 3. Relative contributions of weak lensing and intrinsic alignmentsto the spectrum C++

i j (`) as a function of mixing angle α: weak gravita-

tional lensing Cγi j(`) (black), Ce,GI

i j (`) (green), Cs,IIi j (`) (red line) and Ce,II

i j (`)(blue). Dashed lines correspond to a q = 0.5, while solid lines have q = 0.7.At α = 3π/4, the contributions involving weak lensing vanish for ns = ne.

pend on the particular lensing and intrinsic alignment spectra. Forcompleteness, evaluating the remaining correlations yields

C−−i j (`) = (ns sinα − ne cosα)2 Cγi j(`)

+ 2(n2e cos2 α − nsne cosα sinα)Ce,GI

i j (`)

+ n2s sin2 αCs,II

i j (`) + n2e cos2 αCe,II

i j (`) ,

(33)

and

C+−i j (`) =

(nsne(cos2 α − sin2 α) + cosα sinα(n2

e − n2s))

Cγi j(`)

+(nsne(cos2 α − sin2 α) + 2n2

e cosα sinα)

Ce,GIi j (`)

+ cosα sinα(n2

eCe,IIi j (`) − n2

sCs,IIi j (`)

),

(34)

showing that there identical optimised choices for α from C−−i j (`).Likewise, the contribution Nt(`) to the covariance matrix due toshape noise changes under rotations by α, implying for the entries

N++i j (`) = σ2

ε nbin

(ns cos2 α + ne sin2 α

), (35)

as well as

N−−i j (`) = σ2ε nbin

(ns sin2 α + ne cos2 α

), (36)

and finally

N+−i j (`) = σ2

ε nbin (ne − ns) cosα sinα , (37)

which effectively corresponds to the expressions for C++i j (`), C−−i j (`)

and C+−i j (`) if one sets the lensing effect to zero and replaces the

intrinsic alignment spectra with a Poissonian noise term. With thespectra reflecting the number of galaxy pairs and the shape mea-surement noise being proportional to the number of galaxies on re-tains a correct relative normalisation of all spectra and noise terms.In this work, we will be considering q = 0.7, i.e. 70 per cent of thetotal galaxy sample are spirals.

4.2 Misclassification

Imperfections in the classification of galaxies can be incorporatedby introducing conditional probabilities of the type p(b|r) indicat-ing that an elliptical galaxy (with the label r, due to the red colour)

MNRAS 000, 1–13 (2018)

8 T.M. Tugendhat, R. Reischke, B.M. Schäfer

is wrongly classified as a spiral galaxy (labelled b, due to the bluecolour). Consequently, the number of observed spiral galaxies nb

and observed elliptical galaxies nr is given in terms of the true num-ber of spiral galaxies ns and elliptical galaxies ne by

nb = p(b|b)ns + p(b|r)ne, (38)

nr = p(r|b)ns + p(r|r)ne, (39)

where the normalisation conditions p(b|r) + p(r|r) = 1 and p(r|b) +

p(b|b) = 1 make sure that the total number of galaxies is conserved,n = nb + nr = ns + ne. A misclassification would take place if p(b|r)and p(r|b) were not equal to zero, so in fact only two of the con-ditional probabilities are independent variables. As a consequenceof the misclassification one would obtain (i) wrong amplitudes forthe ellipticity correlations of a given galaxy type, (ii) additionalcontributions from the respective other galaxy type with possiblecross-correlations with the lensing signal, and (iii) one would esti-mate these ellipticity spectra on the basis of the wrongly inferrednumber of galaxy pairs. The observed ellipticity fields in the to-mographic bin i now contain contributions from wrongly classifiedgalaxies,

eb,i = γi + p(b|b)εs,i + p(b|r)εe,i, (40)

er,i = γi + p(r|b)εs,i + p(r|r)εe,i. (41)

Consequently, the covariance matrix C f (`) = S f (`) + N f (`) reads

C f (`) =

(Cbb

i j (`) Cbri j′ (`)

Crbi′ j(`) Crr

i′ j′ (`)

), (42)

whose signal part S f (`) contains contributions from wrongly classi-fied galaxies. The components of matrix in the colour basis assumethe following form

S bbf (`) = n2

b[Cγ

i j(`) + 2p(b|r)Ce,GIi j (`)

+ p(b|b)2C s,IIi j (`) + p(b|r)2Ce,II

i j (`)]

S brf (`) = nbnr

[Cγ

i j′ (`) + Ce,GIi j′ (`)

+ p(b|b)p(r|b)C s,IIi j′ (`) + p(b|r)p(r|r)Ce,II

i j′ (`)]

S rbf (`) = nbnr

[Cγ

i′ j(`) + Ce,GIi′ j (`)

+ p(b|b)p(r|b)C s,IIi′ j (`) + p(b|r)p(r|r)Ce,II

i′ j (`)]

S rrf (`) = n2

r[Cγ

i′ j′ (`) + 2p(r|r)Ce,GIi′ j′ (`)

+ p(r|b)2C s,IIi′ j′ (`) + p(r|r)2Ce,II

i′ j′ (`)]

(43)

and whose noise part N f (`) needs to be updated because the num-bers of blue and red galaxies are not equal to the numbers of spiralsand ellipticals if p(r|b) , 0 or p(b|r) , 0,

N f (`) = σ2ε nbin

(nbδi j 00 nrδi′ j′

). (44)

It should be noted that equations (43) and (44) consistently re-duce to equations (26) and (27) if there is no misclassification,p(r|b) = p(b|r) = 0, such that the covariances are identical,Ct(`) = C f (`). Finally one would again rotate C f (`) into the newbasis with the orthogonal matrix U. We will use for illustrationpurposes misidentification rates of p(b|r) = p(r|b) = 0.1. In Fig-ure 4 we show the influence of the misclassification: the dashedline shows a case with all galaxies being identified correctly, whilethe solid curves have misclassification of 10 %. Clearly the lensingsignal itself is unaffected as it does not depend on the colour of thegalaxy. The other contributions however change in amplitude anddependence on α. It is worth noting that the position where lensingand the GI term vanish does not depend on the misclassification

0 14π

12π

34π π

mixing angle α

−0.2

0.0

0.2

0.4

0.6

rela

tive

contr

ibu

tion

toS

++

Cγij(`)

Ce,GIij (`)

Cs,IIij (`)

Ce,IIij (`)

Figure 4. Relative contributions of weak lensing and intrinsic alignmentsto the spectrum C++

i j (`) as a function of mixing angle α. The colour schemeis the same as in Figure 3. Here q = 0.7 for both dashed and solid curves.However, here, dashed curves have a misclassification rate p(b|r) = p(r|b) =

0.1, while the solid curves represent all galaxies identified correctly.

rate but only on the ratio of elliptical and spiral galaxies in the sur-vey.

4.3 Separating out intrinsic alignments

The signal to noise-ratio Σ of the isolated intrinsic alignmentsC s,II

i j (`) + Ce,IIi j (`) can be determined to be Σ ' 60 for Euclid:

Focusing on the contribution e+,i after setting α = 3π/4 for thecase ns = ne defines the signal covariance S ++

i j (`) as the upper leftnbin × nbin-block of S t(`). Consequently, the significance for mea-suring the intrinsic alignment contribution C s,II

i j (`)+Ce,IIi j (`) is given

by

Σ2 = fsky

∑`

2` + 12

tr(C−1

++(`)S ++(`) C−1++(`)S ++(`)

). (45)

By isolating the C++ components we throw away the informationcontained in the other components of the covariance.

Figure 5 shows in detail the attainable significance Σ for the in-trinsic alignment contribution, both cumulatively and differentially,as a function of multipole moment ` and for different number oftomographic bins. Clearly, intrinsic alignments generate a signifi-cant signal in Euclid’s survey and the magnitude of the alignments,both for spiral and elliptical galaxies, allows a determination of pa-rameters of the alignment model at percent precision. It is also in-teresting to see that most of the signal is picked up at relativelylow multipoles, showing that the assumption of Gaussian randomfields can indeed be used. The reason for this is that the intrinsicalignment signal only probes the auto-correlation of different binsand thus the shape-noise dominates the high multipole moments,which becomes dominant earlier compared to the II alignmentsdue to their lower amplitude. Tomography is a large factor in at-taining a high signal to noise ratio and therefore, in investigatingintrinsic alignment models, as Σ increases from 35 with 2 bins toover 60 with 7 bins. These numbers are in agreement with forecastson alignment amplitudes: As the alignment signal is proportional toA or D, the signal to noise-ratio corresponds to the inverse statisti-cal error on these prefactors, such that one should achieve %-level

MNRAS 000, 1–13 (2018)

Separation of gravitational lensing and IAs 9

101 102 103

`

0

5

10

15

20

25

30

dΣ

2/d`

0

10

20

30

40

50

60

70

80

Σ(`

)

Figure 5. Cumulative Σ(`) (solid lines) and differential signal to noise-ratiodΣ/d` (dashed lines) for the measurement of the intrinsic alignment spectraCs,II

i j (`) + Ce,IIi j (`) as defined by equation 45 for a tomographic ellipticity

measurement for nbin = 2 . . . 7 bins for a survey like Euclid’s.

precision from a measurement yielding 100σ of statistical signif-icance. It is remarkable that this level can be reached even if oneconsiders a combination of cosmological probe and a complex cos-mological model (Kitching et al. 2014; Merkel & Schaefer 2017).Figure 5 shows in detail the attainable significance Σ for the intrin-sic alignment contribution, both cumulatively and differentially, asa function of multipole moment ` and for different number of to-mographic bins. Clearly, intrinsic alignments generate a significantsignal in Euclid’s survey and the magnitude of the alignments, bothfor spiral and elliptical galaxies, allows a determination of param-eters of the alignment model at percent precision. It is also inter-esting to see that most of the signal is picked up at relatively lowmultipoles, showing that the assumption of Gaussian random fieldscan indeed be used. The reason for this is that the II-part of in-trinsic alignment signal only probes the auto-correlation of differ-ent bins (since IA is a small scale effect) and thus the shape-noisedominates the high multipole moments, which becomes dominantearlier compared to the II alignments due to their lower amplitude.Tomography is a large factor in attaining a high signal to noise ratioand therefore, in investigating intrinsic alignment models, as Σ in-creases from 35 with 2 bins to over 60 with 7 bins. These numbersare in agreement with forecasts for Euclid-like surveys on align-ment amplitudes (e.g. Tugendhat & Schaefer 2018): As the align-ment signal is proportional to A or D, the signal to noise-ratio corre-sponds to the inverse statistical error on these prefactors, such thatone should achieve %-level precision from a measurement yield-ing 100σ of statistical significance. It is remarkable that this levelcan be reached even if one considers a combination of cosmologi-cal probe and a complex cosmological model (Kitching et al. 2014;Merkel & Schaefer 2017)

We quantify the difference between the ideal spectra and theone containing wrongly classified galaxies with the average 〈∆χ2〉

between the two models. For that purpose, we identify the C++i j (`)

contribution from the rotated covariance matrix Ct(`) and the corre-sponding X++

i j (`) from the rotated covariance matrix C f (`) includ-ing shape correlations of wrongly identified galaxies, by settingα = 3π/4.

101 102

`

10−1

100

101

102

103

104

105

d/d`〈

∆χ

2〉

10−1

100

101

102

103

104

105

〈∆χ

2〉

Figure 6. Difference 〈∆χ2〉 between the covariances C++(`) (where allgalaxies are correctly identified) and X++(`) (with assumed misidentifi-cation rates of p(b|r) = p(r|b) = 0.1) as a function of multipole ` fornbin = 2 . . . 7 tomographic spectra.

The 〈χ2t 〉-value of the true model is on average given by

〈χ2t 〉 = fsky

∑`

(2` + 1) tr(ln C++(`) + id

). (46)

Here, id stands for the identity matrix. The average 〈χ2f 〉 of the

wrongly classified model can be computed with

〈χ2f 〉 = fsky

∑`

(2` + 1) tr(ln X++(`) + X−1

++(`)C++(`)), (47)

such that the difference 〈∆χ2〉 = 〈χ2f − χ

2t 〉 between the true and

false model in the light of the on average expected data Ct yields

〈∆χ2〉 = fsky

∑`

(2`+1)(ln

(det X++(`)det C++(`)

)+ tr

(X−1

++(`)C++(`))− nbin

),

(48)

where we again assume statistical homogeneity of the e+,i and e−,i-fields, and a number of 2` + 1 statistically uncorrelated modes ona given angular scale ` as a consequence of statistical isotropy. Wescale the resulting χ2-values or signal to noise-ratios Σ by

√fsky if

the sky coverage is incomplete. If X++(`) = C++(`), then 〈∆χ2〉 = 0because tr

(X−1

++C++)

= nbin. In the relations above, id refers to theunit matrix in nbin dimensions.

First of all we expect the obtainable Σ2(`) to be smaller in thecase where galaxies have been identified wrongly, since the relativecontribution is smaller as it can be seen in Figure 4. This will lead toa difference 〈∆χ2〉 between the true model and the model contain-ing wrongly identified galaxies by expressing the difference in thespectra in units of the cosmic variance, which we show in Figure 6.At ` >

∼ 200, the difference in χ2 continues to grow steadily due tothe difference in the noise contributions, as the noise is dependenton the galaxy counts, which naturally differ if there is a misclassifi-cation of galaxy types. Furthermore, the loss of Gaussianity in thecorrelations at larger ` will lead to complicated cross–correlationsbetween the alignment effects that have not been included in thecalculation of 〈∆χ2〉 or indeed in the considerations for a misclas-sified signal S f (`) (cf. Equation 43).

MNRAS 000, 1–13 (2018)

10 T.M. Tugendhat, R. Reischke, B.M. Schäfer

4.4 Boosting the weak lensing spectrum

The weak lensing contribution Cγi j(`) can be boosted relative to the

intrinsic alignments C s,IIi j (`) and Ce,II

i j (`) by choosing a value of α inthe vicinity of π/4. Unlike the previous case, there is no completecancellation of the intrinsic alignment contribution and one canonly hope to optimise the measurement. This optimisation, how-ever, does depend on the particular values of the lensing and align-ment spectra and is therefore not model-independent. Because thisis inevitably the case, there are two competing effects to be takencare of: Firstly, the amplitude of the contribution of the weak lens-ing spectrum to C++

i j (`, α) changes as a function of α, giving rise toa change in the statistical precision of the measurement encoded inthe Fisher-matrix,

Fµν(α) = fsky

∑`

2` + 12

tr(C−1

++(`, α)∂µS ++(`) C−1++(`, α)∂νS ++(`)

),

(49)

and resulting in changes in the marginalised statistical errorsσ2µ(α) =

(F−1(α)

)µµ

. We approximate derivatives ∂µS ++i j (`) with the

derivatives at α = π/4 because weak lensing dominates the spec-trum and the derivatives with respect to the cosmological parame-ters are identical for every choice of α in this limit.

Secondly, the relative contribution of the intrinsic alignmentspectra change as well with α such that there is a varying amountof contamination of intrinsic alignments and a corresponding sys-tematic error δµ(α). This systematic error is computed using the for-malism of Schaefer & Heisenberg (2012) as an extension to Taburetet al. (2009) and (Amara & Réfrégier 2008) from

δµ(α) =∑ν

(G−1(α)

)µν

aν(α) (50)

with the matrix Gµν

Gµν(α) =

= fsky

∑`

2` + 12

tr[C−1

++(α) ∂2µνC

++(`)(C−1

++(α)C++(`) − id)]

− tr[C−1

++(α)∂µC++(`) C−1++(α)∂νC++(`)

(2C−1

++(α)C++(`) − id)],

(51)

and the vector aµ

aµ(α) = fsky

∑`

2` + 12

tr[C−1

++(α)∂µC++(`)(id −C−1

++(α)C++(`))](52)

which are both functions of the mixing angle α. We collect multi-poles up to `max = 2500.

A convenient way for expressing the magnitude of the system-atic error in units of the statistical error is the figure of bias Q(α),

Q2(α) =∑µν

Fµν(α) δµ(α)δν(α), (53)

which is related to the Kullback-Leibler divergence DKL for Gaus-sian likelihoods under the assumption of constant covariances,

DKL =Q2

2. (54)

Figure 7 shows Q as a function of α: Clearly, there is an optimisedchoice of α that reduces the intrinsic alignment contribution in or-der to yield an unbiased measurement of the cosmological parame-ter set, which we demonstrate by computing Q in scanning through

0 14π

12π

34π π

mixing angle α

100

101

102

Fig

ure

ofB

iasQ

F0

Fα

−0.2

0.0

0.2

0.4

0.6

0.8

rela

tive

contr

ibu

tion

s

Figure 7. Total bias Q(α) in units of the statistical error, Q2 =∑µν Fµνδµδν

as a function of the mixing angle α, where the minimum indicates the valueof α that is able to yield the smallest possible systematic error correspondingto the cleanest weak lensing spectrum. F0 (black) corresponds to a deriva-tive at α = π/4, whereas Fα in blue shows the same with derivative takenat the corresponding α. The translucent lines in the background are takenfrom Figure 3 as a guideline where IAs vanish and where they dominate.

all possible choices of α. This optimal choice of α is where thecurves dip between 0 and π/4, where the bias is roughly halved.We calculate the figure of bias Q both with the systematic errorstaken in relation to the statistical errors at α = 0, F0, in black andwith the Fisher matrix taken at the respective α, Fα, in blue. It be-comes clear that as long as there is a significant lensing contribu-tion, up until α ' π/2, the differences are negligible. As soon as thecontributions from intrinsic alignments become comparable to theone from weak lensing, the figure of bias forks: while comparingthe systematic errors to a co–evolving statistical error, the figure ofbias becomes increasingly small. This is not due to a diminishingbias but rather due to extremely large statistical errors as the in-formation from lensing disappears. For a constant statistical errortaken at α = 0, the curve is therefore more representative of the ac-tual precision of the measurement. As α approaches π, the curvesstart to merge again.

Typically, relative contributions of order ten percent of the in-trinsic alignment signal to weak lensing cause figures of bias Qof the order up to a few hundred, which can be controlled by ourtechnique. It would be interesting to propagate intrinsic alignmentswith other systematic effects through the parameter estimation pro-cess as outlined by Cardone et al. (2014).

Lastly, we aim to constrain the parameters of the alignmentmodels for the two types of galaxies, specifically the alignment am-plitude D that describes the elasticity of elliptical galaxies and theamplitude A which describes the magnitude A of the misalignmentbetween tidal shear and inertia which is responsible for angularmomentum generation in spiral galaxies. If one suppresses lens-ing, including GI-alignment, Figure 8 suggests that both alignmentparameters can be constrained at the percent level with the Eucliddata set without having to worry about biases due to gravitationallensing. These numbers result from a Fisher-matrix analysis for theparameters A and D with a choice of α that eliminates lensing, andfitting the corresponding intrinsic ellipticity spectra to the remain-der, while taking account of cosmic variance and shape noise con-

MNRAS 000, 1–13 (2018)

Separation of gravitational lensing and IAs 11

0.92

0.94

0.96

0.98

D ×106

0.22

0.23

0.24

0.25

0.26

0.27

0.28

A

Figure 8. Fisher-constraints of the model parameters A and D for thequadratic and linear alignment models respectively for nbin = 7 in Euclid.

tributions and keeping all cosmological parameters fixed to theirfiducial values.

5 SUMMARY

Subject of this investigation was the use of colour information todifferentiate between intrinsic ellipticity correlations of galaxiesand the weak gravitational lensing signal. We operate under the as-sumptions that (i) the large-scale structure follows Gaussian statis-tics on large scales, (ii) intrinsic shapes of elliptical galaxies followsthe linear tidal shearing model, (iii) intrinsic shapes of spiral galax-ies follows the quadratic tidal torquing model, (iv) gravitationallensing is universal and linear in the tidal shear, and (v) it is possi-ble to classify galaxies on the basis of colour information to obeyone of the two alignment mechanisms. (vi) shape and redshift mea-surements for the cosmic shear analysis are under control. Based onthe fact that lensing is universal and affects all galaxy shapes in anidentical way irrespective of galaxy type it is possible to find linearcombinations of ellipticity fields measured for spiral and ellipti-cal galaxies that do not contain any lensing signal and only retainshape correlations due to intrinsic alignments. It should be empha-sised that this is possible without any assumption about the detailsof gravitational lensing or of the intrinsic alignment models. Intrin-sic alignments that have been separated in this way from the lensingsignal, can be measured on the basis of Euclid’s weak lensing dataset with a high statistical significance. The subject of this investiga-tion was the use of colour information to differentiate between in-trinsic ellipticity correlations of galaxies and the weak gravitationallensing signal. We operate under the assumptions that (i) the large-scale structure follows Gaussian statistics on large scales, (ii) in-trinsic shapes of elliptical galaxies follows the linear tidal shearingmodel, (iii) intrinsic shapes of spiral galaxies follows the quadratictidal torquing model, (iv) gravitational lensing is universal and lin-ear in the tidal shear, and (v) it is possible to classify galaxies on

the basis of colour information to obey one of the two alignmentmechanisms. Based on the fact that lensing is universal and affectsall galaxy shapes in an identical way irrespective of galaxy type itis possible to find linear combinations of ellipticity fields measuredfor spiral and elliptical galaxies that do not contain any lensing sig-nal and only retain shape correlations due to intrinsic alignments. Itshould be emphasised that this is possible without any assumptionabout the details of gravitational lensing or of the intrinsic align-ment models. Intrinsic alignments that have been separated in thisway from the lensing signal, can be measured on the basis of Eu-clid’s weak lensing data set with a high statistical significance.

(i) As alignment models we consider tidal shearing for ellip-tical galaxies and tidal torquing for spiral galaxies, and computethe resulting tomographic angular ellipticity spectra including thenon-zero cross-correlation between the shape of elliptical galaxiesand weak lensing. Both models have a single free parameter each,which we determined from weak lensing data and from numericalsimulations, respectively. In analysing data, we showed that Euclidwill allow their measurement at a level of a few percent. In that,we assume constant and scale-independent alignment parameters.The forecasted statistical precision would allow the investigationof alignment models with Euclid’s weak lensing data set, and shedlight on (i) the average misalignment of angular momenta with tidalshear fields and (ii) the reaction of a virialised structure to externaltidal shear fields.

(ii) We develop a statistical method which allows the separa-tion between weak lensing and both alignment types on the basisof colour or morphological information: We assume that ellipti-cal galaxies, if they are correctly identified, obey exclusively lineartidal shearing as their alignment mechanism, while spiral galaxiesare described by the quadratic tidal torquing model. Gravitationallensing is universal as it affects the shapes of spiral and ellipticalgalaxies identically.

(iii) All mitigation and suppression techniques have in commonthat statistical precision is traded for systematical accuracy, and ourmethod is no exception: Starting from shape catalogues measuredfor different types of galaxies it is possible to find linear combi-nations of the shape measurements that contain no spiral align-ment, no elliptical alignment or no gravitational lensing, includingin this case no lensing-alignment cross correlation either. With thatin mind, it is possible to find a linear combination that eliminateslensing from the data set and leaves only contributions to the shapecorrelations that differ between elliptical and spiral galaxies, i.e. in-trinsic alignments. Taking these cleaned correlation functions, theyallow a measurement of the intrinsic alignment signal without anymodel assumptions to ∼ 60σ of statistical precision for a tomo-graphic survey such as Euclid’s. For other surveys, such as LSST,we expect a significantly lower contributions of IA, since they area) deeper and b) less wide than Euclid. The effect from a) is that thesurvey gathers information from redshifts where IA is much lower,as structure formation is less developed at those stages as well asthe lensing efficiency dominates even more strongly. From b) weexpect a lower statistical significance, as there is more shot noise,which also washes out IA contributions. Therefore, only a surveythat clearly makes out IA contamination to a significant amount isrelevant for this method.

(iv) An independent verification of the validity of the separationof lensing and IA contributions could possibly be done by measur-ing B-mode spectra and comparing those to the predicted B-modes(if any) of the underlying intrinsic alignment models. This is notparticular to our choice of models but can in principle be done for

MNRAS 000, 1–13 (2018)

12 T.M. Tugendhat, R. Reischke, B.M. Schäfer

any assumption of IA models, especially to verify or improve themodel parameter measurement.

(v) Suppression of intrinsic alignment contributions for achiev-ing a bias-free weak lensing measurement is possible, but not com-pletely. With an optimised choice of the mixing angle α one canreduce the systematical bias in units of the statistical error by alarge margin, and we show that biases are reduced to amount totypically a few σ for the full ΛCDM parameter set. We quantifythe magnitude of the systematic error in units of the statistical er-ror by the figure of bias Q2 =

∑µν Fµνδµδν, which takes care of the

orientation of the systematic error δµ with respect to the statisticaldegeneracies encoded in the Fisher-matrix Fµν. Incidentially, Q2/2corresponds to the Kullback-Leibler-divergence between the biasedand unbiased likelihood L.

(vi) Misclassification, i.e. non-zero probabilities p(r|b) andp(b|r) do not affect our conclusions strongly even for very highprobabilities of misclassification on the order of ten per cent. Thisgenerous limit for misclassifications is also chosen to show thatadditional IA mechanisms that can be treated as misclassificationswould not change our results meaningfully. We quantify this bycomputing the difference between the resulting covariance in termsof the χ2-statistic, where instrumental noise and cosmic varianceare present as noise sources. At accessible scales below a few hun-dred in `, i.e. before the instrumental noise dominates, the inte-grated ∆χ2 is a few ten, which, given the sensitivity of a weak lens-ing survey with respect to parameters such as σ8 or Ωm, would giverise to biases. We estimate that the misidentification probabilitieswould need to be controlled to well below ten per cent for the bi-ases not to exceed 1σ in terms of the statistical error in this case.

Precursing to the work presented here, there are quite a fewother mitigation techniques: Catelan et al. (2001) proposed tomo-graphic methods to reduce the II-alignment signal, by avoidingspatially close galaxies. This exploits the large correlation length ofthe weak lensing signal, compared to the intrinsic alignment signal.The drawback of this method is the increased cosmic variance andshape noise and thus a loss of statistical power. This technique wasused by several authors (e.g. Heymans & Heavens 2003; Heymanset al. 2004; King & Schneider 2002; Heymans et al. 2004; Takada& White 2004; Joachimi et al. 2013a; Heymans et al. 2013).

Another method is to construct a different weighting of thecosmic shear signal, to reduce the contamination by GI-alignments.This nulling technique was discussed by Huterer & White (2005);Joachimi & Schneider (2010). It has been proposed as well tonull and boost magnification (Heavens & Joachimi 2011; Schnei-der 2014), which can also bias the parameter inference process.Alternatively, one can also use self-calibration techniques, mak-ing use of additional information and the cross-correlation of cos-mic shear and galaxy clustering, as well as galaxy clustering auto-correlations, which has been used by Bernstein & Jain (2004);Bernstein (2009); Zhang (2010) or use the large difference in theamplitudes of E- and B-mode spectra of intrinsic alignments andweak lensing (Crittenden et al. 2002; Schaefer & Merkel 2015).Our method could as well be extended bispectra of the ellipticityfield (Shi et al. 2010; Merkel & Schaefer 2014; Larsen & Challi-nor 2016), either in order to make the method less dependent onthe assumption of Gaussianity of the tidal shear field or by relax-ing on the relationship that the observable shape depends on thetidal shear field in a linear or quadratic way. Taking this idea fur-ther, it would be very interesting to see if other cross-correlationmeasurements, for instance correlations between the CMB-lensingfield and galaxy shapes at higher redshifts than the ones considered

here, would constraint alignment processes as well (Hall & TaylorNRAS).

In summary, there is a wealth of information that helps to dif-ferentiate intrinsic alignments and weak lensing, even without sac-rificing statistical precision for accuracy. While our investigationassumes that it is possible to assign an alignment model to a givengalaxy on the basis of its colour (or other morphological informa-tion), inclusion of higher-order statistics, E/B-mode decomposi-tion or redshift weighting scheme should enable a thorough under-standing of shape correlations. We would consider the possibilityof eliminating the lensing signal including the GI-terms from theellipticity correlation through a suitable choice of α very interestingfor the model-independent detection of intrinsic alignments.

ACKNOWLEDGEMENTS

TMT acknowledges a doctoral fellowship from AstronomischesRechen-Institut in Heidelberg, and RFR’s work is supported by astipend of Landesgraduiertenförderung Baden-Württemberg fromthe graduate college Astrophysics of cosmological probes of grav-ity. BMS would like to thank the University of Auckland, in par-ticular S. Hotchkiss and R. Easthers for their hospitality during theearly stages of this project, and A. Nusser and V. Desjacques fortheir hospitality at the Technion in Haifa during the final writeup.We would like to thank H. Hildebrandt for a suggestion for val-ues of p(b|r) and p(r|b). We would furthermore like to thank theanonymous referee for their invaluable corrections, suggestions forimprovement, and patience.

REFERENCES

Amara A., Réfrégier A., 2008, MNRAS, 391, 228Amendola L., Kunz M., Sapone D., 2008, JCAP, 4, 13Bardeen J. M., Bond J. R., Kaiser N., Szalay A. S., 1986, ApJ, 304, 15Barnes J., Efstathiou G., 1987, ApJ, 319, 575Bartelmann M., Schneider P., 2001, Physics Reports, 340, 291Bernstein G. M., 2009, ApJ, 695, 652Bernstein G., Jain B., 2004, ApJ, 600, 17Blazek J., McQuinn M., Seljak U., 2011, JCAP, 2011, 010Blazek J., Mandelbaum R., Seljak U., Nakajima R., 2012, JCAP, 2012, 041Blazek J., Vlah Z., Seljak U., 2015, JCAP, 08, 15Blazek J., MacCrann N., Troxel M. A., Fang X., 2017, ArXiv e-prints

1708.09247Capranico F., Merkel P. M., Schäfer B. M., 2013, MNRAS, 435, 194Cardone V. F., Martinelli M., Calabrese E., Galli S., Huang Z., Maoli R.,

Melchiorri A., Scaramella R., 2014, MNRAS, 439, 202Carron J., Amara A., Lilly S., 2011, ArXiv e-prints 1107.0726Casarini L., La Vacca G., Amendola L., Bonometto S. A., Macciò A. V.,

2011, JCAP, 3, 26Casarini L., Bonometto S. A., Borgani S., Dolag K., Murante G., Mezzetti

M., Tornatore L., La Vacca G., 2012, A & A, 542, A126Catelan P., Porciani C., 2011, MNRAS, 323, 713Catelan P., Theuns T., 1996, MNRAS, 282, 436Catelan P., Theuns T., 1997, MNRAS, 292, 225Catelan P., Kamionkowski M., Blandford R. D., 2001, MNRAS, 320, L7Codis S., et al., 2015, MNRAS, 448, 3391Crittenden R. G., Natarajan P., Pen U.-L., Theuns T., 2001, ApJ, 559, 552Crittenden R. G., Natarajan P., Pen U.-L., Theuns T., 2002, ApJ, 568, 20Croft R. A., Metzler C. A., 2000, ApJ, 545, 561Er X., et al., 2018, MNRAS, 476, 5645Hall A., Taylor A., MNRAS, 2014, 443, L119Heavens A. F., Joachimi B., 2011, MNRAS, 415, 1681Heymans C., Heavens A., 2003, MNRAS, 339, 711

MNRAS 000, 1–13 (2018)

Separation of gravitational lensing and IAs 13

Heymans C., Brown M., Heavens A., Meisenheimer K., Taylor A., Wolf C.,2004, MNRAS, 347, 895

Heymans C., et al., 2013, MNRAS, 432, 2433Hirata C. M., Seljak U., 2004, PRD, 70, 063526Hirata C. M., Seljak U., 2010, PRD, 82Hoekstra H., Jain B., 2008, Annual Review of Nuclear and Particle Science,

58, 99Hu W., 1999, ApJL, 522, L21Hu W., White M., 2001, ApJ, 554, 67Huterer D., 2002, PrD, 65, 063001Huterer D., 2010, General Relativity and Gravitation, 42, 2177Huterer D., White M., 2005, PRD, 72, 043002Joachimi B., Schneider P., 2010, ArXiv e-prints 1009.2024Joachimi B., Semboloni E., Bett P. E., Hartlap J., Hilbert S., Hoekstra H.,

Schneider P., Schrabback T., 2013a, MNRAS, 431, 477Joachimi B., Semboloni E., Hilbert S., Bett P. E., Hartlap J., Hoekstra H.,

Schneider P., 2013b, MNRAS, 436, 819Joachimi B., et al., 2015, Space Sci. Rev.,Kiessling A., et al., 2015, preprint, (arXiv:1504.05546)Kilbinger M., 2015, Reports on Progress in Physics, 78, 086901King L., Schneider P., 2002, A & A, 396, 411Kirk D., Bridle S., Schneider M., 2010, MNRAS, 408, 1502Kirk D., Laszlo I., Bridle S., Bean R., 2011, MNRAS, 430Kirk D., Rassat A., Host O., Bridle S., 2012, MRNAS, 424, 1647Kirk D., et al., 2015, preprint, (arXiv:1504.05465)Kitching T. D., Heavens A. F., Das S., 2014, ArXiv e-prints 1408.7052Krause E., Eifler T., Blazek J., 2016, MNRAS, 456, 207Larsen P., Challinor A., 2016, MNRAS, 461, 4343Laszlo I., Bean R., Kirk D., Bridle S., 2012, MNRAS, 423, 1750Laureijs R., et al., 2011, ArXiv e-prints 1110.3193Lee J., Pen U.-L., 2001, ApJ, 555, 106Leonard C. D., Mandelbaum R., 2018, MNRAS, 479, 1412Limber D. N., 1954, ApJ, 119, 655Linder E. V., Jenkins A., 2003, MNRAS, 346, 573Mackey J., White M., Kamionkowski M., 2002, MNRAS, 332, 788Mandelbaum R., Hirata C. M., Ishak M., Seljak U., Brinkmann J., 2006,

MNRAS, 367, 611Mandelbaum R., et al., 2011, MNRAS, 410, 844Martinelli M., Calabrese E., de Bernardis F., Melchiorri A., Pagano L.,

Scaramella R., 2011, PRD, 83, 023012Matilla J. M. Z., Haiman Z., Petri A., Namikawa T., 2017, ArXiv e-prints

1706.05133Mellier Y., Annual Review of Astronomy and Astrophysics, 1999, 37, 127Merkel P. M., Schaefer B. M., 2013, MNRAS, 434, 1808Merkel P. M., Schaefer B. M., 2014, MNRAS, 445, 2918Merkel P. M., Schaefer B. M., 2017, MNRAS, 469, 2760Munshi D., Valageas P., van Waerbeke L., Heavens A., 2008, Physics Re-

ports, 462, 67Natarajan P., Crittenden R. G., Pen U.-L., Theuns T., 2001, PASA, 18, 198Piras D., Joachimi B., Schäfer B. M., Bonamigo M., Hilbert S., van Uitert

E., 2017, ArXiv e-prints 1707.06559Planck Collaboration et al., 2015, preprint, (arXiv:1502.01590)Refregier A., 2003, Annual Review of Astronomy and Astrophysics, 41,

645Schaefer B. M., 2009, IJMPD, 18, 173Schaefer B. M., Heisenberg L., 2012, MNRAS, 423, 3445Schaefer B. M., Merkel P. M., 2012, MNRAS, 421, 2751Schaefer B. M., Merkel P. M., 2015, ArXiv e-prints 1506.07366Schneider M. D., 2014, Physical Review Letters, 112, 061301Shi X., Joachimi B., Schneider P., 2010, A & A, 523, A60Singh S., Mandelbaum R., More S., 2015, MNRAS, 450, 2195Smith R. E., et al., 2003, MNRAS, 341, 1311Stewart K. R., Brooks A. M., Bullock J. S., Maller A. H., Diemand J., Wad-

sley J., Moustakas L. A., 2013, ApJ, 769, 74Taburet N., Aghanim N., Douspis M., Langer M., 2009, MNRAS, 392, 1153Takada M., White M., 2004, ApJ, 601, L1Theuns T., Catelan P., 1997, in Persic M., Salucci P., eds, Astronomical

Society of the Pacific Conference Series Vol. 117, Dark and VisibleMatter in Galaxies and Cosmological Implications. pp 431–+

Troxel M. A., Ishak M., 2015, Physics Reports, 558, 1Tugendhat T. M., Schaefer B. M., 2018, MNRAS, 476, 3460Vanderveld R. A., Mortonson M. J., Hu W., Eifler T., 2012, PRD, 85,

103518White S. D. M., 1984, ApJ, 286, 38Zhang P., 2010, MNRAS, 406, L95

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