Weak lensing corrections to tSZ-lensing cross correlation
PPrepared for submission to JCAP
Weak lensing corrections totSZ-lensing cross correlation
Tilman Tr¨oster, Ludovic Van Waerbeke
Department of Physics and Astronomy, The University of British Columbia, 6224 Agricul-tural Road, Vancouver, B.C., V6T 1Z1, CanadaE-mail: [email protected], [email protected]
Abstract.
The cross correlation between the thermal Sunyaev-Zeldovich (tSZ) effect andgravitational lensing in wide field has recently been measured. It can be used to probe thedistribution of the diffuse gas in large scale structure, as well as inform us about the miss-ing baryons. As for any lensing-based quantity, higher order lensing effects can potentiallyaffect the signal. Here, we extend previous higher order lensing calculations to the case oftSZ-lensing cross correlations. We derive terms analogous to corrections due to the Bornapproximation, lens-lens coupling, and reduced shear up to order O (Φ ) in the Newtonianpotential. Redshift distortions and vector modes are shown to be negligible at this order.We find that the dominant correction due to the reduced shear exceeds percent-level only atmultipoles of (cid:96) (cid:38) Keywords: gravitational lensing, Sunyaev-Zeldovich effect, power spectrum a r X i v : . [ a s t r o - ph . C O ] N ov ontents y parameter 63.1.3 Cross correlations 73.2 Reduced shear 83.3 Redshift distortions 113.4 Vector modes 12 Even though both the direct detection of dark matter and its microscopic description haveproven to be elusive so far, its macroscopic behaviour is thought to be well understood. Thelarge scale clustering of dark matter has been observed though gravitational lensing and hasbeen found to agree with theoretical predictions, see [1] for a review and [2] for an overview ofthe recent results with the Canada France Hawaii Telescope Lensing Survey (CFHTLenS). Onsmall scales its clustering behaviour has been modelled with N-body simulations to relativelyhigh precision [3]. Conversely, even though the microscopic behaviour of baryons is fullyunderstood, half of the universe’s baryon content is in a hitherto unobserved state [4, 5]; themissing baryon problem.A significant fraction of these missing baryons might reside in a warm, low density phasebeyond galactic halos[6]. By cross correlating thermal Sunyaev-Zeldovich (tSZ) effect maps,which traces warm electrons, and mass maps derived from weak gravitational lensing data,there is now observational support for the possibility that a significant fraction of the baryonsindeed reside in such a phase [7–9].Future surveys with large sky coverage [10, 11] will produce data whose precision war-rants a more sophisticated theoretical treatment than has been necessary so far. In this workwe investigate the effect of higher order lensing terms on the tSZ-lensing cross correlation.There has been considerable effort to characterize higher order contributions to correlationsof lensing observables [12–23]. Some of these higher order effects, like the rotation powerspectrum, have been successfully observed in high-resolution ray-tracing simulations [24, 25].– 1 –uilding on this corpus of previous work, we derive analogous contributions to the tSZ-lensingcross correlation.In section 2 we introduce the notation and recapitulate the first order results. In section3 we investigate the higher order corrections. The terms related to the Born approximation,i.e., the evaluation of the integrals along the unperturbed photon path, and lens-lens couplingare derived in section 3.1. The observed quantity in weak gravitational lensing is the reducedshear. Corrections due to its the non-linear relation to the shear and convergence are derivedin section 3.2. We also consider redshift distortions in section 3.3 and vector modes in section3.4 and show that they do not contribute to the order we are considering.
In the Newtonian gauge the perturbed Robertson-Walker (RW) metric without anisotropicstresses can be written asd s = a ( η ) (cid:2) − (1 + 2Φ)d η + (1 − (cid:0) d χ + d A ( χ ) dΩ (cid:1)(cid:3) , (2.1)with d A ( χ ) the comoving angular diameter distance and χ the comoving radial distance. Wewill henceforth work in units where c = 1. The potential Φ is assumed to be small, i.e.,Φ <<
1. The first order solution to the geodesic deviation equation at a comoving distance χ from the observer is then [1, 12, 26] x i ( (cid:126)θ, χ ) = d A ( χ ) θ i − (cid:90) χ d χ (cid:48) d A ( χ − χ (cid:48) )Φ ,i ( (cid:126)x ( (cid:126)θ, χ (cid:48) ) , χ (cid:48) ) , (2.2)where (cid:126)θ represents the angle between the perturbed and fiducial ray at the observer. Vectorquantities are denoted by lowercase Latin indices and partial derivatives with respect tocomoving transverse coordinates, i.e., those perpendicular to the line-of-sight, are denotedby a comma. We make use of the sum convention where repeated indices are summed over.Unless otherwise noted, this sum only includes the two transverse directions. The Jacobimap is defined as the derivative of the deflection angle (cid:126)x ( (cid:126)θ,χ ) d A ( χ ) with respect to (cid:126)θ , i.e., A ij ( (cid:126)θ, χ ) = ∂x i ( (cid:126)θ, χ ) d A ( χ ) ∂θ j = δ ij − (cid:90) χ d χ (cid:48) d A ( χ − χ (cid:48) ) d A ( χ (cid:48) ) d A ( χ ) Φ ,ik ( (cid:126)x ( (cid:126)θ, χ (cid:48) ) , χ (cid:48) ) A kj ( (cid:126)θ, χ (cid:48) ) . (2.3)It can be expressed in terms of the convergence κ , shear γ , γ , and rotation ω as A ij = (cid:18) − κ − γ − γ − ω − γ + ω − κ + γ (cid:19) = δ ij − ψ ij . (2.4)Here we have introduced the distortion tensor ψ ij as a measure of the deviation from theidentity map. The convergence is then given by the trace of the distortion tensor κ = 12 ψ ii . (2.5)Using (2.3) and (2.5), we find for the first order convergence κ (1) ( (cid:126)θ, χ S ) = (cid:90) χ S d χ (cid:48) K ( χ S , χ (cid:48) )Φ ,ii ( d A ( χ (cid:48) ) (cid:126)θ, χ (cid:48) ) , (2.6)– 2 –here we have defined the kernel K ( χ S , χ (cid:48) ) = d A ( χ S − χ (cid:48) ) d A ( χ (cid:48) ) d A ( χ S ) Θ( χ S − χ (cid:48) ) . (2.7)Equation (2.6) describes the convergence due to a single source at a comoving distance χ S = χ ( z S ) from the observer. The convergence of a population of sources with redshiftdistribution n ( z )d z is found by averaging over the sources with n ( z ) as the weighting factor.One then finds κ (1) ( (cid:126)θ ) = (cid:90) ∞ d χ ( z ) p ( z ) d z d χ κ (1) ( (cid:126)θ, χ ( z )) = (cid:90) ∞ d χW κ ( χ )Φ ,ii ( d A ( χ ) (cid:126)θ, χ ) , (2.8)with the kernel given by W κ ( χ ) = (cid:90) ∞ χ d χ (cid:48) p ( z ) d z d χ (cid:48) K ( χ (cid:48) , χ ) . (2.9)The tSZ effect involves the inverse Compton scattering of CMB photons off relativisticelectrons [27]. This introduces a frequency dependent temperature shift ∆ T in the observedCMB temperature. The temperature shift at position (cid:126)θ on the sky and frequency ν can beparameterized as ∆ TT ( (cid:126)θ, ν ) = y ( (cid:126)θ ) S SZ ( ν ) , (2.10)where the Compton y ( (cid:126)θ ) parameter encodes the spatial and S SZ ( ν ) the spectral dependence.The Compton y parameter is defined as the line-of-sight integral over the electron pressure.In this work we adapt the constant bias model of [7] to simplify the analysis. It has beenshown in [8] that the constant bias model is consistent with a halo model approach, thusjustifying the use of the simpler model. For a constant bias, the y parameter can be writtenas an integral over the density contrast δ , i.e., y ( (cid:126)θ ) = (cid:90) χ H d χ ˜ W y ( χ ) δ ( (cid:126)x ( (cid:126)θ, χ ) , χ ) , (2.11)where χ H is the comoving distance to the surface of last scattering. We express the densitycontrast in terms of the Newtonian potential through the Poisson equation as y ( (cid:126)θ ) = (cid:90) χ H d χ ˜ W y ( χ ) 2 a ∆Φ( (cid:126)x ( (cid:126)θ, χ ) , χ )3 H Ω m = (cid:90) χ H d χW y ( χ )∆Φ( (cid:126)x ( (cid:126)θ, χ ) , χ ) , (2.12)where we have absorbed the factors from the Poisson equation in the new kernel W y ( χ ).Ultimately, we are interested in the angular cross power spectrum C yκ(cid:96) . In this workwe assume that the convergence is derived from shear measurements of galaxy surveys. Thesky coverage of these surveys is still relatively small, although this will not be the case offuture surveys such as Euclid and LSST, allowing the analysis to proceed in the flat-skyapproximation. Using the definition of the 2d Fourier transform (A.2), we can write theangular cross power spectrum of ˆ y ( (cid:126)(cid:96) ) and ˆ κ ( (cid:126)(cid:96) ) as (cid:104) ˆ y (1) ( (cid:126)(cid:96) )ˆ κ (1) ( (cid:126)(cid:96) ) (cid:105) = (2 π ) δ ( (cid:126)(cid:96) + (cid:126)(cid:96) ) C (2) (cid:96) = (cid:90) χ H d χ d χ (cid:48) W y ( χ ) W κ ( χ (cid:48) ) | (cid:126)(cid:96) | d A ( χ ) | (cid:126)(cid:96) | d A ( χ (cid:48) ) (cid:104) ˆ φ ( (cid:126)(cid:96) , χ ) ˆ φ ( (cid:126)(cid:96) , χ (cid:48) ) (cid:105) , (2.13)– 3 –here we have dropped the contributions of the derivatives along the line-of-sight in (2.12).Under the Limber approximation [28, 29] one assumes that the transverse modes are muchlarge than the longitudinal modes, i.e., (cid:96)d A ( χ ) (cid:29) k . This ceases to be true on large scales,where extensions to the Limber approximation such as [30] or an exact full-sky treatment haveto be employed. For the scales of interest in this work the Limber approximation is sufficientthough. In fact, it has been shown in [22] that the lowest order Limber approximation is anexcellent fit down to multipoles of (cid:96) ∼
20. Expressing the two-point function in (2.13) interms of the power spectrum (A.5) we find for the y - κ cross power spectrum (cid:104) ˆ y (1) ( (cid:126)(cid:96) )ˆ κ (1) ( (cid:126)(cid:96) ) (cid:105) = (2 π ) δ (cid:16) (cid:126)(cid:96) + (cid:126)(cid:96) (cid:17) (cid:90) χ H d χW y ( χ ) W κ ( χ ) | (cid:126)(cid:96) | d A ( χ ) P Φ (cid:32) | (cid:126)(cid:96) | d A ( χ ) , χ (cid:33) . (2.14)Note that upon replacing the kernel for the y parameter W y with W κ , one recovers the wellknown expression for the angular power spectrum of the convergence. To consistently treat fourth order corrections to the cross spectrum we need to include termsup to third order in Φ of the Compton y parameter and convergence κ [13]. Indeed, it has beenshown in [19] that divergences in second-second order cross terms cancel with correspondingdivergences in first-third order cross terms. It is thus important to find expressions for thetwo fields y and κ up to third order. A full sky treatment of lensing observables to evensecond order is already a formidable task [20–22]; a full sky derivation to third order wouldbe beyond the scope of this work. Fortunately, the calculations can be simplified greatlyby restricting ourselves to small scales. We follow [15] to identify the terms that contributedominantly at small scales and those that can be neglected.Broadly speaking, on small scales terms with the most angular derivatives are expectedto dominate. At second order this are the well known Born approximation, lens-lens coupling,and reduced shear contributions [12, 13, 15, 16, 18, 19, 31]. Third order terms derived fromthe aforementioned have at least the same number of angular derivatives and are thereforeexpected to be the dominant third order contributions. We discuss these contributions insection 3.1 and section 3.2.Recent work by [23] found that contributions from peculiar velocities to the convergencecan be as large as the primary contribution from scalar modes in certain redshift ranges.Even though peculiar velocities formally affect the convergence at first order, they affectthe shear only at second order [32]. In the case where the convergence is derived from shearmeasurements, as we assume in this work, the effect of peculiar velocities enters only at secondorder. We investigate the effect of peculiar velocities in section 3.3. Vector modes inducedby second order perturbations have been shown to yield corrections of similar magnitude astraditional Born and lens-lens terms [23]. We show that vector modes do not contribute tothe y - κ cross spectrum at fourth order in section 3.4.To distinguish the different corrections to the convergence we denote them by subscripts; κ std refers to corrections due to Born approximation and lens-lens coupling, κ rs to correctionsdue to the reduced shear, κ z , and κ v to corrections due to redshift distortions and vectormodes, respectively. – 4 – .1 Born approximation and lens-lens coupling For the derivation of the Born and lens-lens coupling terms we roughly follow [19], in thatwe expand the solution to the geodesic deviation equation (2.2) systematically in powers ofΦ. Alternatively, one could expand the terms in the distortion matrix (2.3), which makes thephysical meaning of the terms more apparent.We expand the comoving transverse displacement in powers of the potential Φ as (cid:126)x = (cid:126)x (0) + (cid:126)x (1) + (cid:126)x (2) + (cid:126)x (3) + O (Φ ) , (3.1)where the superscript in parentheses denotes the order of the expansion. The zeroth and firstorder contributions are given by (cid:126)x (0) = d A ( χ ) (cid:126)θ, x (1) i ( (cid:126)θ, χ ) = − (cid:90) χ d χ (cid:48) d A ( χ − χ (cid:48) )Φ ,i ( d A ( χ (cid:48) ) (cid:126)θ, χ (cid:48) ) . (3.2)The higher order contributions can be found by Taylor expanding Φ( (cid:126)x ) in (2.2) around thezeroth order solution (cid:126)x (0) ( (cid:126)θ, χ ) = d A ( χ ) (cid:126)θ . The potential can then be expanded as Φ =Φ (1) + Φ (2) + Φ (3) + O (Φ ), withΦ (1) ( (cid:126)x ) = Φ( (cid:126)x (0) )Φ (2) ( (cid:126)x ) = Φ ,i ( (cid:126)x (0) ) x (1) i Φ (3) ( (cid:126)x ) = 12 Φ ,ij ( (cid:126)x (0) ) x (1) i x (1) j + Φ ,i ( (cid:126)x (0) ) x (2) i . (3.3)By replacing the Φ with Φ (2) in (2.2), we can write the second order deflection angle as x (2) i ( (cid:126)θ, χ ) d A ( χ ) = − (cid:90) χ d χ (cid:48) d A ( χ − χ (cid:48) ) d A ( χ ) Φ (2) ,i ( (cid:126)x ( (cid:126)θ, χ (cid:48) ) , χ (cid:48) )= 4 (cid:90) χ d χ (cid:48) (cid:90) χ (cid:48) d χ (cid:48)(cid:48) K ( χ, χ (cid:48) ) K ( χ (cid:48) , χ (cid:48)(cid:48) ) d A ( χ (cid:48)(cid:48) ) Φ ,ij ( χ (cid:48) )Φ ,j ( χ (cid:48)(cid:48) ) , (3.4)where we have dropped the angular dependence of the potentials for brevity. We adapt thisshorthand for the rest of this work, i.e., unless otherwise noted Φ( (cid:126)x (0) ( (cid:126)θ, χ ) , χ ) is written asΦ( χ ). Analogously, the third order deflection angle can be written as x (3) i ( (cid:126)θ, χ ) d A ( χ ) = − (cid:90) χ d χ (cid:48) (cid:90) χ (cid:48) d χ (cid:48)(cid:48) (cid:90) χ (cid:48) d χ (cid:48)(cid:48)(cid:48) K ( χ, χ (cid:48) ) K ( χ (cid:48) , χ (cid:48)(cid:48) ) K ( χ (cid:48) , χ (cid:48)(cid:48)(cid:48) ) d A ( χ (cid:48) ) d A ( χ (cid:48)(cid:48) ) d A ( χ (cid:48)(cid:48)(cid:48) ) × Φ ,ijk ( χ (cid:48) )Φ ,j ( χ (cid:48)(cid:48) )Φ ,k ( χ (cid:48)(cid:48)(cid:48) ) − (cid:90) χ d χ (cid:48) (cid:90) χ (cid:48) d χ (cid:48)(cid:48) (cid:90) χ (cid:48)(cid:48) d χ (cid:48)(cid:48)(cid:48) K ( χ, χ (cid:48) ) K ( χ (cid:48) , χ (cid:48)(cid:48) ) K ( χ (cid:48)(cid:48) , χ (cid:48)(cid:48)(cid:48) ) d A ( χ (cid:48)(cid:48)(cid:48) ) × Φ ,ij ( χ (cid:48) )Φ ,jk ( χ (cid:48)(cid:48) )Φ ,k ( χ (cid:48)(cid:48)(cid:48) ) . (3.5) Equipped with second and third order expressions for the deflection angle it is straightforwardto derive expressions for the convergence. Using the relation of the convergence to the trace of– 5 –he distortion tensor (2.5), we can readily write down the second and third order expressionsfor the convergence. At second order this is κ (2) std ( (cid:126)θ, χ S ) = − (cid:90) χ S d χ (cid:48) (cid:90) χ (cid:48) d χ (cid:48)(cid:48) K ( χ, χ (cid:48) ) K ( χ (cid:48) , χ (cid:48)(cid:48) ) × (cid:18) d A ( χ (cid:48) ) d A ( χ (cid:48)(cid:48) ) Φ ,iij ( χ (cid:48) )Φ ,j ( χ (cid:48)(cid:48) ) + Φ ,ij ( χ (cid:48) )Φ ,ji ( χ (cid:48)(cid:48) ) (cid:19) . (3.6)The first term in the bracket is the well known Born term, while the second is the lens-lens coupling term. The extra factors of the comoving angular distance d A ( χ ) arise becausethe derivative in (2.3) is with respect to the angular deviation (cid:126)θ , whereas the potential is afunction of the comoving transverse distance (cid:126)x (0) = d A ( χ ) (cid:126)θ .The third order expression for the convergence is analogously found to be κ (3) std ( (cid:126)θ, χ S ) = 2 (cid:90) χ S d χ (cid:48) (cid:90) χ (cid:48) d χ (cid:48)(cid:48) (cid:90) χ (cid:48) d χ (cid:48)(cid:48)(cid:48) K ( χ, χ (cid:48) ) K ( χ (cid:48) , χ (cid:48)(cid:48) ) K ( χ (cid:48) , χ (cid:48)(cid:48)(cid:48) ) × (cid:18) d A ( χ (cid:48) ) d A ( χ (cid:48)(cid:48) ) d A ( χ (cid:48)(cid:48)(cid:48) ) Φ ,iijk ( χ (cid:48) )Φ ,j ( χ (cid:48)(cid:48) )Φ ,k ( χ (cid:48)(cid:48)(cid:48) ) (cid:19) + 4 (cid:90) χ S d χ (cid:48) (cid:90) χ (cid:48) d χ (cid:48)(cid:48) (cid:90) χ (cid:48) d χ (cid:48)(cid:48)(cid:48) K ( χ, χ (cid:48) ) K ( χ (cid:48) , χ (cid:48)(cid:48) ) K ( χ (cid:48) , χ (cid:48)(cid:48)(cid:48) ) × (cid:18) d A ( χ (cid:48) ) d A ( χ (cid:48)(cid:48)(cid:48) ) Φ ,ijk ( χ (cid:48) )Φ ,ji ( χ (cid:48)(cid:48) )Φ ,k ( χ (cid:48)(cid:48)(cid:48) ) (cid:19) + 4 (cid:90) χ S d χ (cid:48) (cid:90) χ (cid:48) d χ (cid:48)(cid:48) (cid:90) χ (cid:48)(cid:48) d χ (cid:48)(cid:48)(cid:48) K ( χ, χ (cid:48) ) K ( χ (cid:48) , χ (cid:48)(cid:48) ) K ( χ (cid:48)(cid:48) , χ (cid:48)(cid:48)(cid:48) ) d A ( χ (cid:48)(cid:48)(cid:48) ) × ∂∂θ i (cid:0) Φ ,ij ( χ (cid:48) )Φ ,jk ( χ (cid:48)(cid:48) )Φ ,k ( χ (cid:48)(cid:48)(cid:48) ) (cid:1) . (3.7)The term on line 2 corresponds a second order Born correction, the term on line 4 to amixed Born-lens-coupling, and the three terms on line 6 to a second order Born correction,Born-lens-coupling, second order lens-lens coupling, respectively. y parameter The second and third order contributions to the Compton y parameter are somewhat easierto derive, as there are no lens-lens coupling terms. As in the case of the convergence, wereplace Φ in (2.12) by its expansion (3.3). The second order contribution to the y parameteris then y (2) ( (cid:126)θ ) = − (cid:90) χ H d χ (cid:48) (cid:90) χ (cid:48) d χ (cid:48)(cid:48) W y ( χ (cid:48) ) K ( χ (cid:48) , χ (cid:48)(cid:48) ) d A ( χ (cid:48) ) d A ( χ (cid:48)(cid:48) ) Φ ,iij ( χ (cid:48) )Φ ,j ( χ (cid:48)(cid:48) ) . (3.8)The third order term follows analogously and is given by y (3) ( (cid:126)θ ) = 2 (cid:90) χ H d χ (cid:48) (cid:90) χ (cid:48) d χ (cid:48)(cid:48) (cid:90) χ (cid:48) d χ (cid:48)(cid:48)(cid:48) W y ( χ (cid:48) ) K ( χ (cid:48) , χ (cid:48)(cid:48) ) K ( χ (cid:48) , χ (cid:48)(cid:48)(cid:48) ) d A ( χ (cid:48) ) d A ( χ (cid:48)(cid:48) ) d A ( χ (cid:48)(cid:48)(cid:48) ) × Φ ,iijk ( χ (cid:48) )Φ ,j ( χ (cid:48)(cid:48) )Φ ,k ( χ (cid:48)(cid:48)(cid:48) )+ 4 (cid:90) χ H d χ (cid:48) (cid:90) χ (cid:48) d χ (cid:48)(cid:48) (cid:90) χ (cid:48)(cid:48) d χ (cid:48)(cid:48)(cid:48) W y ( χ (cid:48) ) K ( χ (cid:48) , χ (cid:48)(cid:48) ) K ( χ (cid:48)(cid:48) , χ (cid:48)(cid:48)(cid:48) ) d A ( χ (cid:48) ) d A ( χ (cid:48)(cid:48)(cid:48) ) × Φ ,iij ( χ (cid:48) )Φ ,jk ( χ (cid:48)(cid:48) )Φ ,k ( χ (cid:48)(cid:48)(cid:48) ) . (3.9)– 6 –oth terms are due to the Born approximation. The term on the second line stems from the Φ ,ij ( (cid:126)x (0) ) x (1) i x (1) j term in the third order contribution to Φ in (3.3), whereas the term online 4 in (3.9) is due to the Φ ,i ( (cid:126)x (0) ) x (2) i term in (3.3). The second-second order contribution to the angular y - κ cross power spectrum due to Bornand lens-lens terms can be derived by taking the ensemble average of the product of theFourier space expressions ˆ y (2) ( (cid:126)(cid:96) ) and ˆ κ (2) ( (cid:126)(cid:96) ). Using the results from appendix A, we find (cid:104) ˆ y (2) ( (cid:126)(cid:96) )ˆ κ (2) std ( (cid:126)(cid:96) ) (cid:105) = 4 (cid:90) χ H d χ y d χ κ (cid:90) χ y d χ (cid:48) y (cid:90) χ κ d χ (cid:48) κ W y ( χ y ) W κ ( χ κ ) K ( χ y , χ (cid:48) y ) K ( χ κ , χ (cid:48) κ ) d A ( χ y ) d A ( χ (cid:48) y ) d A ( χ κ ) d A ( χ (cid:48) κ ) × (cid:90) d (cid:126)(cid:96) (cid:48) d (cid:126)(cid:96) (cid:48)(cid:48) (2 π ) | (cid:126)(cid:96) (cid:48) | (cid:126)(cid:96) (cid:48) ( (cid:126)(cid:96) − (cid:126)(cid:96) (cid:48) ) (cid:20) | (cid:126)(cid:96) (cid:48)(cid:48) | (cid:126)(cid:96) (cid:48)(cid:48) ( (cid:126)(cid:96) − (cid:126)(cid:96) (cid:48)(cid:48) ) + (cid:16) (cid:126)(cid:96) (cid:48)(cid:48) ( (cid:126)(cid:96) − (cid:126)(cid:96) (cid:48)(cid:48) ) (cid:17) (cid:21) × (cid:104) ˆ φ ( (cid:126)(cid:96) (cid:48) , χ y ) ˆ φ ( (cid:126)(cid:96) − (cid:126)(cid:96) (cid:48) , χ (cid:48) y ) ˆ φ ( (cid:126)(cid:96) (cid:48)(cid:48) , χ κ ) ˆ φ ( (cid:126)(cid:96) − (cid:126)(cid:96) (cid:48)(cid:48) , χ (cid:48) κ ) (cid:105) , (3.10)where we used the kernel W κ for a source distribution n ( z ) instead of a single source atredshift z S . The four-point function on the last line is made up of one connected and threeunconnected terms. The connected term is proportional to the trispectrum (A.7). Underthe Limber approximation this introduces a product of delta functions δ D ( χ y − χ (cid:48) y ) δ D ( χ y − χ κ ) δ D ( χ y − χ (cid:48) κ ), setting all comoving distances along the line-of-sight equal. The kernel K ( χ, χ (cid:48) ) is zero for χ ≤ χ (cid:48) , thus eliminating the contribution from the connected part ofthe correlation function. The unconnected part can be decomposed into three productsof two-point functions by Wick’s theorem. Each of the two-point functions yields a deltafunction times a power spectrum. The term proportional to δ D ( χ y − χ (cid:48) y ) δ D ( χ κ − χ (cid:48) κ ) is zerobecause K ( χ, χ ) = 0. The term proportional to δ D ( χ y − χ (cid:48) κ ) δ D ( χ (cid:48) y − χ κ ) is zero because K ( χ, χ (cid:48) ) K ( χ (cid:48) , χ ) ≡
0. The only surviving term is proportional to δ D ( χ y − χ κ ) δ D ( χ (cid:48) y − χ (cid:48) κ ),and upon evaluating the integrals gives for the angular cross power spectrum (cid:104) ˆ y (2) ( (cid:126)(cid:96) )ˆ κ (2) std ( (cid:126)(cid:96) ) (cid:105) = 4(2 π ) δ ( (cid:126)(cid:96) + (cid:126)(cid:96) ) (cid:90) χ H d χ (cid:90) χ d χ (cid:48) W y ( χ ) W κ ( χ ) K ( χ, χ (cid:48) ) d A ( χ ) d A ( χ (cid:48) ) (cid:90) d (cid:126)(cid:96) (cid:48) (2 π ) × | (cid:126)(cid:96) (cid:48) | (cid:126)(cid:96) (cid:126)(cid:96) (cid:48) (cid:16) (cid:126)(cid:96) (cid:48) ( (cid:126)(cid:96) − (cid:126)(cid:96) (cid:48) ) (cid:17) P Φ (cid:32) | (cid:126)(cid:96) (cid:48) | d A ( χ ) , χ (cid:33) P Φ (cid:32) | (cid:126)(cid:96) − (cid:126)(cid:96) (cid:48) | d A ( χ (cid:48) ) , χ (cid:48) (cid:33) . (3.11)The derivation for the first-third order contributions proceeds similarly. The connected corre-lation function drops out for the same reason as in the second-second order case. Furthermore,terms in the third order expressions for y and κ that include a line-of-sight kernel propor-tional to K ( χ (cid:48) , χ (cid:48)(cid:48) ) K ( χ (cid:48)(cid:48) , χ (cid:48)(cid:48)(cid:48) ), i.e., line 5 in (3.7) and line 3 in (3.9), do not contribute tothe power spectra because the kernel is zero for all possible contractions of the correlationfunction.The contribution from the second term in (3.7), i.e., line 4, to the cross power spectrumis proportional to (cid:90) d (cid:126)(cid:96) (cid:48) (2 π ) (cid:16) (cid:126)(cid:96) (cid:126)(cid:96) (cid:48) (cid:17) P Φ (cid:32) | (cid:126)(cid:96) | d A ( χ ) , χ (cid:33) P Φ (cid:32) | (cid:126)(cid:96) (cid:48) | d A ( χ (cid:48) ) , χ (cid:48) (cid:33) , – 7 –hich is zero due to the antisymmetry of the integral under the transformation (cid:126)(cid:96) (cid:48) → − (cid:126)(cid:96) (cid:48) [19].Hence, only the first Born term in (3.7) contributes to (cid:104) ˆ y (1) ˆ κ (3) (cid:105) . We find (cid:104) ˆ y (1) ( (cid:126)(cid:96) )ˆ κ (3) std ( (cid:126)(cid:96) ) (cid:105) = − π ) δ ( (cid:126)(cid:96) + (cid:126)(cid:96) ) (cid:90) χ H d χ (cid:90) χ d χ (cid:48) W y ( χ ) W κ ( χ ) K ( χ, χ (cid:48) ) d A ( χ ) d A ( χ (cid:48) ) × (cid:90) d (cid:126)(cid:96) (cid:48) (2 π ) | (cid:126)(cid:96) | (cid:16) (cid:126)(cid:96) (cid:126)(cid:96) (cid:48) (cid:17) P Φ (cid:32) | (cid:126)(cid:96) | d A ( χ ) , χ (cid:33) P Φ (cid:32) | (cid:126)(cid:96) (cid:48) | d A ( χ (cid:48) ) , χ (cid:48) (cid:33) . (3.12)Since the only contribution to (cid:104) ˆ y (3) ˆ κ (1) (cid:105) comes from the first term in (3.9), which is identicalto the first term in (3.7) up to an interchange of the kernels W y ( χ ) and W κ ( χ ), the crosspower spectra (cid:104) ˆ y (3) ˆ κ (1) (cid:105) and (cid:104) ˆ y (1) ˆ κ (3) (cid:105) are identical. At first order the shear and convergence are related byˆ κ ( (cid:126)(cid:96) ) = T I ( (cid:126)(cid:96) )ˆ γ I ( (cid:126)(cid:96) ) , T ( (cid:126)(cid:96) ) = cos 2 φ (cid:96) , T ( (cid:126)(cid:96) ) = sin 2 φ (cid:96) , (3.13)where φ (cid:96) is the angle between the two-dimensional wave-vector (cid:126)(cid:96) and some fixed referenceaxis. The components of the shear and other polar quantities are labeled by capital Latinindices. It can be shown that this relation holds exactly up to second order and under theLimber approximation up to third order; see appendix B for details. In the weak lensingregime, the measured quantity is not the shear itself but the reduced shear, conventionallydefined as g I = γ I − κ , I = 1 , . (3.14)The definition (3.14) of the reduced shear is based on the assumption that the Jacobi map(2.3) is symmetric. In general the Jacobi map is not symmetric however, because lens-lenscouplings generate the anti-symmetric contribution ω . Including the anti-symmetric terms inthe Jacobi map, the generalized reduced shear in complex notation is given by (see appendixC) g = γ + ıγ − κ + ıω . (3.15)Accounting for the reduced shear in the relation (3.13) amounts to replacing the shear γ I with the reduced shear g I . Using (C.8) and expanding systematically in Φ to third order wecan express the observed convergence asˆ κ obs = T I ˆ g I = T I (cid:16) ˆ γ (1) I + (ˆ γ (2) std ) I + (ˆ γ (3) std ) I + ˆ γ (1) I ∗ ˆ κ (1) + ˆ γ (1) I ∗ ˆ κ (1) ∗ ˆ κ (1) + ˆ γ (1) I ∗ ˆ κ (2) std + (ˆ γ (2) std ) I ∗ ˆ κ (1) + R (ˆ ω (2) std ) IJ ∗ ˆ γ (1) J (cid:17) + O (Φ ) , (3.16)where ∗ stands for a convolution in Fourier space. As shown in appendix B, the first lineis equivalent to ˆ κ (1) + ˆ κ (2) std + ˆ κ (3) std , where ˆ κ (2) std and ˆ κ (3) std denote the corrections due to Bornapproximation and lens-lens coupling. The second line includes the well known contributionsfrom the reduced shear [12, 16, 18, 19], while the third line is a novel contribution due to– 8 –econd order induced rotations. The sole second order correction due to reduced shear to theconvergence is ˆ κ (2) rs ( (cid:126)(cid:96) ) = [ T I ( (cid:126)(cid:96) )ˆ γ (1) I ∗ ˆ κ (1) ]( (cid:126)(cid:96) ) = (cid:90) d (cid:126)(cid:96) (cid:48) (2 π ) T I ( (cid:126)(cid:96) )ˆ γ (1) I ( (cid:126)(cid:96) (cid:48) )ˆ κ (1) ( (cid:126)(cid:96) − (cid:126)(cid:96) (cid:48) )= (cid:90) d (cid:126)(cid:96) (cid:48) (2 π ) cos(2 φ (cid:96) (cid:48) − φ (cid:96) )ˆ κ (1) ( (cid:126)(cid:96) (cid:48) )ˆ κ (1) ( (cid:126)(cid:96) − (cid:126)(cid:96) (cid:48) ) , (3.17)where we used the identity T I ( (cid:126)(cid:96) ) T I ( (cid:126)(cid:96) (cid:48) ) = cos(2 φ (cid:96) (cid:48) − φ (cid:96) ). Since the reduced shear is an intrin-sic lensing effect, it does not affect the Compton y parameter. The lowest order contributionto the cross power spectrum is therefore formed by the first order y parameter (2.12) andsecond order reduced shear correction (3.17), i.e., (cid:104) ˆ y (1) ( (cid:126)(cid:96) )ˆ κ (2) rs ( (cid:126)(cid:96) ) (cid:105) = − (2 π ) δ ( (cid:126)(cid:96) + (cid:126)(cid:96) ) (cid:90) d χ W y ( χ ) ( W κ ( χ )) d A ( χ ) (cid:90) d (cid:126)(cid:96) (cid:48) (2 π ) × | (cid:126)(cid:96) | | (cid:126)(cid:96) (cid:48) | | (cid:126)(cid:96) − (cid:126)(cid:96) (cid:48) | cos(2 φ (cid:96) (cid:48) − φ (cid:96) ) B Φ (cid:32) | (cid:126)(cid:96) | d A ( χ ) , | (cid:126)(cid:96) (cid:48) | d A ( χ ) , | (cid:126)(cid:96) − (cid:126)(cid:96) (cid:48) | d A ( χ ) (cid:33) , (3.18)where we used the definition (A.6) of the bispectrum. Unlike in the case of the Born and lens-lens terms, there is a third order contribution to the cross power spectrum. As a consistencycheck, one can show that upon replacing W y by W κ , and using the fact that to first orderthe convergence is the same as the E-mode of the shear, one recovers the expression for thecorrection to the E-mode shear due to reduced shear in [16].To analyze the first-third order contributions, we split the third order contribution tothe convergence due to the reduced shear in (3.16) into three componentsˆ κ (3 ,A ) rs ( (cid:126)(cid:96) ) = T I ( (cid:126)(cid:96) )[ˆ γ (1) I ∗ ˆ κ (1) ∗ ˆ κ (1) ]( (cid:126)(cid:96) ) , (3.19a)ˆ κ (3 ,B ) rs ( (cid:126)(cid:96) ) = T I ( (cid:126)(cid:96) )[ˆ γ (1) I ∗ ˆ κ (2) std + (ˆ γ (2) std ) I ∗ ˆ κ (1) ]( (cid:126)(cid:96) ) , (3.19b)ˆ κ (3 ,C ) rs ( (cid:126)(cid:96) ) = T I ( (cid:126)(cid:96) )[ R (ˆ ω (2) std ) IJ ∗ ˆ γ (1) J ]( (cid:126)(cid:96) ) . (3.19c)The cross power spectrum of ˆ κ (3 ,A ) rs with y is then (cid:104) ˆ y (1) ( (cid:126)(cid:96) )ˆ κ (3 ,A ) rs ( (cid:126)(cid:96) ) (cid:105) = (cid:90) d χ y W y ( χ y ) d A ( χ y ) (cid:89) i =1 d χ i W κ ( χ i ) d A ( χ i ) (cid:90) d (cid:126)(cid:96) (cid:48) d (cid:126)(cid:96) (cid:48)(cid:48) (2 π ) cos(2 φ (cid:96) − φ (cid:96) (cid:48) ) × | (cid:126)(cid:96) | | (cid:126)(cid:96) (cid:48) | | (cid:126)(cid:96) (cid:48)(cid:48) | | (cid:126)(cid:96) − (cid:126)(cid:96) (cid:48) − (cid:126)(cid:96) (cid:48)(cid:48) | × (cid:104) ˆ φ ( (cid:126)(cid:96) , χ y ) ˆ φ ( (cid:126)(cid:96) (cid:48) , χ ) ˆ φ ( (cid:126)(cid:96) (cid:48)(cid:48) , χ ) ˆ φ ( (cid:126)(cid:96) − (cid:126)(cid:96) (cid:48) − (cid:126)(cid:96) (cid:48)(cid:48) , χ ) (cid:105) . (3.20)Because the line-of-sight integral does not include the kernel K ( χ, χ (cid:48) ), like in the case of thethird order cross power spectrum (3.18), the connected part of the four-point function doesnot vanish. The connected and unconnected contributions to the cross power spectrum are– 9 –ound to be (cid:104) ˆ y (1) ( (cid:126)(cid:96) )ˆ κ (3 ,A ) rs ( (cid:126)(cid:96) ) (cid:105) c = (2 π ) δ (cid:16) (cid:126)(cid:96) + (cid:126)(cid:96) (cid:17) (cid:90) d χ W y ( χ ) W κ ( χ ) d A ( χ ) (cid:90) d (cid:126)(cid:96) (cid:48) d (cid:126)(cid:96) (cid:48)(cid:48) (2 π ) cos(2 φ (cid:96) − φ (cid:96) (cid:48) ) × | (cid:126)(cid:96) | | (cid:126)(cid:96) (cid:48) | | (cid:126)(cid:96) (cid:48)(cid:48) | | (cid:126)(cid:96) + (cid:126)(cid:96) (cid:48) + (cid:126)(cid:96) (cid:48)(cid:48) | T Φ (cid:32) (cid:126)(cid:96) d A ( χ ) , (cid:126)(cid:96) (cid:48) d A ( χ ) , (cid:126)(cid:96) (cid:48)(cid:48) d A ( χ ) , − (cid:126)(cid:96) + (cid:126)(cid:96) (cid:48) + (cid:126)(cid:96) (cid:48)(cid:48) d A ( χ ) , χ (cid:33) (3.21a) (cid:104) ˆ y (1) ( (cid:126)(cid:96) )ˆ κ (3 ,A ) rs ( (cid:126)(cid:96) ) (cid:105) g = (cid:104) ˆ y (1) ( (cid:126)(cid:96) )ˆ κ (1) ( (cid:126)(cid:96) ) (cid:105) (cid:90) d (cid:126)(cid:96) (cid:48) (2 π ) C κκ, (cid:96) (cid:48) , (3.21b)where the connected and unconnected parts are denoted by the subscript c and g, respectively.For the derivation of the connected part we have used the definition of the trispectrum A.7.Replacing the kernel W y by W κ we recover again the same expression as found in [19].The two other third order contributions to the convergence due the reduced shear (3.19b)and (3.19c) involve second order Born and lens-lens corrections, i.e., include the couplingkernel K ( χ, χ (cid:48) ) in their line-of-sight integrals. Hence, only their unconnected parts contributeto the cross power spectrum. The derivation proceeds as for the other terms discussed sofar, albeit with somewhat more complicated expressions, as there are now two contractionsof the four-point function that survive. The contribution involving ˆ κ (3 ,B ) rs is (cid:104) ˆ y (1) ( (cid:126)(cid:96) )ˆ κ (3 ,B ) rs ( (cid:126)(cid:96) ) (cid:105) = − π ) δ (cid:16) (cid:126)(cid:96) + (cid:126)(cid:96) (cid:17) (cid:90) d χ d χ (cid:48) W y ( χ ) W κ ( χ (cid:48) ) d A ( χ ) d A ( χ (cid:48) ) (cid:90) d (cid:126)(cid:96) (cid:48) (2 π ) × | (cid:126)(cid:96) | | (cid:126)(cid:96) (cid:48) | (cid:126)(cid:96) (cid:126)(cid:96) (cid:48) P Φ (cid:32) | (cid:126)(cid:96) | d A ( χ ) , χ (cid:33) P Φ (cid:32) | (cid:126)(cid:96) (cid:48) | d A ( χ (cid:48) ) , χ (cid:48) (cid:33) × (cid:110) cos(2 φ (cid:96) − φ (cid:96) (cid:48) ) (cid:126)(cid:96) (cid:126)(cid:96) (cid:48) (cid:2) W κ ( χ ) K ( χ, χ (cid:48) ) + W κ ( χ (cid:48) ) K ( χ (cid:48) , χ ) (cid:3) cos(2 φ (cid:96) − φ (cid:126)(cid:96) (cid:48) + (cid:126)(cid:96) ) (cid:104) (cid:126)(cid:96) ( (cid:126)(cid:96) (cid:48) + (cid:126)(cid:96) ) W κ ( χ ) K ( χ, χ (cid:48) ) + (cid:126)(cid:96) (cid:48) ( (cid:126)(cid:96) (cid:48) + (cid:126)(cid:96) ) W κ ( χ (cid:48) ) K ( χ (cid:48) , χ ) (cid:105)(cid:111) . (3.22)The azimuthal integral of the third line can be done analytically and is equal to π . Notethat our result differs from that obtained in [19] by an extra factor of cos(2 φ (cid:96) − φ (cid:126)(cid:96) (cid:48) + (cid:126)(cid:96) ) onfourth line.Using the definition of the matrix R ( ω ) in (C.9), the contribution ˆ κ (3 ,C ) rs can be writtenas ˆ κ (3 ,C ) rs ( (cid:126)(cid:96) ) = T ( (cid:126)(cid:96) )[ γ (1)2 ∗ ˆ ω (2) std ]( (cid:126)(cid:96) ) − T ( (cid:126)(cid:96) )[ γ (1)1 ∗ ˆ ω (2) std ]( (cid:126)(cid:96) )= (cid:90) d (cid:126)(cid:96) (cid:48) (2 π ) sin(2 φ (cid:96) (cid:48) − φ (cid:96) )ˆ κ (1) ( (cid:126)(cid:96) (cid:48) )ˆ ω (2) std ( (cid:126)(cid:96) − (cid:126)(cid:96) (cid:48) ) . (3.23)The cross power spectrum is therefore (cid:104) ˆ y (1) ( (cid:126)(cid:96) )ˆ κ (3 ,C ) rs ( (cid:126)(cid:96) ) (cid:105) = − π ) δ (cid:16) (cid:126)(cid:96) + (cid:126)(cid:96) (cid:17) (cid:90) d χ d χ (cid:48) W y ( χ ) W κ ( χ (cid:48) ) d A ( χ ) d A ( χ (cid:48) ) (cid:90) d (cid:126)(cid:96) (cid:48) (2 π ) × | (cid:126)(cid:96) | | (cid:126)(cid:96) (cid:48) | (cid:126)(cid:96) (cid:126)(cid:96) (cid:48) sin( φ (cid:96) − φ (cid:96) (cid:48) ) sin(2 φ (cid:96) (cid:48) − φ (cid:96) ) P Φ (cid:32) | (cid:126)(cid:96) | d A ( χ ) , χ (cid:33) P Φ (cid:32) | (cid:126)(cid:96) (cid:48) | d A ( χ (cid:48) ) , χ (cid:48) (cid:33) × (cid:2) W κ ( χ ) K ( χ, χ (cid:48) ) + W κ ( χ (cid:48) ) K ( χ (cid:48) , χ ) (cid:3) . (3.24)– 10 –he mode coupling term can be reduced to | (cid:126)(cid:96) | | (cid:126)(cid:96) (cid:48) | φ (cid:96) (cid:48) − φ (cid:96) ) . The azimuthal integralthen evaluates to π . The contributions from ˆ κ (3 ,C ) rs and from the first term in ˆ κ (3 ,B ) rs aretherefore identical.Finally, we find the only second-second order contribution due to the reduced shear tothe cross power spectrum to be (cid:104) ˆ y (2) ( (cid:126)(cid:96) )ˆ κ (2) rs ( (cid:126)(cid:96) ) (cid:105) = − π ) δ (cid:16) (cid:126)(cid:96) + (cid:126)(cid:96) (cid:17) (cid:90) d χ d χ (cid:48) W y ( χ ) K ( χ, χ (cid:48) ) W κ ( χ ) W κ ( χ (cid:48) ) d A ( χ ) d A ( χ (cid:48) ) (cid:90) d (cid:126)(cid:96) (cid:48) (2 π ) × | (cid:126)(cid:96) (cid:48) | | (cid:126)(cid:96) − (cid:126)(cid:96) (cid:48) | (cid:104) (cid:126)(cid:96) (cid:48) (cid:16) (cid:126)(cid:96) − (cid:126)(cid:96) (cid:48) (cid:17)(cid:105) P Φ (cid:32) | (cid:126)(cid:96) (cid:48) | d A ( χ ) , χ (cid:33) P Φ (cid:32) | (cid:126)(cid:96) − (cid:126)(cid:96) (cid:48) | d A ( χ (cid:48) ) , χ (cid:48) (cid:33) × (cid:104) cos(2 φ (cid:96) − φ (cid:126)(cid:96) (cid:48) ) + cos(2 φ (cid:96) − φ (cid:126)(cid:96) − (cid:126)(cid:96) (cid:48) ) (cid:105) . (3.25)The dominant contribution is the third order correction (3.18), by virtue of being of a lowerorder than the other contributions considered in this work, which are all of fourth order. The comoving line-of-sight distance to a source is usually not an observable quantity. Insteadit is derived from the measured redshift, which is affected by the peculiar motions of the sourceand observer, Sachs-Wolf, and integrated Sachs-Wolf effects. The second order contributionto the convergence due to a perturbation of the cosmological redshift is κ (2) z ( χ ) = d κ (1) ( χ )d z δz (1) = d κ (1) ( χ )d χ d χ d z δz (1) . (3.26)The dependence of the convergence on comoving distance of the source isd κ (1) ( χ )d χ = 1 d A ( χ ) (cid:90) χ d χ (cid:48) Φ ,ii ( χ (cid:48) ) d A ( χ (cid:48) ) , (3.27)while the redshift perturbation due to peculiar motion of the source, Sachs-Wolf, and inte-grated Sachs-Wolf effects is given by [20] δz (1) = 1 a (cid:18) − (cid:90) χ d χ (cid:48) ∂ Φ( χ (cid:48) ) ∂χ (cid:48) + Φ( χ ) − n i v (1) i ( χ ) (cid:19) , (3.28)where the potential at the observer and the peculiar motion of the observer have been set tozero, as they would only contribute at the very large scales. The peculiar motion from firstorder perturbation theory is [22] v (1) i ( χ ) = − a H Ω m ∂ i (cid:18) − ∂ Φ( χ (cid:48) ) ∂χ (cid:48) + H ( χ )Φ( χ ) (cid:19) . (3.29)From (3.28) and (3.29) we can already see that only the term corresponding to the peculiarmotion would contribute appreciably as it involves an angular derivative. Restricting our-selves to the contribution due to the peculiar motion, the second order convergence can bewritten as κ (2) z ( χ ) = − n i v (1) i ( χ ) H ( χ ) d A ( χ ) (cid:90) χ d χ (cid:48) Φ ,ii ( χ (cid:48) ) d A ( χ (cid:48) ) . (3.30)– 11 –he photon trajectory (cid:126)n projects the peculiar velocity along the line-of-sight, i.e., the angularderivatives are projected out. Thus all redshift distortions that contribute to (3.26) have onlytwo angular derivatives and can be safely neglected on small scales. In [23] it was shown that fourth order contributions from vector modes to lensing observablescan be of comparable magnitude as other fourth order contributions considered in this work.It would thus be conceivable that there are large third order contributions involving vectormodes. The lowest order cross correlation that includes vector modes is (cid:104) ˆ y (1) ( (cid:126)(cid:96) )ˆ κ (2) v ( (cid:126)(cid:96) ) (cid:105) ,where the second order contribution to the convergence is [23, 33] κ (2) v ( χ ) = (cid:90) χ d χ (cid:48) K ( χ, χ (cid:48) ) n j V j,ii ( χ (cid:48) ) . (3.31)The contraction of the line-of-sight direction (cid:126)n with the vector potential V i is proportionalto n i V i ( χ ) ∝ sin ϑ e ± ıϕ , (3.32)where ϑ and ϕ denote the spherical coordinates on the sky. This expression is manifestlyof odd parity and does not contribute if one correlates it with the even parity field y . Thelowest order vector contribution to the cross power spectrum has to be quadratic in thevector potential. Since the vector potential is already of second order in the scalar potentialΦ, and the lowest vector contribution to y is of third order, there are no fourth order vectorcontribution to the cross power spectrum. In figure 1 we have plotted the cross power spectrum (2.14) and the various higher ordercontributions considered in this work. The underlying non-linear matter power spectrumwas computed with CAMB , using the best fit Planck cosmological parameters [34]. Forthe source redshift distribution n ( z ) we use the fitting formula of the redshift distributionof CFHTLenS [35]. We computed the non-linear bispectrum using the fitting formulae ofboth [36] and [37]. It was found in [38] that the fitting formula [37] slightly overestimatesthe bispectrum on small scales compared to [36]. For clarity, we only show the reduced shearcontribution computed with the fitting formula [36] in figure 1. The relative contributionsto the cross power spectrum due to the third order term (3.18) with both fitting formulae isshown in figure 2. We find that the third order contribution (3.18) gives the largest correctionto the cross power spectrum. At multipoles of (cid:96) ∼ (cid:0) ( C yκ(cid:96) ) + C yy(cid:96) C κκ(cid:96) (cid:1) / (2 (cid:96) + 1) [39] and reaches multiple percents of the second orderresult (2.14) at multipoles of (cid:96) ∼ . The fourth order contributions are over an order ofmagnitude lower at small scales. Furthermore, the difference between the two fitting formulaefor the bispectrum [36] and [37] are at least an order of magnitude larger on small scales thanthe fourth order corrections. It is thus justified to approximate the fourth order contribution(3.20) by its unconnected part, as it is expected to dominate over the connected part at allbut the smallest scales [19]. http://camb.info – 12 – ‘ − − − − − − − − ‘ ( ‘ + ) C y κ ‘ / π C yκ‘ Cosmic varianceReduced shear O (Φ )Reduced shear O (Φ )Born apprioximation& lens-lens coupling Figure 1 . The different contributions to the angular cross power spectrum C yκ(cid:96) . The first order result(2.14)(bold blue), third order reduced shear (3.18)(red), fourth order Born and lens-lens terms (3.11)and (3.12)(cyan), and fourth order reduced shear contributions (3.21), (3.22), (3.24), (3.25) (green). ‘ δ C y κ ‘ / C y κ ‘ Reduced shear SC01Reduced shear GM12Cosmic variance
Figure 2 . The third order contribution (3.18) to the cross power spectrum computed using thefitting formulae for the bispectrum from [36] and [37]. The corrections begin to dominate over cosmicvariance above (cid:96) ∼ We have calculated all contributions up to fourth order due to weak lensing to the tSZ-lensingcross correlation in the small angle approximation. We have found that only the third orderterm 3.18 due to the reduced shear contributes appreciably. At multipoles of (cid:96) ∼ and Euclid , a full-sky treatment will be necessary. As is http://sci.esa.int/euclid/ – 13 –vident from the large amount of terms in even the second order shear in [20, 22], a derivationto the same order as considered in this work will be a formidable task.Even though the simple bias model employed in this work is compatible with a halomodel approach [8], a treatment of the corrections considered in this work in the context ofthe halo model would be of interest. It should be noted that even within the framework ofthe halo model there is still considerable uncertainty in the modelling of the pressure profile,exemplifying the complications one encounters once baryonic physics are introduced.Our work can be used to calculate high order lensing corrections to cross correlationsignals using a continuous map other than tSZ. For instance one can envision measuring thecross correlation between the Cosmic Infrared Background and lower redshift structures. Acknowledgments
We thank Alex Hall for helpful comments. TT is supported by the Natural Sciences andEngineering Research Council of Canada (NSERC). LVW is supported by NSERC and theCanadian Institute for Advanced Research (CIfAR).
A Fourier space identities
Adapting the notation of [40], we define the 2d Fourier transform on the plane perpendicularto the line-of-sight as Φ( d A ( χ ) (cid:126)θ, χ ) = (cid:90) d (cid:126)(cid:96) (cid:48) (2 π ) ˆ φ ( (cid:126)(cid:96) (cid:48) , χ )e ı(cid:126)(cid:96) (cid:48) (cid:126)θ , (A.1)with the angular transform of the field Φ given byˆ φ ( (cid:126)(cid:96), χ ) = (cid:90) d k π d A ( χ ) ˆΦ (cid:32) (cid:126)(cid:96)d A ( χ ) , k (cid:33) e ık χ . (A.2)The higher order expressions for y and κ involve products of the potential Φ. In Fourierspace, these products become convolutions. For two fields F and G we have (cid:91) [ F G ]( (cid:126)(cid:96) ) = (cid:104) ˆ F ∗ ˆ G (cid:105) ( (cid:126)(cid:96) ) = (cid:90) d (cid:126)(cid:96) (cid:48) (2 π ) ˆ F ( (cid:126)(cid:96) (cid:48) ) ˆ G ( (cid:126)(cid:96) − (cid:126)(cid:96) (cid:48) ) . (A.3)This generalizes straightforwardly to the case of three fields F,G, and K as (cid:92) [ F GK ]( (cid:126)(cid:96) ) = (cid:104) ˆ F ∗ ˆ G ∗ ˆ K (cid:105) ( (cid:126)(cid:96) ) = (cid:90) d (cid:126)(cid:96) (cid:48) d (cid:126)(cid:96) (cid:48)(cid:48) (2 π ) ˆ F ( (cid:126)(cid:96) (cid:48) ) ˆ G ( (cid:126)(cid:96) (cid:48)(cid:48) ) ˆ K ( (cid:126)(cid:96) − (cid:126)(cid:96) (cid:48) − (cid:126)(cid:96) (cid:48)(cid:48) ) . (A.4)The two point correlation function of the fields ˆ φ ( (cid:126)(cid:96), χ ) is directly related to the power spec-trum of the potential Φ. Assuming homogeneity, isotropy, and using the Limber approxima-tion [28, 29], i.e., assuming that | (cid:126)(cid:96) | (cid:29) k , thus justifying neglecting the longitudinal modes,the two point function can be written as (cid:104) ˆ φ ( (cid:126)(cid:96) , χ ) ˆ φ ( (cid:126)(cid:96) , χ ) (cid:105) = (2 π ) δ D ( χ − χ ) δ D ( (cid:126)(cid:96) + (cid:126)(cid:96) ) C φφ(cid:96) = (2 π ) δ D ( χ − χ ) δ (cid:16) (cid:126)(cid:96) + (cid:126)(cid:96) (cid:17) d A ( χ ) P Φ (cid:32) | (cid:126)(cid:96) | d A ( χ ) , χ (cid:33) . (A.5)– 14 –imilarly, the three-point function of ˆ φ ( (cid:126)(cid:96), χ ) is related to the bispectrum B Φ ( (cid:126)k , (cid:126)k , (cid:126)k , χ ) by (cid:104) ˆ φ ( (cid:126)(cid:96) , χ ) ˆ φ ( (cid:126)(cid:96) , χ ) ˆ φ ( (cid:126)(cid:96) , χ ) (cid:105) = (2 π ) δ D ( χ − χ ) δ D ( χ − χ ) δ (cid:16) (cid:126)(cid:96) + (cid:126)(cid:96) + (cid:126)(cid:96) (cid:17) d A ( χ ) × B Φ (cid:32) | (cid:126)(cid:96) | d A ( χ ) , | (cid:126)(cid:96) | d A ( χ ) , | (cid:126)(cid:96) | d A ( χ ) , χ (cid:33) . (A.6)The four-point function can be expressed in terms of the trispectrum as (cid:104) ˆ φ ( (cid:126)(cid:96) , χ ) ˆ φ ( (cid:126)(cid:96) , χ ) ˆ φ ( (cid:126)(cid:96) , χ ) ˆ φ ( (cid:126)(cid:96) , χ ) (cid:105) = (2 π ) δ D ( χ − χ ) δ D ( χ − χ ) δ D ( χ − χ ) × δ (cid:16) (cid:126)(cid:96) + (cid:126)(cid:96) + (cid:126)(cid:96) + (cid:126)(cid:96) (cid:17) d A ( χ ) × T Φ (cid:32) | (cid:126)(cid:96) | d A ( χ ) , | (cid:126)(cid:96) | d A ( χ ) , | (cid:126)(cid:96) | d A ( χ ) , | (cid:126)(cid:96) | d A ( χ ) , χ (cid:33) . (A.7)Using the definition of the Fourier transform (A.1), partial derivatives with respect to co-moving transverse coordinates can be written asΦ ,i ...i N ( d A ( χ ) (cid:126)θ, χ ) = (cid:90) d (cid:126)(cid:96) (cid:48) (2 π ) ı N d A ( χ ) N (cid:96) (cid:48) i . . . (cid:96) (cid:48) i N ˆ φ ( (cid:126)(cid:96) (cid:48) , χ )e ı(cid:126)(cid:96) (cid:48) (cid:126)θ . (A.8) B Convergence - shear relation
In this appendix we show that the relation (3.13) holds even beyond first order, justifying theuse of the convergence as the fundamental quantity instead of the shear. The second orderexpressions of the convergence and shear due to Born approximation and lens-lens couplingsin Fourier space areˆ κ (2) std ( (cid:126)(cid:96), χ ) = − (cid:90) χ d χ (cid:48) (cid:90) χ (cid:48) d χ (cid:48)(cid:48) K ( χ, χ (cid:48) ) K ( χ (cid:48) , χ (cid:48)(cid:48) ) d A ( χ (cid:48) ) d A ( χ (cid:48)(cid:48) ) × (cid:90) d (cid:126)(cid:96) (cid:48) (2 π ) | (cid:126)(cid:96) (cid:48) || (cid:126)(cid:96) | cos( φ (cid:96) − φ (cid:96) (cid:48) ) (cid:104) (cid:126)(cid:96) (cid:48) ( (cid:126)(cid:96) − (cid:126)(cid:96) (cid:48) ) (cid:105) ˆ φ ( (cid:126)(cid:96) (cid:48) , χ (cid:48) ) ˆ φ ( (cid:126)(cid:96) − (cid:126)(cid:96) (cid:48) , χ (cid:48)(cid:48) ) (B.1)and (ˆ γ (2) std ) I ( (cid:126)(cid:96), χ ) = − (cid:90) χ d χ (cid:48) (cid:90) χ (cid:48) d χ (cid:48)(cid:48) K ( χ, χ (cid:48) ) K ( χ (cid:48) , χ (cid:48)(cid:48) ) d A ( χ (cid:48) ) d A ( χ (cid:48)(cid:48) ) × (cid:90) d (cid:126)(cid:96) (cid:48) (2 π ) | (cid:126)(cid:96) (cid:48) || (cid:126)(cid:96) | U I ( (cid:126)(cid:96), (cid:126)(cid:96) (cid:48) ) (cid:104) (cid:126)(cid:96) (cid:48) ( (cid:126)(cid:96) − (cid:126)(cid:96) (cid:48) ) (cid:105) ˆ φ ( (cid:126)(cid:96) (cid:48) , χ (cid:48) ) ˆ φ ( (cid:126)(cid:96) − (cid:126)(cid:96) (cid:48) , χ (cid:48)(cid:48) ) , (B.2)where the couplings U I are given by U ( (cid:126)(cid:96), (cid:126)(cid:96) (cid:48) ) = cos( φ (cid:96) + φ (cid:96) (cid:48) ) , U ( (cid:126)(cid:96), (cid:126)(cid:96) (cid:48) ) = sin( φ (cid:96) + φ (cid:96) (cid:48) ) . (B.3)Using the identity T I ( (cid:126)(cid:96) ) U I ( (cid:126)(cid:96), (cid:126)(cid:96) (cid:48) ) = cos( φ (cid:96) − φ (cid:96) (cid:48) ) , (B.4)– 15 –here T I is given in (3.13), we have thus shown that ˆ κ (2) std = T I (ˆ γ (2) std ) I .Generally, the relation does not hold anymore at third order. As we are only concernedwith correlation functions in this work, it is sufficient to show that the relation holds withincorrelation functions, i.e., (cid:104) ˆ y (1) ˆ κ (3) std (cid:105) = (cid:104) ˆ y (1) T I (ˆ γ (3) std ) I (cid:105) . To do so we first note that the thirdterm in the third order expression for the convergence (3.7) does not contribute to thecorrelation function under the Limber approximation. This also applies to the third ordershear, since the line-of-sight integrals are the same for both the convergence and the shear.The mode coupling terms of the angular cross power spectra are (cid:104) ˆ y (1) ( (cid:126)(cid:96) )ˆ κ (3) std ( (cid:126)(cid:96) ) (cid:105) ∝ (cid:90) d (cid:126)(cid:96) (cid:48) (2 π ) (cid:16) | (cid:126)(cid:96) | + 2 (cid:126)(cid:96) (cid:126)(cid:96) (cid:48) (cid:17) | (cid:126)(cid:96) | (cid:104) (cid:126)(cid:96) (cid:126)(cid:96) (cid:48) (cid:105) (B.5)and (cid:104) ˆ y (1) ( (cid:126)(cid:96) ) T I ( (cid:126)(cid:96) )(ˆ γ (3) std ) I ( (cid:126)(cid:96) ) (cid:105) ∝ T I ( (cid:126)(cid:96) ) (cid:90) d (cid:126)(cid:96) (cid:48) (2 π ) (cid:16) | (cid:126)(cid:96) | T I ( (cid:126)(cid:96) ) − | (cid:126)(cid:96) || (cid:126)(cid:96) (cid:48) | U I ( (cid:126)(cid:96) , (cid:126)(cid:96) (cid:48) ) (cid:17) | (cid:126)(cid:96) | (cid:104) (cid:126)(cid:96) (cid:126)(cid:96) (cid:48) (cid:105) . (B.6)Applying the two identities T I ( (cid:126)(cid:96) ) T I ( (cid:126)(cid:96) (cid:48) ) = cos(2 φ (cid:96) (cid:48) − φ (cid:96) ) and (B.4) we see that the two aboveexpressions are equal. We have thus proven that it is justified to use the convergence in crosspower spectra instead of terms of the from T I ( (cid:126)(cid:96) )ˆ γ I ( (cid:126)(cid:96) ), up to third order. To second order,the relation ˆ κ = T I ˆ γ I even holds exactly. C Induced rotation
Let S ( (cid:126)θ ) be the surface brightness distribution of an extended source. The first and secondmoment of the brightness distributions are then defined as [41] θ i = (cid:82) d (cid:126)θ θ i S ( (cid:126)θ ) (cid:82) d (cid:126)θ S ( (cid:126)θ ) (C.1a) Q ij = (cid:82) d (cid:126)θ ( θ i − θ i )( θ j − θ j ) S ( (cid:126)θ ) (cid:82) d (cid:126)θ S ( (cid:126)θ ) . (C.1b)Following [42], we introduce the complex ellipticity parameter (cid:15) = Q − Q + 2 ıQ Q + Q + 2 (cid:112) Q Q − Q . (C.2)For an elliptical source with semi major and minor axis a and b , rotated by an angle α withrespect to a fixed coordinate system, the ellipticity parameter (C.2) is given by (cid:15) = a − ba + b e ıα . (C.3)The Jacobi map (2.3) relates an infinitesimal distance on the source plane to an infinitesimaldistance on the image plane by d (cid:126)θ S = A ( (cid:126)θ S ) d (cid:126)θ O . Assuming the source is sufficientlysmall such that the Jacobi map does not vary over the extend of the source, the secondbrightness moment (C.1b) of the source Q S can be approximately related to the observedsecond brightness moment Q O by Q S = A Q O A T . (C.4)– 16 –e generalize previous work by allowing A ( (cid:126)θ ) to have an anti-symmetric part. This anti-symmetric contribution ω can be thought of as a rotation induced by lens-lens coupling.Given an elliptical source, the observed ellipticity can be written as (cid:15) O = g + (cid:15) S (cid:48) g ∗ (cid:15) S (cid:48) , (C.5)where the generalized reduced shear g and rotated source ellipticity (cid:15) S (cid:48) are given by g = γ + ıγ − κ + ıω , (cid:15) S (cid:48) = (cid:15) S e − ıϑ , tan ϑ = ω − κ . (C.6)The reduced shear now includes a contribution from the anti-symmetric term ω of the generalJacobi map. Furthermore, the source ellipticity is rotated by an angle ϑ . However, assumingthe sources are distributed isotropically, this rotation is not observable. In particular, theensemble average (cid:104) (cid:15) O (cid:105) = g , i.e. the observed ellipticity remains an unbiased estimator of thereduced shear despite the rotation ω . In the limit of a symmetric Jacobi map ω → g = (cid:60) ( g ) = γ (1 − κ ) + γ ω (1 − κ ) + ω , g = (cid:61) ( g ) = γ (1 − κ ) − γ ω (1 − κ ) + ω . (C.7)Because ω is necessarily of at least second order, the generalized reduced shear to third ordercan be written as g I = γ I − κ + R ( ω ) IJ γ J + O (Φ ) , (C.8)where the matrix R ( ω ) is defined as R ( ω ) IJ = (cid:18) ω − ω (cid:19) . (C.9)This can be understood as an infinitesimal rotation of the shear by an angle ω . References [1] M. Bartelmann,
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