Rotation of the CMB polarisation by foreground lensing
Enea Di Dio, Ruth Durrer, Giuseppe Fanizza, Giovanni Marozzi
RRotation of the CMB polarisation by foreground lensing
Enea Di Dio a,b , Ruth Durrer c , Giuseppe Fanizza d and Giovanni Marozzi e a Physics Division, Lawrence Berkeley National Laboratory, Cyclotron Rd, Berkeley, CA 94720 b Berkeley Center for Cosmological Physics and Department of Physics, University of California, Berkeley, CA 94720 c Universit´e de Gen`eve, D´epartement de Physique Th´eorique and CAP,24 quai Ernest-Ansermet, CH-1211 Gen`eve 4, Switzerland d Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy e Dipartimento di Fisica, Universit`a di Pisa, and Istituto Nazionale di Fisica Nucleare,Sezione di Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy (Dated: July 11, 2019)We investigate the weak lensing corrections to the CMB polarization anisotropies. We concentrateon the effect of rotation and show that the rotation of polarisation is a true physical effect which hasto be taken into account at second order in perturbation theory. We clarify inconsistencies on thetreatment of this rotation in the recent literature. We also show that at first order in perturbationtheory there is no rotation of polarisation also for vector and tensor modes.
PACS numbers: 98.80.-k, 98.80.Es
1. INTRODUCTION
Cosmology has made enormous progress in the lasttwenty years from an order of magnitude into a precisionscience. At a closer look, much of this is due to the Cos-mic Microwave Background (CMB). There is a wealth ofhigh precision data on the anisotropies and the polarisa-tion of the CMB, to a large extent dominated at presentby the Planck data [1, 2]. Moreover, there are also otherimportant experiments which are more precise on smallerscales [3–6]. CMB data are so precious because we un-derstand them very well, see e.g. [7], which comes fromthe fact that we can use, to a large extent, linear pertur-bation theory to interpret them. Since the initial pertur-bation spectrum from inflation is simple, we can estimatethe cosmological parameters, on which the linear transferfunctions sensitively depend, with high precision.The main non-linearity which is relevant in our un-derstanding of the CMB is lensing: due to the presenceof massive foreground structures, CMB photons are de-flected and arrive at the observer from a direction whichdoes not agree with the direction of emission, see [8] fora review of CMB lensing. Not that if the CMB wouldbe perfectly isotropic, δT ≡
0, lensing would have no ef-fect of the CMB. In this sense CMB lensing goes beyondlinear perturbation theory, we need both, temperaturefluctuations and fluctuations in the foreground geome-try. On small scales lensing is quite important and itchanges the inferred fluctuation and polarisation spectraby 10% and more at harmonics (cid:96) > ∼ θ < π/ (cid:39) (cid:48) . This lensing ofCMB anisotropies and polarisation has been observed inseveral experiments, see e.g. [9–12].The importance of the effect has prompted several ofus to investigate whether higher order contributions to itmight be relevant for future, high precision S4 type [13]experiments or future satellites [14–16]. In the standardtreatment the contributions from the first order deflec-tion angle are ’summed up’ assuming Gaussianity. In- cluding this non-linearity, which is standard in presentCMB codes like CAMB [17] or class [18, 19], is relevantfor a precise analysis of recent experiments like Planck.However, in this treatment, the deflection angle is alwayscalculated in the so-called Born approximation, i.e. byintegrating the lensing potential along the unperturbedphoton geodesic. At second order this is no longer cor-rect and a treatment beyond the Born approximation isin principle requested. Recent works [20–30] have con-sidered this and other effects including, in several steps,most higher order contributions to CMB lensing.In most of their calculations, presented in Refs. [20–24, 26], the results of the two groups involved in the ana-lytic and numerical evaluation of the higher-order effectsare in reasonable agreement, but there is one exceptionwhich is the subject of the present work: in principle,parallel transport can lead to a rotation of the Sachs ba-sis, i.e. the orthonormal basis on the ’screen’ normal tothe photon direction and to the four velocity of the ob-server, by an angle which we can call α . In this casethe polarisation tensor rotates by − α and changes thecomplex polarisation P = Q + iU by P (cid:55)→ exp( − iα ) P which affects the polarisation spectrum and especiallyinduces B -polarisation from an original E -polarisationspectrum. B -polarisation is already induced by the ef-fect of re-mapping by lensing at first order and this hasbeen measured by several experiments [4, 11, 31]. Thisrotation could reduce the de-lensing efficiency of gradientbased methods [32], and therefore the sensitivity of nextgeneration of CMB experiments to the tensor-to-scalarratio.In particular, using the so-called geodesic light-cone(GLC) gauge [33–35], in Ref. [23] it has been estimatedthat the contribution of B -modes from rotation is at thepercent level for (cid:96) > r for fu-ture experiments [13, 14]. If this is correct, this rotationis of uttermost importance for the analysis of these ex- a r X i v : . [ a s t r o - ph . C O ] J u l periments. However, in Ref. [26] the authors show that,in longitudinal gauge, the Sachs basis does not rotate.In fact, we shall show that in this gauge, a spatial vectornormal to the photon direction does not rotate at any or-der when parallel transported along the photon geodesicin a quasi-Newtonian gravitational potential. The au-thors of Ref. [26] therefore argue that higher order lens-ing effects on the CMB polarisation are very small andcan be safely neglected in the analysis of planned ex-periments. At first sight, this suggests that the rotationangle of the polarization tensor parallel transported fromthe last surface scattering to the observer may be gauge-dependent.This is the present state of affairs. The CMB powerspectra, however, are observables and cannot depend onthe coordinate system which is used to compute them.Therefore, either the conclusion of Ref. [26] or the oneof Ref. [23] (or both) must be wrong. The importantquestion is: rotation with respect to what is relevant forthe CMB spectra? As already discussed in [23] (see be-ginning of Section VI), this cannot be the rotation withrespect to some arbitrarily chosen coordinate system, butit must be a physically defined rotation.In the next section we show that the relevant rotationis the one of the Sachs basis with respect to the directionof a vector connecting neighbouring geodesics. Since thisgeodesic deviation vector is Lie transported along thephoton geodesic, this means that the relevant rotationangle α is the change in the angle between a Lie trans-ported and a parallel transported vector in the screen. Inmore detail, as we shall clearly show in the next section,in an arbitrary coordinate system the physical angle α isgiven by the sum β + ω , where β is the rotation of theSachs basis with respect to an arbitrary fixed basis while ω gives the rotation of the geodesic deviation vector inthe fixed basis (hence the change of the angle between theparallel transported polarisation direction and the direc-tion of the geodesic deviation vector is − β − ω = − α ). Wealso show that, for scalar perturbation, this is exactly theangle β calculated in [23] to second order . In longitudi-nal gauge, the parallel transported Sachs basis does notrotate but the geodesic deviation vector of neighbouringphoton geodesics rotates at second order by the angle ω given by the amplification matrix, ( ∂ n /∂ n (cid:48) ) (where n isthe incoming photon direction and n (cid:48) is the source direc-tion) of the lens map at second order. Denoting this anglein longitudinal gauge by ω LG , this is consistent with thefinding α GLC = ω LG of [23].The remainder of this work is structured as follows: inSection 2 we derive the relation between the angles ω , β and the CMB polarisation power spectra. In Section 3we show that in first order perturbation theory β + ω vanishes not only for scalar but also for vector and ten-sor perturbations. In Section 4 we calculate this rotation According to the definition made here in [23] α has been identi-fied with β (see Sect. 2). angle to second order for scalar perturbation in longi-tudinal gauge where we find again the result derived inRef. [23]. In Section 5 we briefly discuss our findings andconclude. In Appendix A we give some technical details,while in Appendix B we show the equivalence betweenthe derivation presented in this manuscript and the pre-vious one in Ref. [23]. In Appendix C we present analyticapproximations for the slope of the spectra at high andlow (cid:96) .All the calculations are performed in the flat sky ap-proximation, see [7, 8], which is sufficient if we are inter-ested in spherical harmonics with (cid:96) >
50. Furthermore,numerical results are obtained within the Limber approx-imation [36, 37], which works very well for CMB lensing(CMB lensing is appreciable only for (cid:96) ≥ (cid:96) rotation is the dominant effecton the B -polarisation spectrum. We also do not discusshere the dominant lensing terms which can be obtainedwithin the Born approximation as these are well known,see e.g. [7, 8] and there is no controversy concerning theseterms.
2. CMB SPECTRA
Throughout we shall work in the flat sky approxima-tion which is fully sufficient for harmonic modes (cid:96) > x and x (cid:48) . Theymay have a slightly different temperature and differentpolarisation. Here we are interested in the polarisation.Since Thomson scattering only produces linear polarisa-tion, we expect the Stokes parameter V to vanish andintroduce the complex polarisation P ( x ) = Q ( x ) + iU ( x ) (2.1)which of course depends on the orientation of our co-ordinate system. P has helicity 2 and transforms un-der a rotation of the basis by an angle θ as P ( x ) (cid:55)→ exp( − iθ ) P ( x ). In Fourier space we can express P , andits complex conjugate P ∗ = Q − iU , in terms of E and B polarisations as [38] P ( x ) = Q ( x ) + iU ( x )= − (cid:90) d (cid:96) π [ E ( (cid:96) ) + iB ( (cid:96) )] e iϕ (cid:96) e i (cid:96) · x , (2.2) P ∗ ( x ) = Q ( x ) − iU ( x )= − (cid:90) d (cid:96) π [ E ( (cid:96) ) − iB ( (cid:96) )] e − iϕ (cid:96) e i (cid:96) · x , (2.3)where ϕ (cid:96) is the polar angle of the 2d vector (cid:96)(cid:96) . Inversely E ( (cid:96)(cid:96) ) + iB ( (cid:96)(cid:96) ) = − (cid:90) d x π P ( x ) e − iϕ x e − i (cid:96) · x , (2.4) E ( (cid:96)(cid:96) ) − iB ( (cid:96)(cid:96) ) = − (cid:90) d x π P ∗ ( x ) e iϕ x e − i (cid:96) · x , (2.5)where ϕ x is the polar angle of x (see, e.g. [7]). Inthe above integrations one can of course fix the non-integrated variable along the abscissa axis so that ϕ de-notes the angle between (cid:96)(cid:96) and x in both cases.In the flat sky approximation the power spectra of the E - and B -polarisation are defined by (cid:104) E ( (cid:96) ) E ∗ ( (cid:96) (cid:48) ) (cid:105) = δ ( (cid:96) − (cid:96) (cid:48) ) C E(cid:96) , (2.6) (cid:104) B ( (cid:96) ) B ∗ ( (cid:96) (cid:48) ) (cid:105) = δ ( (cid:96) − (cid:96) (cid:48) ) C B(cid:96) . (2.7)The Dirac delta function is a consequence of statisticalisotropy (which correponds to statistical homogeneity onthe flat sky) and we request statistical parity invarianceso that the correlations between E and B vanish.We now want to correlate P ( x ) with P ( x (cid:48) ) which, bystatistical isotropy depends only on s = x − x (cid:48) . In orderto define a correlation function which is independent ofthe orientation of the basis ( e , e ), we determine thepolarisation with respect to a new basis ( e (cid:48) , e (cid:48) ) with e (cid:48) =ˆ s , the unit vector in direction s . This new polarisationis then given by P s ( x ) = e − iϕ s P ( x ) , (2.8)where ϕ s is the polar angle of s with respect to the origi-nal basis ( e , e ). With respect to the new intrinsic basis( e (cid:48) , e (cid:48) ) we now define ξ + ( s ) = (cid:104) P ∗ s ( x ) P s ( x (cid:48) ) (cid:105) = (cid:104) P ∗ ( x ) P ( x (cid:48) ) (cid:105) == (cid:104) Q ( x ) Q ( x (cid:48) ) (cid:105) + (cid:104) U ( x ) U ( x (cid:48) ) (cid:105) , (2.9) ξ − ( s ) = (cid:104) P s ( x ) P s ( x (cid:48) ) (cid:105) = (cid:104) e − iϕ s P ( x ) P ( x (cid:48) ) (cid:105) = (cid:104) Q s ( x ) Q s ( x (cid:48) ) (cid:105) − (cid:104) U s ( x ) U s ( x (cid:48) ) (cid:105) . (2.10)The terms (cid:104) Q s ( x ) U s ( x (cid:48) ) (cid:105) vanish since they change signunder parity, s → − s and we assume statistical parityinvariance of the signal. In terms of the E and B powerspectra we obtain ξ + ( s ) = 12 π (cid:90) ∞ d(cid:96)(cid:96) (cid:2) C E(cid:96) + C B(cid:96) (cid:3) J ( (cid:96)s ) , (2.11) ξ − ( s ) = 12 π (cid:90) ∞ d(cid:96)(cid:96) (cid:2) C E(cid:96) − C B(cid:96) (cid:3) J ( (cid:96)s ) . (2.12)For this we simply use Eqs. (2.2, 2.3) and (2.6, 2.7), aswell as the identity( − i ) n π (cid:90) π dϕ e niϕ + iy cos ϕ = J n ( y ) , where J n denotes the Bessel function [39] of order n .Eqs. (2.11, 2.12) are readily inverted to C E(cid:96) + C B(cid:96) = 2 π (cid:90) ∞ ds s ξ + ( s ) J ( (cid:96)s ) , (2.13) C E(cid:96) − C B(cid:96) = 2 π (cid:90) ∞ ds s ξ − ( s ) J ( (cid:96)s ) . (2.14) x e ′ = ̂ s e ′ s φ s e e e e x e ′ = ̂ s e ′ s φ s ωβ = 0 α = β + ω Without LensingParallel transport Lie transport ≠ With Lensing
FIG. 1. We show schematically the rotation of the polariza-tion for scalar perturbations in Poisson gauge. The polariza-tion vector (orange) is parallel transported from the sourceto the observer plane and does not rotate wrt the arbitrarybasis { e , e } , here chosen to align with an unlensed ellipticalimage ( β = 0). The image (green) and the separation vector s (with the basis { e (cid:48) , e (cid:48) } ) are Lie transported and they ro-tate by an angle ω . The physical effect is due to the gaugeinvariant angle α = β + ω which is generated by the differentrotations induced by a parallel transport with respect to a Lietransport. We first concentrate on the rotation of the polarisationonly. We shall see that this contribution dominates. Letus assume that at position x lensing rotates the polar-isation basis by parallel transport by some angle β ( x ),( e + i e ) (cid:55)→ exp( iβ ( x ))( e + i e ). Furthermore, vectorsor tensors have to be transformed with the Jacobian ofthe lens map . Here we only consider the rotation, ω , By this we mean the standard fact that under a transformation by which the unlensed tangent vectors have to be trans-formed. Hence the system ( e (cid:48) + i e (cid:48) ) describing the basisalong the unlensed direction s at x is rotated by ω ( x )( e (cid:48) + i e (cid:48) ) (cid:55)→ exp( iω ( x ))( e (cid:48) + i e (cid:48) ). Introducing the rel-ative angle α = β + ω , we infer from Eq. (2.8) that thepolarisation oriented with respect to ( e (cid:48) , e (cid:48) ) is rotatedby − α (and correspondingly for x (cid:48) ). The different anglesare represented in Fig. 1 for the case of scalar perturba-tions in Poisson gauge (see Sections 3 and 4). Denotingthe rotated polarisation by ˜ P r we have˜ P rs ( x ) = e − iα ( x ) e − iϕ s P ( x ) (2.15)˜ P rs ( x (cid:48) ) = e − iα ( x (cid:48) ) e − iϕ s P ( x (cid:48) ) . (2.16)Correspondingly, the lensed correlation functions ξ ± pick up a factor exp( − i ( α ( x ) ∓ α ( x (cid:48) ))) from a rotation.It is this and only this angle α = ω + β which is truly ob-servable. The angle by which a tangent vector is rotatedby the lens map, i.e. the rotation of a vector connect-ing two neighbouring geodesics, which is Lie transported,and the Sachs basis which is parallel transported. Notethat for clarity we take into account here only the rota-tion induced by lensing, neglecting all other effects fromlensing even though they may be significantly larger.More precisely˜ ξ r + ( s ) = (cid:104) e +2 i ( α ( x ) − α ( x (cid:48) )) (cid:105) ξ + ( s ) , (2.17)˜ ξ r − ( s ) = (cid:104) e − i ( α ( x )+ α ( x (cid:48) )) (cid:105) ξ − ( s ) . (2.18) Here we assume that we can neglect the correlations ofthe unlensed polarisation (mainly produced at the lastscattering surface) and the deflection angles (generatedby foreground structures). We now use a relation whichis strictly true only for Gaussian variables, but we as-sume that the non-Gaussianity of α gives a subdominantcontribution with respect to the leading order effect. As-suming Gaussianity we can set (cid:104) e +2 i ( α ( x ) − α ( x (cid:48) )) (cid:105) (cid:39) e − (cid:104) ( α ( x ) − α ( x (cid:48) )) (cid:105) = e − C α (0) − C α ( s )) , (2.19) (cid:104) e − i ( α ( x )+ α ( x (cid:48) )) (cid:105) (cid:39) e − (cid:104) ( α ( x )+ α ( x (cid:48) )) (cid:105) = e − C α (0)+ C α ( s )) , (2.20)where we have introduced the correlation function of theangle α , C α ( s ) = (cid:104) α ( x ) α ( x (cid:48) ) (cid:105) , s = | x − x (cid:48) | . Using Eqs. (2.13, 2.14), we then find the following resultfor rotation induced B -spectrum˜ C Br(cid:96) = π (cid:90) ∞ ds s [ ˜ ξ r + ( s ) J ( (cid:96)s ) − ˜ ξ r − ( s ) J ( (cid:96)s )] = π e − C α (0) (cid:90) ∞ ds s [ ξ + ( s ) J ( (cid:96)s ) e C α ( s ) − ξ − ( s ) J ( (cid:96)s ) e − C α ( s ) ]= 12 e − C α (0) (cid:90) ∞ ds s (cid:90) ∞ d(cid:96) (cid:48) (cid:96) (cid:48) C E(cid:96) (cid:48) [ J ( (cid:96) (cid:48) s ) J ( (cid:96)s ) e C α ( s ) − J ( (cid:96) (cid:48) s ) J ( (cid:96)s ) e − C α ( s ) ] ≈ (cid:90) ∞ ds s (cid:90) ∞ d(cid:96) (cid:48) (cid:96) (cid:48) C E(cid:96) (cid:48) [ J ( (cid:96) (cid:48) s ) J ( (cid:96)s ) e C α ( s ) − J ( (cid:96) (cid:48) s ) J ( (cid:96)s ) e − C α ( s ) ] (2.21)where we have assumed no primordial B -modes. Thesecond line of Eq. (2.21) shows that the correction dueto the variance of α , C α (0) factorizes and contributes toa constant shift in the spectrum of the B -modes. Thisconfirms what has been found in [23] at the leading or-der, where this variance has been estimated to be of order10 − − − depending on the spectrum used. This justi-fies the approximation made in the last line of Eq. (2.21).For a practical calculations, the exponentials in the inte-grand of Eq. (2.21) can be further expanded, giving the x (cid:55)→ φ ( x ), vectors transform with v ( x ) (cid:55)→ φ ∗ ( x ) v ( φ ( x )), where φ ∗ denotes the tangent map. leading result˜ C Br(cid:96) =2 (cid:90) ∞ dssC α ( s ) (cid:90) ∞ d(cid:96) (cid:48) (cid:96) (cid:48) C E(cid:96) (cid:48) [ J ( (cid:96)s ) J ( (cid:96) (cid:48) s )+ J ( (cid:96)s ) J ( (cid:96) (cid:48) s )] , (2.22)where we have used (cid:90) ∞ dssJ n ( (cid:96)s ) J n ( (cid:96) (cid:48) s ) = 1 (cid:96) δ ( (cid:96) − (cid:96) (cid:48) ) . (2.23)With similar manipulations we find the rotation of the E -spectrum,˜ C Er(cid:96) = [1 − C α (0)] C E(cid:96) +2 (cid:90) ∞ dssC α ( s ) (cid:90) ∞ d(cid:96) (cid:48) (cid:96) (cid:48) C E(cid:96) (cid:48) [ J ( (cid:96)s ) J ( (cid:96) (cid:48) s ) − J ( (cid:96)s ) J ( (cid:96) (cid:48) s )] . (2.24)From our derivation it becomes clear that the polar-isation P s which enters the calculation of the E and B power spectra depends on α and is rotated by it when s or/and e is rotated. Whether one or the other or bothare rotated depends on the chosen coordinate system, butthe sum of the two rotation angles, the total rotation, isphysical. In previous calculations of one group [20, 22, 26]longitudinal gauge (LG) was used where, as we shall see, β vanishes but ω is non-zero at second order. In thecalculations of the other group [21, 23, 24] geodesic light-cone (GLC) gauge [33] was used, where directly the angle α of the rotation of the Sachs basis, ( e , e ), with respectto the incoming photon direction is determined. Thereit has been shown that α GLC = ω LG at second order forscalar perturbations.Indeed, according to what we have shown so far, therotation between the displacement vector of two nullgeodesics ξ µ and the polarisation vector (cid:15) µ of a photonis given by cos α = ˆ ξ µ ˆ (cid:15) µ = ˆ ξ A ˆ (cid:15) A (2.25)where hatted quantities are normalized vectors and ˆ ξ A ≡ ˆ ξ µ e Aµ and ˆ (cid:15) A ≡ ˆ (cid:15) µ e Aµ and e Aµ is the Sachs basis. Becauseboth ˆ (cid:15) µ and e Aµ are parallel transported, we have that ˆ (cid:15) A remains constant while traveling along the geodesic. Thismeans that any change in α can only be due to the rota-tion of ˆ ξ A . Because photons on the same light-cone travelat fixed angular coordinates and start at the same time,their displacement vector will be ξ µ = δ µa C a , with C a constant and a = 1 , ξ A = e Aa C a . FromEq. (2.25) and due to the constancy of ˆ (cid:15) , it follows thatin GLC α changes according to the rotation of the Sachsbasis. This angle is what has been found in Appendix Cof [23].Let us underline that α is a gauge invariant quantitysuch that its value is independent of the coordinate sys-tem. The equality between its value and the rotation ofthe basis is a peculiarity of the GLC gauge where ξ µ issomehow trivial.It is convenient to rewrite Eq. (2.22) by introducingthe angular power spectrum of the rotation angle α as C αα(cid:96) = 2 π (cid:90) dssC α ( s ) J ( s(cid:96) ) . (2.26)Simply using the inverse relation C α ( s ) = 12 π (cid:90) d(cid:96)(cid:96)C αα(cid:96) J ( s(cid:96) ) , (2.27) The constancy of these angular coordinates leads to equal themwith the photon’s incoming direction as seen by the observer.This explicit identification has been recently discussed and im-plemented in [40, 41] by exploiting the residual gauge freedom ofthe GLC coordinates. we find ˜ C Br(cid:96) = 1 π (cid:90) d(cid:96) (cid:48) d(cid:96) (cid:48)(cid:48) (cid:96) (cid:48) (cid:96) (cid:48)(cid:48) C αα(cid:96) (cid:48)(cid:48) C E(cid:96) (cid:48) F (cid:96)(cid:96) (cid:48) (cid:96) (cid:48)(cid:48) , (2.28)where we have introduced the geometrical factor (see Ap-pendix A for further details) F (cid:96)(cid:96) (cid:48) (cid:96) (cid:48)(cid:48) = (cid:90) dssJ ( (cid:96) (cid:48)(cid:48) s ) [ J ( (cid:96)s ) J ( (cid:96) (cid:48) s ) + J ( (cid:96)s ) J ( (cid:96) (cid:48) s )] . (2.29)Before we go on, we also want to find the effect fromthe rotation of the position x . For this we write therotated position as ˜ x = x + ∇ ∧ Ω . (2.30)Here Ω is the potential of the rotation angle and α =∆Ω /
2. Note that in 2 dimensions( ∇ ∧ Ω) a = (cid:15) ab ∇ b Ω , where (cid:15) ab is the totally antisymmetric symbol in 2 dimen-sions and Ω is a (pseudo-)scalar. In the literature this hasbeen considered mainly in longitudinal gauge where forscalar perturbations α = ω . But as we have argued be-fore, this expression is not gauge invariant and we haveto consider the rotation of x with respect to the Sachsbasis which is given by the angle α . As we shall showin the next section, this implies that the curl componentof the deflection field can not be sourced by any linearperturbation, including vector and tensor perturbationscontrarily to what claimed in the past literature, startingfrom Refs. [42, 43]. We denote the so displaced polari-sation by ˜ P d ( x ). To lowest order in α and hence Ω thischanges the polarization into˜ P d ( x ) = P (˜ x ) = P ( x ) + (cid:15) ab ∇ b Ω( x ) ∇ a P ( x ) . (2.31)Inserting this in the expressions (2.9) and (2.10) for ξ + ( s )and ξ − ( s ), and Fourier transforming, we find˜ ξ d + ( (cid:96) ) = (cid:90) d (cid:96) (cid:48) (2 π ) ( (cid:96)(cid:96) ∧ (cid:96)(cid:96) (cid:48) ) C Ω | (cid:96) − (cid:96) (cid:48) | C E(cid:96) (cid:48) , (2.32)˜ ξ d − ( (cid:96) ) = − (cid:90) d (cid:96) (cid:48) (2 π ) ( (cid:96) ∧ (cid:96) (cid:48) ) C Ω | (cid:96) − (cid:96) (cid:48) | C E(cid:96) (cid:48) e iϕ (cid:96) , (cid:96) (cid:48) . (2.33)Here we use the notation x ∧ y = (cid:15) ab x a y b for the vec-tor product in 2 dimensions. Note that in two dimen-sions a vector product is a (pseudo-)scalar (the lengthof the corresponding radial vector in 3d). Parity invari-ance ξ ( s ) = ξ ( − s ) for all correlation functions impliesthat power spectra are real. The imaginary part abovetherefore cannot contribute and we may replace the ex-ponential by its real part, For the purpose of this derivation we neglect the gradient partin the deflection angle. Due to the different parity the gradientand the curl components of the deflection field are uncorrelated. Re (cid:0) e iϕ (cid:96) , (cid:96) (cid:48) (cid:1) = cos(4 ϕ (cid:96) , (cid:96) (cid:48) ) = 1 − (2 ϕ (cid:96) , (cid:96) (cid:48) ) . With this we find˜ C Bd(cid:96) = 12 (cid:16) ˜ ξ d + ( (cid:96) ) − ˜ ξ d − ( (cid:96)(cid:96) ) (cid:17) = (cid:90) d (cid:96) (cid:48) (2 π ) ( (cid:96) ∧ (cid:96) (cid:48) ) C Ω | (cid:96) − (cid:96) (cid:48) | C E(cid:96) (cid:48) sin (2 ϕ (cid:96) , (cid:96) (cid:48) ) . (2.34)This term agrees exactly with the corresponding term in[32], (see Eq. (9)) or [26] (see Eq. (B9)).Interestingly also the cross term between the rotationof polarisation and the curl deflection does not vanish.Taking the first non-vanishing order in both rotation ofpolarisation and the curl deflection of position we find˜ P rd ( x ) = P ( x ) + 2 iα ( x ) P ( x ) + (cid:15) ab ∇ b Ω( x ) ∇ a P ( x ) . (2.35)This leads to the following cross terms in ˜ ξ + ( (cid:96) ) and ˜ ξ − ( (cid:96) ),˜ ξ rd + ( (cid:96) ) = 0 , (2.36)˜ ξ rd − ( (cid:96) ) = 4 i (cid:90) d (cid:96) (cid:48) (2 π ) ( (cid:96) ∧ (cid:96) (cid:48) ) e iϕ (cid:96) , (cid:96) (cid:48) C α Ω | (cid:96) − (cid:96) (cid:48) | C E(cid:96) (cid:48) = − (cid:90) d (cid:96) (cid:48) (2 π ) ( (cid:96) ∧ (cid:96) (cid:48) ) sin(4 ϕ (cid:96) , (cid:96) (cid:48) ) C α Ω | (cid:96) − (cid:96) (cid:48) | C E(cid:96) (cid:48) . (2.37)In (2.37) we have again only considered the non-vanishing real part. The B -spectrum hence acquires the cross term,˜ C Brd(cid:96) = − (cid:90) d (cid:96) (cid:48) (2 π ) ( (cid:96) ∧ (cid:96) (cid:48) ) C α Ω | (cid:96) − (cid:96) (cid:48) | C E(cid:96) (cid:48) sin(4 ϕ (cid:96) , (cid:96) (cid:48) ) . (2.38)Also this expression would agree with the one in [26](Eq. (B9)) if their rotation angle β would agree with our α which is not the case. Their β is much smaller thanour α and actually is due to an effect which we neglectin our treatment.If we do not perform the integration over angles in(2.22) and replace the correlation function C α ( s ) withthe corresponding power spectrum, or if we simply com-pute the α - α contribution to the B -polarisation spectrumstarting from (2.35), we can write the effect from rota-tion to lowest order in a similar way. Following the samesteps as for ˜ C Bd we find˜ C Br(cid:96) = 4 (cid:90) d (cid:96) (cid:48) (2 π ) C α | (cid:96) − (cid:96) (cid:48) | C E(cid:96) (cid:48) cos (2 ϕ (cid:96) , (cid:96) (cid:48) ) . (2.39)This result agrees also with (B.2) (up to a variable trans-form (cid:96) (cid:48) (cid:55)→ (cid:96) − (cid:96) (cid:48) ). Adding all the terms together and using2∆Ω = − α hence C Ω (cid:96) = 4 (cid:96) − C α(cid:96) and C α Ω (cid:96) = 2 (cid:96) − C α(cid:96) , wecan write the total B − spectrum induced by rotation, tolowest order in the rotation angle α , as∆ ˜ C B(cid:96) = 4 (cid:90) d (cid:96) (cid:48) (2 π ) C α | (cid:96) − (cid:96) (cid:48) | C E(cid:96) (cid:48) (cid:20) cos (2 ϕ (cid:96) , (cid:96) (cid:48) ) − (cid:96)(cid:96) (cid:48) | (cid:96) − (cid:96) (cid:48) | sin(4 ϕ (cid:96) , (cid:96) (cid:48) ) sin( ϕ (cid:96) , (cid:96) (cid:48) ) + ( (cid:96)(cid:96) (cid:48) ) | (cid:96) − (cid:96) (cid:48) | sin (2 ϕ (cid:96) , (cid:96) (cid:48) ) sin ( ϕ (cid:96) , (cid:96) (cid:48) ) (cid:21) . (2.40)In section 5 when we present numerical results for thecontribution to the B -spectrum from rotation for secondorder scalar perturbations, we shall see that the termfrom the rotation of the polarisation dominates the totalresult on all scales.
3. LIE TRANSPORT AND PARALLELTRANSPORT AT FIRST ORDER
In this section we calculate to first order the relevantangle α = ω + β . To determine ω we study the propa-gation of neighbouring photons in an infinitesimal lightbeam (sometimes called a null congruence) which is givenby the so called Jacobi matrix D , see e.g. [44–46]. De-noting the geodesic deviation vector by X we find fromthe geodesic deviation equation¨ X α = − R αβµν k β X µ k ν . (3.41)Note that the geodesic deviation equation together withthe geodesic equation for k implies that X is Lie trans- ported . Decomposing X into a part parallel to k , a partparallel to the observer/emitter 4-velocity u and a part inthe ’screen’ normal to k and u with basis ( e , e ), we de-note by D the map which maps directions at the observergiven by the screen basis ( e , e ) to a distance vector Y on the screen of the emitter. Since (3.41) is linear, D is alinear map which expresses the vector Y in terms of theSachs basis at the emitter, i.e. the basis ( e , e ) whichis parallel transported backwards from the final to theinitial screen, Y a = D ab e b + y a k + z a u , This is easily seen in coordinate free notation where ¨ X = ∇ k ∇ k X and − R αβµν k β X µ k ν = ( − R ( X, k ) k ) α = ( ∇ k ∇ X −∇ X ∇ k ) k = ∇ k ∇ X k , where we have used ∇ k k = 0. Hence (3.41)implies ∇ k ( ∇ k X − ∇ X k ) = ∇ k ([ k, X ]) = ∇ k L k X = 0, where L k denotes the Lie derivative in direction k . But in the sourceplane, λ = λ in , we can choose X to denote a coordinate direction, X = ∂ s and k = ∂ λ so that at λ in we have [ k, X ] = [ ∂ λ , ∂ s ] = 0,so that constancy of L k X implies L k X = 0 along the photongeodesic. where ( a, b ) take the values 1 and 2 and Y a starts out as0. The 2 × D can be written as D = R ( ω ) R ( χ ) (cid:18) D + D − (cid:19) R ( − χ ) (3.42)where R ( γ ) = (cid:18) cos γ − sin γ sin γ cos γ (cid:19) denotes a rotation by an angle γ . The matrixΣ = R ( χ ) (cid:18) D + D − (cid:19) R ( − χ )is symmetric and describes the area distance given by √ det Σ = (cid:112) D + D − and the shear which is parametrizedby D + − D − and χ . The latter rotates the coordinateaxis into the principle axes of the shear tensor. The angle ω describes a rotation of the image.In an unperturbed Friedmann Universe we have D = D A where D A denotes the background angular diame-ter distance. At the perturbative level, we can thereforeparametrize it as D = D A [ + A ] with A = (cid:18) − κ − γ − γ − ω − γ + ω − κ + γ (cid:19) . (3.43)At first order κ describes the convergence of light rays, γ ≡ γ + iγ describes their shear and ω their rotation.The matrix A is also called the amplification matrix. Thedeterminant det D − is proportional to the luminosityof the image so that, to first order in the perturbations1 + 2 κ is the magnification [44].In GLC gauge the Jacobi map is expressed in the Sachsbasis [34] so that there is no intrinsic distinction between β and ω and one calculates directly the physical angle α which describes the rotation of an image with respect tothe Sachs Basis.Even though algebraically somewhat more involved,GLC gauge is geometrically more intuitive. Neverthe-less, in the following we shall perform all the calculationsin longitudinal gauge which is more commonly known. A. Scalar perturbations
As the rotation angle α is an observable (hence gaugeinvariant) we can perform our calculations in any gauge.In longitudinal gauge, ds = − a ( η )(1 + 2Ψ) dη + a ( η )(1 − δ ij dx i dx j , (3.44)where Φ and Ψ are the so-called Bardeen potentials, η denotes conformal time and, for simplicity, we have setspatial curvature to zero. The Jacobi map for scalar per-turbations at first order can be expressed in terms of thelensing potential ψ given by ψ ( x , z ) = 12 (cid:90) r ( z )0 dr r ( z ) − rr ( z ) r (Φ + Ψ) , (3.45) where the Bardeen potentials are to be evaluated alongthe (unperturbed) photon geodesic and r ( z ) is the co-moving distance to redshift z . The Jacobi map at firstorder in these coordinates is very well known, but forcompleteness we here repeat the result found in the lit-erature, see e.g. [7] κ = (cid:52) ψ , (3.46) γ = [( ∇ ∇ − ∇ ∇ ) + 2 i ∇ ∇ ] ψ , (3.47) ω = 0 . (3.48)To determine the rotation β of the Sachs basis we haveto integrate the geodesic transport equation. A shortcalculation gives that for a vector normal to the photondirection we have de ia dλ = e ia d Φ dλ + k i ( ∇ Φ · e a ) . (3.49)The first term just ensures that e remains normalizedand the second term ensures the constancy of the scalarproduct of k and e . But the basis vector e a does notacquire any component in direction e b . Hence there is norotation of the Sachs basis in longitudinal coordinates, β = 0. This result remains true for a quasi-Newtoniangravitational potential, Ψ = Φ non-perturbatively whenreplacing 1 + 2Ψ (cid:55)→ exp(2Φ) and 1 − (cid:55)→ exp( − ds = − e x ) dη + e − x ) δ ij dx i dx j . (3.50)The (exact) non-vanishing Christoffel symbols of thismetric are Γ i = ∂ i Φ , Γ i = ∂ i Φ , Γ ijm = δ jm ∂ i Φ − δ ij ∂ m Φ − δ im ∂ j Φ . (3.51)Denoting the photon 4-vector by k = ν ( − , n ) and e =( e , e ) with e · n = 0, parallel transport, ∇ k e = 0 implies de i dλ = e i ( ∇ Φ · n ) + n i ( ∇ Φ · e ) , (3.52) de dλ = ( ∇ Φ · e ) . (3.53)The first term ensures that the length of e remains con-stant and the second term together with the second equa-tion ensure that the scalar product k µ e µ remains con-stant, and we have already made use of δ ij n i e j = 0.But clearly, e does not rotate in the plane normal to u ∝ ∂ η and n . Actually, the polarisation is not par-allel transported but we have to project (3.52) into theplane normal to u and k which simply removes the com-ponent in direction n and the component e . Thereforethe true evolution equation for the polarisation in longi-tudinal gauge is de i dλ = e i ( ∇ Φ · n ) = e i d Φ dλ . (3.54)To summarize, α = ω + β is equal to zero for scalarperturbation to first order. This agrees with the resultobtained in [23] where the GLC gauge is used and theresult α (1) GLC = 0 is obtained directly.
B. Vector and Tensor perturbations
We now consider linear vector and tensor perturba-tions. As photon geodesics are conformally invariant, wecan ignore the scale factor of the expanding universe inthis calculation and consider a perturbed Minkowski met-ric. Vector and tensor perturbations in the metric thenare given by ds = − dη − B i dx i dη + ( δ ij + 2 h ij ) dx i dx j . (3.55)where h ij = ∂ ( i F j ) + H ij , with B i and F i pure transversevector perturbations and H ij are the symmetric, tracelessand transverse tensor perturbations and the parenthesesin ∂ ( i F j ) denote symmetrization. The condition for thegeodesic transport of the polarization (cid:15) can be written infull generality as (assuming (cid:15) = 0) d(cid:15) i dλ = − k µ Γ µji (cid:15) j ≡ − K ij (cid:15) j (3.56)At linear order for the metric (3.55) we obtain d(cid:15) i dλ = − (cid:34) dh ij dλ + k m (cid:0) ∂ j h im − ∂ i h jm (cid:1) + k (cid:0) ∂ i B j − ∂ j B i (cid:1)(cid:21) ¯ (cid:15) j = − δ il (cid:20) dh lj dλ + k m ( ∇ j h ml − ∇ l h jm )+ k ∇ l B j − ∇ j B l ) (cid:21) ¯ (cid:15) j ≡ − K ij ¯ (cid:15) j (3.57)where ¯ (cid:15) j is the background direction of the polarizationand K now denotes the linearized expression. Noticethat in the last equal sign above we changed the ordi-nary derivatives with the covariant ones which does notaffect the result due to the antisymmetric structure of theinvolved terms. Without perturbations, the polarizationwill not rotate. The rotation we are interested in can beevaluated (always at linear order, i.e. for small angles) as β = (cid:15) ijm ¯ (cid:15) i (cid:15) j k m (3.58)where the affine parameter λ is normalized such that k µ = ( − , n i ) is the background direction of propaga-tion. Then the evolution equation for β is dβdλ = (cid:15) ijm ¯ (cid:15) i d(cid:15) j dλ k m = − (cid:15) ijm ¯ (cid:15) i K jl ¯ (cid:15) l k m . (3.59) The first term of Eq. (3.57), ( dh ij /dλ )¯ e j , just integratesto h fin ij ¯ e j − h in ij ¯ e j . While this may induce a rotation itis very small, much smaller than term involving spa-tial derivatives and we neglected that in our treatment .Here we only consider the terms with the highest numberof transversal derivatives since only these can contributeappreciably. With this additional approximation K be-comes anti-symmetric and we have K jm = k l (cid:16) ∇ m h jl − ∇ j h ml (cid:17) + k (cid:0) ∇ j B m − ∇ m B j (cid:1) . (3.60)It is more convenient to write the result in polar coordi-nates, where k i = δ ir and ¯ (cid:15) i = δ ia (latin indices a, b, c, d denoting angular directions) such that we have dβdλ = − (cid:15) abr ¯ (cid:15) a K bc ¯ (cid:15) c . (3.61)Because K bc is antisymmetric we can write K bc = (cid:15) bc (cid:15) da K ad /
2. This leads to¯ (cid:15) i k l (cid:15) jil K jm ¯ (cid:15) m = ¯ (cid:15) a (cid:15) ba (cid:15) bc ¯ (cid:15) c (cid:15) de K ed / (cid:15) da K ad / , (3.62)for these equalities we use that both ¯ (cid:15) and ¯ k i are nor-malized to 1 and ¯ (cid:15) is orthogonal to ¯ k i = δ ir . Hence, therotation angle of the polarization is simply given by dβdλ = − (cid:18) (cid:15) da ∇ d h ar + 12 (cid:15) da ∇ d B a (cid:19) , (3.63)where, in polar coordinates h rr = ∇ r F r + H rr ,h ra = ∇ ( a F r ) + H ra . (3.64)On the other hand, the image is Lie transported andthe related rotation can be evaluated as the leading partin the number of spatial derivatives of the antisymmetricpart of amplification matrix. This is given by ω = 12 (cid:15) ca A ac = 12 (cid:15) ca ∇ c θ a , (3.65)where [50–52] θ a = (cid:90) λ s dλ (cid:34) B a + 2 h ra + ¯ γ ab ∂ b (cid:90) λ dλ (cid:48) ( h rr + B r ) (cid:35) = (cid:90) λ s dλ (cid:34) B a + 2 h ra + ¯ γ ab ∇ b (cid:90) λ dλ (cid:48) ( h rr + B r ) (cid:35) (3.66)with a, b denote the angular coordinates and r is the ra-dial index and (¯ γ ab ) = r − diag (cid:0) , sin − θ (cid:1) . The doubleintegral gives a symmetric contribution to ∇ c θ a and does This is in line with the results found in [48, 49]. therefore not contribute to the rotation so that ω is givenby ω = (cid:90) λ s dλ (cid:20) (cid:15) ca ∇ c B a + (cid:15) ca ∇ c h ar (cid:21) (3.67)or, equivalently dωdλ = (cid:18) (cid:15) ca ∇ c B a + (cid:15) ca ∇ c h ar (cid:19) (3.68)which agrees with the result for − β given in Eq. (3.63).With the initial condition β (0) = ω (0) = 0, this impliesthat α = β + ω = 0 for linear vector and tensor pertur-bations.As for the scalar case, this result can also be obtainedusing GLC gauge. As mentioned above, in this gaugewe directly evaluate α (1) which can easily been shownto vanish also for vector and tensor perturbations, seeAppendix C of [23], where this is shown in general, with-out decomposition into scalar vector and tensor pertur-bations.This result disagrees with the analysis presented inRef. [51], while it is in line with Refs. [53, 54] and, re-garding tensor perturbations, with Ref. [48]. Indeed, inRef. [51] the Author expresses the polarisation rotation β with respect to some global coordinate basis. Never-theless this arbitrary coordinate basis is not Lie trans-ported from the last scattering surface to the observerand, therefore, β alone does not represent a physical,measurable rotation angle. In the analysis of Ref. [51],the contribution of ω has not been taken into account.
4. LIE TRANSPORT AND PARALLELTRANSPORT AT SECOND ORDER
The value for α was already computed in [23] for scalarperturbation up to second order. Here we show the com-putation in longitudinal gauge for convenience of thereader and also to demonstrate the gauge invariance ofthe result.As we have seen in the previous section, in longitudi-nal gauge parallel transport does not lead to any rotation.However, the geodesic deviation equation which is equiv-alent to Lie transport does induce a non-vanishing ω inlongitudinal gauge.The evaluation of ω to second order for scalar pertur-bations has already been presented in the literature. Forexample, considering Eqs. (C.35)-(C.40) of [23] we findthe following expression for ωω (2) ( x ) = 2(2 π ) (cid:90) r s dr r s − rr s r (cid:90) r dr r − r rr (cid:90) d (cid:96) d (cid:96) n · ( (cid:96) ∧ (cid:96) ) ( (cid:96) · (cid:96) ) Φ W ( z, (cid:96) ) Φ W ( z , (cid:96) ) e − i ( (cid:96) + (cid:96) ) · x . (4.69)Here Φ W = (Φ + Ψ) / z and z denote the redshift out to comoving distance r and r respectively. The comoving distance to the last scattering surface is denoted r s . Fourier transforming Eq. (4.69) wefind ω (2) ( (cid:96) ) = 2(2 π ) (cid:90) r s dr r s − rr s r (cid:90) r dr r − r rr (cid:90) d (cid:96) n · ( (cid:96) ∧ (cid:96) ) (cid:0) (cid:96) · (cid:96) − (cid:96) (cid:1) Φ W ( z ( r ) , (cid:96) ) Φ W ( z ( r ) , (cid:96) − (cid:96) ) . (4.70)From this we can compute the power spectrum of the rotation angle at second order (cid:104) ω (2) ( (cid:96) ) ω (2) (cid:0) (cid:96) (cid:48) (cid:1) (cid:105) = 4(2 π ) (cid:90) r s dr r s − rr s r (cid:90) r dr r − r rr (cid:90) r s dr r s − r (cid:48) r s r (cid:48) (cid:90) r (cid:48) dr (cid:48) r (cid:48) − r (cid:48) r (cid:48) r (cid:48) (cid:90) d (cid:96) d (cid:96) n · ( (cid:96) ∧ (cid:96) ) (cid:0) (cid:96) · (cid:96) − (cid:96) (cid:1) n · (( − (cid:96) ) ∧ (cid:96) ) (cid:0) − (cid:96) · (cid:96) − (cid:96) (cid:1)(cid:104) C W(cid:96) ( z, z (cid:48) ) C W | (cid:96) − (cid:96) | ( z , z (cid:48) ) δ D ( (cid:96) + (cid:96) ) δ D (cid:0) (cid:96) + (cid:96) (cid:48) (cid:1) + C W(cid:96) ( z, z (cid:48) ) C W | (cid:96) − (cid:96) | ( z , z (cid:48) ) δ D ( (cid:96) − (cid:96) − (cid:96) ) δ D (cid:0) (cid:96) + (cid:96) (cid:48) (cid:1)(cid:105) = δ D (cid:0) (cid:96) + (cid:96) (cid:48) (cid:1) π ) (cid:90) r s dr r s − rr s r (cid:90) r dr r − r rr (cid:90) r s dr r s − r (cid:48) r s r (cid:48) (cid:90) r (cid:48) dr (cid:48) r (cid:48) − r (cid:48) r (cid:48) r (cid:48) (cid:90) d (cid:96) (cid:2) n · ( (cid:96) ∧ (cid:96) ) (cid:0) (cid:96) · (cid:96) − (cid:96) (cid:1)(cid:3) (cid:104) C W(cid:96) ( z, z (cid:48) ) C W | (cid:96) − (cid:96) | ( z , z (cid:48) ) − C W(cid:96) ( z, z (cid:48) ) C W | (cid:96) − (cid:96) | ( z , z (cid:48) ) (cid:105) . (4.71)Inserting (cid:104) ω (2) ( (cid:96) ) ω (2) (cid:0) (cid:96) (cid:48) (cid:1) (cid:105) = δ D (cid:0) (cid:96) + (cid:96) (cid:48) (cid:1) C ωω(cid:96) and denoting the transfer function of the Weyl potential T Φ+Ψ ( k, z ),0we obtain [23], with the help of the Limber approximation [36, 37], the result C ωω(cid:96) = 14 (2 π ) (cid:90) r s drr (cid:90) r dr r (cid:18) r − r rr (cid:19) (cid:18) r s − rr s r (cid:19) (cid:90) d (cid:96) (cid:2) n · ( (cid:96) ∧ (cid:96) ) (cid:0) (cid:96) · (cid:96) − (cid:96) (cid:1)(cid:3) (cid:20) T Φ+Ψ (cid:18) (cid:96) + 1 / r , z (cid:19) T Φ+Ψ (cid:18) | (cid:96) − (cid:96) | + 1 / r , z (cid:19)(cid:21) P R (cid:18) (cid:96) + 1 / r (cid:19) P R (cid:18) | (cid:96) − (cid:96) | + 1 / r (cid:19) = C αα(cid:96) , (4.72)where P R ( k ) is the primordial curvature power spectrum. For the last equal sign we used that β = 0 in longitudinalgauge. × - × - × - × - × - × - ℓ ℓ C ℓ αα LinearHalofit κκαα
10 50 100 500 1000 5000 10 - - - - ℓ C ℓ FIG. 2. Top panel: we plot the angular power spectrum ofthe rotation angle α = ω ( LG ) . In blue by using the linearpower spectrum and in red with Halofit. Bottom panel: ascomparison we show the angular spectrum of the rotationangle (red) together with the spectrum of the convergence κ (green). The first is related to the curl potential as C αα(cid:96) = (cid:96) C ΩΩ (cid:96) /
4, while the latter to the lensing potential φ through C κκ(cid:96) = (cid:96) C φφ(cid:96) / The numerical results for ˜ C ( Br ) (cid:96) have been generated byperforming the double integral (2.28) with F (cid:96)(cid:96) (cid:48) (cid:96) (cid:48)(cid:48) given in(A.5) using the same cosmological parameters as Ref. [23]for comparison purpose. Namely h = 0 . h Ω cdm = ω cdm = 0 .
12, Ω b h = ω b = 0 .
022 and vanishing curva-ture. The primordial curvature power spectrum has theamplitude A s = 2 . × − at the pivot scale k pivot =0 . − , the spectral index n s = 0 .
96 and no runningis assumed. The transfer function for the Bardeen po-
New approachOld approachLimit solution50 100 500 100010 - - - - - ℓ ℓ ( ℓ + ) / ( π ) Δ C ℓ BB [ μ K ] B - mode New approachOld approach0 500 1000 1500 2000 2500 3000 35000.0000.0050.0100.0150.020 ℓ Δ C ℓ BB / C ℓ BB B - mode FIG. 3. We show the effect induced by rotation on the angular B -mode power spectrum. In the top panel we show the goodaccuracy of the low- (cid:96) limit solution derived in Eq. (B.3). Inthe bottom panel we show the relative amplitude comparedto the first order lensed B -mode. The red line is the presentresult according to Eq. (A.5), the dashed blue line refers toour previous result [23] recomputed by integrating Eq. (B.2)and the dotted black line is the limit solution described byEq. (B.3). tentials, T Φ+Ψ has been computed with class [19] usingthe linear power spectrum and Halofit [55].From Fig. 2 we see that the lensing spectrum increasesby about a factor 5 on small scales when using the non-linear Halofit spectrum and (cid:96)C α(cid:96) decays very slowly with (cid:96) . In Appendix B we also show the formal equivalence of1the expression (2.28) and the result obtained in [23]. InFig. 3 we plot the B -mode power spectrum induced fromrotation of polarisation (top panel). In the lower panelwe plot the relative contribution to the first order lensing B -spectrum. As a numerical cross-check we show alsothe results by integrating the double integral given byEq. (B.2) and the low (cid:96) approximation given in Eq. (B.3). deflection angle - mixedpolarizationtotal50 100 500 1000 500010 - - - - ℓ ℓ ( ℓ + ) / ( π ) Δ C ℓ BB [ μ K ] B - mode deflection angle - mixedpolarizationtotal500 1000 20000.000050.000100.000150.000200.00025 ℓ ℓ ( ℓ + ) / ( π ) Δ C ℓ BB [ μ K ] B - mode FIG. 4. We show all three contributions to ∆ C B(cid:96) , the polari-sation rotation (blue line), which is also shown in Fig. 3, thecurl deflection (red line) and the negative of the mixed term(green line). Their sum is indicated as dashed black line. Thebottom plot is a magnification of the gray region of the toppanel, in order to compare the different effects at the scaleswhere they are comparable.
In Fig. 4 we show the different contributions includ-ing also the curl-type deflection angle term computed in(2.34) and the mixed term (2.38). Cleary the two addi-tional terms are relevant mainly around (cid:96) ∼
5. DISCUSSION AND CONCLUSION
In this paper we clarify an issue concerning the rotationof polarisation under the parallel transport of CMB pho-tons in the clustered Universe. We show that the relevant angle is the one between parallel transported vectors andgeodesic deviation vectors which are Lie transported. Or,in other words, the rotation of the geodesic deviation vec-tor in the Sachs basis. This well defined geometric anglewhich we call α vanishes at first order, but not at secondorder. Its second order value is therefore gauge-invariantas a consequence of the Stewart-Walker lemma [56] andits generalization to higher-order [57]. Denoting the an-gle of rotation of the Sachs basis (with respect to somearbitrary coordinate basis) by β and the one of geodesicdeviation vectors (with respect to the same arbitrary ba-sis) by ω we have α = β + ω . For scalar perturbation,we have shown that in longitudinal gauge, β = 0 at allorders. The gauge invariance of α (2) is confirmed by thefinding that ω (2) LG = α (2) GLC .Even if observers measure polarisation with respect toa fixed observer coordinate system (they measure theStokes parameters), they then combine the coordinatedependent Stokes parameters into the coordinate inde-pendent E - and B -polarisation spectra and these are af-fected by rotation in the way computed here.This result is important for polarisation measurementswith high sensitivity, like CMB S4 [13], which want todetect primordial gravitational waves with a tensor-to-scalar ratio as small as r ∼ − . To correctly subtractthe lensing contribution to the B -polarisation this re-quires a precision of better than 0.1% for the lensingspectrum in the crucial (cid:96) range which is used for de-lensing, namely 1000 ≤ (cid:96) ≤ (cid:96) rangethe contribution from rotation increases up to 1% andtherefore has to be considered.The amplitude of the effects induced by the curl com-ponent Ω (with α = ∆Ω /
2) could reduce the efficiencyof de-lensing gradient based methods [32]. This may setan accuracy limit in the search for primordial B-modesand, in general, weaken the constraints on cosmologicalparameters strongly sensitive to the sharpness of BAOpeaks in the CMB power spectrum (that are smearedout by lensing), like e.g. neutrino masses.Furthermore, even if r is much smaller than what anexperiment can ever reach, measuring the rotation of po-larisation is a measurement of frame dragging on cosmo-logical scales which would represent a formidable test ofGeneral Relativity on these scales. ACKNOWLEDGEMENTS
We thank Camille Bonvin, Anthony Challinor, ChrisClarkson, Pierre Fleury, Alex Hall, Martin Kunz, AntonyLewis, Roy Maartens, Miguel Vanvlasselaer and GabrieleVeneziano for helpful and clarifying discussions. RDis grateful for the hospitality and for financial supportof the Physics Department and INFN section of Pisa.GF is grateful for the hospitality of the Physics Depart-ment of the University of Bari. ED (No. 171494) andRD acknowledge support from the Swiss National Sci-ence Foundation. GF and GM are supported in part by2INFN under the program TAsP (Theoretical Astroparti-cle Physics).
Appendix A: Details on the geometrical factor F k(cid:96)q In order to evaluate analytically the geometrical factor F defined in Eq. (2.29), we use the following identity [58],see also [59], (cid:90) ∞ dssJ ( qs ) J n ( ks ) J n ( (cid:96)s ) = Re (cid:18) cos ( nθ ) πk(cid:96) sin θ (cid:19) , (A.1)where cos θ = (cid:96) + k − q k(cid:96) . (A.2)The real part ’Re’ ensures that the integral vanishes if ( q, k, (cid:96) ) do not satisfy the triangle inequality, and θ is the anglebetween the sides of lengths k and (cid:96) in the triangle formed by ( q, k, (cid:96) ). In particular, we are interested in the followingintegrals (cid:90) ∞ dssJ ( qs ) J ( ks ) J ( (cid:96)s ) = 2Re π (cid:32) (cid:112) ( q − ( k − (cid:96) ) )(( k + (cid:96) ) − q ) (cid:33) , (A.3) (cid:90) ∞ dssJ ( qs ) J ( ks ) J ( (cid:96)s ) = (cid:16) k − k q + k (cid:0) q − (cid:96) q (cid:1) − k q (cid:0) (cid:96) − q (cid:1) + (cid:0) (cid:96) − q (cid:1) (cid:17) πk (cid:96) × Re (cid:32) (cid:112) ( q − ( k − (cid:96) ) )(( k + (cid:96) ) − q ) (cid:33) . (A.4)At the boundaries of the triangle equality, i.e. q = | k ± (cid:96) | , the integrals diverge and they need to be interpreted as adistribution within integral (2.28). With this identity we can rewrite the geometrical factor (2.29) as F k(cid:96)q = (cid:0) k − q (cid:0) k + (cid:96) (cid:1) + (cid:96) + q (cid:1) Re (cid:18) √ ( q − ( k − (cid:96) ) )(( k + (cid:96) ) − q ) (cid:19) πk (cid:96) . (A.5) Appendix B: Equivalence with the previous calculation
In this appendix we show that the expression for the B -mode induced by rotation from our previous paper (seeEq. (6.17) in Ref. [23]) is equal to Eq. (2.28). We start with Eq. (6.17) of Ref. [23]∆ C B (2 , (cid:96) ≡ (cid:104) ∆ (cid:0) C E (cid:96) + C B (cid:96) (cid:1) (2 , − ∆ (cid:0) C E(cid:96) − C B(cid:96) (cid:1) (2 , (cid:105) = 16 (cid:90) d (cid:96) (2 π ) (cid:90) d (cid:96) (2 π ) [ n · ( (cid:96) ∧ (cid:96) ) ( (cid:96) · (cid:96) )] (cid:90) r s dr r s − rr s r (cid:90) r dr r − r r r × (cid:90) r s dr r s − r r s r (cid:90) r dr r − r r r (cid:2) C W(cid:96) ( z, z ) C W(cid:96) ( z , z ) − C W(cid:96) ( z, z ) C W(cid:96) ( z , z ) (cid:3) × (cid:110) C E | (cid:96) − (cid:96) − (cid:96) | ( z s ) cos (cid:2) (cid:0) ϕ (cid:96) − ϕ | (cid:96) − (cid:96) − (cid:96) | (cid:1)(cid:3) + C B | (cid:96) − (cid:96) − (cid:96) | ( z s ) sin (cid:2) (cid:0) ϕ (cid:96) − ϕ | (cid:96) − (cid:96) − (cid:96) | (cid:1)(cid:3)(cid:111) , (B.1)then, by making a change of variable (cid:96) = (cid:96) (cid:48) − (cid:96) and using Eqs. (4.72), we obtain (in absence of primordial B -mode)∆ C B (2 , (cid:96) = 4 (cid:90) d (cid:96) (cid:48) (2 π ) C ωω(cid:96) (cid:48) C E | (cid:96) − (cid:96) (cid:48) | cos (cid:0) ϕ (cid:96) − ϕ | (cid:96) − (cid:96) (cid:48) | (cid:1) . (B.2)3For low (cid:96) we can approximate the contribution induced by the rotation to the B -mode as follows∆ C B (2 , (cid:96) (cid:39) (cid:90) d(cid:96) (cid:48) π (cid:96) (cid:48) C ωω(cid:96) (cid:48) C E(cid:96) (cid:48) + O (cid:0) (cid:96) (cid:1) = (cid:90) d ln (cid:96) (cid:48) π (cid:96) (cid:48) C ωω(cid:96) (cid:48) C E(cid:96) (cid:48) + O (cid:0) (cid:96) (cid:1) ∼ × − µK . (B.3)As we see from the upper panel of Fig. 3, this limiting white noise contribution fully captures the power-law dependenceinduced by the rotation up to scale (cid:96) ∼ C (cid:96) is independent of direction we may rotate (cid:96) suchthat ϕ (cid:96) = 0. We then find∆ C B (2 , (cid:96) = 4 (cid:90) d (cid:96) (cid:48)(cid:48) (2 π ) C ωω(cid:96) (cid:48)(cid:48) C E | (cid:96) − (cid:96) (cid:48)(cid:48) | cos (cid:0) ϕ ( (cid:96) − (cid:96) (cid:48)(cid:48) ) (cid:1) = 4 (cid:90) d (cid:96) (cid:48)(cid:48) (2 π ) d(cid:96) (cid:48) δ D (cid:0) (cid:96) (cid:48) − | (cid:96) − (cid:96) (cid:48)(cid:48) | (cid:1) C ωω(cid:96) (cid:48)(cid:48) C E(cid:96) (cid:48) cos (cid:0) ϕ ( (cid:96) − (cid:96) (cid:48)(cid:48) ) (cid:1) . (B.4)We rewrite the Dirac delta distribution as δ D (cid:0) (cid:96) (cid:48) − | (cid:96) − (cid:96) (cid:48)(cid:48) | (cid:1) = δ D (cid:18) (cid:96) (cid:48) − (cid:113) (cid:96) + (cid:96) (cid:48)(cid:48) − (cid:96)(cid:96) (cid:48)(cid:48) cos ϕ (cid:96) (cid:48)(cid:48) (cid:19) = (cid:88) i =1 δ D ( ϕ (cid:96) (cid:48)(cid:48) − ϕ i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:112) (cid:96) + (cid:96) (cid:48)(cid:48) − (cid:96)(cid:96) (cid:48)(cid:48) cos ϕ i (cid:96)(cid:96) (cid:48)(cid:48) sin ϕ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (B.5)where ϕ , = ± arccos (cid:32) (cid:96) − (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) (cid:96)(cid:96) (cid:48)(cid:48) (cid:33) , (B.6)if | (cid:96) − (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) | < (cid:96)(cid:96) (cid:48)(cid:48) . If the triangle equality is not satisfied, the integral vanishes. The triangle inequality can beexplicitly enforced by writing (cid:90) dϕ (cid:96) (cid:48)(cid:48) f ( ϕ (cid:96) (cid:48)(cid:48) ) δ D (cid:0) (cid:96) (cid:48) − | (cid:96) − (cid:96) (cid:48)(cid:48) | (cid:1) == 2 (cid:96) (cid:48) Re (cid:113) (cid:96) (cid:48) − ( (cid:96) − (cid:96) (cid:48)(cid:48) ) (cid:113) ( (cid:96) + (cid:96) (cid:48)(cid:48) ) − (cid:96) (cid:48) (cid:90) dϕ (cid:96) (cid:48)(cid:48) f ( ϕ (cid:96) (cid:48)(cid:48) ) [ δ D ( ϕ (cid:96) (cid:48)(cid:48) − ϕ ) + δ D ( ϕ (cid:96) (cid:48)(cid:48) − ϕ )] . (B.7)The angular integral over ϕ (cid:96) (cid:48)(cid:48) can now be performed analytically, (cid:90) dϕ (cid:96) (cid:48)(cid:48) cos (cid:0) ϕ ( (cid:96) − (cid:96) (cid:48)(cid:48) ) (cid:1) [ δ D ( ϕ (cid:96) (cid:48)(cid:48) − ϕ ) + δ D ( ϕ (cid:96) (cid:48)(cid:48) − ϕ )] == 2 (cid:90) dϕ (cid:96) (cid:48)(cid:48) (cid:16) (cid:96) − (cid:96)(cid:96) (cid:48)(cid:48) cos( ϕ (cid:96) (cid:48)(cid:48) ) + (cid:96) (cid:48)(cid:48) cos(2 ϕ (cid:96) (cid:48)(cid:48) ) (cid:17) (cid:0) (cid:96) − (cid:96)(cid:96) (cid:48)(cid:48) cos( ϕ (cid:96) (cid:48)(cid:48) ) + (cid:96) (cid:48)(cid:48) (cid:1) δ D ( ϕ (cid:96) (cid:48)(cid:48) − ϕ ) = (cid:16) (cid:96) − (cid:96) (cid:48)(cid:48) (cid:16) (cid:96) + (cid:96) (cid:48) (cid:17) + (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) (cid:17) (cid:96) (cid:96) (cid:48) . (B.8)Inserting this in (B.4) we find∆ C B (2 , (cid:96) = 4 (cid:90) d (cid:96) (cid:48)(cid:48) (2 π ) C ωω(cid:96) (cid:48)(cid:48) C E | (cid:96) − (cid:96) (cid:48)(cid:48) | cos (cid:0) ϕ | (cid:96) − (cid:96) (cid:48)(cid:48) | (cid:1) == 1 π (cid:90) d(cid:96) (cid:48) d(cid:96) (cid:48)(cid:48) (cid:96) (cid:48) (cid:96) (cid:48)(cid:48) C ωω(cid:96) (cid:48)(cid:48) C E(cid:96) (cid:48) × (cid:16) (cid:96) − (cid:96) (cid:48)(cid:48) (cid:16) (cid:96) + (cid:96) (cid:48) (cid:17) + (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) (cid:17) π(cid:96) (cid:96) (cid:48) Re (cid:113) (cid:96) (cid:48) − ( (cid:96) − (cid:96) (cid:48)(cid:48) ) (cid:113) ( (cid:96) + (cid:96) (cid:48)(cid:48) ) − (cid:96) (cid:48) = 1 π (cid:90) d(cid:96) (cid:48) d(cid:96) (cid:48)(cid:48) (cid:96) (cid:48) (cid:96) (cid:48)(cid:48) C ωω(cid:96) (cid:48)(cid:48) C E(cid:96) (cid:48) F (cid:96)(cid:96) (cid:48) (cid:96) (cid:48)(cid:48) , (B.9)where we have used the definition (A.5) in the last equality. Appendix C: The shape of the spectra
To discuss the form of the spectra shown in Fig. 4,we plot the full convolution C con (cid:96) = (cid:82) d (cid:96) C α | (cid:96) − (cid:96) | C E(cid:96) in Fig. 5. This spectrum starts off as white noise and de-cays roughly like (cid:96) − for (cid:96) > ≡ (cid:96) max . This comes4from the same behavior of the C α(cid:96) spectrum for large (cid:96) .Together with equation (2.40) this explains the growth ∝ (cid:96) of (cid:96) ˜ C Br(cid:96) at high (cid:96) and the decay of (cid:96) ˜ C Bd(cid:96) like (cid:96) − due to the additional pre-factor which reduces to 1 /(cid:96) athigh (cid:96) . For small (cid:96) (cid:28) (cid:96) max , (cid:96) ˜ C Br(cid:96) ∝ (cid:96) behaves likewhite noise, while (cid:96) ˜ C Bd(cid:96) has an additional suppressionfactor of roughly ( (cid:96)/(cid:96) max ) due to the (cid:96) -dependent pre-factor. The amplitude at (cid:96) ∼ (cid:96) max ∼ is of the orderof (cid:96) × − / (2 π ) ∼ − µK which is in the right bullpark.The mixed spectrum, ˜ C Brd(cid:96) is somewhat more intri-
50 100 500 1000 50002.0 × - × - × - × - × - × - ℓ ∫ d ℓ C ℓ EE C ℓ - ℓ αα [ μ K ] FIG. 5. We show the convolution spectrum C con (cid:96) = (cid:82) d (cid:96) C α(cid:96) − (cid:96) C E(cid:96) . cate. The mixed term acquires naively a factor (cid:96)/(cid:96) (cid:48) for low (cid:96) , but the true spectrum scales as (cid:96) at low (cid:96) . This is due to an additional cancellation comingfrom positive and negative contributions in the angu-lar integral of sin(4 ϕ (cid:96) (cid:48) (cid:96) ) sin ϕ (cid:96) (cid:48) (cid:96) and requires a more sub-tle analysis: The pure integral (cid:82) dϕ sin(4 ϕ ) sin( ϕ ) = 0,hence the mixed contribution does not vanish only dueto the angular dependence of C α | (cid:96) − (cid:96) (cid:48) | . Approximating C α | (cid:96) − (cid:96) (cid:48) | by a polynomial in | (cid:96) − (cid:96) (cid:48) | /(cid:96) (cid:48) for small (cid:96) , thefirst non-vanishing contribution in the angular integralcomes from ( (cid:96) cos ϕ ) , which increases as (cid:96) for small (cid:96) .This leads to a behavior of (cid:96) ˜ C Brd(cid:96) ∝ (cid:96) for small (cid:96) . Thepeak at (cid:96) ∼ (cid:96) max is again determined by the ’peak’ of C α | (cid:96) − (cid:96) (cid:48) | C E(cid:96) (cid:48) at roughly this scale. For large (cid:96) (cid:29) (cid:96) max ,the spectrum C α | (cid:96) − (cid:96) (cid:48) | goes approximately as | (cid:96) − (cid:96) (cid:48) | − .Including also the pre-factor (cid:96)/ | (cid:96) − (cid:96) (cid:48) | , again the firstnon-vanishing term in the angular integral comes from(( (cid:96) (cid:48) /(cid:96) ) cos ϕ ) /(cid:96) . 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