Thesis Presented to the Faculty of Sciences of the University of Geneva for a degree of Master in Physics

Gravitational Lensing of the CMB by Tensor Perturbations

SUPERVISORS: Prof. Ruth Durrer Dr. Julian Adamek Vittorio Tansella

April 2015 Abstract

We investigate the properties of Gravitational Lensing by tensor perturbations and the e↵ect on the CMB temperature field. After a first introductory chapter and a second chapter where the formalism of Lensing by density perturbations is introduced as a reference for comparison, in the third chapter we discuss some aspects of lensing by tensor perturbations. The displacement vector of a null geodesic travelling through a tensor perturbed FRW universe is computed. This displacement vector has the unique feature of a curl- component at the linear level and so it is decomposed into its gradient-mode and curl-mode lensing potentials. We compute the power spectra of the two lensing potentials and we get a qualitative understanding of their properties by giving plots for a matter dominated universe. In the last section the modifications induced by this type of lensing to the temperature fluctuations field of the CMB are investigated and we find them to 7 be of the order of 10 , smaller than the cosmic variance of intrinsic CMB anisotropies for every observable multipole. Contents

1 Introduction 2 1.1 HistoricalRemarks...... 2 1.2 TheUnlensedCMB ...... 3 1.3 SimpleExample:TheSchwarzschildLens ...... 5 1.4 LensMapping...... 7

2 Lensing by Scalar Perturbations 9 2.1 Perturbed Photon Path and Lensing Potential ...... 9 2.2 TheCMBLensedPowerSpectrum ...... 13

3 Lensing by Tensor Perturbations 15 3.1 PerturbedPhotonPath ...... 15 3.2 EulerRotationmethod...... 16 3.3 Scalar and Vector Lensing Potential ...... 18 3.3.1 Angular decomposition of and $ ...... 21 3.4 Curl-mode&Gradient-modePowerSpectra ...... 24 3.5 TheCMBLensedPowerSpectrum ...... 26 3.5.1 Full-Sky - Small angle ...... 27 3.5.2 Flat-Sky - Arbitrary angle ...... 28

4 Summary 32

Appendices 34 AppendixA-StatisticsoftheCMB ...... 34 ↵ Appendix B - µ⌫ fortensorperturbations...... 37 AppendixC-Sphericalharmonics ...... 39 AppendixD-Besselfunctions ...... 43 AppendixE-LimberApproximation...... 45 Appendix F - Total deflection angle power ...... 46

1 Chapter 1

Introduction

1.1 Historical Remarks

Figure 1.1: Extract from Einstein’s notebook showing a rough sketch of a lens system and a few formulae. Image from The Collected Papers of Albert Einstein, Volume 3, http://www.einstein-online.info/.

The idea that massive bodies could act upon light through gravity can be traced back long before Ein- stein’s theory of General Relativity. Isaac Newton was the first to have the suspicion that gravity influences the behavior of light, as he states in the first edition of Opticks in 1704. After that the idea was not carried further for almost a century but, in the meantime, through the work of Peter Simon Laplace, another aspect of this interaction was discovered: by computing the escape velocity 2 of light from a massive body, he introduced what today we call Schwarzschild radius RS =2GM/c and anticipated the existence of black-holes. In the XIX century Johann Georg von Soldner published a paper in which he considered the error induced by the deflection of light in the determination of the angular position of stars and, as the small angle limit of the classical mechanic result, he used a deflection angle ↵ = Rs/r for a light ray with impact parameter r. The same result was computed by Einstein in 1911 without using the full formalism of General Relativity, not yet developed. As the times were right for a breakthrough in the field, history got in the way: first, World War I broke out and spoiled Freundlich’s expedition to verify experimentally the deflection of light and second, even when Einstein used the full equations of General Relativity to derive the correct expression of the deflec- tion angle ↵ =2Rs/r, the growing anti-Semitism in Berlin made it hard for him to be taken seriously [1, 2, 3].

Was Arthur Eddington that than took the reins of the subject, not only with his famous expedition to observe the solar eclipse of 29 May 1919, but also as the first to point out that light deflection in the universe can generate multiple images of the same object, introducing the phenomenon of Gravitational Lenses. The newborn field of Gravitational Lensing (GL) grew through the work of Zwicky, who understood the

2 possibility of reconstructing the mass of clusters of galaxies using their lensing properties [4], but it was considered a fairly esoteric field until 1979 when Walsh, Carswell and Weymann discovered and described the first GL candidate: two images of a quasar at z 1.4 lensed by a galaxy at z 0.34 [5] (nowadays hundreds of GL have been detected, see e.g. http://www.cfa.harvard.edu/castles/⇠ ⇠). In the mean time cosmologist were developing cosmological perturbation theory and it was not long until they realized that light propagating through an inhomogeneous universe is lensed and di↵erent from light propagating through an unperturbed Friedmann-Lemaitre universe. This, together with the discovery of the Cosmic Microwave Background in 1964, made GL an important subject for modern cosmology: in fact, as cosmologists realized the importance of CMB anisotropies and as the experimental precision in the measure of these anisotropies grew, it became important to consider the e↵ect of lensing on the CMB.

1.2 The Unlensed CMB

Preliminaries The universe is described by a four-dimensional spacetime ( ,g). The Cosmological principle dictates the universe to be isotropic and homogeneous and this means thatM the spacetime admits a slicing into maximally symmetric 3-spaces. There is a preferred coordinate ⌧, the cosmic time1, such that the 3-spaces of constant time ⌃⌧ are indeed maximally symmetric and hence of constant curvature K. The generic form of the metric g is then

ds2 = g dxµdx⌫ = d⌧ 2 + a2(⌧) dxidxj = a2(t)( dt2 + dxidxj) , (1.1) µ⌫ ij ij d⌧ where we have introduced the conformal time dt a and ij is the metric of the 3-spaces of constant curvature. Among the many ways to describe we⌘ should write it in spherical coordinates since it is the most convenient for this work

dxidxj = dr2 2(r) d✓2 +sin2 ✓d2 , (1.2) ij where

r Flat 3-space K = 0, 1 sin( K r) Spherical 3-space K>0 (r)=8p K | | (1.3) > | | <> 1 sinh(p K r) Hyperbolic 3-space K<0. p K | | | | > p The symmetry of space time:> allows only an energy-momentum of the form ⇢g 0 (T )= 00 , (1.4) µ⌫ 0 Pg ✓ ij◆ where ⇢ and P are the energy density and the pressure of the components of the universe. The Einstein’s equations, including the cosmological constant ⇤, are then written

a˙ 2 8⇡G a2⇤ + K = a2⇢ + , a 3 3 ✓ ◆ 2 (1.5) a˙ • a˙ 2 + + K = 8⇡Ha2P + a2⇤ , a a ✓ ◆ ✓ ◆ where = d and H = a˙ is the Hubble parameter. • dt a2 1⌧ is the proper time of an observer who experiences an isotropic and homogeneous universe.

3 The Cosmic Microwave Background A complete treatment of the thermal history of the universe is beyond our purposes and can be found, for example, in [6] or [11]. Here we just give a general overview of recombination and decoupling. At early times in the history of the universe reaction rates for particle interactions were much faster than the expansion rate so that the cosmic plasma was in thermal equilibrium. As the universe expands it also 1 cools adiabatically following T a . As long as the temperature is above the ionization energy of neutral hydrogen T>1Ry = = 13.6eV/ all hydrogens atoms that forms through e– +p+ H+ are rapidly dissociated but, at a temperature of about T 4000K 0.4eV, the number density of! photons with energies above drops below the baryon density of the⇠ Universe.⇠ As a consequence e– and p+ begin to (re)combine to neutral hydrogen. This behavior can be described with the Saha equation

2 3/2 xe 45 me /T = e (1.6) 1 x 4⇡2 2⇡T e – ⇣ ⌘ 2 where xe is the e ionization fraction and is the entropy per baryon . We can see from Figure 1.2 that for – + T Trec 0.4eV we have xe 1, meaning that far behind recombination epoch e and p are not in form ⇠ ⇠ – of neutral hydrogen, while for T Trec the ionization fraction xe 0 meaning that almost all the e in the universe are bonded in H atoms.⌧ Defining the temperature of recombination⇠ as T T (x =0.5) we find3 rec ⌘ e Trec = 3757K.

1.0

0.8

0.6 2 Wb h =0.01 e

x 2 Out[17]= Wb h =0.02 0.4 2 Wb h =0.03

0.2

0.0 0.25 0.30 0.35 0.40 0.45 T eV

– 2 Figure 1.2: The e ionization fraction as a function of temperatureH L for di↵erent values of ⌦bh ,togetherwiththelinexe =0.5which is our definition of recombination temperature.

Photons and baryons are tightly coupled before recombination by Thompson scattering of electrons. During recombination the free (ionized) e– density drops as in Eq. 1.6 and the mean free path of the photons grows larger than the Hubble scale. Usually one says that when the scattering rate of on e– falls below the expansion rate of the Universe

2 s 4⇡2 T 3 The entropy per baryon is = = ,wheres is the photon entropy density and nB is the conserved baryon nB 45 nB number density so that is a constant. ⇣ ⌘ 3 We point out that there is a physical reason for the fact that Trec . The condition T isnotsucienttotrigger ⌧ ' recombination because n nb and since there are so many more photons than baryons in the Universe, even at a temperature much below there are still enough photons in the high energy tail of the Plank distribution to keep the Universe ionized. Protons and electrons can start (re)combining into hydrogen when n (E>)

4 fluctuations and polarization are fundamental to observational cosmology, since they give the possibility to directly observe the early universe.

The power spectrum of the CMB temperature fluctuations is what we are most interested in. Of course the fact that there are fluctuations in the temperature of the CMB means that the description of the uni- verse as homogeneous and isotropic given in the previous paragraph is, at least, incomplete: it is believed that quantum fluctuations of the inflaton field, brought to cosmic scales by inflation, laid down perturbations on all scales4. The small initial perturbations have grown via gravitational instability and the evolution can be studied in linear theory. This mechanism is responsible for the fluctuations ⇥(ˆn ) of the CMB

T (ˆn ) T T (ˆn ) ⇥(ˆn )= 0 (1.7) T0 ⌘ T0

where T0 2.7K is the mean temperature today and ˆn is the direction of observation. T (ˆn )isa ⇠ 2 function on the sphere since it is evaluated today (t0) and here (x0) so that ˆn S . We expand ⇥(ˆn )in spherical harmonics 2

⇥(ˆn )= ✓lm(x0)Ylm(ˆn ) (1.8) Xlm and define the power spectrum as

⇥ ✓lm✓⇤ = ll mm C (1.9) h l0m0 i 0 0 l where the o↵-diagonal correlators vanish since we assume the process generating the initial perturbations to be statistically isotropic (see Appendix A). § 1.3 Simple Example: The Schwarzschild Lens

After the very brief introduction to the CMB, it is useful to start the discussion of Gravitational Lensing with the simple example of the Schwarzschild Lens, both for pedagogical and historical reasons (Eddington expedition in 1919 measured essentially the deflection angle of the sun). The Schwarzschild metric describes a static, spherically symmetric solution of Einstein’s equations in the vacuum Tµ⌫ = 0. That is to say the exterior of a static star. In spherical coordinates R R 1 ds2 = 1 S dt2 1 S dr2 r2(d✓2 +sin2 ✓d2). (1.10) r r ⇣ ⌘ ⇣ ⌘ Since we are interested in the light path near the star (null-geodesic) we consider the Lagrangian of a massless particle = 1 g x˙ µx˙ ⌫ = 0. The radial Euler-Lagrange equation gives L 2 µ⌫ 2 2 2 E 3 (u0) + u = + R u , (1.11) L2 s d 1 ˙ Rs 2 ˙ where 0 = d , u = r and we highlighted the two constants of motion E = t 1 r and L = r . After di↵erentiating with respect to we obtain

3 2 u00 + u = R u . (1.12) 2 S Using the fact that the impact parameter will be much bigger than the Schwarzschild radius we can treat 3 2 the term 2 RSu as a perturbation and substitute it by zero to lowest order

u00 + u 0 . (1.13) ' 4Fluctuations of the CMB and perturbations of the Universe are observational facts, independently of Inflation. Inflation is the best theory we have to explain where these fluctuations come from.

5 (-0.31, 7.38) 7 S η

6

5

4

M ξ

3 α

Figure 1.3: The geometry of a 2 Schwarzschild lens: The mass M is lo- cated at a distance Dd from the observer O.ThesourceS is at distance Ds from β O.Theangle is the angular separation

1 of the source from the mass in absence of lensing and ↵ is the deflection angle of a O light ray with impact parameter ⇠. (10.47, 0.21)

For an impact parameter ⇠ Eq. 1.13 has solution u =sin/⇠. Inserting this in the full Eq. 1.12, which 3 2 contains the perturbation 2 RSu , and solving at first order, we find 1 3R 1 u = sin + s 1+ cos 2 (1.14) ⇠ 4⇠2 3 ⇣ ⌘ Assuming a small deflection angle and considering that in the limit r the angle becomes very small, we get !1 2R ↵ =2 = S (1.15) | 1| ⇠ For the sun 1.75 ↵ = 00 (1.16) ⇠/R The validity of this results in the gravitational field of the sun has been confirmed with radio-interferometric methods with less than 1% uncertainty [7]. This simple result for the deflection angle is enough to demon- strate the e↵ects of the simplest gravitational lens configuration: the Schwarzschild lens. As shown in Figure 1.3 a mass M is located at a distance Dd from the observer O and the source S is at distance Ds from O. The angle is the angular separation of the source from the mass in absence of lensing and ↵ is the deflection angle of a light ray with impact parameter ⇠ RS (in this sense the mass M is called a point mass ). The condition under which the ray meets the observer in O is obtained only from geometrical consideration

Ds 2RS Ds = ⇠ Dds . (1.17) Dd ⇠ Even though, for simplicity, we will stick throughout this work to a flat geometry K = 0, this is a good moment to point out that in general Dds = Ds Dd (since in the definition of angular diameter distance one has to account for the curvature of space6 with trigonometric functions, the equality holds only for K = 0). The angular separation between the mass M and the deflected ray (i.e. the image) is denoted by ✓ = ⇠/Dd and we can define a characteristic angle

D ↵ = 2R ds , (1.18) 0 S D D r d s

6 the meaning of which will be clear in a moment. The lens equation 1.17 is then rewritten

✓2 ✓ ↵2 =0. (1.19) 0 The solutions are the positions of the images

1 2 2 ✓ = 4↵0 + . (1.20) ± 2 ± q The term “image” (as the term “lens” actually)⇣ is abused in⌘ the sense that the rays are not focused at the observer, nevertheless the angular separation between the two images’ positions is ✓ = ✓+ ✓ = 4↵2 + 2 2↵ . We can see that 2↵ is the minimum of the angular separation, this happens when = 0. 0 0 0 This situation is called Einstein ring, the solutions are ✓ = ↵0 and, due to symmetry, the whole ring p ± ± of angular radius ✓ = ↵0 is a solution: a point source placed exactly on the line mass-observer is seen as a circle around the lens. For a detailed discussion see [8] and [3].

1.4 Lens Mapping

We have discussed the Schwarzschild lens example and computed the deflection angle in Eq. 1.15. We point out that, for geometrically-thin lenses, the deflection angles of several point masses simply add, such that one finds (recalling RS =2GM) ⇠ ⇠ ↵(⇠)= 4Gm i , (1.21) i ⇠ ⇠ 2 i i X | | where, in the lens plane, ⇠ is the position of the light ray while ⇠i are the positions of the masses mi (or the projected positions of the masses on the lens plane). This can be trivially extended to the continuum limit

2 ⇠ ⇠0 ↵(⇠)=4G d ⇠0 ⌃(⇠0) 2 , (1.22) ⇠ ⇠0 Z | | where we have defined the surface mass density distribution ⌃(⇠0) and the integral extends over the whole lens plane. The lens equation 1.17 can be written in terms of the distance ⌘ = Ds from the source to the optical axis

Ds ⌘ = ⇠ Dds ↵(⇠) . (1.23) Dd Given the matter distribution and the position of the source, the lens equation may have more than one solution5 ⇠. The lens equation describes a mapping ⇠ ⌘ from the lens plane to the source plane and this ! mapping is obtained straightforwardly for any mass distribution ⌃(⇠0). On the other hand the inversion of this mapping can be done analytically only for very simple mass distributions since ⇠ ⌘ is not linear. It is useful to scale the problem to dimensionless variables, by defining ! ⇠ ⌘ x = , y = (1.24) ⇠0 ⌘0

where ⇠0 is a length scale in the lens plane and ⌘0 = ⇠0Ds/Dd. Defining the dimensionless surface mass density

⌃(⇠0x) Ds (x)= with ⌃c = , (1.25) ⌃c 4⇡GDdDds 5It turns out actually than any transparent mass distribution with finite total mass and with a weak gravitational field produces an odd number of images [3].

7 the scaled deflection angle is obtained as

1 2 x x0 ↵ˆ(x)= d x0 (x0) , (1.26) ⇡ x x 2 Z | 0| and the lens equation reduces to

y = x ↵ˆ(x) . (1.27) x Furthermore since ln x = x 2 we have ↵ˆ = ,where r | | | | r

1 2 2 (x)= d x0 (x0)lnx x0 (1.28) ⇡ | | Z is the lensing potential associated with the surface density (x). The dimensionless mapping x y is a gradient mapping !

1 y = x2 (x) (1.29) r 2 ✓ ◆ which can also be expressed in terms of the scalar function 1 (x, y)= (x y)2 (x) (1.30) 2 as

(x, y)=0. (1.31) r What we are introducing here is the so called scalar formalism. The amplification factor6 is defined through the scalar function (x, y)

1 @2(x, y) I = det . (1.32) @x @x  i j The potential can be used to understand intuitively lens geometries: if we consider the graph x (x, y), for fixed y, this is a surface such that images of the source at position y, appear at the minima,! maxima and saddle points of this surface. Furthermore the amplification factor can be understood as the curvature of this surface. For more details see [9] and [10].

6The amplification factor is given by the ration I = S/S˜ where S˜ and S are respectively the flux of the source in absence of lensing and considering the lensing e↵ects.

8 Chapter 2

Lensing by Scalar Perturbations

2.1 Perturbed Photon Path and Lensing Potential

Since our aim is to discuss the e↵ect of GL on the CMB, the first step is to compute the deflection of a light ray in a FL universe perturbed by scalar (density) perturbations. We will closely follow the derivation in [11] and [12]. We first note that the lensed CMB temperature field in a direction nˆ, ⇥˜ (nˆ), is given by the unlensed temperature in the deflected direction: ⇥˜ (nˆ)=⇥(nˆ + ↵), where ↵ is the displacement vector. The perturbation is included in the line element using the Bardeen potentials1:and

ds2 = a2(t)( (1 + 2 )dt2 +(1 2) dxidxj) , (2.1) ij where, as usual (for K = 0),

i j 2 2 2 2 2 ijdx dx = dr + r (d✓ +sin ✓d) . (2.2) In computing the deflection of light we are only interested in null-geodesics and we may as well consider the conformally related metric

ds˜2 = (1 + 4 )dt2 + dxidxj , (2.3) W ij 1 where we have introduced the Weyl potential W = 2 ( +). We now consider a photon moving on the unperturbed trajectory towards the observer (that we set in x = 0): the photon will come radially with unperturbed velocity⌘ ¯µ =(1, 1, 0, 0) = (1, ~n). The perturbed velocity is then written ⌘µ =(1+n0, ~n + ~n). The Christo↵el’s symbols for the conformally related metric ds˜2 are easily computed at first order (see Appendix B) and we can write the geodesic equations § d2xµ dx↵ dx +µ = 0 (2.4) ds2 ↵ ds ds µ at first order in W and in n 2 2 ✓¨ = @ + ✓˙ , (2.5) r2 ✓ W r 2 2 ¨ = @ W + ˙ , (2.6) r2 sin2 ✓ r where = d/ds. We can also rearrange these equations using the fact that, at lowest order, for a radial geodesicr ˙·= 1 and ✓˙ = ˙ =0 1For a discussion on how the Bardeen potentials are defined in the gauge invariant perturbation theory see [11] pp. 57-80.

9 (✓˙ r2)=˙ 2 @ , (2.7) ✓ W 2 2 (˙ r )=˙ 2sin ✓@ . (2.8) W We integrate these equations using another lowest order equality: ds = dt, and we obtain

t 2 r (t t)✓˙(t)=2 dt0 @ (t0,t t0,✓ , ) , (2.9) 0 ✓ W 0 0 0 Zt0 t 2 dt0 r (t0 t)˙(t)=2 @ W (t0,t0 t0,✓0,0) , (2.10) sin2 ✓ Zt0 0 where t0 is the time at which the observer in x = 0 receives the photon and the integration constant is 2 fixed consistently with the fact that r (t = t0) = 0. Integrating once more

t ⇤ t t ✓(t )=✓0 +2 dt ⇤ @✓ W (t, t0 t, ✓0,0) , (2.11) ⇤ t (t0 t )(t0 t) Z 0 ⇤ t ⇤ dt t t ⇤ (t )=0 +2 2 @ W (t, t0 t, ✓0,0) , (2.12) ⇤ t sin ✓0 (t0 t )(t0 t) Z 0 ⇤ where t is the initial (emission) time. We can finally⇤ define the displacement vector on the sphere (deflection angle) as

✓ ✓ ↵ = 0 (2.13) sin ✓ ( ) ✓ 0 0 ◆ so that

t0 t t ↵ = 2 dt ⇤ W (t, t0 t, ✓0,0) , (2.14) t (t0 t )(t0 t)r Z ⇤ ⇤ 1 where =(@ , sin ✓@ ) is the gradient on the sphere. Of course when we will study the CMB we r ✓ will set t = tdec. It is also important to point out that working at first order allows us to use the Born approximation⇤ and compute the integral on the unperturbed path.

Lensing Potential Looking at Eq. 2.14 is clear that we can rewrite it as ↵ = ⌥, through the definition of the Lensing Potential r t0 t t ⌥= 2 dt ⇤ W (t, t0 t, ✓0,0) . (2.15) t (t0 t )(t0 t) Z ⇤ ⇤ A single 2D map of the lensing potential ⌥contains all the required information since recombination is approximated as instantaneous (a single hyper-surface source at t = tdec) and in general the e↵ects of late-time sources, including reionization, are neglected. ⇤ As for the CMB temperature field ⇥(nˆ), we are interested in the angular power spectrum of the lensing potential ⌥(ˆn ). This will be written in terms of the power spectrum of the gravitational potential P .The lensing potential can be expanded in spherical harmonics since it lives on the sphere

⌥(ˆn )= lmYlm(ˆn ) (2.16) Xlm ⌥ and we are interested in computing ⇤ = C . The gravitational potential is a 3D field lm l0m0 ll0 mm0 l and it has to be expanded in Fourier spaceh i

10 3 d k ik x (x; t)= (k; t) e · . (2.17) W (2⇡)3 W Z We define the power spectrum of a statistically isotropic variable f

3 f(k; t)f ⇤(k0; t0) =(2⇡) P (k; t, t0) (k k0) , (2.18) h i f such that

3 (k; t) ⇤ (k0; t0) =(2⇡) P (k; t, t0) (k k0) . (2.19) h W W i Using Eq. 2.15 and 2.19 the angular correlation function for the lensing potential is

t t ⇤ ⇤ t t t0 t ⌥(ˆn )⌥(ˆn 0) =4 dt dt0 ⇤ ⇤ h i t t (t0 t )(t0 t) (t0 t )(t0 t0) Z 0 Z 0 ⇤ ⇤ (2.20) 3 d k 3 ik x(t) ik x0(t) (2⇡) P (k; t, t0) e · e · , ⇥ (2⇡)6 Z

where x(t)=ˆn (t0 t) and x0(t)=ˆn 0(t0 t0). We can now make explicit the angular dependence by using the Rayleigh’s expansion

ik x ik n(t t) l e · = e · 0 =4⇡ i j (k(t t))Y (ˆn )Y ⇤ (kˆ) (2.21) l 0 lm lm Xlm and performing the angular integral over k using the properties of spherical harmonics (see Appendix C) we find §

t t 8 ⇤ ⇤ t t t0 t ⌥(ˆn )⌥(ˆn 0) = dt dt0 ⇤ ⇤ h i ⇡ t0 t0 (t0 t )(t0 t) (t0 t )(t0 t0) ll0Xmm0 Z Z ⇤ ⇤ (2.22) 2 dk k jl(k(t0 t))jl (k(t0 t0))P (k; t, t0)Ylm(ˆn )Y ⇤ (ˆn 0)ll mm , ⇥ 0 l0m0 0 0 Z 2 where we’ve made use of the spherical Bessel function j . Now we can simply recall ⇤ = l lm l0m0 ⌥ h i ll0 mm0 Cl and read o↵the spherical harmonics components of Eq. 2.22 to find

t t ⌥ 8 2 ⇤ ⇤ t t t0 t Cl = dk k dt dt0 ⇤ ⇤ jl(k(t0 t))jl(k(t0 t0))P (k; t, t0) . ⇡ t t (t0 t )(t0 t) (t0 t )(t0 t0) Z Z 0 Z 0 ⇤ ⇤ 2 Here we can introduce the primordial dimensionless power spectrum (k) which is related to the power spectrum P through the transfer function T (k, t)

2 2⇡ 2 P (k; t, t0)= (k)T (k, t)T (k, t0) . (2.23) k3 This allow us to recast

t 2 ⌥ dk 2 ⇤ t t Cl = 16⇡ (k) dt ⇤ jl(k(t0 t)) T (k, t0 t) . (2.24) k t (t0 t )(t0 t) Z Z 0 ⇤ 2 In terms of the ordinary Bessel functions Jn the spherical Bessel are defined j (x)= ⇡ J (x). For more details see [21], [22] or Appendix D. l 2x l+1/2 § q

11 2 ⌥ Given some primordial power spectrum (k), the Cl coecients can be computed using numerical codes such as CAMB3, CMBFAST4 or CLASS5 [13]. The numerical computation is lighter if we use Limber approximation (see Appendix E) to get § t 2 ⌥ 64 ⇤ t t l +1/2 ⇤ Cl 4 dt P ; t, t . (2.25) ' (2l + 1) t (t0 t )(t0 t) t0 t Z 0 ✓ ⇤ ◆ ✓ ◆ 2 In Figure 2.1 and 2.2 is shown the power spectrum for a ⇤CDM cosmology with h =0.67, ⌦mh =0.14, 2 ⌦bh =0.022 and ⌦K = 0. The primordial dimensionless power spectrum for scalar perturbation is

n 1 k s 2(k)=A , (2.26) s k ✓ ⇤ ◆ 1 9 with k =0.05 Mpc , As(k )=2.215 10 and the spectral index ns =0.962. ⇤ ⇤ ⇥

-3 ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ k < 10 ÊÊÊÊÊÊ ÊÊÊÊÊÊÊÊÊÊÊÊÊ 4 ÊÊÊÊ ÊÊÊÊÊÊÊ 2¥10 ÊÊ ÊÊÊÊÊ -3 -2 ÊÊ ÊÊÊÊÊ < < Ê Ê ÊÊÊÊÊ 10 k 10 Ê ÊÊÊÊÊ -7 Ê ÊÊÊÊÊ -2 -1 ÊÊÊÊÊ 1¥10 Ê ÊÊÊÊ 10 < k < 10 4 Ê ÊÊÊ 1¥10 ÊÊÊ Ê ÊÊÊÊ ÊÊÊÊ ÊÊÊÊ Ê ÊÊÊÊ ÊÊÊ ÊÊÊ 5000 ÊÊÊ Ê ÊÊÊ -8 ÊÊÊ 5¥10 ÊÊÊ ÊÊÊ ÊÊÊ p ÊÊÊ ÊÊÊ 2 Ê ÊÊÊ ê ÊÊ ÊÊÊ l Ê 2000 ÊÊÊ ÊÊÊ L Ê C ÊÊ ÊÊÊ k Ê ÊÊÊ H ÊÊ 2 Ê ÊÊÊ L Ê P ÊÊ 1000 -8 ÊÊÊ 1 Ê 2¥10 ÊÊÊ ÊÊ + Ê ÊÊÊ l Ê ÊÊÊ H Ê ÊÊ ÊÊ 2 Ê 500 ÊÊ l ÊÊ ÊÊ ÊÊ ÊÊ -8 ÊÊ ¥ ÊÊ 1 10 ÊÊ ÊÊ ÊÊ ÊÊ 200 ÊÊ ÊÊ ÊÊ ÊÊ ÊÊ ÊÊ ÊÊ -9 ÊÊ 100 ÊÊ 5¥10 ÊÊ ÊÊ ÊÊ ÊÊ ÊÊ ÊÊ ÊÊ ÊÊ ÊÊ ÊÊ 10-5 10-4 0.001 0.01 0.1 1 5 10 50 100 500 1000 Ê k h Mpc multipole l

Figure 2.1: Left: The matter densityH ê fluctuationsL power spectrum today: the dots are the values of CLASS while the dot dashed line is the plot obtained by evolving the primordial spectrum in Eq. 2.26 using the analytic approximation of [14] for the matter transfer function. Right: The power spectrum of the lensing potential: the values of CLASS (dots), the Limber Approximation (dot dashed) 1 and the contribution of di↵erent wavenumbers k in hMpc (dashed).

10-7 1¥10-7 5¥10-8 10-8 p p -8

¥ 2 - 2 1 10 9 ê ê 10 l l -9 C C 5¥10 2 2 L L - z<•

1 10 1 10 + +

l < l z 0.2 H H -9 2 2 ¥

1 10 l

l z<0.5 -10 z>0 -11 < 5¥10 z>0.5 10 z 1 z>1 z<3 z>5 z<10 z>10 10-12 1¥10-10 z>20

5 10 50 100 500 1000 5 10 50 100 500 1000 multipole l multipole l

Figure 2.2: The contribution of di↵erent redshifts (dot dashed) to the power spectrum of the lensing potential (solid).

3http://camb.info 4http://lambda.gsfc.nasa.gov/toolbox/tb_cmbfast_ov.cfm 5http://class-code.net/

12 0.10 10-10

p -11

2 10 Q ê l C

Q 0.05 ê l -12

C 10 Q L

Q l

1 Unlensed C

l C +

l -13 D H 10 l Q Lensed Cl 0.00 10-14

10-15 200 500 1000 2000 5000 0 500 1000 1500 2000 2500 multipole l multipole l

Figure 2.3: Left: The lensed temperature power spectrum (solid) and the unlensed spectrum (dashed). Right: The fractional change 2 2 in the power spectrum due to lensing. Both plots are for a typical concordance ⇤CDM model with h =0.67, ⌦mh =0.14, ⌦bh =0.022 and ⌦K =0.

2.2 The CMB Lensed Power Spectrum

As staten before GL a↵ects also the temperature field of the CMB and we now want to determine how, by ⌥ computing the lensed CMB power spectrum at lowest order in Cl . We shall assume that temperature fluctuations are statistically isotropic and we work in harmonic space using the flat sky 2D Fourier transform convention

2 2 dl il x dx il x ⇥(x)= ⇥(l)e · , ⇥(l)= ⇥(x)e · . (2.27) 2⇡ 2⇡ Z Z For the assumption of isotropy the correlation function ⇠ can only depend on the separation between two points ⇥(x)⇥(x0) = ⇠( x x0 ), so that h i | |

2 2 d x d x0 il x il0 x0 ⇥(l)⇥⇤(l0) = e · e · ⇠( x x0 ) h i 2⇡ 2⇡ | | Z 2 Z 2 d x d r i(l0 l) x il0 r = e · e · ⇠(r) (2.28) 2⇡ 2⇡ Z Z 2 il r = (l0 l) d r e · ⇠(r) , Z where r = x x0. We can now read from Eq. 2.28

⇥ 2 il r ilrcos(l r) Cl = d r e · ⇠(r)= rdr dre ⇠(r)=2⇡ rdrJ0(lr)⇠(r). (2.29) Z Z Z Z Here we have used the Bessel function J0(r) that arises from the integration. Of course the same equa- ⌥ tions also relate the lensing potential correlation function to its power spectrum Cl .

Lensing re-maps the temperature according to

⇥˜ (x)=⇥(x + ⌥) r 1 (2.30) ⇥(x)+ a⌥(x) ⇥(x)+ a⌥(x) b⌥(x) ⇥(x)+... ' r ra 2r r rarb This series expansion is not likely to be accurate on all scales: when l & 3000 the deflection angle is comparable to the wavelength of the unlensed field [11], so the Taylor expansion is no longer a good approximation. Using

13 2 2 d l il x d l il x ⌥(x)=i l ⌥(l)e · , ⇥(x)=i l ⇥(l)e · (2.31) ra 2⇡ ra 2⇡ Z Z and assuming that ⌥and ⇥are uncorrelated ⌥(ˆn )⇥(ˆn ) = 0, we can obtain the Fourier components of ⇥˜ (l) h i

2 d l0 ⇥˜ (l) ⇥(l) l0 (l l0)⌥(l l0)⇥(l0) ⇠ 2⇡ · Z 2 2 (2.32) 1 d l1 d l2 l (l + l l)l l ⇥(l )⌥(l )⌥⇤(l + l l) . 2 2⇡ 2⇡ 1 · 1 2 1 · 2 1 2 1 2 Z Z

The covariance for the lensed field is still statistically isotropic so that is diagonal: ⇥˜ (l)⇥˜ ⇤(l0) = ˜⇥ ⌥ h ˜ i (l l0)Cl . At lowest order in Cl there are contributions from the square of the first order term in ⇥, and also a cross-term from the zeroth and second order terms. In the end, once can show [11],[15]

2 ˜⇥ 2 ⌥ ⇥ d l0 2 ⌥ ⇥ Cl (1 l R )Cl + [l0 (l l0)] C l l Cl , (2.33) ⇠ 2⇡ · | 0| 0 Z where we have defined half of the total deflection angle power 1 1 R⌥ = ⌥ 2 = dl l3C⌥ . (2.34) 2 h|r | i 4⇡ l Z The e↵ect of GL on the CMB power spectrum can be understood in Figure 2.3. This e↵ect is several percent at l & 1000. On small scales where there is little power in the unlensed CMB, GL transfers power from large scales to small scales to increase the small-scale power. For a complete treatment of lensing of the CMB by scalar perturbations, including a discussion on the validity of the Taylor expansion, the small-scale limit, the correlation function for the flat-sky limit and lensing of polarization one can see [11] or [12]. For a full-sky treatment see [15].

14 Chapter 3

Lensing by Tensor Perturbations

3.1 Perturbed Photon Path

Following the same path used in the previous chapter, we compute the deflection of a light ray in a FL Universe perturbed by Gravitational Waves (GW). The perturbation is included into the line element

ds2 = a2(t)( dt2 +( + h )dxidxj) (3.1) ij ij and it is parametrized by hij 1 a symmetric, traceless and divergence-free 3 3 tensor. Working in the conformally related metric| |⌧ ⇥

ds˜2 =( dt2 +( + h )dxidxj) (3.2) ij ij will be more convenient. In spherical coordinates, for a flat universe K = 0, the metric is written

10 0 0 01+hrr rhr✓ r sin ✓hr g˜µ⌫ = 2 2 2 3 . (3.3) 0 rhr✓ r (1 + h✓✓) r sin ✓h✓ 6 0 r sin ✓h r2 sin ✓h r2 sin2 ✓ (1 + h )7 6 r ✓ 7 Similarly to the scalar perturbations4 computation we consider a photon with5 unperturbed radial tra- jectory towards the observer and hence with perturbed velocity ⌘µ =(1+n0, ~n + ~n)=(1+n0, 1+ nr,n✓,n) and the aim is to solve the geodesic equations

d2xµ dx↵ dx +µ =0, (3.4) ds2 ↵ ds ds at first order in the perturbation and in nµ. We will only need the geodesics equations for µ = ,✓ to compute the deflection angle (the displacement along the line of sight leads to a time delay which is not relevant in computing ↵)

d 2 1 d (r ✓˙)= @✓hrr + r hr✓ hr✓ ds 2 ds (3.5) d 2 1 1 r d (r ˙)= @hrr hr + hr . ds 2sin2 ✓ sin ✓ sin ✓ ds Integrating these equations, using the fact that at lowest order dt = ds, gives

15 t 2 t 1 t [r ✓˙] = dt0 @ h (t0,t t0,✓ , ) +[rh (t0,t t0,✓ , )] , t0 2 ✓ rr 0 0 0 r✓ 0 0 0 t0 Zt0 ✓ ◆ t 2 ˙ t 1 1 1 t [r ]t0 = 2 dt0 @hrr(t0,t0 t0,✓0,0) + [rhr(t0,t0 t0,✓0,0)]t0 . sin ✓ 2 sin ✓0 0 Zt0 ✓ ◆

Here t0 is, as before, the time of arrival at x = 0. Given that r(t0) = 0 and ✓˙(t0)=˙(t0) = 0 we get

1 t 1 1 ✓˙(t)= dt0 @ h (t0,t t0,✓ , ) + h (t0,t t0,✓ , ) , (t t )2 2 ✓ rr 0 0 0 (t t ) r✓ 0 0 0 0 Zt0 ✓ ◆ 0 1 t 1 1 ˙(t)= dt0 @ h (t0,t t0,✓ , ) + h (t0,t t0,✓ , ) . 2 2 rr 0 0 0 r 0 0 0 sin ✓ (t t ) 2 sin ✓0(t t0) 0 0 Zt0 ✓ ◆ Integrating this equations once more leads to

t ⇤ hr✓(t, t0 t, ✓0,0) 1 t t ✓(t )=✓0 + dt + ⇤ @✓hrr(t, t0 t, ✓0,0) , ⇤ t (t t0) 2 (t t0)(t0 t ) Z 0  ⇤

t ⇤ hr(t, t0 t, ✓0,0) 1 t t ⇤ (t )=0 + dt + 2 @hrr(t, t0 t, ✓0,0) . ⇤ t sin ✓0(t t0) 2sin ✓0 (t t0)(t0 t ) Z 0  ⇤ 1 The gradient on the sphere is =(@✓, sin ✓@) and recalling the definition in Eq. 2.13 we can write the displacement vector ↵ in a compactr form

t ⇤ hri(t, t0 t, ✓0,0) 1 t t ↵i = dt + ⇤ ihrr(t, t0 t, ✓0,0) . (3.6) t (t t0) 2 (t t0)(t0 t )r Z 0  ⇤ 3.2 Euler Rotation method

Having found an expression for the displacement vector we now have to work out a way to describe tensor perturbations: in this section we compute the Fourier components of the tensor field in the frame (r, e✓, e) which we have used in the displacement vector. We start by defining the Fourier transform of tensor metric perturbation as

3 d k ik x h (x,t)= e · h(t)e(k)+h⌦(t)e⌦(k) , (3.7) ij (2⇡)3 k ij k ij Z ⇥ ⇤ where we have introduced two time-independent polarization tensors eij(k) and eij⌦(k) which can be expressed in terms of vectors of the orthonormal basis (kˆ, e1, e2)

1 1 1 2 2 e(k)= [e (k)e (k) e (k)e (k)] , ij p2 i j i j (3.8) 1 1 2 2 1 e⌦(k)= [e (k)e (k) e (k)e (k)] . ij p2 i j i j

The polarization tensors depend on k since the frame (kˆ, e1, e2), which reveals the 2 physical d.o.f. of the GW in Eq. 3.7, is clearly k dependent. In this frame the GW has non zero components only in the plane (e1, e2) and, for fixed k, we can always find a basis in which, in the plane to kˆ ?

16 Figure 3.1: Euler rotation: The two vectors k and r are showed together with the three Euler angles (↵,,)necessarytoturn (e1, e2, k)into(e✓ , e, r).

10 01 (e)= , (e⌦)= (3.9) ij 0 1 ij 10 ✓ ◆ ✓ ◆ such that the tensor perturbation can be written

h h⌦ h˜(k)= k k . (3.10) h⌦ h ✓ k k ◆ Introducing the helicity basis

1 1 2 e± = (e ie ) , (3.11) p2 ⌥ we can write the tensor perturbation in this basis

1 + ˜ hk ihk⌦ 0 hk 0 + + hE(k)= = = hk eij + hkeij , (3.12) p2 0 h + ih⌦ 0 h ✓ k k ◆ ✓ k ◆ 1 where we have defined e± = e±e±. We have also set h± (h ih⌦) and we shall work with these ij i j k ⌘ p2 k ⌥ k from now on1. We now want to write the polarization tensors in the spherical basis of equation 3.2. In order 1 2 to do that we perform a rotation with Euler angles (↵,,)toturn(k, e , e )into(r, e✓, e).

The rotation proceeds as follows (see Figure 3.1):

1. Rotation around k by angle to align e1 with the node line connecting k & r 2. Rotation around e2 by angle to align k and r

3. Rotation around k = r by angle ↵ to align e1 & e✓ and e2 & e.

For the three Euler rotations we write

h˜ = R ( ↵)R ( )R () h˜ RT ()RT ( )RT ( ↵) , (3.13) S 1 3 1 1 3 1 1 , Working with hk± instead of the usual hk ⌦ is more convenient and, in the end, it will not make any di↵erence since that, when substituting the power spectra: + + 1 h h ⇤ = ( hh⇤ + h⌦h⌦⇤ )= hh⇤ h k k i 2 h k k i h k k i h k k i

17 ˜ 1 2 ˜ where h is the tensor perturbation in the frame (k, e , e ) which is a function of hk± and hS is the perturbation in the frame (r, e✓, e). Performing the transformation we can write the Fourier components in (r, e✓, e) as

rr 1 2 2i + 2i h = sin (e h + e h) k 2 k k 2i + 2i (3.14) r 1 r✓ r sin() (cos() 1)e hk + (cos() 1)e hk i↵ hk± (hk ihk )= ⌥ ± e . ⌘ p2 ⌥ 2p2 We point out that ↵, and are function of both kˆ and ˆn . Writing Eq. 3.6 in Fourier components we obtain

t 3 r ⇤ h ± 1 d k k ik(t t0)cos ↵ = (↵✓ i↵)= dt 3 e ± p2 ⌥ t (2⇡) t t0 Z 0 Z ⇣ (3.15) 1 t t rr ik(t t0)cos + ⇤ ( ✓ i )(hk e ) , 2p2 (t t0)(t0 t ) r ⌥ r ⇤ ⌘ ik r ikr cos where we used the fact that is the angle between k and r such that e · = e . From this, using Eq. 3.14 we can write ↵ in terms of the modes h±. ± k 3.3 Scalar and Vector Lensing Potential

In the case of Scalar Perturbation it is clear that the deflection angle ↵ can be written as a gradient of a single scalar lensing potential ↵ = ⌥. When considering tensor perturbations this is no longer true and we have to decompose ↵ generically,r in its gradient-mode and curl-mode potentials. We write

↵ = + A~ (3.16) r r⇥ Since we want ↵ S2, we allow the curl-mode potential A~ to have only a radial component A~ =($, 0, 0) 2 in this way A~ will not generate a radial part which is not possible for the deflection angle2. In spherical coordinatesr⇥

1 @✓ ~ sin ✓ @$ = 1 , A = (3.18) r @ r⇥ @✓$ ✓ sin ✓ ◆ ✓ ◆ explicitly

1 ↵✓ = @✓ + @$, sin ✓ (3.19) 1 ↵ = @ @ $. sin ✓ ✓ 2This is actually an intuitive explanation while the mathematics behind is more complicated and involves the Hodge dual (?) and the exterior derivative (d). Given that, in a n-dimensional space, ? : k-form (n k)-form and d : k-form (k +1)-form, the curl can be generally defined as ! !

( !)=?(d!) r⇥ (3.17) ( ):k-form (n k 1)-form r⇥ ! n Since the space of k-form has dimension k ,in3dimensionthecurltakesa1-form(a3Dvector)intoanother1-form(a3D vector). In 2 dimension however the curl takes a 0-form (a scalar) into a 1-form (a 2D vector): ( !)i =(? !)i = "ij !. r⇥ r rj

18 Let us simply introduce the spin raising and lowering operators @/ and @/⇤ i @/ = s cot ✓ @ @ , (3.20) s ✓ sin ✓ ⇣ ⌘ i @/⇤ = s cot ✓ @ + @ . (3.21) s ✓ sin ✓ Later we discuss their relations with the⇣ spin-s Spherical Harmonics.⌘ In the helicity basis we can write

1 ↵+ = @/⇤( + i$) , p2 (3.22) 1 ↵ = @/( i$) . p2

We can rewrite the explicit expressions for ↵ in terms of = t t0 ± 3 r ⇤ d k hk± ik cos 1 rr ik cos ⇤ ↵ = d 3 e ( ✓ i )(hk e ) ± 0 (2⇡) 2p2 r ⌥ r Z Z ✓ ⇤ ◆ ⇤ and observing that = p2 e ,e+ = ( ✓ i )=@/,@/ we can write two di↵erential equations for and $ r r ± r

3 r+ ⇤ d k hk ik cos 1 rr ik cos /⇤ p ⇤ /⇤ @ ( + i$)= 2 d 3 e + @ (hk e ) , 0 (2⇡) 2p2 Z Z ✓ ⇤ ◆ 3 r ⇤ d k hk ik cos 1 rr ik cos / p ⇤ / @( i$)= 2 d 3 e + @(hk e ) . 0 (2⇡) 2p2 Z Z ✓ ⇤ ◆ Our aim is to get an expression for and $ and the idea behind that is to write the equations above as @/⇤( + i$)=@/⇤([...]) and @/( i$)=@/([...]). Basically we wish to recast everything on the RHS inside the rr spin lowering and raising operators. We first observe that the hk part is already in the desired form and we r + have to worry only about the hk± parts. To illustrate the procedure we do it explicitly for the hk part of ↵+; for the rest one can follow the same steps. Using Eq. 3.14 we write explicitly this part in terms of the Euler angles

3 ⇤ d k 1 sin ( 1 + cos ) i↵ 2i ik cos + ↵+ = dt e e e h +[...]h + ... (3.23) (2⇡)3 2p2 k k Z0 Z ✓ ◆ and we make the ansatz

1 sin ( 1 + cos ) ei↵e2i eik cos = @/⇤ g()e2i eik cos (3.24) 2p2 ⇣ ⌘ To understand this assumption, one should know that, from geometrical considerations, we have

@✓(✓,)= cos ↵ (3.25) @(✓,)=sin↵ sin ✓

and we then expect the ↵ dependance ei↵ to be generated by @/⇤. Solving the di↵erential equation 3.24, we find

i(i + k + k cos ) tan2 ik cos 2 2 g()=Ce tan (3.26) 2 2p2 k23 ⇣ ⌘ ✓ ◆

19 + It turns out that the hk part of ↵ has the same solution g(), while the hk part of ↵+ and the hk part of ↵ have solution (1 + ik ik cos ) cot2 ik cos 2 2 µ()=De cot + (3.27) 2 2p2 k23 ⇣ ⌘ ✓ ◆ ik ik The constants C and D are fixed to C = e and D = e by requiring and $ to be 2p2 k23 2p2 k23 square integrable on the sphere. We now have two equations in the form

3 ⇤ d k 2i + 2i 1 rr ik cos /⇤ p /⇤ ⇤ @ [ + i$]= 2 @ d 3 (g()e hk + µ()e hk)+ hk e 0 (2⇡) 2p2 Z Z ⇤ h 3 ⇣ ⌘ i ⇤ d k 2i + 2i 1 rr ik cos / p / ⇤ @[ i$]= 2 @ d 3 (µ()e hk + g()e hk)+ hk e 0 (2⇡) 2p2 hZ Z ⇣ ⇤ ⌘ i so that 3 ⇤ d k 2i + 2i 1 rr ik cos p ⇤ + i$ = d 3 2(g()e hk + µ()e hk)+ hk e 0 (2⇡) 2 Z Z ⇤ 3 ⇣ ⌘ ⇤ d k 2i + 2i 1 rr ik cos p ⇤ i$ = d 3 2(µ()e hk + g()e hk)+ hk e 0 (2⇡) 2 Z Z ⇣ ⇤ ⌘ by simply adding and subtracting we find the scalar and the vector potentials. Inserting the results from Eq. 3.26, 3.27 and 3.14 we find

ik ik(cos()+1) 2 2 4 4 2ik e 4e 2csc ()+ik cos() 1 8csc () sin 2 + cos 2 e ˜ = 0 ⇣ 8k23 ⇣ ⇣ ⌘ ⇣ ⌘ ⌘⌘ @ 1 2 ik cos 2i + 2i + ⇤ sin ()e (he + h e ) , 4 k k ⇤ ◆

ik 2 4 4 2ik ik(cos()+1) e 2i csc () sin + cos e + e (k 2i cot()csc()) 2 2 2i + 2i $˜ = (he +h e ) , ⇣ ⇣ ⇣ ⌘ ⇣ ⌘2k23⌘ ⌘ k k where we have written

3 3 ⇤ d k ⇤ d k = d ˜ and $ = d $˜ (2⇡)3 (2⇡)3 Z0 Z Z0 Z since it is less heavy to work without the integral signs and all the manipulations which will be done in the following (i.e. the angular decomposition) leave the integrals untouched. The two potentials are both square integrable on the sphere

⇡ d sin ˜ 2 < , 0 | | 1 Z ⇡ (3.28) d sin $˜ 2 < | | 1 Z0 and despite the fact that their are not very illuminating at the moment, their angular expansion will be rather simple.

20 3.3.1 Angular decomposition of and $ We define, as in the scalar perturbation case,

(ˆn )= almYlm(ˆn ) , Xlm (3.29) $(ˆn )= blmYlm(ˆn ) Xlm and we are left with the problem of expanding ˜ and$ ˜ in spherical harmonics. This is not trivial since the expression of the potentials are somewhat lengthy and also they are explicit functions of the Euler angles (↵,,) and not of ˆn =(✓,). To go through the computation one needs to have a basic knowledge of spin-s spherical harmonics sYlm(✓,): we simply write the e↵ect of the spin raising and lowering operator on them

@/( Y )= (l s)(l + s + 1) Y , s lm s+1 lm @/⇤(sYlm)=p (l + s)(l s + 1) s 1Ylm . (3.30) p We also wish to find an operator similar to the Laplacian L2 but for s = 0, the spin-weighted L2 such that 6

2 L(s)(sYlm)=l(l + 1) sYlm . (3.31)

Since Eq. 3.31 and 3.30 imply, for the commutators, [L2 , @/]=[L2 , @/⇤] = 0 one can find (see Ap- (s) (s) § pendix C)

2 2 2 cos ✓ 1 2 2s cos ✓ s L (✓,)= @ + @✓ + @ i @ + . (3.32) (s) ✓ sin ✓ sin2 ✓ sin2 ✓ sin2 ✓ ⇣ ⌘ Another useful formula is derived from Eq. 2.21

ik cos ✓ l e = (2l + 1) i jl(k)Pl(cos ✓) (3.33) Xl and its second derivative with respect to cos ✓

ik cos ✓ 1 l e = (2l + 1) i j (k)P 00(cos ✓) . (3.34) k22 l l Xl Finally, as we pointed out before, since and $ are functions of the Euler angles we will first expand them into sYlm(,↵) and then we will need a formula to write them in terms of Ylm(✓,): the rotation with Euler angles (↵,,) which we have performed to rotate (✓k,k)into(✓,) means, for the spherical harmonics,

4⇡ is sYlm (✓k,k) mY ⇤ (✓,)=sYlm(,↵)e . (3.35) 2l +1 0 lm0 r m X0 With these tools3 we can now start the decomposition of the gradient-mode and curl-mode potentials.

3For an extended treatment see [11],[16]

21 Gradient-mode potential To perform the computation is easier to split ˜ in two parts

ik ik(cos()+1) 2 2 4 4 2ik e 4e 2csc ()+ik cos() 1 8csc () sin + cos e 2 2 ˜1 = , ⇣ 8k23 ⇣ ⇣ ⌘ ⇣ ⌘ ⌘⌘ 1 2 ik cos ˜ = ⇤ sin ()e 2 4 ⇤ such that 3 ⇤ d k 2i + 2i = d ( ˜ + ˜ )(he + h e ) . (3.36) (2⇡)3 1 2 k k Z0 Z The second part is simpler and we should start with this. Using Eq. 3.34

1 2 ik cos 1 2 1 l ˜ = ⇤ sin e = ⇤ sin (2l + 1) i j (k)P 00(cos ) 2 4 4 k22 l l l ⇤ ⇤ X (3.37) 1 l jl(k) 2 = ⇤ (2l + 1) i sin P00(cos ) . 4 k23 l ⇤ Xl Working with the spin raising and lowering operators is trivial to find

2l +1 (l 2)! 2 2Yl0(,↵)= sin Pl00(cos ) , (3.38) ± r 4⇡ s(l + 2)! which can be inserted into Eq. 3.37 to give

p⇡ (l + 2)! l jl(k) ˜2 = ⇤ (2l + 1) i 2Yl0(,↵) . (3.39) 2 s (l 2)! k23 ± ⇤ Xl Leaving this result aside for a moment we can do the same for ˜1. The problem here is that is impossible to “guess” the decomposition of ˜1 into sYlm(,↵) as was done in Eq. 3.38 for ˜2. Luckily one finds that 1 L2 (,↵)[ ˜ ]= ( 3 ik cos )sin2 eik cos . (3.40) (2) 1 2

We can now compute this decomposition and then, when all the (,↵)dependenceisinthesYlm(,↵), act with the inverse Laplacian to come back to ˜1. In doing this, we have to deal with a part proportional 2 ik cos to sin e , which goes exactly like the ˜2 calculation, and the other part proportional to

2 ik cos 2 2 cos sin e cos sin P00(cos ) = cos P (cos ) , / l l m where we have introduced the associated Legendre Polynomials Pl (see Appendix C). Now, we can get rid of the cos by using §

2 2 2 (l 1)Pl+1(cos )+(l + 2)Pl 1(cos ) cos P (cos )= (3.41) l 2l +1 and we are back to the already known problem, which is easy to solve since we have already seen that 2 Pl (cos ) 2Yl0(,↵). / ±

We now have both ˜1 and ˜2 decomposed into 2Yl0(,↵). Going back to Eq. 3.36 we reconstruct is ± and we recognise the sYlm(,↵)e part in Eq. 3.35.

22 2i +2i ˜ = ([...]) h± 2Yl0(,↵)e + 2Yl0(,↵)e . (3.42) k l X 4 This gives us the sum over m and the decomposition into Ylm(ˆn ) from which we can read o↵

3 ⇤ d k l (l 2)! l +1 jl(k) jl+1(k) ⇤ alm = dt 3 2⇡i (l + 2)(l 1) l +2 2 3 2 0 (2⇡) " s(l + 2)! 2 k k Z Z ✓ ✓ ⇤ ◆ ◆ (3.43) ˆ + ˆ 2Ylm(k) hk + +2Ylm(k) hk . ⇥ # ⇣ ⌘ Curl-mode potential $ For the vector potential we have

2 k 2 ik cos + +2i 2i L (,↵)[$ ˜ ]= sin e (h e + he ) . (3.44) (2) 2 k k We go through the same computation as before

2 k 2 ik cos + +2i 2i L (,↵)[$ ˜ ]= sin e (h e + he ) (2) 2 k k 1 l 2 + +2i 2i = (2l + 1)i jl(k)sin P00(cos )(h e + he ) 2k2 l k k (3.45) Xl (l + 2)! l jl(k) + +2i 2i = 4⇡(2l + 1) i 2 2Yl0(,↵)(hk e + hke ) s (l 2)! 2k ! ± Xl and act with the inverse Laplacian to go back to the expansion of$ ˜ . Using Eq. 3.35 we find

l (l + 2)(l 1) jl(k) ˆ ˆ + $˜ = 4⇡i 2 (2Ylm(k) hk + 2Ylm(k) hk )Ylm(ˆn ) . (3.46) s l(l + 1) 2k ! Xlm Again, we can read o↵the spherical harmonics expansion coecients of $

3 ⇤ d k l (l + 2)(l 1) jl(k) + blm = d 4⇡i (2Ylm(kˆ) h + 2Ylm(kˆ) h ) . (3.47) (2⇡)3 l(l + 1) 2k2 k k Z0 Z " s ! #

4 ˜ Inserting again the integral signs so that the almsaretheangularcoecientsof and not of

23 3.4 Curl-mode & Gradient-mode Power Spectra

The Power Spectra of the Scalar and Vector potentials are defined by

$ alma⇤ = ll mm C , blmb⇤ = ll mm C . (3.48) h l0m0 i 0 0 l h l0m0 i 0 0 l

Let us first compute Cl

l (l 2)! l0 (l0 2)! a a⇤ =2⇡i 2⇡i (l + 2)(l 1)(l0 + 2)(l0 1) lm l0m0 h i s(l + 2)! s(l0 + 2)! 3 3 ⇤ ⇤ d k d k0 l +1 jl(k) jl+1(k) ⇤ d d0 3 3 l +2 2 3 2 ⇥ 0 0 (2⇡) (2⇡) 2 k k Z Z Z Z "✓ ✓ ⇤ ◆ ◆

l0 +1 0 0 jl0 (k00) jl0+1(k00) l0 ⇤ +2 2 3 2 ⇥ 2 0 k0 0 k00 ✓ ✓ ⇤ ◆ ◆ + + 2Ylm(✓k,k) hk ()++2Ylm(✓k,k) hk() 2Yl0m0 (✓k0 ,k0 ) hk (0)++2Yl0m0 (✓k0 ,k0 ) hk (0) . ⇥h 0 0 i # (3.49) We now use the definitions

+ + 3 3 h ()h ⇤(0) = h()h⇤(0) =(2⇡) (k k0)Ph(k; ,0) (3.50) h k k0 i h k k0 i 3 to get rid of the d k0 integral. We have assumed the two helicities to have identical spectra and to be + uncorrelated (parity invariance), so that h ()h⇤(0) = 0. The ll mm part of Eq. 3.48 is generated by h k k0 i 0 0 the angular integral over d⌦k, and finally we can write

2 (l + 2)(l 1) dk 2 ⇤ l +1 jl(k) jl+1(k) ⇤ Cl =2⇡ h(k) dTh(k, ) l +2 2 3 2 . (3.51) l(l + 1) k 0 2 k k Z Z ✓ ✓ ⇤ ◆ ◆ Here we have written the power spectrum Ph(k; ,0)using 3 k 2 P (k; ,0)= (k)T (k, )T (k, 0) , (3.52) 2⇡2 h h h h 2 where h(k) is the dimensionless primordial power spectrum and Th(k, ) is the tensor transfer function. The same computation can be done for the lensing curl-mode potential giving

2 (l + 2)(l 1) dk ⇤ j (k) C$ = ⇡ 2 (k) dt T (k, ) l . (3.53) l l(l + 1) k h h k2 Z Z0 For the cross-correlation power spectrum CX we have l X almb⇤ = ll mm C =0. (3.54) h l0m0 i 0 0 l This can be understood as conservation of parity due to the fact that the two perturbation helicities are independent. From a mathematical point of view this arises from the angular integral d⌦k in Eq. 3.49: in the C and C$ cases this gives + while for the cross correlation we get = 0. l l ll0 mm0 ll0 mm0 ll0 mm0 ll0 mm0 We have computed the spectra for the gradient-mode and the curl-mode lensing potential of tensor per- turbation. The derivation is independent from the one developed in [17] where the authors use the total angular momentum method (TAM) [24], but the results are in perfect agreement.

24 10-8 10-10

10-9 10-12 p L

-10 2 k ê -14

H 10

l 10 2 C T 2 L

z=0.1 L k 1 H + 2

-11 l H D 10 z=1 10-16 2

l y z=10 Cl w Cl -12 z=100 10 10-18 PGW

-13 10 10-20 1¥10-4 5¥10-4 0.001 0.005 0.010 0.050 0.100 5 10 50 100 500 k 1 Mpc multipole l

Figure 3.2: Left: The dimensionless power spectrum generated by primordial GWs with r =0.2withtheanalyticmatterdomination H ê L $ transfer function. Right: The angular power spectra for the curl-mode Cl and gradient-mode Cl lensing potentials.

At this point is useful to compute the two power spectra. For simplicity and only to get a qualitative understanding of the behavior of the power spectra we do the computation in a matter dominated (MD) 2 universe with h =0.675, ⌦m = 1, ⌦bh =0.022 and ⌦K = 0. The dimensionless power spectrum for primordial tensor perturbation is assumed to be

k nt 2 (k)=rA , (3.55) h s k ✓ ⇤ ◆ with r =0.2 and the other parameters consistent with the scalar perturbations computation. We will r r also assume the second order consistency relation nt = 8 2 8 ns . The transfer function of tensor perturbations in matter domination is 3j (k(t )) T (k, )= 1 0 . (3.56) h k(t ) 0 The behavior of the power spectra is shown in Figures 3.2, 3.3 & 3.4. Unfortunately Limber approximation $ cannot be used in the computation of Cl and Cl since this transfer function is not slowly varying with respect to the spherical Bessel functions.

w y Cl Cl -3 k < 10-3 k < 10 -3 -3 -3 -3 10-11 10 < k < 2 ¥ 10 10 < k < 2 ¥ 10 -3 -2 2 ¥ 10-3 < k < 10-2 10-12 2 ¥ 10 < k < 10

p 10-13 p 2 2 ê ê l l

C C 10-14 2 2 L - L 1 10 15 1 + + l l H H 2 2 l l 10-16 10-17

-19 10 10-18 2 5 10 20 50 100 5 10 20 50 100 multipole l multipole l

$ Figure 3.3: The angular power spectra of the curl-mode Cl and gradient-mode Cl lensing potentials (solid), together with the 1 contribution of wavenumbers k in Mpc (dashed).

We can make a first remark here: since we have r =0.2 one could naively expect the spectra for the tensor lensing potentials to be 5 times smaller than the one generated by scalar perturbation. This is of course not true (C and C$ ⇠are 5 orders of magnitude smaller than C⌥) since density perturbations l l ⇠ l

25 w y Cl Cl z<300 z<300 10-11 z<100 10-11 z<100 z<20 z<20 z<5 z<5 z<1 z<1 -13

p p 10 2 2 ê ê

l -13 l

C 10 C 2 2 L L

1 1 -15

+ + 10 l l H H 2 2 l l 10-15 10-17

10-17 10-19 2 5 10 20 50 100 5 10 20 50 100 multipole l multipole l

$ Figure 3.4: The contribution of di↵erent redshift z (dot dashed) to the power spectra of the curl-mode Cl and gradient-mode Cl lensing potentials (solid). grow while tensor perturbations are constant outside the horizon and then decay inside the horizon. For this reason we also expect the contribution of high redshift to be the main contribution of the lensing potentials power spectra: in the scalar perturbations case we have seen that the main contribution comes from z<10, in the tensor perturbations case, however, we see that the main contribution comes from high redshift z>100.

We also know that primordial GWs are not the only tensor perturbations in our universe: at second order dark matter induces both a vector and tensor power spectra ([18]). In a previous work by Adamek, Durrer & Kunz [19], such a contribution was computed: the power spectrum of the induced tensor perturbations is

i j 2 2 (2) 100 3 2 (ijq k ) ) 2 2 in in 4 P (k)= d q q T (q)T ( k q ) P (q)P ( k q ) k , (3.57) h 9 k2 | | | | Z ✓ ◆ where P (q) is the power spectrum of the gravitational potential and T (q) is the transfer function already used in Section 2. This is of course a second order quantity that vanishes in linear perturbation theory, nevertheless we can use Eq. 3.53 and 3.51 to check if they have a relevant e↵ect in lensing by tensor perturbations. The lensing potentials power spectra of the induced GWs are many orders of magnitude smaller than the lensing potentials spectra of primordial GWs. This again comes from the fact that induced GWs, since they are induced by density perturbation, are relevant at small redshift and they did not have the time to become an important contribution with respect to the primordial one, whose contribution comes from high z.

3.5 The CMB Lensed Power Spectrum

Having computed the power spectra of the the two lensing potentials of GWs, we are now interested in how ⇥ GL a↵ects the shape of the CMB temperature spectrum Cl defined in Eq. 1.9. This means, as we have ˜⇥ done at the end of section 2, computing the lensed CMB power spectrum Cl . We do that in two ways: first the full-sky method, using a Taylor expansion for the deflection angle, then the flat-sky method which is valid for ↵ not necessarily small5. These two regimes (Full-sky small angle & Flat-sky arbitrary angle) are the relevant two since on large scales, when a full-sky treatment is required, ↵ is much smaller than the wavelength being deflected and the Taylor series is appropriate, while for small scales the flat-sky treatment

5We recall that the angle must be anyway quite small (a few arcminutes [12]) to remain in the Weak GL regime where the Born approximation used in section 2.1 and 3.1 is still valid. We will see later that the deflection angle for GWs is of the order of some arcsec.

26 is sucient but the Taylor expansion fails6.

3.5.1 Full-Sky - Small angle Assuming a small deflection angle ↵ we can Taylor expand ⇥˜ (ˆn )=⇥(ˆn + ↵) to second order 1 ⇥˜ (ˆn )=⇥(ˆn + ↵) ⇥(ˆn )+ a⇥ ↵ + b a⇥ ↵ ↵ + ... (3.58) ' r a 2r r a b Recalling the angular decompositions of Eq. 3.29 and Eq. A.1 this yields to

a 1 b a ✓˜ = ✓ + d⌦ ⇥ ↵ + ⇥ ↵ ↵ Y ⇤ lm lm r a 2r r a b lm Z ✓ ◆ = ✓ + ✓ a I(a) + b I(b) lm l1m1 l2m2 l2m2 (3.59) l1mX1l2m2 ⇣ ⌘ 1 (a) (b) + ✓ a a⇤ K + b b⇤ K , 2 l1m1 l2m2 l3m3 l2m2 l3m3 l1m1lX2m2l3m3 ⇣ ⌘ where for the last equality we have used ↵ = + "b $ and defined a ra arb

(a) a I = d⌦( Yl m Y ⇤ aYl m ) , lml1m1l2m2 r 1 1 lmr 2 2 Z (b) a b I = d⌦ Yl m Y ⇤ " bYl m , lml1m1l2m2 r 1 1 lm ar 2 2 Z (3.60) (a) b a K = d⌦ Yl m Y ⇤ aYl m bY ⇤ , lml1m1l2m2l3m3 r r 1 1 lmr 2 2 r l3m3 Z (b) b a c d K = d⌦ Yl m Y ⇤ " cYl m " dY ⇤ . lml1m1l2m2l3m3 r r 1 1 lm ar 2 2 b r l3m3 Z ⇥˜ Our aim is to compute C = ✓˜ ✓˜⇤ , so we compute the two-point angular correlation function ll0 mm0 l lm l0m0 by substituting Eq. 3.59. This gives h i

˜ 1 C⇥ = C⇥ + C⇥ C ⇧(1a) + C$ ⇧(1b) + C⇥ C ⇧(2a) + C$ ⇧(2b) , (3.61) l l l1 l2 ll1l2 l2 ll1l2 2 l l1 ll1 l1 ll1 Xl1l2 ⇣ ⌘ Xl1 ⇣ ⌘ with

⇧(1a) = I(a) 2 , ⇧(1b) = I(b) 2 ll1l2 | | ll1l2 | | m m m m X1 2 X1 2 (3.62) (2a) (a) (a) (2b) (b) (b) ⇧ = K + K ⇤ , ⇧ = K + K ⇤ . ll1 ll1 m m X1 ⇣ ⌘ X1 ⇣ ⌘

To simplify this expression we have used the fact that ✓lm,alm and blm are gaussian random variables with zero mean and uncorrelated between each other. This means that the bispectrum is zero. We have also neglected the trispectrum of these variables since it is second order and of course any other higher order polyspectra. To compute the integrals in Eq. 3.62 we can use useful tricks for the integration of the Ylmssuch

6As we will see in Chapter 4 the Taylor expansion in the case of tensor perturbations hardly fails: for density perturbations we have ✓rms 2.7arcminandthismeansthatforl 3000 the wavelength being deflected is of the same order of ↵.For ' ⇠ 4 tensor perturbations however ✓rms 7arcsecandtheTaylorexpansionfailsforveryhighmultipolesvaluesl 10 . '

27 2 as integrating by parts to reconstruct Ylm = l(l + 1)Ylm or use the following formulas for the Wigner-3j symbols (for more details see [20]) r

m +s (2l1 + 1)(2l2 + 1)(2l3 + 1) d⌦ Y ⇤ ( Y )( Y )=( ) 1 1 s1 l1m1 s2 l2m2 s3 l3m3 4⇡ Z r l1 l2 l3 l1 l2 l3 ⇥ s1 s2 s3 m1 m2 m3 ✓ ◆✓ ◆ and l l l l l l 1 1 2 3 1 2 3 = . m1 m2 m3 m1 m2 m3 2l +1 m m 3 X1 2 ✓ ◆✓ ◆ In the end one can show

2 (1a) 1 ll1 l2 ⇧ = (2l1 + 1)(2l2 + 1) [l(l + 1) l1(l1 + 1) l2(l2 + 1)] , ll1l2 16⇡ 000 ✓ ✓ ◆◆ 1 2 (1b) l+l1+l2 ll1 l2 (3.63) ⇧ = (2l1 + 1)(2l2 + 1)(l1(l1 + 1)l2(l2 + 1)) [1 ( 1) ] , ll1l2 16⇡ 0 11 ✓ ✓ ◆◆ (2a) (2b) 2l1 +1 ⇧ =⇧ = l(l + 1)l1(l1 + 1) . ll1 ll1 4⇡ Finally, combining all expressions, we get

˜ C⇥ =C⇥ + C⇥ C ⇧(1a) + C$ ⇧(1b) l l l1 l2 ll1l2 l2 ll1l2 l1l2 X ⇣ ⌘ (3.64) ⇥ 2l1 +1 $ l(l + 1)C l1(l1 + 1) C + C , l 8⇡ l1 l1 Xl1 ⇣ ⌘ which is the expression we wanted: the 1st order lensed CMB temperature fluctuations power spectrum in the full-sky, small ↵ regime.

6. ¥10-7

4. ¥10-7

Q -7 l 2. ¥10 C ê Q Out[55]= l 0 C D

-2. ¥10-7

-4. ¥10-7 0 500 1000 1500 2000 multipole l

Figure 3.5: The fractional change in the power spectrum due to lensing by primordial tensor perturbations in the Full-Sky small angle regime. The plot is for a matter dominated universe with ⌦K =0,⌦m =1andh =0.67.

3.5.2 Flat-Sky - Arbitrary angle If we are interested in the calculation of the lensed CMB power spectrum at small scales we might do not trust the Full-sky small angles result since the Taylor expansion is a good approximation only for scales

28 much bigger than ↵7. To determine a formula in this regime we consider the correlation function. As before the e↵ect of GL on the CMB is a remapping of the temperature fluctuations according to ⇥˜ (ˆn )=⇥(ˆn + ↵) where ↵ = + $. For r = x x0, r = r the lensed correlation function ⇠˜(r)iswritten r r⇥ | |

⇠˜(r)= ⇥˜ (x)⇥˜ (x0) = ⇥(x + ↵)⇥(x0 + ↵0) . (3.65) h i h i Working in harmonic space and introducing the flat-sky 2D Fourier transform

2 d l il x ⇥(x)= ⇥(l)e · , (3.66) 2⇡ Z we find

2 2 d l d l0 il (x+↵) il0 (x0+↵0) ⇠˜(r)= e · e · ⇥(l)⇥(l0) 2⇡ 2⇡ h ih i Z 2 Z 2 d l d l0 il r il (↵0 ↵) ⇥ = e · e · C (3.67) 2⇡ 2⇡ h i ll0 l Z 2 Z d l il r il (↵0 ↵) ⇥ = e · e · C , (2⇡)2 h i l Z where for the first equality we have made the assumption that the CMB anisotropies and the deflection angle are uncorrelated and hence the expectation value of the product is the product of the expectation values. We now use the fact that ↵ is a Gaussian variable and so also l (↵0 ↵) is a Gaussian random 2 · variable with mean zero and variance [l (↵0 ↵)] . This allows us to write the expectation value of its exponential as8 h · i

1 2 1 2 il (↵0 ↵) [l (↵0 ↵)] e · = e 2 h · i = e 2 . (3.68) h i To compute the variance of l (↵0 ↵)weintroduce ·

Aij(r)= ↵i(x)↵j(x + r) h i (3.69) = [ (x)+( $(x)) ][ (x + r)+( $(x + r)) ] h ri r⇥ i rj r⇥ j i and we make now use of the formulas

i 2 il x (x)= d l l (l) e · , ra 2⇡ a Z i 2 il x ( $(x)) = d l (l $(l)) e · (3.70) r⇥ a 2⇡ ⇥ a Z i 2 b il x = d l " l $(l) e · , 2⇡ a b Z 7As already stated, this might be a concern in computing the CMB modification due to scalar lensing. In the case of lensing by tensor perturbations the deflection angle is so small that this will not be a problem for any reasonable multipole l. Nevertheless, for completeness, we do the computation also in this regime. 8 For a function g,withp.d.f.fg,ofvariableX,withp.d.f.fX ,wehave

1 1 g(X) = dy yfg(y)= dx g(x)f (x) h i X Z1 Z1 if X is a Gaussian variable and g(X)=eiX

x2 iX 1 1 ix 1 2 e = dx e e 22 = e 2 h i p2⇡ Z1

29 to write

2 d l k p $ ir l A (r)= l l C + " " l l C e · . (3.71) ij (2⇡)2 i j l i j k p l Z ⇣ ⌘ 1 By symmetry the correlation can only depend on ij and the trace-free tensorr ˆirˆj 2 ij and therefore is of the form

1 1 A (r)= A (r) A (r) rˆ rˆ . (3.72) ij 2 0 ij 2 i j 2 ij  To determine the functions A0 and A2 we start by taking the trace of Aij

2⇡ ilr cos ij 1 dl 3 ij k p $ e A0(r)= Aij = l Cl + l "i "j lklpCl d 0 2⇡ 0 2⇡ Z ⇣ ⌘ Z (3.73) 1 dl 3 $ = l Cl + Cl J0(rl) , 0 2⇡ Z ⇣ ⌘ where , in this section, is the angle between r and l. 2⇡ ix cos ! For the last equality we made use of the Bessel function J0(x) arising from the integral 0 d!e = 2⇡J (x). 0 R To compute A2 we contract the correlation withr ˆirˆj

d2l A rˆ rˆ = l2 (cos2 C +sin2 C$) eirl cos ij i j (2⇡)2 l l Z (3.74) 2⇡ 2 2⇡ 2 1 dl cos sin = l3 C d eirlcos + C$ d eirlcos . 2⇡ l 2⇡ l 2⇡ Z0 ✓ Z0 Z0 ◆ We see again that the Bessel functions arise from the integrals

2⇡ 2 ix cos ! d! sin !e = ⇡ (J0(x)+J2(x)) , Z0 2⇡ d! cos2 !eix cos ! = ⇡ (J (x) J (x)) , 0 2 Z0 to give

1 1 dl 3 $ A2(r)= l Cl Cl J2(rl) . (3.75) 2 0 2⇡ Z ⇣ ⌘ We can now compute the required expectation value

2 i j [l (↵0 ↵)] = l l (↵i ↵0 )(↵j ↵0 ) h · i h i j i (3.76) = l2 [A (0) A (r)+A (r) cos(2)] , 0 0 2 which can be inserted in Eq. 3.67 to give

d2l l2 ⇠˜(r)= C⇥ eirlcos Exp (A (0) A (r)+A (r) cos(2)) . (3.77) (2⇡)2 l 2 0 0 2 Z  This is an exact expression and we can actually perform the angular integral using the following tricks. First we use the relation

30 ix cos 1 n e = Jo(x)+2 i Jn(x) cos(n) (3.78) n=1 X and then we use the integral

1 2⇡ dex cos cos(n)=I (x) , (3.79) 4⇡ n Z0 9 where the In(x) are the hyperbolic Bessel functions of the first kind . We obtain

dl l2 ⇠˜(r)= lC⇥ Exp (A (0) A (r)) (2⇡) l 2 0 0 Z  (3.80) 1 I (rl)+2 I (l2A (r)/2)J (rl) . ⇥ 0 n 2 2n n=1 ! X Using the fact that the correlation function is the Fourier transform of the power spectrum

1 2 ⇥ il r ⇠(r)= d l C e · , (3.81) (2⇡)2 l Z we can obtain the lensed power spectrum from ⇠˜(r)

2⇡ ˜⇥ 2 ˜ il r 1 ˜ ilrcos Cl = d r ⇠(r)e · = dr r ⇠(r) de Z Z0 Z0 (3.82) 1 =2⇡ dr r J0(rl) ⇠˜(r)1 . Z0 Sometimes is useful to approximate the exponential in ⇠˜(r) to give (see [11])

2 2 1 dl l l ⇠˜(r) lC⇥ Exp [A (0) A (r)] J (rl)+ A (r)J (rl) . ' 2⇡ l 2 0 0 0 2 2 2 Z0  ✓ ◆ Inserting this in Eq. 3.82 we finally find

2 2 ⇥ 1 1 l0 l0 C˜ = dl0 l0C dr r Exp [A (0) A (r)] J (rl) J (rl0)+ A (r)J (rl0) . (3.83) l l0 2 0 0 0 0 2 2 2 Z0 Z0  ✓ ◆ Recalling

1 dl 3 $ 1 1 dl 3 $ A0(r)= l Cl + Cl J0(rl) ,A2(r)= l Cl Cl J2(rl) 0 2⇡ 2 0 2⇡ Z ⇣ ⌘ Z ⇣ ⌘ Eq. 3.83 is the expression of the lensed CMB power spectrum in the flat-sky arbitrary angle regime, as $ a function of the lensing potentials Cl and Cl .

9 n The hyperbolic Bessel functions of the first kind are defined by In(x)=i Jn(ix). For more information about the Bessel function Jn, In,thesphericalBesseljl and their properties such as Eq. 3.78 or Eq. 3.79, see [21] and [22].

31 Chapter 4

Summary

Subject of this work was Gravitational Lensing by tensor perturbation (i.e. GWs). To compute the displace- ment vector ↵ we followed the standard [11, 12] procedure of the perturbed photon path. Since GWs are best described in the frame kˆ an Euler rotation is necessary to write down the perturbation along the line of sight. The displacement? vector is then decomposed into its gradient-type potential and its curl-type potential $

↵ = + ~$ r r⇥ and the Euler rotations performed plays a central role in the decomposition of the potentials in Spherical Harmonics. The main results we computed is the angular power spectra of the two lensing potentials

2 (l + 2)(l 1) dk 2 ⇤ l +1 jl(k) jl+1(k) ⇤ Cl =2⇡ h(k) dTh(k, ) l +2 2 3 2 l(l + 1) k 0 2 k k Z Z ✓ ✓ ⇤ ◆ ◆ 2 (l + 2)(l 1) dk ⇤ j (k) C$ = ⇡ 2 (k) dt T (k, ) l l l(l + 1) k h h k2 Z Z0 for which we give plots in the case of primordial tensor perturbations in a MD universe. It is interesting to see how, due to the decaying nature of primordial tensor perturbations once entered the horizon, the lensing potentials spectra have contribution from very high redshift while the lensing potential for density perturbations gets its main contribution for z . 10. Another interesting aspect of lensing by tensor perturbations is to consider the scalar-induced tensor pertur- bation: using the result derived in [19] we have been able to estimate the lensing potentials for scalar-induced tensor perturbations and found them to be several orders of magnitude smaller than the primordial ones. This might be surprising at first since, as shown in [18], the spectrum of these second order perturbation might even exceed the primordial one at large scales at the present epoch, but is then quite clear if one thinks that scalar-induced tensor are of course relevant only at small redshift while, as said, the lensing potentials due to primordial tensor get contributions up to very high redshift. In fairness we should point out that a MD calculation might not be the best for estimating the scalar-induced tensor contribution: including the Radiation domination (RD) era has an e↵ect on the primordial spectrum (modes that enter the horizon dur- ing the RD era su↵er from a less severe decay) but has a bigger e↵ect on the scalar-induced tensor spectrum. Primordial II order 5 Nevertheless we expect, even including the RD era, Cl /Cl & r 10 . In the last section of this work we study the e↵ect of primordial tensor perturbation⇥ lensing on the CMB temperature field. To get an idea of this e↵ect we can estimate the rms deflection angle. We compute the total deflection angle power (see Appendix F) §

32 Scalar 1 2 1 2 7 R = ✓ = ⌥ 3 10 , 2 rms 2 h|r | i⇠ ⇥ Tensor 1 2 1 2 9 R = ✓ = + $ 1.2 10 , 2 rms 2 h|r r⇥ | i⇠ ⇥ and obtain ✓ Scalar 2.7 arcmin and ✓ tensor 7 arcsec. We therefore expect a small modification to the CMB temperature due⇠ to tensor lensing. We present⇠ two calculations for the two relevant cases: Full-sky & small angle and Flat-sky & arbitrary angle but, given the result of the rms deflection angle, we expect the first case to be valid up until very high multipole l and we give a plot for this case from which we can see 7 how the modifications to the CMB temperature are of the order of 10 , smaller than the cosmic variance of intrinsic CMB anisotropies for every reasonable multipole

⇥ CV ⇥ GL Cl 2 Cl 7 10 . C⇥ ⇠ 2l +1 C⇥ ⇠ ✓ l ◆ r ✓ l ◆ Let us terminate this work with a remark about the curl part of the displacement vector: at the linear level tensor perturbations produce a curl part of the displacement vector and that is a unique feature of GL by GWs which is not found in lensing by density perturbation. Nevertheless it is important to point out that at second order also the displacement vector induced by density perturbations has a curl-component [23] which, at intermediate and small scales, is probably higher than the primordial tensor one.

33 Appendices

Appendix A - Statistics of the CMB The CMB temperature fluctuations are characterized by ⇥(ˆn) defined in Eq. 1.7, which we consider a Gaussian1 random variable and an isotropic and homogeneous field. This has remarkable e↵ects on its statistic properties.

Fourier Space The characterization of the statistics of random variables is most commonly expressed in terms of the correlation functions. The two-point correlation function is the ensemble expectation value

C = ⇥(k)⇥(k0) k,k0 h i In the absence of any symmetries of ⇥we would only have two properties for the correlation function:

first, since ⇥is real we get ⇥⇤(k)=⇥( k) such that Ck⇤,k = C k, k0 , and, due to the associative nature of 0 the expectation value, we also have Ck,k0 = Ck0,k. Since ⇥is gaussian we have that all the higher order correlation function are either zero (for an odd number of points) or connected to the 2-point correlator through Wick’s theorem

⇥(ki1 )...⇥(ki2n ) = Cj1j2 ....Cj2n 1 j2n h i AllX perm. We now consider the issue of homogeneity. A field is homogeneous if its expectation values (or averages) do not dependend on the spatial points where they are evaluated. In terms of the N-point functions in real space

3 3 d k1...d kN ik x ik x ⇥(x )...⇥(x ) = e 1· 1 ...e N· N ⇥(k )...⇥(k ) h 1 N i (2⇡)3N h 1 N i Z Since the RHS must be function only of the distances between spatial points and not of the points themselves we find that the expectation value in Fourier space must be proportional to (k1 + ... + kN ), so that

⇥(k )...⇥(k ) =(2⇡)3P (k ,...,k ) (k + ... + k ) h 1 N i 1 N 1 N Furthermore for isotropy we must have that the N-point function in real space cannot depend on the direction. We see from the previous equation that this implies the Fourier space correlator to be only dependent on the moduli of the wave vectors, such that we get P (k1,...,kN )=P (k1,...,kN ).

1Arandomvariablex is called Gaussian if it has the p.d.f. in the form

1 (x x )2/22 p(x)= e 0 p2⇡

34 Harmonic space The harmonic representation is most directly related to the observations of the CMB, ⇥(ˆn ), which are taken over the unit sphere. The fluctuation are expanded

⇥(ˆn )= ✓lmYlm(ˆn ) (A.1) Xlm m and, again, since ⇥is a real field we have ✓lm⇤ =( ) ✓l m and this means that each multipole l has 2l + 1 real degrees of freedom. The multipole coecients are given by

✓lm = d⌦⇥(ˆn) Ylm⇤ (ˆn) Z and the harmonic space two-point correlator is given by

✓lm✓⇤ = d⌦ d⌦0 ⇥(ˆn)⇥(ˆn0) Y ⇤ (ˆn)Yl m (ˆn0) h l0m0 i h i lm 0 0 Z If we now use the Rayleigh expansion in Eq. 2.21 we get

3 3 d kd k0 2 l l0 ✓lm✓⇤ = (4⇡) i ( i) jl(k)jl (k0)Ylm(kˆ)Y ⇤ (kˆ0) ⇥(k)⇥⇤(k0) h l0m0 i (2⇡)6 0 lm h i Z where we have written x = ˆn . Under the hypothesis of homogeneity, this expression simplifies, leading to ·

3 d k l l0 ✓lm✓⇤ = 2i ( i) jl(k)jl (k)Ylm(kˆ)Y ⇤ (kˆ) f(k) h l0m0 i ⇡ 0 lm ⇥ Z and, if we also assume isotropy, then f(k) P (k) can only depend on the modulo of k. The angular ! ˆ ˆ integral now involves only the spherical harmonics and gives d⌦k Ylm(k)Ylm⇤ (k)=ll0 mm0 , such that R 2 dk 3 2 ✓lm✓⇤ = ll mm k j (k) P (k) h l0m0 i 0 0 ⇡ k l Z dk = 4⇡ j2(k)2(k) ll0 mm0 k l Z = ll0 mm0 Cl

2 k3 where we have defined the dimensionless power spectrum (k)= 2⇡2 P (k). Therefore, for statistically isotropic random fields, the expansion in terms of spherical harmonics diagonalizes the correlation function. If the underlying fluctuation are Gaussian, the ✓lms are independent Gaussian variables. Using the facts that the spherical Bessel jl peaks when its argument is approximately given by l and that dx 2 1 x jl (x)= 2l(l+1) , we can approximate R 2⇡ C 2(k = l/) l ' l(l + 1) which explains why the quantity that is normally plotted is

l(l + 1) C 2(k = l/) Cl ⌘ 2⇡ l ' We can also construct the angular two-point correlation function in harmonic space

35 ⇥(ˆn)⇥(ˆn0) = ✓lm✓⇤ Ylm(ˆn)Y ⇤ (ˆn0) h i h l0m0 i l0m0 Xlm lX0m0 = C Y (ˆn)Y ⇤ (ˆn0) l ll0 mm0 lm l0m0 Xlm lX0m0 2l +1 = C P (ˆn nˆ0) l 4⇡ l · Xl l 2l+1 where we have used the addition theorem of spherical harmonics m= l Ylm(ˆn)Ylm⇤ (ˆn0)= 4⇡ Pl(ˆn nˆ0). · P

36 ↵ Appendix B - µ⌫ for tensor perturbations

We compute the Christo↵el’s symbols 1 ↵ = g↵(@ g + @ g @ g ) µ⌫ 2 µ ⌫ ⌫ µ µ⌫ for the tensor perturbation metric in Eq. 3.2, at first order in the perturbation

1 r = @ h + 2 rr 2 r rr O 1 r = ( 2h + @ h )+ 2 ✓r 2 r✓ ✓ rr O 1 r = r +(rh rh + r@ h r2@ h )+ 2 ✓✓ rr ✓✓ ✓ r✓ 2 r ✓✓ O 1 r = ( 2sin✓h + @ h )+ 2 r 2 r r O 1 r = (r(cos ✓h +2sin✓h @ h sin ✓@ h + r sin ✓@ h )) + 2 ✓ 2 r ✓ r✓ ✓ r r ✓ O 1 r = r sin2 ✓ + r sin ✓ (2 sin ✓h + 2 cos ✓h 2sin✓h +2@ h r sin ✓@ h )+ 2 2 rr r✓ r r O 1 r = @ h + 2 rt 2 t rr O 1 r = @ h + 2 ✓t 2 t r✓ O 1 r = @ h + 2 t 2 t r O 1 ✓ = (2h @ h +2r@ h )+ 2 rr 2r2 r✓ ✓ rr r r✓ O 1 1 ✓ = + @ h + 2 ✓r r 2 r ✓✓ O 1 ✓ = h + @ h + 2 ✓✓ r✓ 2 ✓ ✓✓ O 1 ✓ = ( cos ✓h + @ h +sin✓ ( @ h + r@ h )) + 2 r 2r r r ✓ r r ✓ O 1 ✓ = ( 2 cos ✓h + @ h )+ 2 ✓ 2 ✓ ✓✓ O 1 ✓ = cos ✓ sin ✓ + sin ✓ (2 sin ✓h + 2 cos ✓h 2 cos ✓h +2@ h sin ✓@ h )+ 2 2 r✓ ✓✓ ✓ ✓ O 1 ✓ = @ h + 2 tr 2r t r✓ O 1 ✓ = @ h + 2 t✓ 2 t ✓✓ O

37 1 ✓ = sin ✓@ h + 2 t 2 t ✓ O 1 = csc ✓ (2h csc ✓@ h +2r@ h )+ 2 rr 2r2 r rr r r O 1 = csc ✓ (cot ✓h csc ✓@ h + @ h + r@ h )+ 2 ✓r 2r r r✓ ✓ r r ✓ O 1 = csc ✓ (2h + 2 cot ✓h csc ✓@ h +2@ h )+ 2 ✓✓ 2 r ✓ ✓✓ ✓ ✓ O 1 1 = + @ h + 2 r r 2 r O 1 = cot ✓ + @ h + 2 ✓ 2 ✓ O 1 =sin✓h + cos ✓h + @ h + 2 r ✓ 2 O 1 = csc ✓@h + 2 tr 2r t r O 1 = csc ✓@h + 2 t✓ 2 t ✓ O 1 = @ h + 2 t 2 t O 1 t = @ h + 2 rr 2 t rr O 1 t = r@ h + 2 r✓ 2 t r✓ O 1 t = r2 @ h + 2 ✓✓ 2 t ✓✓ O 1 t = r sin ✓@ h + 2 r 2 t r O 1 t = r2 sin ✓@ h + 2 ✓ 2 t ✓ O 1 t = r2 sin2 ✓@ h + 2 2 t O

38 Appendix C - Spherical harmonics In this appendix we review the most important results on Spherical harmonics which have been useful in this work. We will follow closely [11] while the proofs of most of the results and more can be found in [30]. We start by defining the Legendre Polynomials and then we will concentrate on spin-0 and spin-s spherical harmonics.

Legendre Polynomials

The Legendre Polynomials Pl(x) are solutions of the di↵erential equation

2 (1 x )P 00 2xP 0 + l(l + 1)P =0 l l l They are a set of orthonormal polynomials on the interval [ 1, 1]. Given the lowest order polynomials P0 = 1 and P1 = x one can derive all the other through the recursion relation

(l + 1)P (x)=(2l + 1)xP (x) lP (x) l+1 l l They obey the normalization condition

1 2 dx Pl(x)Pl0 (x)= ll0 1 2l +1 Z and Rodrigue’s formula

1 dl P (x)= (x2 1)l l 2l l! dxl The associated Legendre polynomials are defined by

dmP (x) 1 dl+m P (x)=(1 x2)m/2 l =(1 x2)m/2 (x2 1)l lm dxm 2l l! dxl+m They are in principle defined for every complex degree l and order m, but we will need them only for integer m, non-negative integer l with m

1 2 (l + m!) dx Plm(x)Pl0m(x)= ll0 1 2l +1(l m)! Z and, from their definition, the parity relation P ( x)=( )l+mP (x). lm lm Spin-0 spherical harmonics The spherical harmonics are functions on the sphere. For a unit vectorn ˆ defined by its polar angles (✓,) the spherical harmonics are given by

m 2l +1(l m)! im Ylm(ˆn)=( ) e Plm(µ) s 4⇡ (l + m)! with µ = cos ✓. The parity and orthogonality relations are derived directly from the Legendre polynomials m properties and they are given by : Yl m =( ) Ylm, and

39 d⌦ Y (ˆn)Y ⇤ = lm l0m0 ll0 mm0 Z Given the Laplacian on the sphere we construct

2 2 2 cos ✓ 1 2 L = = @ + @✓ + @ r ✓ sin ✓ sin2 ✓  (where ~ = 1), and we have

2 L Ylm = l(l + 1)Ylm In the end we simply report the addition theorem of spherical harmonics (proof in [11]) which we have used in this work

2l +1 l P (ˆn nˆ0)= Y (ˆn)Y ⇤ (ˆn0) 4⇡ l · lm lm m= l X Spin-s spherical harmonics 1 We consider a tensor field on the sphere with e1 = e✓ = @✓ and e2 = e = sin ✓ @ and we recall the definition of helicity basis 1 e = (e1 e2) ± p2 ⌥ The helicity basis has the property that under a rotation e cos e sin e , e cos e +sin e transforms as 1 ! 1 2 2 ! 2 1

i e e e + ! + e ei e ! and a tensor field of type (+, )=(s, r) (of rank r + s) in the helicity basis transform under a rotation by

s i(s r) s T e T r ! r + i + i For example, the components of a vector field transform as V e V and V e V . Compo- nents which transform with eis are called components of spin s or helicity! s. To expand! a spin-s component of a tensor field on a sphere, one uses the spin weighted spherical| | harmonics, which are defined by

2l +1(l m)!(l m!) Y (✓,)=( )m eim (sin ✓/2)2l s lm 4⇡ (l + s)!(l s!) s l s l + s l r s 2r+s m ( ) (cot ✓/2) ⇥ r r + s m r X ✓ ◆✓ ◆ The spin weighted spherical harmonics are defined for s l and m l and, similarly to the spin-0 harmonics, we have an orthogonality relation | | | |

d⌦ sYlm(ˆn) sYl0m0 (ˆn)=ll0 mm0 Z

40 Now let R1 be the rotation with Euler angle (1,✓1, 0) which rotates ez into n1 and R2 the rotation with 1 Euler angle (2,✓2, 0) which rotates ez into n2. Let (↵,,) the Euler angle of the rotation R1 R2,wethen have a generalization of the addition theorem for spin weighted spherical harmonics:

4⇡ is sYlm0 (✓2,2) mYlm⇤ (✓1,1)=sYlm(,↵)e (C.1) 2l +1 0 r m X0 We now introduce the spin raising and spin lowering operators @/ and @/⇤. They are defined by

i @/ Y = s cot ✓ @ @ Y s lm ✓ sin ✓ s lm ✓ ◆

sµ 2 i = + 1 µ @µ @ sYlm 1 µ2 1 µ2 ! p p p i @/⇤ Y = s cot ✓ @ + @ Y s lm ✓ sin ✓ s lm ✓ ◆

sµ 2 i = + 1 µ @µ + @ sYlm 1 µ2 1 µ2 ! p p p and with these operators we can construct the spin-s harmonics starting from the 0Ylms. In fact we have

@/ Y = (l s)(l + s + 1) Y s lm s+1 lm @/⇤ sYlm = p (l s)(l s + 1) s 1Ylm and p

(l + 2)! (@/)2 Y = Y lm (l 2)! 2 lm s

2 (l + 2)! (@/⇤) Ylm = 2Ylm (l 2)! s At this point we are interested in find an operator which acts in the same way that L2 acts on the 2 spin-0 harmonics. We call this operator L(s) and we want the spin weighted spherical harmonics to be eigenfunctions of this operator

2 L(s) sYlm = l(l + 1) sYlm To construct this operator we see that, by definition, the commutators satisfy

2 2 ⇤ [L(s), @/]=[L(s), @/ ]=0 and we make the ansatz L2 = 2 + K . The commutation relation [L2 , @/] = 0 implies that (s) r (s) (s) 2 2 L(s+1) @/(s) = @/(s)L(s) which gives

[ 2, @/ ]=@/ K K @/ r (s) (s) (s) (s+1) (s) 2 2 For s = 0 (given that K(0) =0sinceL(s) must reduce to L for s = 0)

41 K @/ =[@/, 2] (1) r explicitly,

i 1 i cos ✓ 2 K(1) @✓ @ = @ +2 @ @✓ +2i cos ✓@@✓ sin ✓ sin2 ✓ sin ✓ sin ✓ ✓ ◆ ✓ ◆ and we find 1 K(1) = (1 2i cos ✓@) sin2 ✓ 2 We can do the same procedure with [L , @/⇤] = 0 to find an expression for K( 1) and we can actually (s) find every K(s) by recursion. At some point will be clear that the expression we are looking for is

2s cos ✓ s2 K(s) = i @ + sin2 ✓ sin2 ✓ to give

2 2 2 cos ✓ 1 2 2s cos ✓ s L (✓,)= @ + @✓ + @ i @ + (s) ✓ sin ✓ sin2 ✓ sin2 ✓ sin2 ✓ ⇣ ⌘

42 Appendix D - Bessel functions

1.0 1.0 J x j0 x 0 J x 0.8 j1 x 1 J2 x j2 HxL H L J3 x 0.6 j3 x 0.5 H L H L J x 4 H L j4 HxL L L H L x x H H 0.4 H L l l j J H L H L

0.2 0.0

0.0

-0.2 -0.5

0 5 10 15 20 0 5 10 15 20 x x

Figure 4.1: Left: The first 5 Spherical Bessel function jn(x)forintegern. Right: The first 5 Bessel function Jn(x)forintegern.

The Bessel functions J⌫ (x) are solutions of the di↵erential equation

d2f df x2 + x +(x2 ⌫2)f =0 dx2 dx

The J⌫ (x) are called Bessel function of the first kind and are finite at the origin x = 0 for integer or positive ⌫ (see Fig. 4.1), and diverge as ⌫ 0 for negative non-integer ⌫. It is possible to define them through the Gamma function !

1 ( 1)m x 2m+⌫ J (x)= ⌫ m!(m + ⌫ + 1) 2 m=0 X ⇣ ⌘ If ⌫ = n is an integer one also has an integral representation

( i)n ⇡ J (x)= d✓eix cos ✓ cos(n✓) n ⇡ Z0 There are many useful recursion relation for the Bessel function ([21],[22]). We give here a couple of examples 2⌫ J⌫ 1 + J⌫+1 = J⌫ x

J⌫ 1 J⌫+1 =2J⌫0 ⌫ J⌫ 1 J⌫ = J⌫0 x

Spherical Bessel

The Spherical Bessel function jn(x) are defined

⇡ j (x)= J (x) n 2x n+1/2 r and they are solutions of the di↵erential equation

d2f df x2 +2x +(x2 n(n + 1))f =0 dx2 dx

43 Among many recursion relations for the spherical Bessel, in this work we made use of

jn 1 = (jn 1 + jn+1) x 2n +1 1 jn0 = (njn 1 (n + 1)jn+1) 2n +1

to rewrite in a nice way the expression of Cl . Another useful relation that we have been using many times in this work comes from noticing that the functions jl(rk)Ylm(ˆn) are solutions of the equation

( 2 + k2) f =0 r ik x and since also e · is a solution and spherical harmonics form a complete set of functions on the sphere it has to be possible an expansion of the type

ik x ikrkˆ ˆn e · = e · = clm jl(rk)Ylm(ˆn) Xlm and this expression is precisely (see [30])

ikrkˆ ˆn l e · = (2l + 1)i j (rk)P (ˆn kˆ) l l · Xlm

44 Appendix E - Limber Approximation The Limber approximation [31] and its generalization to Fourier space [32] is a commonly used technique to simplify numerical computation (it reduces the dimension of the integral to compute). The Limber approximation is valid when one assumes small angular separations (l 1) and in the situation where some of the functions being integrated (typically the power spectra) are slowly varying with respect to others functions in the integral (typically the spherical Bessels). We start from the large order behavior of the jl ([33])

⇡ j (x) (l +1/2 x) l ' 2l +1 r and using the properties of the Dirac delta

1 (x x ) (ax)= (x); dx (y x)(z x)=(y z); dx f(x)(g(x)) = i a g0(xi) Z Z xi g(xi)=0 | X | | we can write a not-so-formal derivation for the Limber approximation

1 l +1/2 l +1/2 2 0 2 dk (l +1/2 k)(l +1/2 k0)= ( )= ( )= (0 ) 2 2l +1 2l +1 Z 0 0 such that

2 ⇡ 2 ⇡ l k dk f(k) j (k)j (k0) k dk f(k) (l +1/2 k)(l +1/2 k0) f ( 0) l l ' 2l +1 ' 22 Z Z ✓ ◆ or, equivalently

2⇡ l +1/2 dk f(k) j (k)j (k0) f ( 0) l l ' (2l + 1)2 Z ✓ ◆ This expression was used in the case of lensing by density perturbations. However in the case of lensing by tensor perturbation the Transfer function Th(k, ) is not slowly varying with respect to the spherical Bessel and this approximation fails. In the same way one can find a formula useful when the spherical Bessel are not of the same order

2⇡ 1 l l dk f(k) jl(k)jl0 (k0) 2 f ( 0) ' (2l + 1)(2l + 1) (2l0 + 1) l0 Z 0 ✓ ◆ so that p 2⇡ 1 l l dk f(k) jl(k)jl 1(k0) f ( 0) ± ' (2l + 1) (2(l 1) + 1)3/2 l 1 Z ± ✓ ◆ ± p

45 Appendix F - Total deflection angle power We want to compute the total deflection angle power of density & tensor perturbations. 1 R ↵ 2 ⌘ 2 h| | i For density perturbation this yields to

⌥ 1 2 1 2 1 R = ↵ = ⌥ = lm⇤ Ylm(ˆn ) Y ⇤ (ˆn ) 2 h| | i 2 h|r | i 2 h l0m0 ir r l0m0 lmlX0m0 1 ⌥ = ll mm C Ylm(ˆn ) Y ⇤ (ˆn ) 2 0 0 l r r l0m0 lmlX0m0 Now, we make use of the relation [24]

l(l + 1) Ylm = (1Ylme+ + 1Ylme ) r r 2 to find

⌥ 1 ⌥ l(l + 1) R = Cl (1Ylm 1Ylm⇤ + 1Ylm 1Ylm⇤ ) 2 2 Xlm This can be further simplified using the Generalized addition relation C.1 in the particular case where (✓1,1)=(✓2,2)

2l +1 s1 Ylm(ˆn) s2 Ylm⇤ (ˆn)= s2 Yl s1 (0) 4⇡ m r X In our case we need to use

2l +1 1Ylm(ˆn) 1Ylm(ˆn)= 1Yl 1(0) = ± ± ± ⌥ 4⇡ m r X to give 1 2l +1 R⌥ = l(l + 1) C⌥ 2 4⇡ l Xl The computation for tensor perturbations goes in the same way, with the exception that we also have to use the formula

l(l + 1) Ylm = ( 1Ylme+ 1Ylme ) r⇥ r 2 to give 1 1 2l +1 RTensor = + $ 2 = l(l + 1) (C + C$) 2 h|r r⇥ | i 2 4⇡ l l Xl

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48