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 CMBtemperature field. After a first introductory chapter and a second chapter where the formalism of Lensingby density perturbations is introduced as a reference for comparison, in the third chapter we discuss someaspects of lensing by tensor perturbations. The displacement vector of a null geodesic travelling througha 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 theirproperties by giving plots for a matter dominated universe. In the last section the modifications induced bythis type of lensing to the temperature fluctuations field of the CMB are investigated and we find them tobe of the order of 10�7, smaller than the cosmic variance of intrinsic CMB anisotropies for every observablemultipole.

Contents

1 Introduction 21.1 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 The Unlensed CMB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Simple Example: The Schwarzschild Lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Lens Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Lensing by Scalar Perturbations 92.1 Perturbed Photon Path and Lensing Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 The CMB Lensed Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Lensing by Tensor Perturbations 153.1 Perturbed Photon Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Euler Rotation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Scalar and Vector Lensing Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3.1 Angular decomposition of and $ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.4 Curl-mode & Gradient-mode Power Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.5 The CMB Lensed Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.5.1 Full-Sky - Small angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.5.2 Flat-Sky - Arbitrary angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 Summary 32

Appendices 34Appendix A - Statistics of the CMB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Appendix B - �↵µ⌫ for tensor perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Appendix C - Spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Appendix D - Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Appendix E - Limber Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Appendix 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 influencesthe 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 ofPeter Simon Laplace, another aspect of this interaction was discovered: by computing the escape velocityof light from a massive body, he introduced what today we call Schwarzschild radius RS = 2GM/c2 andanticipated the existence of black-holes.In the XIX century Johann Georg von Soldner published a paper in which he considered the error inducedby the deflection of light in the determination of the angular position of stars and, as the small angle limitof 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 brokeout and spoiled Freundlich’s expedition to verify experimentally the deflection of light and second, evenwhen 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 toobserve the solar eclipse of 29 May 1919, but also as the first to point out that light deflection in the universecan 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 wasconsidered a fairly esoteric field until 1979 when Walsh, Carswell and Weymann discovered and describedthe first GL candidate: two images of a quasar at z ⇠ 1.4 lensed by a galaxy at z ⇠ 0.34 [5] (nowadayshundreds 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 untilthey realized that light propagating through an inhomogeneous universe is lensed and di↵erent from lightpropagating through an unperturbed Friedmann-Lemaitre universe. This, together with the discovery of theCosmic Microwave Background in 1964, made GL an important subject for modern cosmology: in fact, ascosmologists realized the importance of CMB anisotropies and as the experimental precision in the measureof 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 (M, g). The Cosmological principle dictates theuniverse to be isotropic and homogeneous and this means that the spacetime admits a slicing into maximallysymmetric 3-spaces. There is a preferred coordinate ⌧ , the cosmic time1, such that the 3-spaces of constanttime ⌃⌧ are indeed maximally symmetric and hence of constant curvature K. The generic form of the metricg is then

ds2 = gµ⌫dxµdx⌫ = �d⌧2 + a2(⌧)�ijdx

idxj = a2(t)(�dt2 + �ijdxidxj) , (1.1)

where we have introduced the conformal time dt ⌘ d⌧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 themost convenient for this work

�ijdxidxj = dr2 � �2(r)

�d✓2 + sin2 ✓d�2

�, (1.2)

where

�(r) =

8>><

>>:

r Flat 3-space K = 0,1p|K|

sin(p|K|r) Spherical 3-space K > 0

1p|K|

sinh(p|K|r) Hyperbolic 3-space K < 0.

(1.3)

The symmetry of space time allows only an energy-momentum of the form

(Tµ⌫) =

✓�⇢g00

00 Pgij

◆, (1.4)

where ⇢ and P are the energy density and the pressure of the components of the universe. The Einstein’sequations, including the cosmological constant ⇤, are then written

✓a

a

◆2

+K =8⇡G

3a2⇢+

a2⇤

3,

2

✓a

a

◆•+

✓a

a

◆2

+K = �8⇡Ha2P + a2⇤ ,

(1.5)

where • = ddt and H = a

a2 is the Hubble parameter.

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, forexample, 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 thanthe expansion rate so that the cosmic plasma was in thermal equilibrium. As the universe expands it alsocools adiabatically following T / a�1. As long as the temperature is above the ionization energy of neutralhydrogen T > 1Ry = � = 13.6eV all hydrogens atoms that forms through e– + p+ ��! H + � are rapidlydissociated but, at a temperature of about T ⇠ 4000K ⇠ 0.4eV, the number density of photons with energiesabove � drops below the baryon density of the Universe. As a consequence e– and p+ begin to (re)combineto neutral hydrogen. This behavior can be described with the Saha equation

x2

e

1� xe=

45�

4⇡2

⇣ me

2⇡T

⌘3/2

e��/T (1.6)

where xe is the e– ionization fraction and � is the entropy per baryon2. We can see from Figure 1.2 that forT � T

rec

⇠ 0.4eV we have xe ⇠ 1, meaning that far behind recombination epoch e– and p+ are not in formof neutral hydrogen, while for T ⌧ T

rec

the ionization fraction xe ⇠ 0 meaning that almost all the e– in theuniverse are bonded in H atoms. Defining the temperature of recombination as T

rec

⌘ T (xe = 0.5) we find3

Trec = 3757K.

Out[17]=

0.25 0.30 0.35 0.40 0.450.0

0.2

0.4

0.6

0.8

1.0

THeVL

x e

Wbh2=0.01

Wbh2=0.02

Wbh2=0.03

Figure 1.2: The e

–ionization fraction as a function of temperature for di↵erent values of ⌦

b

h2, together with the line x

e

= 0.5 which

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 photonsgrows larger than the Hubble scale. Usually one says that when the scattering rate � of � on e– falls belowthe expansion rate of the Universe � < H, photons become free to propagate without (almost) any interac-tion. This epoch is called decoupling (zdec ⇠ 1100). The radiation of free photons with a perfect black-bodyspectrum (due to thermal equilibrium before recombination) free streaming after decoupling is the CMB, its

2The entropy per baryon is � =

s

�

n

B

=

⇣4⇡2

45

⌘T

3

n

B

, where s

�

is the photon entropy density and n

B

is the conserved baryon

number density so that � is a constant.

3We point out that there is a physical reason for the fact that Trec ⌧ �. The condition T ' � is not su�cient to trigger

recombination because n

�

� n

b

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 > �) < n

b

. In other words we would have Trec ' �

if � ' 1, but the entropy per baryon is approximately � ' 10

10, and delays recombination significantly.

4

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

The power spectrum of the CMB temperature fluctuations is what we are most interested in. Of coursethe 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 believedthat quantum fluctuations of the inflaton field, brought to cosmic scales by inflation, laid down perturbationson all scales4. The small initial perturbations have grown via gravitational instability and the evolution canbe studied in linear theory. This mechanism is responsible for the fluctuations ⇥(n) of the CMB

⇥(n) =T (n)� T

0

T0

⌘ �T (n)

T0

(1.7)

where T0

⇠ 2.7K is the mean temperature today and n is the direction of observation. �T (n) is afunction on the sphere since it is evaluated today (t

0

) and here (x0

) so that n 2 S2. We expand ⇥(n) inspherical harmonics

⇥(n) =X

lm

✓lm(x0

)Ylm(n) (1.8)

and define the power spectrum as

h✓lm✓⇤l0m0i = �ll0�mm0C⇥l (1.9)

where the o↵-diagonal correlators vanish since we assume the process generating the initial perturbationsto 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 Lensingwith the simple example of the Schwarzschild Lens, both for pedagogical and historical reasons (Eddingtonexpedition in 1919 measured essentially the deflection angle of the sun).The Schwarzschild metric describes a static, spherically symmetric solution of Einstein’s equations in thevacuum Tµ⌫ = 0. That is to say the exterior of a static star. In spherical coordinates

ds2 =⇣1� RS

r

⌘dt2 �

⇣1� RS

r

⌘�1

dr2 � r2(d✓2 + sin2 ✓d�2). (1.10)

Since we are interested in the light path near the star (null-geodesic) we consider the Lagrangian of amassless particle L = 1

2

gµ⌫ xµx⌫ = 0. The radial Euler-Lagrange equation gives

(u0)2 + u2 =E2

L2

+Rsu3 , (1.11)

where 0 = dd� , u = 1

r and we highlighted the two constants of motion E = t�1� R

s

r

�and L = r2�. After

di↵erentiating with respect to � we obtain

u00 + u =3

2RSu

2 . (1.12)

Using the fact that the impact parameter will be much bigger than the Schwarzschild radius we can treatthe term 3

2

RSu2 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

1

2

3

4

5

6

7

(-0.31, 7.38)

(10.47, 0.21)

α

ξ

ηS

M

O

β

Figure 1.3: The geometry of a

Schwarzschild lens: The mass M is lo-

cated at a distance Dd

from the observer

O. 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 ⇠.

For an impact parameter ⇠ Eq. 1.13 has solution u = sin�/⇠. Inserting this in the full Eq. 1.12, whichcontains the perturbation 3

2

RSu2, and solving at first order, we find

u =1

⇠sin�+

3Rs

4⇠2

⇣1 +

1

3cos 2�

⌘(1.14)

Assuming a small deflection angle and considering that in the limit r ! 1 the angle � becomes verysmall, we get

↵ = 2|�1| = 2RS

⇠(1.15)

For the sun

↵� =1.7500

⇠/R�(1.16)

The validity of this results in the gravitational field of the sun has been confirmed with radio-interferometricmethods 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 inFigure 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 apoint mass). The condition under which the ray meets the observer in O is obtained only from geometricalconsideration

�Ds =Ds

Dd⇠ � 2RS

⇠Dds . (1.17)

Even though, for simplicity, we will stick throughout this work to a flat geometry K = 0, this is a goodmoment to point out that in general Dds 6= Ds�Dd (since in the definition of angular diameter distance onehas to account for the curvature of space 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

↵0

=

r2RS

Dds

DdDs, (1.18)

6

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

✓2 � �✓ � ↵2

0

= 0 . (1.19)

The solutions are the positions of the images

✓± =1

2

⇣� ±

q4↵2

0

+ �2

⌘. (1.20)

The term “image” (as the term “lens” actually) is abused in the sense that the rays are not focusedat the observer, nevertheless the angular separation between the two images’ positions is �✓ = ✓

+

� ✓� =p4↵2

0

+ �2 � 2↵0

. We can see that 2↵0

is the minimum of the angular separation, this happens when � = 0.This situation is called Einstein ring, the solutions are ✓± = ±↵

0

and, due to symmetry, the whole ringof angular radius ✓ = ↵

0

is a solution: a point source placed exactly on the line mass-observer is seen as acircle 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 pointout that, for geometrically-thin lenses, the deflection angles of several point masses simply add, such thatone finds (recalling RS = 2GM)

↵(⇠) =X

i

4Gmi⇠ � ⇠i

|⇠ � ⇠i|2 , (1.21)

where, in the lens plane, ⇠ is the position of the light ray while ⇠i are the positions of the masses mi (orthe projected positions of the masses on the lens plane). This can be trivially extended to the continuumlimit

↵(⇠) = 4G

Zd2⇠0 ⌃(⇠0)

⇠ � ⇠0

|⇠ � ⇠0|2 , (1.22)

where we have defined the surface mass density distribution ⌃(⇠0) and the integral extends over the wholelens plane.The lens equation 1.17 can be written in terms of the distance ⌘ = Ds� from the source to the optical axis

⌘ =Ds

Dd⇠ �Dds ↵(⇠) . (1.23)

Given the matter distribution and the position of the source, the lens equation may have more than onesolution5 ⇠. The lens equation describes a mapping ⇠ ! ⌘ from the lens plane to the source plane and thismapping is obtained straightforwardly for any mass distribution ⌃(⇠0). On the other hand the inversion ofthis 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 =⇠

⇠0

, y =⌘

⌘0

(1.24)

where ⇠0

is a length scale in the lens plane and ⌘0

= ⇠0

Ds/Dd. Defining the dimensionless surface massdensity

(x) =⌃(⇠

0

x)

⌃cwith ⌃c =

Ds

4⇡GDdDds, (1.25)

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

↵(x) =1

⇡

Zd2x0 (x0)

x� x0

|x� x0|2 , (1.26)

and the lens equation reduces to

y = x� ↵(x) . (1.27)

Furthermore since rln|x| = x

|x|2 we have ↵ = r , where

(x) =1

⇡

Zd2x0 (x0) ln|x� x0|2 (1.28)

is the lensing potential associated with the surface density (x). The dimensionless mapping x ! y is agradient mapping

y = r✓1

2x2 � (x)

◆(1.29)

which can also be expressed in terms of the scalar function

�(x,y) =1

2(x� y)2 � (x) (1.30)

as

r�(x,y) = 0 . (1.31)

What we are introducing here is the so called scalar formalism. The amplification factor6 is definedthrough the scalar function �(x,y)

I =

det

@2�(x,y)

@xi@xj

��1

. (1.32)

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 thecurvature 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 lightray 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 n, ⇥(n), is given by theunlensed temperature in the deflected direction: ⇥(n) = ⇥(n+↵), 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�)�ijdxidxj) , (2.1)

where, as usual (for K = 0),

�ijdxidxj = dr2 + r2(d✓2 + sin2 ✓ d�2) . (2.2)

In computing the deflection of light we are only interested in null-geodesics and we may as well considerthe conformally related metric

ds2 = �(1 + 4 W )dt2 + �ijdxidxj , (2.3)

where we have introduced the Weyl potential W = 1

2

( + �). We now consider a photon moving onthe unperturbed trajectory towards the observer (that we set in x = 0): the photon will come radially withunperturbed 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 ds2 are easily computed at first order(see § Appendix B) and we can write the geodesic equations

d2xµ

ds2+ �µ↵�

dx↵

ds

dx�

ds= 0 (2.4)

at first order in W and in �nµ

✓ = � 2

r2@✓ W +

2

r✓ , (2.5)

� = � 2

r2 sin2 ✓@� W +

2

r� , (2.6)

where · = d/ds. We can also rearrange these equations using the fact that, at lowest order, for a radialgeodesic r = �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 @✓ W , (2.7)

(� r2) = �2 sin�2 ✓ @� W . (2.8)

We integrate these equations using another lowest order equality: ds = dt, and we obtain

r2(t0

� t)✓(t) = 2

Z t

t0

dt0 @✓ W (t0, t0

� t0, ✓0

,�0

) , (2.9)

r2(t0

� t)�(t) = 2

Z t

t0

dt0

sin2 ✓0

@� W (t0, t0

� t0, ✓0

,�0

) , (2.10)

where t0

is the time at which the observer in x = 0 receives the photon and the integration constant isfixed consistently with the fact that r2(t = t

0

) = 0. Integrating once more

✓(t⇤) = ✓0

+ 2

Z t⇤

t0

dtt� t⇤

(t0

� t⇤)(t0 � t)@✓ W (t, t

0

� t, ✓0

,�0

) , (2.11)

�(t⇤) = �0

+ 2

Z t⇤

t0

dt

sin2 ✓0

t� t⇤(t

0

� t⇤)(t0 � t)@� W (t, t

0

� t, ✓0

,�0

) , (2.12)

where t⇤ is the initial (emission) time.We can finally define the displacement vector on the sphere (deflection angle) as

↵ =

✓✓ � ✓

0

sin ✓0

(�� �0

)

◆(2.13)

so that

↵ = �2

Z t0

t⇤

dtt� t⇤

(t0

� t⇤)(t0 � t)r W (t, t

0

� t, ✓0

,�0

) , (2.14)

where r = (@✓, sin�1 ✓ @�) is the gradient on the sphere. Of course when we will study the CMB we

will set t⇤ = tdec. It is also important to point out that working at first order allows us to use the Bornapproximation and compute the integral on the unperturbed path.

Lensing Potential

Looking at Eq. 2.14 is clear that we can rewrite it as ↵ = r⌥, through the definition of the Lensing Potential

⌥ = �2

Z t0

t⇤

dtt� t⇤

(t0

� t⇤)(t0 � t) W (t, t

0

� t, ✓0

,�0

) . (2.15)

A single 2D map of the lensing potential ⌥ contains all the required information since recombinationis approximated as instantaneous (a single hyper-surface source at t⇤ = tdec) and in general the e↵ects oflate-time sources, including reionization, are neglected.

As for the CMB temperature field ⇥(n), we are interested in the angular power spectrum of the lensingpotential ⌥(n). This will be written in terms of the power spectrum of the gravitational potential P

. Thelensing potential can be expanded in spherical harmonics since it lives on the sphere

⌥(n) =X

lm

�lmYlm(n) (2.16)

and we are interested in computing h�lm�⇤l0m0i = �ll0�mm0C⌥l . The gravitational potential is a 3D fieldand it has to be expanded in Fourier space

10

W (x; t) =

Zd3k

(2⇡)3 W (k; t) eik·x . (2.17)

We define the power spectrum of a statistically isotropic variable f

hf(k; t)f⇤(k0; t0)i = (2⇡)3 Pf (k; t, t0) �(k� k0) , (2.18)

such that

h W (k; t) ⇤W (k0; t0)i = (2⇡)3 P

(k; t, t0) �(k� k0) . (2.19)

Using Eq. 2.15 and 2.19 the angular correlation function for the lensing potential is

h⌥(n)⌥(n0)i = 4

Z t⇤

t0

dt

Z t⇤

t0

dt0t� t⇤

(t0

� t⇤)(t0 � t)

t0 � t⇤(t

0

� t⇤)(t0 � t0)

⇥Z

d3k

(2⇡)6(2⇡)3 P

(k; t, t0) eik·x(t)e�ik·x0(t) ,

(2.20)

where x(t) = n(t0

� t) and x0(t) = n0(t00

� t0). We can now make explicit the angular dependence byusing the Rayleigh’s expansion

eik·x = eik·n(t0�t) = 4⇡X

lm

iljl(k(t0 � t))Ylm(n)Y ⇤lm(k) (2.21)

and performing the angular integral over k using the properties of spherical harmonics (see § Appendix C)we find

h⌥(n)⌥(n0)i = 8

⇡

X

ll0 mm0

Z t⇤

t0

dt

Z t⇤

t0

dt0t� t⇤

(t0

� t⇤)(t0 � t)

t0 � t⇤(t

0

� t⇤)(t0 � t0)

⇥Z

dk k2 jl(k(t0 � t))jl0(k(t0 � t0))P

(k; t, t0)Ylm(n)Y ⇤l0m0(n0)�ll0�mm0 ,

(2.22)

where we’ve made use of the spherical Bessel2 function jl. Now we can simply recall h�lm�⇤l0m0i =�ll0�mm0C⌥l and read o↵ the spherical harmonics components of Eq. 2.22 to find

C⌥l =8

⇡

Zdk k2

Z t⇤

t0

dt

Z t⇤

t0

dt0t� t⇤

(t0

� t⇤)(t0 � t)

t0 � t⇤(t

0

� t⇤)(t0 � t0)jl(k(t0 � t))jl(k(t0 � t0))P

(k; t, t0) .

Here we can introduce the primordial dimensionless power spectrum �2

(k) which is related to the powerspectrum P

through the transfer function T

(k, t)

P

(k; t, t0) =2⇡2

k3�2

(k)T

(k, t)T

(k, t0) . (2.23)

This allow us to recast

C⌥l = 16⇡

Zdk

k�2

(k)

����Z t⇤

t0

dtt� t⇤

(t0

� t⇤)(t0 � t)jl(k(t0 � t))T

(k, t0

� t)

����2

. (2.24)

2In terms of the ordinary Bessel functions J

n

the spherical Bessel are defined

j

l

(x) =

q⇡

2xJl+1/2(x). For more details see [21], [22] or § Appendix D.

11

Given some primordial power spectrum �2

(k), the C⌥l coe�cients can be computed using numericalcodes such as CAMB3, CMBFAST4 or CLASS5 [13]. The numerical computation is lighter if we useLimber approximation (see § Appendix E) to get

C⌥l ' 64

(2l + 1)4

Z t⇤

t0

dt

✓t� t⇤

(t0

� t⇤)(t0 � t)

◆2

P

✓l + 1/2

t0

� t; t, t

◆. (2.25)

In Figure 2.1 and 2.2 is shown the power spectrum for a ⇤CDM cosmology with h = 0.67, ⌦mh2 = 0.14,⌦bh2 = 0.022 and ⌦K = 0. The primordial dimensionless power spectrum for scalar perturbation is

�2(k) = As

✓k

k⇤

◆ns

�1

, (2.26)

with k⇤ = 0.05 Mpc�1, As(k⇤) = 2.215⇥ 10�9 and the spectral index ns = 0.962.

10-5 10-4 0.001 0.01 0.1 1

100

200

500

1000

2000

5000

1¥104

2¥104

k HhêMpcL

PHkL

Ê

ÊÊ

ÊÊÊÊ Ê

ÊÊÊÊÊÊÊÊÊÊÊ

ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ5 10 50 100 500 1000

5¥10-9

1¥10-8

2¥10-8

5¥10-8

1¥10-7

multipole l

l2Hl+1L2 C

lê2p

10-2< k < 10-110-3< k < 10-2k < 10-3

Figure 2.1: Left: The matter density fluctuations 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)

and the contribution of di↵erent wavenumbers k in hMpc

�1(dashed).

5 10 50 100 500 1000

1¥10-10

5¥10-101¥10-9

5¥10-91¥10-8

5¥10-81¥10-7

multipole l

l2Hl+1L2 C

lê2p

z>20z>10z>5z>1z>0.5z>0

5 10 50 100 500 1000

10-12

10-11

10-10

10-9

10-8

10-7

multipole l

l2Hl+1L2 C

lê2p

z<10z<3z<1z<0.5z<0.2z<•

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

200 500 1000 2000 500010-15

10-14

10-13

10-12

10-11

10-10

multipole l

lHl+1LC

lQê2p

0 500 1000 1500 2000 2500

0.00

0.05

0.10

multipole l

DC lQêC lQ

Lensed ClQ

Unlensed ClQ

Figure 2.3: Left: The lensed temperature power spectrum (solid) and the unlensed spectrum (dashed). Right: The fractional change

in the power spectrum due to lensing. Both plots are for a typical concordance ⇤CDM model with h = 0.67, ⌦m

h2= 0.14, ⌦

b

h2= 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, bycomputing the lensed CMB power spectrum at lowest order in C⌥l .We shall assume that temperature fluctuations are statistically isotropic and we work in harmonic spaceusing the flat sky 2D Fourier transform convention

⇥(x) =

Zdl2

2⇡⇥(l)eil·x , ⇥(l) =

Zdx2

2⇡⇥(x)e�il·x. (2.27)

For the assumption of isotropy the correlation function ⇠ can only depend on the separation between twopoints h⇥(x)⇥(x0)i = ⇠(|x� x0|), so that

h⇥(l)⇥⇤(l0)i =Z

d2x

2⇡

Zd2x0

2⇡e�il·xeil

0·x0⇠(|x� x0|)

=

Zd2x

2⇡

Zd2r

2⇡ei(l

0�l)·xe�il0·r⇠(r)

= �(l0 � l)

Zd2r e�il·r⇠(r) ,

(2.28)

where r = x� x0. We can now read from Eq. 2.28

C⇥l =

Zd2r e�il·r⇠(r) =

Zr dr

Zd�

r

e�i lr cos(�l��r)⇠(r) = 2⇡

Zr drJ

0

(lr)⇠(r). (2.29)

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 C⌥l .

Lensing re-maps the temperature according to

⇥(x) = ⇥(x+r⌥)' ⇥(x) +ra⌥(x)ra⇥(x) +

1

2ra⌥(x)rb⌥(x)rarb⇥(x) + ...

(2.30)

This series expansion is not likely to be accurate on all scales: when l & 3000 the deflection angleis comparable to the wavelength of the unlensed field [11], so the Taylor expansion is no longer a goodapproximation. Using

13

ra⌥(x) = i

Zd2l

2⇡l⌥(l)eil·x , ra⇥(x) = i

Zd2l

2⇡l⇥(l)eil·x (2.31)

and assuming that ⌥ and ⇥ are uncorrelated h⌥(n)⇥(n)i = 0, we can obtain the Fourier components of⇥(l)

⇥(l) ⇠⇥(l)�Z

d2l0

2⇡l0 · (l� l0)⌥(l� l0)⇥(l0)

� 1

2

Zd2l

1

2⇡

Zd2l

2

2⇡l1

· (l1

+ l2

� l)l1

· l2

⇥(l1

)⌥(l2

)⌥⇤(l1

+ l2

� l) .

(2.32)

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

C⇥l ⇠ (1� l2R⌥)C⇥l +

Zd2l0

2⇡[l0 · (l� l0)]2C⌥|l�l

0| C⇥

l0 , (2.33)

where we have defined half of the total deflection angle power

R⌥ =1

2h|r⌥|2i = 1

4⇡

Zdl l3C⌥l . (2.34)

The e↵ect of GL on the CMB power spectrum can be understood in Figure 2.3. This e↵ect is severalpercent at l & 1000. On small scales where there is little power in the unlensed CMB, GL transfers powerfrom 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 thevalidity of the Taylor expansion, the small-scale limit, the correlation function for the flat-sky limit andlensing 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 FLUniverse perturbed by Gravitational Waves (GW). The perturbation is included into the line element

ds2 = a2(t)(�dt2 + (�ij + hij)dxidxj) (3.1)

and it is parametrized by |hij | ⌧ 1 a symmetric, traceless and divergence-free 3⇥ 3 tensor. Working inthe conformally related metric

ds2 = (�dt2 + (�ij + hij)dxidxj) (3.2)

will be more convenient. In spherical coordinates, for a flat universe K = 0, the metric is written

gµ⌫ =

2

664

�1 0 0 00 1 + hrr rhr✓ r sin ✓ hr�

0 rhr✓ r2(1 + h✓✓) r2 sin ✓ h✓�0 r sin ✓ hr� r2 sin ✓ h✓� r2 sin2 ✓ (1 + h��)

3

775 . (3.3)

Similarly to the scalar perturbations computation we consider a photon with 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

ds2+ �µ↵�

dx↵

ds

dx�

ds= 0 , (3.4)

at first order in the perturbation and in �nµ. We will only need the geodesics equations for µ = �, ✓ tocompute the deflection angle (the displacement along the line of sight leads to a time delay which is notrelevant in computing ↵)

d

ds(r2✓) =

1

2@✓hrr + r

d

dshr✓ � hr✓

d

ds(r2�) =

1

2 sin2 ✓@�hrr � 1

sin ✓hr� +

r

sin ✓

d

dshr� .

(3.5)

Integrating these equations, using the fact that at lowest order dt = ds, gives

15

[r2✓]tt0 =

Z t

t0

dt0✓1

2@✓hrr(t

0, t0

� t0, ✓0

,�0

)

◆+ [r hr✓(t

0, t0

� t0, ✓0

,�0

)]tt0 ,

[r2�]tt0 =1

sin2 ✓0

Z t

t0

dt0✓1

2@�hrr(t

0, t0

� t0, ✓0

,�0

)

◆+

1

sin ✓0

[r hr�(t0, t

0

� t0, ✓0

,�0

)]tt0 .

Here t0

is, as before, the time of arrival at x = 0. Given that r(t0

) = 0 and ✓(t0

) = �(t0

) = 0 we get

✓(t) =1

(t� t0

)2

Z t

t0

dt0✓1

2@✓hrr(t

0, t0

� t0, ✓0

,�0

)

◆+

1

(t� t0

)hr✓(t

0, t0

� t0, ✓0

,�0

) ,

�(t) =1

sin2 ✓0

(t� t0

)2

Z t

t0

dt0✓1

2@�hrr(t

0, t0

� t0, ✓0

,�0

)

◆+

1

sin ✓0

(t� t0

)hr�(t

0, t0

� t0, ✓0

,�0

) .

Integrating this equations once more leads to

✓(t⇤) = ✓0

+

Z t⇤

t0

dt

hr✓(t, t0 � t, ✓

0

,�0

)

(t� t0

)+

1

2

t� t⇤(t� t

0

)(t0

� t⇤)@✓hrr(t, t0 � t, ✓

0

,�0

)

�,

�(t⇤) = �0

+

Z t⇤

t0

dt

hr�(t, t0 � t, ✓

0

,�0

)

sin ✓0

(t� t0

)+

1

2 sin2 ✓0

t� t⇤(t� t

0

)(t0

� t⇤)@�hrr(t, t0 � t, ✓

0

,�0

)

�.

The gradient on the sphere is r = (@✓, sin�1 ✓@�) and recalling the definition in Eq. 2.13 we can write

the displacement vector ↵ in a compact form

↵i =

Z t⇤

t0

dt

hri(t, t0 � t, ✓

0

,�0

)

(t� t0

)+

1

2

t� t⇤(t� t

0

)(t0

� t⇤)rihrr(t, t0 � t, ✓

0

,�0

)

�. (3.6)

3.2 Euler Rotation method

Having found an expression for the displacement vector we now have to work out a way to describe tensorperturbations: 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 metricperturbation as

hij(x, t) =

Zd3k

(2⇡)3eik·x

⇥h�k

(t)e�ij(k) + h⌦k

(t)e⌦ij(k)⇤, (3.7)

where we have introduced two time-independent polarization tensors e�ij(k) and e⌦ij(k) which can be

expressed in terms of vectors of the orthonormal basis (k, e1, e2)

e�ij(k) =1p2[e1i (k)e

1

j (k)� e2i (k)e2

j (k)] ,

e⌦ij(k) =1p2[e1i (k)e

2

j (k)� e2i (k)e1

j (k)] .(3.8)

The polarization tensors depend on k since the frame (k, e1, e2), which reveals the 2 physical d.o.f. ofthe 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 k

16

Figure 3.1: Euler rotation: The two vectors k and r are showed together with the three Euler angles (↵, �, �) necessary to turn

(e1, e2,k) into (e

✓

, e�

, r).

(e�ij) =

✓1 00 �1

◆, (e⌦ij) =

✓0 11 0

◆(3.9)

such that the tensor perturbation can be written

h(k) =

✓h�k

h⌦k

h⌦k

�h�k

◆. (3.10)

Introducing the helicity basis

e± =1p2(e1 ⌥ ie2) , (3.11)

we can write the tensor perturbation in this basis

hE(k) =1p2

✓h�k

� ih⌦k

00 h�

k

+ ih⌦k

◆=

✓h+

k

00 h�

k

◆= h+

k

e+ij + h�k

e�ij , (3.12)

where we have defined e±ij = e±i e±j . We have also set h±

k

⌘ 1p2

(h�k

⌥ ih⌦k

) and we shall work with these

from now on1. We now want to write the polarization tensors in the spherical basis of equation 3.2. In orderto do that we perform a rotation with Euler angles (↵,�, �) to turn (k, e1, e2) 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

hS = R1

(�↵)R3

(��)R1

(�) h RT1

(�)RT3

(��)RT1

(�↵) , (3.13)

1Working with h

±k instead of the usual h

�,⌦k is more convenient and, in the end, it will not make any di↵erence since that,

when substituting the power spectra:

hh

+k h

+⇤k i =

12 (hh

�k h

�⇤k i+ hh

⌦k h

⌦⇤k i) = hh

�k h

�⇤k i

17

where h is the tensor perturbation in the frame (k, e1, e2) which is a function of h±k

and hS is theperturbation in the frame (r, e✓, e�). Performing the transformation we can write the Fourier componentsin (r, e✓, e�) as

hrrk

=1

2sin2 � (e2i� h+

k

+ e�2i� h�k

)

hr±k

⌘ 1p2(hr✓

k

⌥ ihr�k

) =sin(�)

�(cos(�)⌥ 1)e2i� h+

k

+ (cos(�)± 1)e�2i� h�k

�

2p2

ei↵ .(3.14)

We point out that ↵, � and � are function of both k and n. Writing Eq. 3.6 in Fourier components weobtain

↵± =1p2(↵✓ ⌥ i↵�) =

Z t⇤

t0

dt

Zd3k

(2⇡)3

⇣ hr±k

t� t0

eik(t�t0) cos �

+1

2p2

t� t⇤(t� t

0

)(t0

� t⇤)(r✓ ⌥ ir�)(h

rrk

eik(t�t0) cos �)⌘,

(3.15)

where we used the fact that � is the angle between k and r such that eik·r = eikr cos � . From this, usingEq. 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 asingle scalar lensing potential ↵ = r⌥. When considering tensor perturbations this is no longer true andwe have to decompose ↵ generically, in its gradient-mode and curl-mode potentials.We write

↵ = r +r⇥ ~A (3.16)

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

r =

✓@✓ 1

sin ✓@�

◆, r⇥ ~A =

✓1

sin ✓@�$�@✓$◆

(3.18)

explicitly

↵✓ = @✓ +1

sin ✓@�$ ,

↵� =1

sin ✓@� � @✓$ .

(3.19)

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

(r⇥ !) = ?(d!)

(r⇥) : k-form ! (n� k � 1)-form

(3.17)

Since the space of k-form has dimension

�n

k

�, in 3 dimension the curl takes a 1-form (a 3D vector) into another 1-form (a 3D

vector). In 2 dimension however the curl takes a 0-form (a scalar) into a 1-form (a 2D vector): (r⇥ !)

i

= (?r!)

i

= "

ij

r

j

!.

18

Let us simply introduce the spin raising and lowering operators /@ and /@⇤

/@s =⇣s cot ✓ � @✓ � i

sin ✓@�⌘, (3.20)

/@⇤s =

⇣�s cot ✓ � @✓ +

i

sin ✓@�⌘. (3.21)

Later we discuss their relations with the spin-s Spherical Harmonics. In the helicity basis we can write

↵+

= � 1p2/@⇤( + i$) ,

↵� = � 1p2/@( � i$) .

(3.22)

We can rewrite the explicit expressions for ↵± in terms of � = t� t0

↵± =

Z �⇤

0

d�

Zd3k

(2⇡)3

✓hr±k

�eik� cos � � 1

2p2

�� �⇤��⇤

(r✓ ⌥ ir�)(hrrk

eik� cos �)

◆

and observing that = �p2 r

e�,e+ = �(r✓ ± ir�) = /@, /@⇤we can write two di↵erential equations for and

$

/@⇤( + i$) = �

p2

Z �⇤

0

d�

Zd3k

(2⇡)3

✓hr+k

�eik� cos � +

1

2p2

�� �⇤��⇤

/@⇤(hrr

k

eik� cos �)

◆,

/@( � i$) = �p2

Z �⇤

0

d�

Zd3k

(2⇡)3

✓hr�k

�eik� cos � +

1

2p2

�� �⇤��⇤

/@(hrrk

eik� cos �)

◆.

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

spin lowering and raising operators. We first observe that the hrrk

part is already in the desired form and wehave to worry only about the hr±

k

parts. To illustrate the procedure we do it explicitly for the h+

k

part of↵+

; for the rest one can follow the same steps. Using Eq. 3.14 we write explicitly this part in terms of theEuler angles

↵+

=

Z �⇤

0

dt

Zd3k

(2⇡)3

✓1

�

sin�(�1 + cos�)

2p2

ei↵e2i� eik� cos � h+

k

+ [...]h�k

+ ...

◆(3.23)

and we make the ansatz

1

�

sin�(�1 + cos�)

2p2

ei↵e2i� eik� cos � = /@⇤⇣g(�)e2i� eik� cos �

⌘(3.24)

To understand this assumption, one should know that, from geometrical considerations, we have

@✓�(✓,�) = � cos↵

@��(✓,�) = sin↵ sin ✓(3.25)

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

we find

g(�) = C e�ik� cos � tan2✓�

2

◆�

i(i+ k�+ k� cos�) tan2⇣�2

⌘

2p2 k2�3

(3.26)

19

It turns out that the h�k

part of ↵� has the same solution g(�), while the h�k

part of ↵+

and the h+

k

partof ↵� have solution

µ(�) = D e�ik� cos � cot2✓�

2

◆+

(1 + ik�� ik� cos�) cot2⇣�2

⌘

2p2 k2�3

(3.27)

The constants C and D are fixed to C = � e�ik�

2

p2 k2�3 and D = � eik�

2

p2 k2�3 by requiring and $ to be

square integrable on the sphere.We now have two equations in the form

/@⇤[ + i$] = �

p2 /@

⇤hZ �⇤

0

d�

Zd3k

(2⇡)3

⇣(g(�)e2i�h+

k

+ µ(�)e�2i�h�k

) +1

2p2

�� �⇤��⇤

hrrk

⌘eik� cos �

i

/@[ � i$] = �p2 /@

hZ �⇤

0

d�

Zd3k

(2⇡)3

⇣(µ(�)e2i�h+

k

+ g(�)e�2i�h�k

) +1

2p2

�� �⇤��⇤

hrrk

⌘eik� cos �

i

so that

+ i$ = �Z �⇤

0

d�

Zd3k

(2⇡)3

⇣p2(g(�)e2i�h+

k

+ µ(�)e�2i�h�k

) +1

2

�� �⇤��⇤

hrrk

⌘eik� cos �

� i$ = �Z �⇤

0

d�

Zd3k

(2⇡)3

⇣p2(µ(�)e2i�h+

k

+ g(�)e�2i�h�k

) +1

2

�� �⇤��⇤

hrrk

⌘eik� cos �

by simply adding and subtracting we find the scalar and the vector potentials. Inserting the results fromEq. 3.26, 3.27 and 3.14 we find

=�0

@e�ik�

⇣4eik�(cos(�)+1)

�2 csc2(�) + ik� cos(�)� 1

�� 8 csc2(�)⇣sin4

⇣�2

⌘+ cos4

⇣�2

⌘e2ik�

⌘⌘

8k2�3

+1

4

�� �⇤��⇤

sin2(�)eik� cos �

◆(h�

k

e�2i� + h+

k

e2i�) ,

$ =e�ik�

⇣2i csc2(�)

⇣� sin4

⇣�2

⌘+ cos4

⇣�2

⌘e2ik�

⌘+ eik�(cos(�)+1)(k�� 2i cot(�) csc(�))

⌘

2k2�3

(h�k

e�2i�+h+

k

e2i�) ,

where we have written

=

Z �⇤

0

d�

Zd3k

(2⇡)3 and $ =

Z �⇤

0

d�

Zd3k

(2⇡)3$

since it is less heavy to work without the integral signs and all the manipulations which will be done inthe following (i.e. the angular decomposition) leave the integrals untouched. The two potentials are bothsquare integrable on the sphere

Z ⇡

0

d� sin� | |2 < 1 ,

Z ⇡

0

d� sin� |$|2 < 1(3.28)

and despite the fact that their are not very illuminating at the moment, their angular expansion will berather simple.

20

3.3.1 Angular decomposition of and $

We define, as in the scalar perturbation case,

(n) =X

lm

almYlm(n) ,

$(n) =X

lm

blmYlm(n)(3.29)

and we are left with the problem of expanding and $ in spherical harmonics. This is not trivial sincethe expression of the potentials are somewhat lengthy and also they are explicit functions of the Euler angles(↵,�, �) and not of n = (✓,�).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

/@(sYlm) =p

(l � s)(l + s+ 1) s+1

Ylm ,

/@⇤(sYlm) = �

p(l + s)(l � s+ 1) s�1

Ylm . (3.30)

We also wish to find an operator similar to the Laplacian L2 but for s 6= 0, the spin-weighted L2 suchthat

L2

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

Since Eq. 3.31 and 3.30 imply, for the commutators, [L2

(s), /@] = [L2

(s), /@⇤] = 0 one can find (see § Ap-

pendix C)

L2

(s)(✓,�) = �⇣@2✓ +

cos ✓

sin ✓@✓ +

1

sin2 ✓@2�

⌘� i

2s cos ✓

sin2 ✓@� +

s2

sin2 ✓. (3.32)

Another useful formula is derived from Eq. 2.21

eik� cos ✓ =X

l

(2l + 1) iljl(k�)Pl(cos ✓) (3.33)

and its second derivative with respect to cos ✓

eik� cos ✓ = � 1

k2�2

X

l

(2l + 1) iljl(k�)P00l (cos ✓) . (3.34)

Finally, as we pointed out before, since and $ are functions of the Euler angles we will first expandthem into sYlm(�,↵) and then we will need a formula to write them in terms of Ylm(✓,�): the rotationwith Euler angles (↵,�, �) which we have performed to rotate (✓k,�k) into (✓,�) means, for the sphericalharmonics,

r4⇡

2l + 1

X

m0sYlm0(✓k,�k) mY ⇤

lm0(✓,�) = sYlm(�,↵)e�is� . (3.35)

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

1

=�e�ik�

⇣4eik�(cos(�)+1)

�2 csc2(�) + ik� cos(�)� 1

�� 8 csc2(�)⇣sin4

⇣�2

⌘+ cos4

⇣�2

⌘e2ik�

⌘⌘

8k2�3

,

2

=� 1

4

�� �⇤��⇤

sin2(�)eik� cos �

such that

=

Z �⇤

0

d�

Zd3k

(2⇡)3(

1

+ 2

)(h�k

e�2i� + h+

k

e2i�) . (3.36)

The second part is simpler and we should start with this. Using Eq. 3.34

2

= �1

4

�� �⇤��⇤

sin2 � eik� cos � =1

4

�� �⇤��⇤

sin2 �1

k2�2

X

l

(2l + 1) iljl(k�)P00l (cos�)

=1

4

�� �⇤�⇤

X

l

(2l + 1) iljl(k�)

k2�3

sin2 � P 00l (cos�) .

(3.37)

Working with the spin raising and lowering operators is trivial to find

±2

Yl0(�,↵) =

r2l + 1

4⇡

s(l � 2)!

(l + 2)!sin2 � P 00

l (cos�) , (3.38)

which can be inserted into Eq. 3.37 to give

2

=

p⇡

2

�� �⇤�⇤

X

l

s

(2l + 1)(l + 2)!

(l � 2)!iljl(k�)

k2�3

±2

Yl0(�,↵) . (3.39)

Leaving this result aside for a moment we can do the same for 1

. The problem here is that is impossibleto “guess” the decomposition of

1

into sYlm(�,↵) as was done in Eq. 3.38 for 2

. Luckily one finds that

L2

(2)

(�,↵)[ 1

] =1

2�(�3� ik� cos�) sin2 � eik� cos � . (3.40)

We can now compute this decomposition and then, when all the (�,↵) dependence is in the sYlm(�,↵),act with the inverse Laplacian to come back to

1

. In doing this, we have to deal with a part proportionalto sin2 � eik� cos � , which goes exactly like the

2

calculation, and the other part proportional to

cos� sin2 � eik� cos � / cos� sin2 � P 00l (cos�) = cos� P 2

l (cos�) ,

where we have introduced the associated Legendre Polynomials Pml (see § Appendix C). Now, we can

get rid of the cos� by using

cos� P 2

l (cos�) =(l � 1)P 2

l+1

(cos�) + (l + 2)P 2

l�1

(cos�)

2l + 1(3.41)

and we are back to the already known problem, which is easy to solve since we have already seen thatP 2

l (cos�) / ±2

Yl0(�,↵).

We now have both 1

and 2

decomposed into ±2

Yl0(�,↵). Going back to Eq. 3.36 we reconstruct and we recognise the sYlm(�,↵)e�is� part in Eq. 3.35.

22

=X

l

([...])h±k

�2

Yl0(�,↵)e�2i� + �2

Yl0(�,↵)e+2i�

�. (3.42)

This gives us the sum over m and the decomposition into Ylm(n) from which we can read o↵4

alm =

Z �⇤

0

dt

Zd3k

(2⇡)3

"2⇡il

s(l � 2)!

(l + 2)!(l + 2)(l � 1)

✓l + 1

2

✓l�� �⇤�⇤

+ 2

◆jl(k�)

k2�3

� jl+1

(k�)

k�2

◆

⇥⇣�2

Ylm(k)h+

k

++2

Ylm(k)h�k

⌘#.

(3.43)

Curl-mode potential $

For the vector potential we have

L2

(2)

(�,↵)[$] =k

2sin2 � eik� cos � (h+

k

e+2i� + h�k

e�2i�) . (3.44)

We go through the same computation as before

L2

(2)

(�,↵)[$] =k

2sin2 � eik� cos � (h+

k

e+2i� + h�k

e�2i�)

=1

2k�2

X

l

(2l + 1)iljl(k�) sin2 � P 00

l (cos�) (h+

k

e+2i� + h�k

e�2i�)

=X

l

s

4⇡(2l + 1)(l + 2)!

(l � 2)!iljl(k�)

2k�2

!

±2

Yl0(�,↵) (h+

k

e+2i� + h�k

e�2i�)

(3.45)

and act with the inverse Laplacian to go back to the expansion of $. Using Eq. 3.35 we find

$ =X

lm

4⇡ il

s(l + 2)(l � 1)

l(l + 1)

jl(k�)

2k�2

!(2

Ylm(k)h�k

+ �2

Ylm(k)h+

k

)Ylm(n) . (3.46)

Again, we can read o↵ the spherical harmonics expansion coe�cients of $

blm =

Z �⇤

0

d�

Zd3k

(2⇡)3

" 4⇡ il

s(l + 2)(l � 1)

l(l + 1)

jl(k�)

2k�2

!(2

Ylm(k)h�k

+ �2

Ylm(k)h+

k

)

#. (3.47)

4Inserting again the integral signs so that the a

lm

s are the angular coe�cients of and not of

˜

23

3.4 Curl-mode & Gradient-mode Power Spectra

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

halma⇤l0m0i = �ll0�mm0C l , hblmb⇤l0m0i = �ll0�mm0C$

l . (3.48)

Let us first compute C l

halma⇤l0m0i = 2⇡il

s(l � 2)!

(l + 2)!2⇡il

0

s(l0 � 2)!

(l0 + 2)!(l + 2)(l � 1)(l0 + 2)(l0 � 1)

⇥Z �⇤

0

d�

Z �⇤

0

d�0Z

d3k

(2⇡)3

Zd3k0

(2⇡)3

"✓l + 1

2

✓l�� �⇤�⇤

+ 2

◆jl(k�)

k2�3

� jl+1

(k�)

k�2

◆

⇥✓l0 + 1

2

✓l0�0 � �0

⇤�0⇤

+ 2

◆jl0(k0�0)

k02�0

3

� jl0+1

(k0�0)

k0�02

◆

⇥ h��2

Ylm(✓k,�k)h+

k

(�) ++2

Ylm(✓k,�k)h�k

(�)� �

�2

Yl0m0(✓k0 ,�k0)h+

k

0(�0) ++2

Yl0m0(✓k0 ,�k0)h�k

0(�0)�i#.

(3.49)

We now use the definitions

hh+

k

(�)h+⇤k

0 (�0)i = hh�k

(�)h�⇤k

0 (�0)i = (2⇡)3�3(k� k0)Ph(k;�,�0) (3.50)

to get rid of the d3k0 integral. We have assumed the two helicities to have identical spectra and to beuncorrelated (parity invariance), so that hh+

k

(�)h�⇤k

0 (�0)i = 0. The �ll0�mm0 part of Eq. 3.48 is generated bythe angular integral over d⌦

k

, and finally we can write

C l = 2⇡

(l + 2)(l � 1)

l(l + 1)

Zdk

k�2

h(k)

����Z �⇤

0

d� Th(k,�)

✓l + 1

2

✓l�� �⇤�⇤

+ 2

◆jl(k�)

k2�3

� jl+1

(k�)

k�2

◆����2

. (3.51)

Here we have written the power spectrum Ph(k;�,�0) using

k3

2⇡2

Ph(k;�,�0) = �2

h(k)Th(k,�)Th(k,�0) , (3.52)

where �2

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

C$l = ⇡

(l + 2)(l � 1)

l(l + 1)

Zdk

k�2

h(k)

����Z �⇤

0

dt Th(k,�)jl(k�)

k�2

����2

. (3.53)

For the cross-correlation power spectrum CXl we have

halmb⇤l0m0i = �ll0�mm0CXl = 0 . (3.54)

This can be understood as conservation of parity due to the fact that the two perturbation helicities areindependent. From a mathematical point of view this arises from the angular integral d⌦

k

in Eq. 3.49: inthe C

l and C$l cases this gives �ll0�mm0+�ll0�mm0 while for the cross correlation we get �ll0�mm0��ll0�mm0 = 0.

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 totalangular momentum method (TAM) [24], but the results are in perfect agreement.

24

1¥10-4 5¥10-4 0.001 0.005 0.010 0.050 0.10010-13

10-12

10-11

10-10

10-9

10-8

k H1êMpcL

D2HkLT

2HkL

5 10 50 100 50010-20

10-18

10-16

10-14

10-12

10-10

multipole l

l2Hl+1L2 C

lê2p

ClwCly

PGW

z=100

z=10

z=1

z=0.1

Figure 3.2: Left: The dimensionless power spectrum generated by primordial GWs with r = 0.2 with the analytic matter domination

transfer function. Right: The angular power spectra for the curl-mode C$l

and gradient-mode C l

lensing potentials.

At this point is useful to compute the two power spectra. For simplicity and only to get a qualitativeunderstanding of the behavior of the power spectra we do the computation in a matter dominated (MD)universe with h = 0.675, ⌦m = 1, ⌦bh2 = 0.022 and ⌦K = 0.

The dimensionless power spectrum for primordial tensor perturbation is assumed to be

�2

h(k) = r As

✓k

k⇤

◆nt

, (3.55)

with r = 0.2 and the other parameters consistent with the scalar perturbations computation. We willalso assume the second order consistency relation nt = � r

8

�2� r

8

� ns

�. The transfer function of tensor

perturbations in matter domination is

Th(k,�) =3j

1

(k(t0

� �))

k(t0

� �). (3.56)

The behavior of the power spectra is shown in Figures 3.2, 3.3 & 3.4. Unfortunately Limber approximationcannot be used in the computation of C

l and C$l since this transfer function is not slowly varying with

respect to the spherical Bessel functions.

2 5 10 20 50 10010-19

10-17

10-15

10-13

10-11

multipole l

l2Hl+1L2 C

lê2p

5 10 20 50 10010-18

10-16

10-14

10-12

multipole l

l2Hl+1L2 C

lê2p

2 ¥ 10-3< k < 10-210-3< k < 2 ¥ 10-3k < 10-3Clw

2 ¥ 10-3< k < 10-210-3< k < 2 ¥ 10-3k < 10-3Cly

Figure 3.3: The angular power spectra of the curl-mode C$l

and gradient-mode C l

lensing potentials (solid), together with the

contribution of wavenumbers k in Mpc

�1(dashed).

We can make a first remark here: since we have r = 0.2 one could naively expect the spectra for thetensor lensing potentials to be ⇠ 5 times smaller than the one generated by scalar perturbation. This isof course not true (C

l and C$l are ⇠ 5 orders of magnitude smaller than C⌥l ) since density perturbations

25

2 5 10 20 50 10010-17

10-15

10-13

10-11

multipole l

l2Hl+1L2 C

lê2p

5 10 20 50 10010-19

10-17

10-15

10-13

10-11

multipole l

l2Hl+1L2 C

lê2p

z<1z<5z<20z<100z<300Clw

z<1z<5z<20z<100z<300Cly

Figure 3.4: The contribution of di↵erent redshift z (dot dashed) to the power spectra of the curl-mode C$l

and gradient-mode C l

lensing potentials (solid).

grow while tensor perturbations are constant outside the horizon and then decay inside the horizon. For thisreason we also expect the contribution of high redshift to be the main contribution of the lensing potentialspower 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 orderdark 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

P (2)

h (k) =100

9

Zd3q

✓q2 � (�ijqikj)2)

k2

◆2

T 2

(q)T 2

(|k� q|)P in

(q)P in

(|k� q|) k�4 , (3.57)

where P

(q) is the power spectrum of the gravitational potential and T

(q) is the transfer functionalready used in Section 2. This is of course a second order quantity that vanishes in linear perturbationtheory, nevertheless we can use Eq. 3.53 and 3.51 to check if they have a relevant e↵ect in lensing by tensorperturbations. The lensing potentials power spectra of the induced GWs are many orders of magnitudesmaller than the lensing potentials spectra of primordial GWs. This again comes from the fact that inducedGWs, since they are induced by density perturbation, are relevant at small redshift and they did not havethe time to become an important contribution with respect to the primordial one, whose contribution comesfrom 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 howGL a↵ects the shape of the CMB temperature spectrum C⇥l defined in Eq. 1.9. This means, as we havedone at the end of section 2, computing the lensed CMB power spectrum C⇥l . We do that in two ways:first the full-sky method, using a Taylor expansion for the deflection angle, then the flat-sky method whichis 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 thewavelength 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 su�cient but the Taylor expansion fails6.

3.5.1 Full-Sky - Small angle

Assuming a small deflection angle ↵ we can Taylor expand ⇥(n) = ⇥(n+↵) to second order

⇥(n) = ⇥(n+↵) ' ⇥(n) +ra⇥↵a +1

2rbra⇥↵a↵b + ... (3.58)

Recalling the angular decompositions of Eq. 3.29 and Eq. A.1 this yields to

✓lm = ✓lm +

Zd⌦

✓ra⇥↵a +

1

2rbra⇥↵a↵b

◆Y ⇤lm

= ✓lm +X

l1m1l2m2

✓l1m1

⇣al2m2I

(a) + bl2m2I(b)⌘

+1

2

X

l1m1l2m2l3m3

✓l1m1

⇣al2m2a

⇤l3m3

K(a) + bl2m2b⇤l3m3

K(b)⌘

,

(3.59)

where for the last equality we have used ↵a = ra + "barb$ and defined

I(a)lml1m1l2m2=

Zd⌦ (raYl1m1Y

⇤lmraYl2m2) ,

I(b)lml1m1l2m2=

Zd⌦�raYl1m1Y

⇤lm"

barbYl2m2

�,

K(a)lml1m1l2m2l3m3

=

Zd⌦�rbraYl1m1Y

⇤lmraYl2m2rbY

⇤l3m3

�,

K(b)lml1m1l2m2l3m3

=

Zd⌦�rbraYl1m1Y

⇤lm"

carcYl2m2"

dbrdY

⇤l3m3

�.

(3.60)

Our aim is to compute �ll0�mm0C˜

⇥

l = h✓lm✓⇤l0m0i, so we compute the two-point angular correlation functionby substituting Eq. 3.59. This gives

C˜

⇥

l = C⇥l +X

l1l2

C⇥l1

⇣C

l2⇧(1a)

ll1l2+ C$

l2 ⇧(1b)ll1l2

⌘+

1

2C⇥l

X

l1

⇣C

l1⇧(2a)

ll1+ C$

l1 ⇧(2b)ll1

⌘, (3.61)

with

⇧(1a)ll1l2

=X

m1m2

|I(a)|2 , ⇧(1b)ll1l2

=X

m1m2

|I(b)|2

⇧(2a)ll1

=X

m1

⇣K(a) +K(a)⇤

⌘, ⇧(2b)

ll1=X

m1

⇣K(b) +K(b)⇤

⌘.

(3.62)

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

6As we will see in Chapter 4 the Taylor expansion in the case of tensor perturbations hardly fails: for density perturbations

we have ✓rms ' 2.7 arcmin and this means that for l ⇠ 3000 the wavelength being deflected is of the same order of ↵. For

tensor perturbations however ✓rms ' 7 arcsec and the Taylor expansion fails for very high multipoles values l � 10

4.

27

as integrating by parts to reconstruct r2Ylm = �l(l+1)Ylm or use the following formulas for the Wigner-3jsymbols (for more details see [20])

Zd⌦�s1Y

⇤l1m1

�(s2Yl2m2) (s3Yl3m3) =(�)m1+s1

r(2l

1

+ 1)(2l2

+ 1)(2l3

+ 1)

4⇡

⇥✓l1

l2

l3

s1

�s2

�s3

◆✓l1

l2

l3

�m1

m2

m3

◆

and X

m1m2

✓l1

l2

l3

m1

m2

m3

◆✓l1

l2

l3

m1

m2

m3

◆=

1

2l3

+ 1.

In the end one can show

⇧(1a)ll1l2

=1

16⇡(2l

1

+ 1)(2l2

+ 1)

✓[l(l + 1)� l

1

(l1

+ 1)� l2

(l2

+ 1)]

✓l l

1

l2

0 0 0

◆◆2

,

⇧(1b)ll1l2

=1

16⇡(2l

1

+ 1)(2l2

+ 1)(l1

(l1

+ 1)l2

(l2

+ 1))

✓[1� (�1)l+l1+l2 ]

✓l l

1

l2

0 �1 1

◆◆2

,

⇧(2a)ll1

= ⇧(2b)ll1

= �l(l + 1)l1

(l1

+ 1)2l

1

+ 1

4⇡.

(3.63)

Finally, combining all expressions, we get

C˜

⇥

l =C⇥l +X

l1l2

C⇥l1

⇣C

l2⇧(1a)

ll1l2+ C$

l2 ⇧(1b)ll1l2

⌘

� l(l + 1)C⇥lX

l1

l1

(l1

+ 1)2l

1

+ 1

8⇡

⇣C

l1+ C$

l1

⌘,

(3.64)

which is the expression we wanted: the 1st order lensed CMB temperature fluctuations power spectrumin the full-sky, small ↵ regime.

Out[55]=

0 500 1000 1500 2000

-4.¥10-7

-2.¥10-7

0

2.¥10-7

4.¥10-7

6.¥10-7

multipole l

DClQêC lQ

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

= 1 and h = 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 nottrust 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 beforethe e↵ect of GL on the CMB is a remapping of the temperature fluctuations according to ⇥(n) = ⇥(n+↵)where ↵ = r +r⇥$. For r = x� x0, r = |r| the lensed correlation function ⇠(r) is written

⇠(r) = h⇥(x)⇥(x0)i = h⇥(x+↵)⇥(x0 +↵0)i . (3.65)

Working in harmonic space and introducing the flat-sky 2D Fourier transform

⇥(x) =

Zd2l

2⇡⇥(l)eil·x , (3.66)

we find

⇠(r) =

Zd2l

2⇡

Zd2l0

2⇡heil·(x+↵)e�il0·(x0

+↵0)i h⇥(l)⇥(l0)i

=

Zd2l

2⇡

Zd2l0

2⇡eil·r heil·(↵0�↵)i �ll0C⇥l

=

Zd2l

(2⇡)2eil·r heil·(↵0�↵)iC⇥l ,

(3.67)

where for the first equality we have made the assumption that the CMB anisotropies and the deflectionangle are uncorrelated and hence the expectation value of the product is the product of the expectationvalues. We now use the fact that ↵ is a Gaussian variable and so also l · (↵0 � ↵) is a Gaussian randomvariable with mean zero and variance h[l · (↵0 �↵)]2i. This allows us to write the expectation value of itsexponential as8

heil·(↵0�↵)i = e�12 h[l·(↵

0�↵)]

2i = e�12�

2

. (3.68)

To compute the variance of l · (↵0 �↵) we introduce

Aij(r) = h↵i(x)↵j(x+ r)i= h[ri (x) + (r⇥$(x))i][rj (x+ r) + (r⇥$(x+ r))j ]i

(3.69)

and we make now use of the formulas

ra (x) = � i

2⇡

Zd2l la (l) e

il·x ,

(r⇥$(x))a = � i

2⇡

Zd2l (l⇥$(l))a e

il·x

= � i

2⇡

Zd2l "ba lb$(l) eil·x ,

(3.70)

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.

8For a function g, with p.d.f. f

g

, of variable X, with p.d.f. f

X

, we have

hg(X)i =

Z 1

�1dy yf

g

(y) =

Z 1

�1dx g(x)f

X

(x)

if X is a Gaussian variable and g(X) = e

iX

he

iX

i =

1

p

2⇡�

Z 1

�1dx e

ix

e

� x

2

2�2= e

� 12�

2

29

to write

Aij(r) =

Zd2l

(2⇡)2

⇣liljC

l + "ki "

pj lklpC

$l

⌘eir·l . (3.71)

By symmetry the correlation can only depend on �ij and the trace-free tensor rirj � 1

2

�ij and thereforeis of the form

Aij(r) =1

2A

0

(r)�ij �A2

(r)

rirj � 1

2�ij

�. (3.72)

To determine the functions A0

and A2

we start by taking the trace of Aij

A0

(r) = �ijAij =

Z 1

0

dl

2⇡

⇣l3C

l + l �ij"ki "pj lklpC

$l

⌘Z 2⇡

0

d�ei lr cos�

2⇡

�

=

Z 1

0

dl

2⇡l3⇣C

l + C$l

⌘J0

(rl) ,

(3.73)

where �, in this section, is the angle between r and l.For the last equality we made use of the Bessel function J

0

(x) arising from the integralR2⇡

0

d! eix cos! =2⇡J

0

(x).To compute A

2

we contract the correlation with rirj

Aij rirj =

Zd2l

(2⇡)2l2 (cos2 �C

l + sin2 �C$l ) ei rl cos�

=

Z 1

0

dl

2⇡l3✓C

l

Z2⇡

0

d�cos2 �

2⇡ei rl cos� + C$

l

Z2⇡

0

d�sin2 �

2⇡ei rl cos�

◆.

(3.74)

We see again that the Bessel functions arise from the integrals

Z2⇡

0

d! sin2 ! eix cos! = ⇡ (J0

(x) + J2

(x)) ,

Z2⇡

0

d! cos2 ! eix cos! = ⇡ (J0

(x)� J2

(x)) ,

to give

A2

(r) =1

2

Z 1

0

dl

2⇡l3⇣C

l � C$l

⌘J2

(rl) . (3.75)

We can now compute the required expectation value

h[l · (↵0 �↵)]2i = lilj h(↵i � ↵0i)(↵j � ↵0

j)i= l2 [A

0

(0)�A0

(r) +A2

(r) cos(2�)] ,(3.76)

which can be inserted in Eq. 3.67 to give

⇠(r) =

Zd2l

(2⇡)2C⇥l ei rl cos� Exp

� l2

2(A

0

(0)�A0

(r) +A2

(r) cos(2�))

�. (3.77)

This is an exact expression and we can actually perform the angular integral using the following tricks.First we use the relation

30

eix cos� = Jo(x) + 21X

n=1

in Jn(x) cos(n�) (3.78)

and then we use the integral

1

4⇡

Z2⇡

0

d� ex cos� cos(n�) = In(x) , (3.79)

where the In(x) are the hyperbolic Bessel functions of the first kind9. We obtain

⇠(r) =

Zdl

(2⇡)l C⇥l Exp

� l2

2(A

0

(0)�A0

(r))

�

⇥ I0

(rl) + 21X

n=1

In(l2A

2

(r)/2)J2n(rl)

!.

(3.80)

Using the fact that the correlation function is the Fourier transform of the power spectrum

⇠(r) =1

(2⇡)2

Zd2l C⇥l eil·r , (3.81)

we can obtain the lensed power spectrum from ⇠(r)

C⇥l =

Zd2r ⇠(r)eil·r =

Z 1

0

dr r ⇠(r)

Z2⇡

0

d� ei lr cos�

= 2⇡

Z 1

0

dr r J0

(rl) ⇠(r)1 .

(3.82)

Sometimes is useful to approximate the exponential in ⇠(r) to give (see [11])

⇠(r) 'Z 1

0

dl

2⇡l C⇥l Exp

� l2

2[A

0

(0)�A0

(r)]

�✓J0

(rl) +l2

2A

2

(r)J2

(rl)

◆.

Inserting this in Eq. 3.82 we finally find

C⇥l =

Z 1

0

dl0 l0Cl0

Z 1

0

dr rExp

� l02

2[A

0

(0)�A0

(r)]

�J0

(rl)

✓J0

(rl0) +l02

2A

2

(r)J2

(rl0)

◆. (3.83)

Recalling

A0

(r) =

Z 1

0

dl

2⇡l3⇣C

l + C$l

⌘J0

(rl) , A2

(r) =1

2

Z 1

0

dl

2⇡l3⇣C

l � C$l

⌘J2

(rl)

Eq. 3.83 is the expression of the lensed CMB power spectrum in the flat-sky arbitrary angle regime, asa function of the lensing potentials C$

l and C l .

9The hyperbolic Bessel functions of the first kind are defined by I

n

(x) = i

�n

J

n

(ix). For more information about the Bessel

function J

n

, I

n

, the spherical Bessel j

l

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 arebest described in the frame ? k an Euler rotation is necessary to write down the perturbation along the lineof sight. The displacement vector is then decomposed into its gradient-type potential and its curl-typepotential $

↵ = r +r⇥ ~$

and the Euler rotations performed plays a central role in the decomposition of the potentials in SphericalHarmonics. The main results we computed is the angular power spectra of the two lensing potentials

C l = 2⇡

(l + 2)(l � 1)

l(l + 1)

Zdk

k�2

h(k)

����Z �⇤

0

d� Th(k,�)

✓l + 1

2

✓l�� �⇤�⇤

+ 2

◆jl(k�)

k2�3

� jl+1

(k�)

k�2

◆����2

C$l = ⇡

(l + 2)(l � 1)

l(l + 1)

Zdk

k�2

h(k)

����Z �⇤

0

dt Th(k,�)jl(k�)

k�2

����2

for which we give plots in the case of primordial tensor perturbations in a MD universe. It is interestingto see how, due to the decaying nature of primordial tensor perturbations once entered the horizon, thelensing potentials spectra have contribution from very high redshift while the lensing potential for densityperturbations 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-inducedtensor 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 perturbationmight even exceed the primordial one at large scales at the present epoch, but is then quite clear if one thinksthat scalar-induced tensor are of course relevant only at small redshift while, as said, the lensing potentialsdue to primordial tensor get contributions up to very high redshift. In fairness we should point out that aMD calculation might not be the best for estimating the scalar-induced tensor contribution: including theRadiation 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.Nevertheless we expect, even including the RD era, CPrimordial

l /CII order

l & r ⇥ 105.In the last section of this work we study the e↵ect of primordial tensor perturbation lensing on the CMBtemperature field. To get an idea of this e↵ect we can estimate the rms deflection angle. We compute thetotal deflection angle power (see § Appendix F)

32

R Scalar =1

2✓2rms

=1

2h|r⌥|2i ⇠ 3⇥ 10�7 ,

R Tensor =1

2✓2rms

=1

2h|r +r⇥$|2i ⇠ 1.2⇥ 10�9 ,

and obtain ✓ Scalar ⇠ 2.7 arcmin and ✓ tensor ⇠ 7 arcsec. We therefore expect a small modification to theCMB 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 thefirst case to be valid up until very high multipole l and we give a plot for this case from which we can seehow the modifications to the CMB temperature are of the order of 10�7, smaller than the cosmic varianceof intrinsic CMB anisotropies for every reasonable multipole

✓�C⇥lC⇥l

◆CV

⇠r

2

2l + 1�

✓�C⇥lC⇥l

◆GL

⇠ 10�7 .

Let us terminate this work with a remark about the curl part of the displacement vector: at the linearlevel tensor perturbations produce a curl part of the displacement vector and that is a unique feature of GLby GWs which is not found in lensing by density perturbation. Nevertheless it is important to point outthat 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 ⇥(n) defined in Eq. 1.7, which we consider aGaussian1 random variable and an isotropic and homogeneous field. This has remarkable e↵ects on itsstatistic properties.

Fourier Space

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

Ck,k0 = h⇥(k)⇥(k0)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 C⇤

k,k0 = C�k,�k

0 , and, due to the associative nature ofthe expectation value, we also have C

k,k0 = Ck

0,k.Since ⇥ is gaussian we have that all the higher order correlation function are either zero (for an odd numberof points) or connected to the 2-point correlator through Wick’s theorem

h⇥(ki1)...⇥(ki2n)i =X

All perm.

Cj1j2 ....Cj2n�1 j2n

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 realspace

h⇥(x1

)...⇥(xN )i =Z

d3k1

...d3kN(2⇡)3N

e�ik1·x1 ...e�ikN·xN h⇥(k1

)...⇥(kN )i

Since the RHS must be function only of the distances between spatial points and not of the pointsthemselves we find that the expectation value in Fourier space must be proportional to �(k

1

+ ...+ kN ), sothat

h⇥(k1

)...⇥(kN )i = (2⇡)3P (k1

, ... ,kN ) �(k1

+ ...+ kN )

Furthermore for isotropy we must have that the N-point function in real space cannot depend on thedirection. We see from the previous equation that this implies the Fourier space correlator to be onlydependent on the moduli of the wave vectors, such that we get P (k

1

, ... ,kN ) = P (k1

, ... , kN ) .

1A random variable x is called Gaussian if it has the p.d.f. in the form

p(x) =

1

p

2⇡�

e

�(x�x0)2/2�2

34

Harmonic space

The harmonic representation is most directly related to the observations of the CMB, ⇥(n), which are takenover the unit sphere. The fluctuation are expanded

⇥(n) =X

lm

✓lmYlm(n) (A.1)

and, again, since ⇥ is a real field we have ✓⇤lm = (�)m✓l�m and this means that each multipole l has2l + 1 real degrees of freedom. The multipole coe�cients are given by

✓lm =

Zd⌦ ⇥(n)Y ⇤

lm(n)

and the harmonic space two-point correlator is given by

h✓lm✓⇤l0m0i =Z

d⌦ d⌦0 h⇥(n)⇥(n0)i Y ⇤lm(n)Yl0m0(n0)

If we now use the Rayleigh expansion in Eq. 2.21 we get

h✓lm✓⇤l0m0i =Z

d3k d3k0

(2⇡)6(4⇡)2il(�i)l

0jl(k�)jl0(k

0�)Ylm(k)Y ⇤lm(k0) h⇥(k)⇥⇤(k0)i

where we have written x = n ·�. Under the hypothesis of homogeneity, this expression simplifies, leadingto

h✓lm✓⇤l0m0i =Z

d3k

⇡2il(�i)l

0jl(k�)jl0(k�)Ylm(k)Y ⇤

lm(k)⇥ f(k)

and, if we also assume isotropy, then f(k) ! P (k) can only depend on the modulo of k. The angularintegral now involves only the spherical harmonics and gives

Rd⌦k Ylm(k)Y ⇤

lm(k) = �ll0�mm0 , such that

h✓lm✓⇤l0m0i = �ll0�mm02

⇡

Zdk

kk3j2l (k�)P (k)

= �ll0�mm04⇡

Zdk

kj2l (k�)�

2(k)

= �ll0�mm0 Cl

where we have defined the dimensionless power spectrum �2(k) = k3

2⇡2P (k). Therefore, for statisticallyisotropic 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 thatR

dxx j2l (x) =

1

2l(l+1)

, we can approximate

Cl ' 2⇡

l(l + 1)�2(k = l/�)

which explains why the quantity that is normally plotted is

Cl ⌘ l(l + 1)

2⇡Cl ' �2(k = l/�)

We can also construct the angular two-point correlation function in harmonic space

35

h⇥(n)⇥(n0)i =X

lm

X

l0m0

h✓lm✓⇤l0m0iYlm(n)Y ⇤l0m0(n0)

=X

lm

X

l0m0

Cl �ll0�mm0Ylm(n)Y ⇤l0m0(n0)

=X

l

Cl2l + 1

4⇡Pl(n · n0)

where we have used the addition theorem of spherical harmonicsPl

m=�l Ylm(n)Y ⇤lm(n0) = 2l+1

4⇡ Pl(n · n0).

36

Appendix B - �↵µ⌫

for tensor perturbations

We compute the Christo↵el’s symbols

�↵µ⌫ =1

2g↵�(@µg�⌫ + @⌫g�µ � @�gµ⌫)

for the tensor perturbation metric in Eq. 3.2, at first order in the perturbation

�rrr =1

2@rhrr +O2

�r✓r =1

2(�2hr✓ + @✓hrr) +O2

�r✓✓ = �r + (rhrr � rh✓✓ + r@✓hr✓ � 1

2r2@rh✓✓) +O2

�r�r =1

2(�2 sin ✓ hr� + @�hr�) +O2

�r✓� = �1

2(r(cos ✓ hr� + 2 sin ✓ h✓� � @�hr✓ � sin ✓@✓hr� + r sin ✓ @rh✓�)) +O2

�r�� = �r sin2 ✓ +1

2r sin ✓ (2 sin ✓hrr + 2 cos ✓hr✓ � 2 sin ✓h�� + 2@�hr� � r sin ✓@rh��) +O2

�rrt =1

2@thrr +O2

�r✓t =1

2@thr✓ +O2

�r�t =1

2@thr� +O2

�✓rr =1

2r2(2hr✓ � @✓hrr + 2r@rhr✓) +O2

�✓✓r =1

r+

1

2@rh✓✓ +O2

�✓✓✓ = hr✓ +1

2@✓h✓✓ +O2

�✓�r =1

2r(� cos ✓hr� + @�hr� + sin ✓ (�@✓hr� + r@rh✓�)) +O2

�✓✓� =1

2(�2 cos ✓h✓� + @�h✓✓) +O2

�✓�� = � cos ✓ sin ✓ +1

2sin ✓ (2 sin ✓hr✓ + 2 cos ✓h✓✓ � 2 cos ✓h�� + 2@�h✓� � sin ✓@✓h��) +O2

�✓tr =1

2r@thr✓ +O2

�✓t✓ =1

2@th✓✓ +O2

37

�✓t� =1

2sin ✓@th�✓ +O2

��rr =1

2r2csc ✓ (2hr� � csc ✓@�hrr + 2r@rhr�) +O2

��✓r =1

2rcsc ✓ (cot ✓hr� � csc ✓@�hr✓ + @✓hr� + r@rh✓�) +O2

��✓✓ =1

2csc ✓ (2hr� + 2 cot ✓h✓� � csc ✓@�h✓✓ + 2@✓h✓�) +O2

���r =1

r+

1

2@rh�� +O2

���✓ = cot ✓ +1

2@✓h�� +O2

���� = sin ✓hr� + cos ✓h✓� +1

2@�h�� +O2

��tr =1

2rcsc ✓ @thr� +O2

��t✓ =1

2csc ✓ @th✓� +O2

��t� =1

2@th�� +O2

�trr =1

2@thrr +O2

�tr✓ =1

2r @thr✓ +O2

�t✓✓ =1

2r2 @th✓✓ +O2

�tr� =1

2r sin ✓ @thr� +O2

�t✓� =1

2r2 sin ✓ @th✓� +O2

�t�� =1

2r2 sin2 ✓ @th�� +O2

38

Appendix C - Spherical harmonics

In this appendix we review the most important results on Spherical harmonics which have been useful inthis 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 sphericalharmonics.

Legendre Polynomials

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

(1� x2)P 00l � 2xP 0

l + l(l + 1)Pl = 0

They are a set of orthonormal polynomials on the interval [�1, 1]. Given the lowest order polynomialsP0

= 1 and P1

= x one can derive all the other through the recursion relation

(l + 1)Pl+1

(x) = (2l + 1)xPl(x)� lPl(x)

They obey the normalization condition

Z1

�1

dx Pl(x)Pl0(x) =2

2l + 1�l l0

and Rodrigue’s formula

Pl(x) =1

2l l!

dl

dxl(x2 � 1)l

The associated Legendre polynomials are defined by

Plm(x) = (1� x2)m/2 dmPl(x)

dxm= (1� x2)m/2 1

2l l!

dl+m

dxl+m(x2 � 1)l

They are in principle defined for every complex degree l and order m, but we will need them only forinteger m, non-negative integer l with |m| < l and x 2 [�1, 1]. In this interval and with these values of orderand degree they are singularity free and analytic. The Plm have many recurrence relations ([21],[22]) andthe most important for this work is

xPlm =l +m

2l + 1Pl�1m +

l �m+ 1

2l + 1Pl+1m

Finally we have the orthogonality relation

Z1

�1

dx Plm(x)Pl0m(x) =2

2l + 1

(l +m!)

(l �m)!�ll0

and, from their definition, the parity relation Plm(�x) = (�)l+mPlm(x).

Spin-0 spherical harmonics

The spherical harmonics are functions on the sphere. For a unit vector n defined by its polar angles (✓,�)the spherical harmonics are given by

Ylm(n) = (�)m

s2l + 1

4⇡

(l �m)!

(l +m)!eim� Plm(µ)

with µ = cos ✓. The parity and orthogonality relations are derived directly from the Legendre polynomialsproperties and they are given by : Yl�m = (�)mYlm, and

39

Zd⌦ Ylm(n)Y ⇤

l0m0 = �ll0�mm0

Given the Laplacian on the sphere we construct

L2 = �r2 = �@2✓ +

cos ✓

sin ✓@✓ +

1

sin2 ✓@2�

�

(where ~ = 1), and we have

L2 Ylm = l(l + 1)Ylm

In the end we simply report the addition theorem of spherical harmonics (proof in [11]) which we haveused in this work

2l + 1

4⇡Pl(n · n0) =

lX

m=�l

Ylm(n)Y ⇤lm(n0)

Spin-s spherical harmonics

We consider a tensor field on the sphere with e1

= e✓ = @✓ and e2

= e� = 1

sin ✓@� and we recall the definitionof helicity basis

e± =1p2(e

1

⌥ e2

)

The helicity basis has the property that under a rotatione1

! cos � e1

� sin � e2

, e2

! cos � e2

+ sin � e1

transforms as

e+

! e�i� e+

e� ! ei� e�

and a tensor field of type (+,�) = (s, r) (of rank r + s) in the helicity basis transform under a rotationby

T sr ! ei(s�r)� T s

r

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

sYlm(✓,�) =(�)m

s2l + 1

4⇡

(l �m)!(l �m!)

(l + s)!(l � s!)eim� (sin ✓/2)2l

⇥X

r

✓l � sr

◆✓l + s

r + s�m

◆(�)l�r�s (cot ✓/2)2r+s�m

The spin weighted spherical harmonics are defined for |s| l and |m| l and, similarly to the spin-0harmonics, we have an orthogonality relation

Zd⌦ sYlm(n) sYl0m0(n) = �ll0�mm0

40

Now let R1

be the rotation with Euler angle (�1

, ✓1

, 0) which rotates ez into n1

and R2

the rotation withEuler angle (�

2

, ✓2

, 0) which rotates ez into n2

. Let (↵,�, �) the Euler angle of the rotation R�1

1

R2

, we thenhave a generalization of the addition theorem for spin weighted spherical harmonics:

r4⇡

2l + 1

X

m0sYlm0(✓

2

,�2

)�mY ⇤lm0(✓

1

,�1

) = sYlm(�,↵)e�is� (C.1)

We now introduce the spin raising and spin lowering operators /@ and /@⇤. They are defined by

/@ sYlm =

✓s cot ✓ � @✓ � i

sin ✓@�

◆sYlm

=

sµp1� µ2

+p

1� µ2@µ � ip1� µ2

@�

!

sYlm

/@⇤sYlm =

✓�s cot ✓ � @✓ +

i

sin ✓@�

◆sYlm

=

�sµp1� µ2

+p

1� µ2@µ +ip

1� µ2

@�

!

sYlm

and with these operators we can construct the spin-s harmonics starting from the0

Ylms. In fact we have

/@ sYlm =p(l � s)(l + s+ 1) s+1

Ylm

/@⇤sYlm = �

p(l � s)(l � s+ 1) s�1

Ylm

and

(/@)2 Ylm =

s(l + 2)!

(l � 2)!2

Ylm

(/@⇤)2 Ylm =

s(l + 2)!

(l � 2)!�2

Ylm

At this point we are interested in find an operator which acts in the same way that L2 acts on thespin-0 harmonics. We call this operator L2

(s) and we want the spin weighted spherical harmonics to beeigenfunctions of this operator

L2

(s) sYlm = l(l + 1) sYlm

To construct this operator we see that, by definition, the commutators satisfy

[L2

(s), /@] = [L2

(s), /@⇤] = 0

and we make the ansatz L2

(s) = �r2 +K(s). The commutation relation [L2

(s), /@] = 0 implies that

L2

(s+1)

/@(s) = /@

(s)L2

(s)

which gives

[�r2, /@(s)] = /@

(s)K(s) �K(s+1)

/@(s)

For s = 0 (given that K(0)

= 0 since L2

(s) must reduce to L2 for s = 0)

41

K(1)

/@ = [/@,r2]

explicitly,

K(1)

✓�@✓ � i

sin ✓@�

◆=

1

sin2 ✓

✓ �i

sin ✓@� + 2

cos ✓

sin ✓@2� � @✓ + 2i cos ✓ @�@✓

◆

and we find

K(1)

=1

sin2 ✓(1� 2i cos ✓ @�)

We can do the same procedure with [L2

(s), /@⇤] = 0 to find an expression for K

(�1)

and we can actuallyfind every K

(s) by recursion. At some point will be clear that the expression we are looking for is

K(s) = �i

2s cos ✓

sin2 ✓@� +

s2

sin2 ✓

to give

L2

(s)(✓,�) = �⇣@2✓ +

cos ✓

sin ✓@✓ +

1

sin2 ✓@2�

⌘� i

2s cos ✓

sin2 ✓@� +

s2

sin2 ✓

42

Appendix D - Bessel functions

0 5 10 15 20

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

x

j lHxL

j4 HxLj3 HxLj2 HxLj1 HxLj0 HxL

0 5 10 15 20

-0.5

0.0

0.5

1.0

x

J lHxL

J4 HxLJ3 HxLJ2 HxLJ1 HxLJ0 HxL

Figure 4.1: Left: The first 5 Spherical Bessel function jn

(x) for integer n. Right: The first 5 Bessel function Jn

(x) for integer n.

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

x2

d2f

dx2

+ xdf

dx+ (x2 � ⌫2)f = 0

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

J⌫(x) =1X

m=0

(�1)m

m! �(m+ ⌫ + 1)

⇣x2

⌘2m+⌫

If ⌫ = n is an integer one also has an integral representation

Jn(x) =(�i)n

⇡

Z ⇡

0

d✓ ei x cos ✓ cos(n✓)

There are many useful recursion relation for the Bessel function ([21],[22]). We give here a couple ofexamples

J⌫�1

+ J⌫+1

=2⌫

xJ⌫

J⌫�1

� J⌫+1

= 2J 0⌫

J⌫�1

� ⌫

xJ⌫ = J 0

⌫

Spherical Bessel

The Spherical Bessel function jn(x) are defined

jn(x) =

r⇡

2xJn+1/2(x)

and they are solutions of the di↵erential equation

x2

d2f

dx2

+ 2xdf

dx+ (x2 � n(n+ 1))f = 0

43

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

jnx

=1

2n+ 1(jn�1

+ jn+1

)

j0n =1

2n+ 1(njn�1

� (n+ 1)jn+1

)

to rewrite in a nice way the expression of C l .

Another useful relation that we have been using many times in this work comes from noticing that thefunctions jl(rk)Ylm(n) are solutions of the equation

(r2 + k2) f = 0

and since also eik·x is a solution and spherical harmonics form a complete set of functions on the sphereit has to be possible an expansion of the type

eik·x = eikrˆ

k·n =X

lm

clm jl(rk)Ylm(n)

and this expression is precisely (see [30])

eikrˆ

k·n =X

lm

(2l + 1)il jl(rk)Pl(n · k)

44

Appendix E - Limber Approximation

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

jl(x) 'r

⇡

2l + 1�(l + 1/2� x)

and using the properties of the Dirac delta

�(ax) =1

a�(x) ;

Zdx �(y � x)�(z � x) = �(y � z) ;

Zdx f(x)�(g(x)) =

X

xi

| g(xi

)=0

�(x� xi)

|g0(xi)|

we can write a not-so-formal derivation for the Limber approximation

Zdk �(l + 1/2� k�)�(l + 1/2� k�0) =

1

�2

�(l + 1/2

�� l + 1/2

�0 ) =2

2l + 1�(�0 � �

��0 ) =2

2l + 1�(�0 � �)

such that

Zk2dk f(k) jl(k�)jl(k�

0) ' ⇡

2l + 1

Zk2dk f(k) �(l + 1/2� k�)�(l + 1/2� k�0) ' ⇡

2�2

f

✓l

�

◆�(�� �0)

or, equivalently

Zdk f(k) jl(k�)jl(k�

0) ' 2⇡

(2l + 1)2f

✓l + 1/2

�

◆�(�� �0)

This expression was used in the case of lensing by density perturbations. However in the case of lensingby tensor perturbation the Transfer function Th(k,�) is not slowly varying with respect to the sphericalBessel and this approximation fails.

In the same way one can find a formula useful when the spherical Bessel are not of the same order

Zdk f(k) jl(k�)jl0(k�

0) ' 2⇡p(2l + 1)(2l0 + 1)

1

(2l0 + 1)2f

✓l

�

◆�(�� l

l0�0)

so that

Zdk f(k) jl(k�)jl±1

(k�0) ' 2⇡p(2l + 1)

1

(2(l ± 1) + 1)3/2f

✓l

�

◆�(�� l

l ± 1�0)

45

Appendix F - Total deflection angle power

We want to compute the total deflection angle power of density & tensor perturbations.

R ⌘ 1

2h|↵|2i

For density perturbation this yields to

R⌥ =1

2h|↵|2i = 1

2h|r⌥|2i = 1

2

X

lml0m0

h�lm�⇤l0m0ir Ylm(n)r Y ⇤l0m0(n)

=1

2

X

lml0m0

�ll0�mm0C⌥l r Ylm(n)r Y ⇤l0m0(n)

Now, we make use of the relation [24]

r Ylm =

rl(l + 1)

2(1

Ylme+

+ �1

Ylme�)

to find

R⌥ =1

2

X

lm

C⌥ll(l + 1)

2(1

Ylm 1

Y ⇤lm + �1

Ylm �1

Y ⇤lm)

This can be further simplified using the Generalized addition relation C.1 in the particular case where(✓

1

,�1

) = (✓2

,�2

)

X

m

s1Ylm(n) s2Y⇤lm(n) =

r2l + 1

4⇡s2Yl�s1(0)

In our case we need to use

X

m

±1

Ylm(n) ±1

Ylm(n) = ±1

Yl⌥1

(0) =

r2l + 1

4⇡

to give

R⌥ =1

2

X

l

l(l + 1)2l + 1

4⇡C⌥l

The computation for tensor perturbations goes in the same way, with the exception that we also have touse the formula

r⇥ Ylm =

rl(l + 1)

2(�1

Ylme+

�1

Ylme�)

to give

RTensor =1

2h|r +r⇥$|2i = 1

2

X

l

l(l + 1)2l + 1

4⇡(C

l + C$l )

46

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