Gravitational waves and their effects on CMB

Gravitational waves and their effects on CMB

Daan Mulder

10275037

supervised by prof. Erik Verlinde

Institute for Theoretical Physics - FNWI - Universiteit van Amsterdam

Bachelor thesis 12 ECTS from 07-04-14 to 21-07-14

July 21, 2014

Abstract

As an introduction to the theory behind the BICEP-2 findings, this thesis reviews the derivation of gravitational waves as a solution to Einstein’s field equations, and the basics of inflation. How temperature differences lead to polarization through Thomson scattering is discussed, as well as the difference between E and B modes. Polarization patterns associated with scalar, vector and tensor waves are found using a linear reprentation of the spin two group. However, they do not seem to contain B-modes.

Onder het hemellichaamloos tonaal uitspansel-Waar kentering van ijlte en densiteit massieven Geluid in een bewogen evenwicht doet wankelen En stilte en storm zich voortduren in elkaar verliezen-Lopen de lijnen, die berekende motieven

Rhytmisch verdelen, en hun deinzende gedaanten Geleiden naar uiteindelijke klankgebieden,

Zich strekkend in ’t polair klimaat der dominanten. Dit is het land over welks doodverstarde vlakten-Altijd de greep ontwijkend van abstracte stelsels, Voortvluchtig uit telkens opnieuw vergeefs bedachte Accoustische systemen, tartend het nuchter vreten Van de vergetelheid- de late echo’s zwerven

Van het oud lied der verlangen dat niet wil sterven.

Contents

1 Een vingerafdruk van zwaartekrachtgolven 4

2 Introduction 6

3 Deriving gravitational waves from Einstein’s field

equa-tions 7

3.1 Relevant formulas . . . 7

3.2 Perturbing Einstein’s field equations . . . 7

3.3 Gauge transformations . . . 9

3.4 First order solution in vacuum . . . 10

3.5 Effects on spacetime . . . 10

4 Inflation and quantum fluctuations 13 4.1 Basics of slow roll inflation . . . 13

4.2 Quantum perturbations . . . 14

5 Deriving polarization patterns 16 5.1 Stokes parameters, E modes and B modes . . . 16

5.2 Thomson scattering . . . 20

5.3 Physical effects and spherical harmonics . . . 20

5.4 Rotating spherical harmonics by linear representation . . . 22

5.5 Plot of polarization patterns . . . 25

1

Een vingerafdruk van

zwaartekracht-golven

Sinds Einstein weten we dat ruimte en tijd niet het toneel vormen waarop de dingen gebeuren, maar zelf ook ’een ding’ zijn. Een ding zelfs dat kan vervormen: wie beweegt, ziet tijd en ruimte anders dan iemand die stilstaat. Massa kromt de ruimtetijd, waardoor klokken dichtbij het aardoppervlak langzamer lopen dan diep in het heelal. Einsteins veld-vergelijkingen beschrijven deze krommingen, maar hebben een nadeel: ze zijn vreselijk ingewikkeld. Krommingen beinvloeden de bewegingen van bijvoorbeeld sterren, planeten, waarop hun massa de krommingen weer ve-randert. Deze zogeheten parti¨ele differentiaalvergelijkingen hebben maar zelden een oplossing die je zo op kunt schrijven. Hoe lossen we ze dan toch op?

Stel dat je de golven op de zee precies zou willen beschrijven. Eigenlijk moet je rekening houden met de kromming van het aardoppervlak. Maar wie golven beschrijft maakt het zich een stuk gemakkelijker door te doen alsof de zee gewoon plat is. Zolang die golven geen kilometers hoog zijn, heb je een beste kans dat dat een heel aardige benadering is. Zo gaan we ook met de ruimtetijd te werk. We doen alsof die helemaal niet gekromd is, en dan kijken we hoe een kleine verstoring eruit moet zien. En dan blijkt de ruimtetijd ook te golven: een golf rekt de ene richting uit, en krimpt de andere richting weer in, en dan weer andersom. Maar, zoals golven ten opzichte van de zee, zijn deze golven heel erg klein. Zo klein, dat we nog nooit in staat zijn geweest er een te meten, ondanks de enorme apparaten die we hiervoor hebben gebouwd.

Er was dus behoorlijk wat opwinding dit voorjaar, toen er met de BICEP-2 telescoop effecten van zwaartekrachtsgolven gemeten zouden zijn in po-larisatie van de achtergrondstraling. Achtergrondstraling? Popo-larisatie? Het licht heeft een eindige snelheid: zien we een ster aan de hemel, dan zien we nu het licht dat die ster vele jaren geleden onze richting uit zond. Het licht dat het langst onderweg is geweest is de achtergrondstraling, die ons bereikt vanaf het moment dat het heelal genoeg was afgekoeld om doorzichtig te worden. Zoals alle straling is deze straling een golf in het elektromagnetische veld. Deze golven kunnen gepolariseerd zijn: de trilling vindt dan voornamelijk plaats langs ´e´en as. Als in het vroege heelal een elektron vanuit ´e´en richting meer licht ontvangt dan uit een andere, kan het licht uit gaan zenden dat gepolariseerd is. Vanuit welke richting een elektron het meeste licht ontvangt, wordt bepaald door de temperatuur. Verschillen in temperatuur kunnen zo dus zorgen voor een polarisatie in de achtergrondstraling. Zwaartekrachtsgolven zouden er voor zorgen dat dit signaal niet symmetrisch is, maar een ’twist’ heeft. En die ’twist’, die natuurkundigen een B-mode noemen, zou de BICEP-telescoop nu gemeten hebben.

Niet alleen zijn er dus voor het eerst zwaartekrachtsgolven gevonden, ook klopten de vondsten met de voorspellingen van de ’inflatietheorie’, ´e´en van de theorie¨en die het begin van het heelal beschrijven. Volgens de inflati-etheorie was het heelal ooit heel klein was, en is het toen plots exponen-tieel uit gedijt. Miniscule quantumfluctuaties, waarvan

zwaartekrachts-golven een voorbeeld zijn, kregen zo letterlijk astronomische proporties. Elk model voor het vroege heelal voorspelt weer andere fluctuaties, dus deze te kunnen meten betekent dat we meer informatie krijgen over het allervroegste heelal. Wie hier meer over wil weten, raad ik aan mijn scrip-tie te lezen- maar wees gewaarschuwd: die B-modes heb ik niet terug kunnen vinden.

2

Introduction

It is hard to imagine anyone in the field of physics who missed the an-nouncement in March 2014 that so-called B-modes had been found in the polarization of the Cosmic Microwave Background (CMB) (Ade et al. 2014). These B-modes, an effect similar to the curl in a vector field, would be a proof for the existence of gravitational waves, a huge source of infor-mation to learn more about the physical laws that governed the universe short after the Big Bang, and a test for the inflationary paradigm (Bau-mann 2007).

This thesis focuses on three main questions. Firstly, what is a gravita-tional wave? Using a perturbation method on a few standard equations from general relativity, we can determine the properties of a gravitational wave. This derivation closely follows chapter 7 of the book ’Spacetime and Geometry’ by Sean Carroll (Carroll 2004). Secondly, what is slow roll inflation? Inflation through a scalar field was proposed to solve some problems earlier Big Bang models had. The ’slow roll’ type is one of the many inflation models. We review some of its basics and show how quan-tum perturbations of the scalar field are causing density differences in the early universe. Although the quantum effects that lead to metric pertur-bations and hence primordial gravitational waves are similar, they are not discussed here. This second section is based on a lecture by Daniel Bau-mann (BauBau-mann 2007). The last section- how do gravitational waves lead to polarization?- is an attempt to derive the results presented in (Hu & White 1997) concerning polarization patterns that different types of waves would produce in the CMB. The Stokes parameters and the mathemat-ical definition of E and B modes are introduced. It turns out that only intensity densities around an electon which have Y2,m components can

cause polarization. Using a linear representation of the spin two group, the patterns are found and plotted. Although our results match, I have not been able to derive any B-modes.

I would like to thank my supervisor, professor Erik Verlinde, who put me on track of this subject and greatly helped resolve the problems I encoun-tered.

3

Deriving gravitational waves from

Ein-stein’s field equations

The first order perturbation of Einstein’s field equations turn out to be gravitational waves, the ’ripples in the fabric of spacetime’. These can be derived using a few standard formulas of general relativity, summed up below. Closely following (Carroll 2004) in the way we proceed, a few definitions are made that hugely simplify the problem after choosing a proper gauge transformation. Indeed, the solution turns out to be waves that compress and stretch space, perpendicular to the propagation axis.

3.1

Relevant formulas

Transformation law for the metric of (curved) spacetime gµν

gρσ= gµν0

∂x0µ ∂xρ

∂x0ν

∂xσ (1)

The Christoffel symbols,

Γσµν =

1 2g

σρ

(∂µgνρ+ ∂νgρµ− ∂ρgµν). (2)

The Riemann tensor

Rρσµν= ∂µΓρνσ− ∂νΓρµσ+ Γ ρ µλΓ λ νσ− Γ ρ νλΓ λ µσ. (3)

Einstein’s equation in vacuum:

Rµν−

1

2Rgµν = 8πGTµν = 0, (4)

where R is the trace of Rµν, the Ricci tensor, which is itself the

contrac-tion Rλσλν of the Riemann tensor.

The geodesic deviation equation D2

dτ2S µ

= RµνρσTνTρSσ, (5)

where Sµ _{is the displacement vector between two events, and T}µ _{is the}

tangent vector along a geodesic.

3.2

Perturbing Einstein’s field equations

Gravitational waves turn out to be the first order tensor perturbations hµν

to the flat Minkowski metric that solve Einstein’s equation in vacuum. We start by adding a perturbation to the Minkowski metric, which must be symmetric, since gµν is.

Demanding gµν = gµαgνβgαβ, it is straightforward to derive

gµν = ηµν− hµν

. (7)

To find an expression for the Christoffel symbols, we use the fact that the Minkowski metric is independent of space and time and thus has a vanishing derivative, so the only terms that remain are derivatives of hµν.

We assume them to be of order 1 too, which is reasonable since we want to find a physical, bounded solution. Since we only approximate in order 1, we can use equation 7 to write our approximation in order one of the Christoffel symbol as Γσµν= 1 2η σρ (∂µhνρ+ ∂νhρµ− ∂ρhµν). (8)

Now regarding the Riemann tensor, we can immediately throw away the last two terms, since they consist only of elements of order 2 or higher. Writing out the first two in terms of equation 8, we get

Rρσµν=

1 2η

ρλ

(∂µ∂σhλν− ∂σ∂νhλµ− ∂λ∂µhνσ+ ∂ν∂λhµσ). (9)

If we contract µ and ρ, we find the first-order terms of the Ricci tensor,

Rµν=

1 2(∂λ∂νh

λ

µ+ ∂µ∂λhλν− ∂µ∂νhλλ− ∂λ∂λhµν). (10)

To simplify the upcoming algebra, we use a notation that brings some symmetries of the problem to light and reduces the number of prefactors.

h00= −2Φ, (11)

h0i= wi, (12)

hij= 2(sij− Ψδij), (13)

where sij is traceless and Ψ adds the trace according to

sij= 1 2(hij− 1 3δklhklδij) (14) Ψ = −1 6δijhij. (15)

This notation can be used to write out Rµν. Like in equation 13 and 15,

we use the summation convention for indices written as a 0 and an i term. In that case, indices are summed over when repeated, indepent of being up or down. Furthermore, since every term in the Ricci tensor already has order 1 in hµν, identity 7 makes clear that indices can be brought down

and by using the flat Minkowski metric. We bring all the indices down by adding a minus sign to the 0th component. Lastly, we use that hµν is a

R00= ∂i∂0wi+ 3∂0∂0Ψ + ∂i∂iΦ (16) R0i= 1 2∂i∂jwj− 1 2∂j∂jwi+ 2∂0∂iΨ + ∂0∂jsij (17) Rij= ∂i∂j(Ψ − Φ) − 1 2(∂0∂jwi+ ∂i∂0wj) + (∂0∂0− ∂i∂i)(sij− Ψδij) + ∂j∂kski+ ∂i∂kskj. (18)

3.3

Gauge transformations

We can use coordinate transforms to fix a gauge in which these formulas work out neatly, which we generate by infinitessimal coordinate trans-forms, of the same order as hµν,

x0µ= xµ+ ξµ. (19)

We fill in the transformation law for the metric, discarding terms of order higher than 1. Since the Minkowski metric is independent of space and time, all of the changes in the matric must come from hµν, which is why

ηρσ+hρσ = ηρσ+hρσ0 +ηρν∂σξν+ηµσ∂ρξµimplies h0ρσ= hρσ−∂σξρ−∂ρξσ,

we derive that Ψ, wi, sijand Φ change as

Φ0= Φ + ∂0ξ0, (20) w0i= wi− ∂iξ0− ∂0ξi, (21) s0ij= sij− 1 2(ξi,j+ ξj,i) + 1 3∂kξkδij, (22) Ψ0= Ψ −1 3∂iξi. (23)

The gauge which simplifies the problem is the transverse gauge, which imposes

∂is0ij= 0, (24)

∂iwi0= 0. (25)

It is called transverse since these demands require the motion to be per-pendicular to the wave vectors, which is seen by taking the Fourier trans-form. We can meet these requirements by choosing ξiand ξ0as a solution

to

2∂isij=

1

3∂i∂jξi+ ∂k∂kξj (26) ∂iwi= ∂i∂iξ0+ ∂i∂0ξi. (27)

3.4

First order solution in vacuum

We now finally have the tools to find the Ricci tensor. We fill in equations 16, 17 and 18 in this gauge.

R00= 3∂0∂0Ψ + ∂i∂iΦ (28) R0i= 2∂0∂iΨ − 1 2∂j∂jwi (29) Rij= ∂i∂j(Ψ − Φ) + (∂0∂0− ∂k∂k)(sij− Ψδij) (30) By noting R = Rµν = −R00+ Rii, and Rii= 4∂i∂iΨ − ∂i∂iΦ − 3∂0∂0Ψ

since the trace of sijis 0, we can now simply fill in Einstein’s equations

as R00− 1 2g00R = R00+ 1 2R = 1 2(R00+ Rii) = 4∂i∂iΨ (31) R0i− 1 2g0iR = 2∂0∂iΨ − 1 2∂j∂jwi (32) Rij− 1 2gijR = (∂i∂j− δij∂k∂k)(Ψ − Φ) + (∂0∂0− ∂k∂k)sij+ 2δij∂0∂0Ψ (33) Equation 31 has no bounded solution except for Ψ = 0, which also implies that (32) has no other solution than wj = 0. Since (33) must be 0 for

all combinations ij, the trace must be 0, which is an easy way to derive that 2∂k∂kΦ = 0, and therefore Φ must be 0 too. So after this derivation,

the only terms we are left with are sij, that all have to obey the wave

equation, propagating at the speed of light. We can use the extra limits we have imposed to gain more insight: sijhas to be symmetric and traceless

and orthogonal to ki. If we choose ki to be in the z-direction, returning

to equation 13, we conclude we have a solution of the form

hµν= ei(ωt−kz)     0 0 0 0 0 h+ h× 0 0 h× −h+ 0 0 0 0 0     . (34)

Other solutions have waves running in other direction, or consist of linear combinations of these types of solutions. Since the effect on spacetime is a linear one, as we will see in a moment, the effect of these linear combinations is easily determined.

3.5

Effects on spacetime

By using the geodesic deviation equation, we can determine the relative motion between test particles that stand still compared to each other in the Minkowski metric, but are ’hit’ by a gravitaional wave. Since the Riemann tensor is already of order 1, as we have seen in equation 9, we can write the four velocities on the geodesic in order 0, which means that

Figure 1: Effects on a ring of test particles of a + polarized gravitational wave, taken from (Carroll 2004)

.

the test particles stand still, and Tµ_{= (1, 0, 0, 0). Furthermore, for these}

particles moving slowly, the proper time τ can be replaced by time t. Filling this, using equation 9 and hµ0= 0 in gives

d2 dt2S µ = Rµ00σS σ = 1 2S σ d2 dt2h µ σ. (35)

Regarding only the plus polarization, this can be solved to first order as

S1(t) = (1 +1 2h+e

ikλxλ_)S1_{(0), S}2_{(t) = (1 −}1 2h+e

ikλxλ_)S2_{(0) (36)} This means particles that already have a displacement in x-direction os-cillate back and forth, as does the y displacement, with a phase difference of π. This is drawn in figure 1,

Figure 2: Effects on a ring of test particles of a × polarized gravitational wave, taken from (Carroll 2004)

.

The effect of the × polarization is

S1(t) = (1 +1 2h×e

ikλxλ_)S2_{(0), S}2_{(t) = (1 +}1 2h×e

ikλxλ_)S1_{(0), (37)} which means the oscillation in the x direction scales with the the initial y-component and vice versa. This effect is drawn in figure 2.

Lastly, note that also linear combination of these waves are possible, for example the counter-clockwise rotating _√1

2(h++ ih×), drawn in figure 3.

Figure 3: Effects on a ring of test particles of a √1

2(h++ ih×) polarized

gravi-tational wave, taken from (Carroll 2004) .

4

Inflation and quantum fluctuations

First off, we determine how a scalar field and its potential yield a expo-nentially expanding universe using the Friedmann equations. After that, we work out the behaviour of the perturbations on the scalar field, which will result in temperature differences in the CMB. Both derivations rely heavily on a Princeton lecture by Daniel Baumann (Baumann 2007).

4.1

Basics of slow roll inflation

Assuming that the universe is homogeneous and isotropic, Einstein’s equa-tions are exactly solved by the Friedmann-Robinson-Walker metric, which takes the form ds2= −dt2+ a(t)2ds23, where s

2

3 is a metric for the three

spatial components, and a(t) is the scale factor. Using Einstein’s stress tensor for a perfect fluid Tµν= Diag(−ρ, P, P, P ), ρ being the density, P

the pressure, we can derive the two Friedmann equations that describe the behaviour of the parameter a(t) in a flat universe, and the continuity equation, which is their direct consequence:

 ˙a a 2 = 8πG 3 ρ, (38) ¨ a a = − 4πG 3 (ρ + 3P ), (39) ˙ ρ = −3˙a a(ρ + 3P ). (40)

For ordinary matter or radiation, the second equation leads to a decelera-tion of expansion, since ρ + 3P is always positive. To let infladecelera-tion happen, we introduce a scalar field with the following Lagrangian

Lφ=

1 2g

µν

∂µφ∂νφ − V (φ), (41)

where g is the trace of the metric. Such fields have a stress-energy tensor of the form (carroll 164):

Tµν= ∂µφ∂νφ − gµν(

1 2∂

σ

φ∂σφ + V (φ)). (42)

This tensor is diagonal; any off-diagonal element is 0 since φ is not space dependent, and the FRW-metric is diagonal too. Writing out the 00 and ii components, we see it takes the form of a perfect fluid, where

ρφ= 1 2φ˙ 2 + V (φ) (43) Pφ= 1 2 ˙ φ2− V (φ). (44)

Implementing this in the second Friedmann equation, we see we can get accelerated expansion if V (φ)  1₂φ˙2, in other words, the potential term

dominates the kinetic term. There are of course multiple formulas that satisfy that condition, however, in the case of the slow roll approximation, we limit ourselves to the cases where the first and second derivatives of the potential are small. This, combined with a small initial ”speed” of φ, assures a prolonged period of inflation.

From these assumptions, we deduce that ρφ ≈ V (φ), which is

approxi-mately constant. Therefore, by the first Friedmann equation, the scale factor grows exponentially, a(t) = eHt, where H = ˙a/a is called the Hub-ble constant. This universe with a exponentially growing scale factor is called De Sitter space.

4.2

Quantum perturbations

According to quantum mechanics, the vacuum is never completely empty. Pairs of particles and anti-particles are can appear and annihilate each other again. In De Sitter space, it can happen that a pair is created, but is driven apart by inflation, preventing it from annihilation. Baumann gives a basic mathematical description. We look at perturbations in the scalar field, writing it as φ(t) + δφ(t, ~x). Note that the variations have a spatial dependence, unlike the scalar field itself. The Lagrangian (equation 41) gives the following equation of motion, noting that gii= 1/a2(t)

¨

φ − 1

a2_(t)5 2

φ + 3 ˙φ + V0(φ) = 0. (45)

This equation is linear, except for the potential term. Expanding V0(φ) in its Taylor series, we see we can neglect it, since we have assumed V00(φ) to be small. We can find a solution to this equation if we use conformal time, which is defined by dt/a(t) = dτ . We integrate this, by using the formula for the scale factor derived above, and choose a constant of integration such that τ (t) = −1/Ha(t), or a(t) = −1/Hτ (t). In the conformal time coordinate, the Big Bang took place at τ = −∞, since a(t) = 0 then. If τ = 0, the scale factor blows up, which corresponds to t = ∞, ”the end of the universe”. Since inflation does not stop in this model, we consider τ = 0 to be in the end of inflation. Another property of the conformal time is that it gives a measure for the size of the ’reachable’ universe: since light travels 1/a(t) in a time dt, a photon starting at a time t can only reach a distance equal to |τ (t)|, which is therefore a scale on which causal contact can occur. Note that in an inflating universe, this distance gets smaller in time.

If we then switch to v ≡ a(t)δφ as coordinate, take the Fourier transform of the spatial coordinates, multiply by a3(t) and use conformal time, (45) becomes d2vk dτ2 + (k 2 − 2H2a2(t))vk= d2vk dτ2 + (k 2 − 2 τ2)vk= 0, (46)

which is a harmonic oscillator with time-dependent frequency. The two limit cases of this equation give some more insight in the physics involved. If k2 1/τ2

wavelength, the equation becomes that of a harmonic oscillator in Fourier space, or a wave equation in position space. If the waves fluctuate on a scale larger than the ’visible universe’ τ , we use that we can write 2/τ as a00(τ )/a(τ ), primes denoting derivation with respect to τ . This too is easily solved by filling in vk = a(τ )f (~k), f being any function. Since

the Fourier transform does not affect the temporal coordinate, this means that δφ, which is equal to v/a, can not be time dependent. Equation (46) can even be solved exactly, the solution being

vk(τ ) = α e−ikτ √ 2k(1 − i kτ) + β eikτ √ 2k(1 + i kτ). (47)

Adopting some methods from field theory to find the action for vk,

Bau-mann is able to quantize this harmonic oscillator. From this, vkbecomes

operator ˆvk. Working in the the Heisenberg picture of quantum

mechan-ics, he derives the following statements: if vk(t) is a solution of the classic

equations of motion for a quantum operator ˆvk, h| ˆvk|2i = |vk(t)|2.

Fur-thermore, the vacuum solution of the time-indepent harmonic oscillator is vk(t) = q ¯ h 2ke −ikt

. If we assume the universe to have started (τ = −∞) in this vacuum, this means β = 0, α = √¯h in (47). In that case, the variance in ˆδφk at the end of inflation (τ = 0) is

h ˆδφ2ki = lim τ →0|δφk(τ )| 2 = lim τ →0| vk a(τ )| 2 =¯hH 2 2k3 , (48)

where we have used a(τ ) = −1/Hτ in the last step. Baumann uses the Dirac orthogonality of the δφk’s to arrive at the variance in δφ itself

h ˆδφ2i ∼ Z ~_k φkeikxdk  Z ~_k0 φ0ke ik0x dk0 ∗ ∼ Z kf ki ¯ hH2 2k3 k 2 dk ∼ H2ln(kf ki ). (49)

Baumann cuts off the integral at low and high k to account for the fact that inflation has a begin and an ending. Very low frequencies correspond to wavelengths which have always been larger than τ , which have not been amplified this way. Likewise, high frequencies were not amplified because their wavelengths have always been much smaller than τ .

The scalar field rolling down the potential hill will reach the end of in-flation earlier if δφ is positive. This therefore leads to a higher density compared to points around it, which we can measure as temperature dif-ferences in the CMB. Equation (49) is one of the examples Baumann gives that show CMB measurements could be able to tell us more about the physics of inflation: the variation gives a measure for the speed of inflation H.

Starting from a Lagrangian for a metric perturbation rather than (41), one can derive properties of density perturbations generated by gravitational waves. Measuring these, in combination with (49) would put further re-strictions on the potential V (φ). Moreover, these perturbations cause the B-modes in the CMB polarization.

5

Deriving polarization patterns

The CMB consists of the first photons that could travel freely after re-combination. This light is put forth by an almost perfect black body and therefore unpolarized. However, due to Thomson scattering from electrons, the light can gain a polarization. To examine these effects, the Stokes parameters form a useful framework. It turns out that only quadrupole intensity densities Y2,maround the electrons can cause

polar-ization. For a different value of ±m, there is a different physical wave associated with it: scalar, vector and tensor waves. The polarization pat-tern for any of the quadrupole Y2,m’s over the whole sky can be found

by regarding them from different angles. By this rotation, a pure Y2,m

changes into a linear combination of quadrupoles. In the approach used below, the polarization is then given by the magnitude of component Y2,−2, which can be determined by regarding a linear representation of

the group.

5.1

Stokes parameters, E modes and B modes

We start by introducing the Stokes parameters, given an electric field

Ex= axcos(ω0t − θx), Ey= aycos(ω0t − θy), (50)

we can define the Stokes parameters (Kosowsky 1999)

I := a2x+ a 2 y (51) Q := a2x− a 2 y (52) U := 2axay cos(θx− θy) (53) V := 2axay sin(θx− θy). (54)

I is the intensity of the beam. Q, U and V are measures for the polar-isation: Q being the polarisation parallel to the x-axis (positive Q) and parallel to the y-axis (negative Q), U the polarisation along the system of axis shifted 45 degrees counterclockwise to the x- and y-axis and V the rotation of the polarization. Since Thomson scattering cannot create rotating polarization, V = 0 and will not be regarded further.

One should note that in this definition, the light does have a preferred direction. Unpolarized light can be described by taking the expectation values of ax, ay, θx, θy and ω0, letting them be time dependent. For our

current purposes, this makes no difference.

The Q- and U -polarisation can be interchanged by rotating the system of axes. This property, which is also useful to derive the expression for U , can be summarised as

Another way of looking at the polarization is as a tensor field (Cabella & Kamionkowski 2004). If we write the polarization as a traceless symmetric matric, Pab= Q U U −Q  , (56)

we note that this identity undergoes the transformation described by equa-tion (55) when we work on it with rotaequa-tion matrices.

Figure 4: Graph for the polarization pattern Q = 2xy, U = x2− y2_{, for which}

52_P

E> 0, 52PB = 0.

A vector field admits a decomposition into a field that is the gradient of a scalar density and a field that is the curl of another vector field. Likewise, a traceless symmetric tensor field, which consists of two scalar fields, admits a decomposition as the sum of scalar fields PE and PB,

chosen such that 52_P

E= ∂a∂bPaband 52PB= ac∂b∂cPab, abbeing the

antisymmetric tensor. If we work in Fourier space, derivation becomes multiplication by il, and we can write out PE and PB explicitly as

PE(~l) = (k2 x− k2y)Q(~k) + 2kxkyU (~k) k2 x+ ky2 , (57) PB(~l) = 2kxkyQ(~k) − (kx2− k 2 y)U (~k) k2 x+ ky2 . (58)

This notation underlines the rotational invariance of our decomposition: by rotating the axis and using equation (55), one can see that the expres-sion stays the same. To get a feel for what these equations signify, we take

Figure 5: Graph for the polarization pattern Q = −2xy, U = x2_{− y}2_{, for which}

52_P

E< 0, 52PB = 0.

Figure 6: Graph for the polarization pattern Q = −2xy, U = x2_{− y}2_{, for which}

52_P

E= 0, 52PB < 0.

a fields for which either PEor PBis zero, and draw their polarization plots.

From these four figures, we see what is going on: when 52PB = 0, the

polarization pattern is reflectively symmetric. For any point on the plane, two directions of polarization are unchanged under reflection, which is why we have two different patterns. On the other hand, if 52_P

E= 0, we get

Figure 7: Graph for the polarization pattern Q = 2xy, U = −x2_{+ y}2_{, for which}

52_P

E= 0, 52PB > 0.

their mirror image is their oppposite, and the pattern has a ”handedness”. The nomenclature reflects that the fields for which PB is nonzero seem to

have a curl, which is a property of the magnetic B field in magnetostatics and the PEfield seems to be a gradient, which is a property of the electric

E field.

To work out these derivatives on a curved surface, like the sky, a lengthier derivation is necessary, working with Christoffel symbols and a curvature tensor. This is done by (Cabella & Kamionkowski 2004), but for our purposes, we can use that the spherical surface is approximately flat on small angular scales.

5.2

Thomson scattering

We now fix a coordinate system where the outgoing wave is headed in the z-direction. The incoming wave is placed in the z-y plane, θ being the angle between in- and outgoing radiation. Assuming the incoming beam to be unpolarized, we set out to determine the polarization of the outgoing beam. The following relation is used:

dσ

dΩ∝ |ˆei· ˆee|

2

, (59)

where ˆei and ˆeo are unit vectors aligned with the polarization of the

in-coming and outgoing radiation, respectively.

By means of equation 59 we note that polarization along the x-direction is independent of the angle made with the incoming wave. The y-direction however, depends on the angle with a factor cos2_{(θ). Integrating the}

incoming radiation I0(θ, φ) over both angles, we have to carefully regard which point yields what kind of polarization: x direction (φ = 0) gives −Q, φ =π

4 gives −U , y-direction (φ = π

2) gives positive Q, which we can

account for by using 55, which yields:

I(ˆz) = 3σT 16πσB Z dΩ I0(θ, φ)(1 + cos2(θ)) (60) Q(ˆz) − iU (ˆz) = − 3σT 16πσB Z

dΩ I0(θ, φ) sin2(θ)e−2iφ. (61)

The factor in front properly scales the intensity, σT being the total cross

section and σBthe cross section of scattering in the z-direction. (Kosowsky

1999)

The intensity density I0((θ, φ) can be expanded in the orthonormal basis of spherical harmonics Yl,m: I0((θ, φ) = ∞ X l=0 l X m=−l al,mYl,m(θ, φ). (62)

Noting that sin2_(θ)e2iφ_{= 4}q2π 15Y

∗

2,2(θ, φ), equation 61 can be interpreted

as an inner product, and thus

Q(ˆz) − iU (ˆz) = − 3σT 4πσB

r 2π

15a2,2. (63)

5.3

Physical effects and spherical harmonics

From a larger view, equation 63 represents the polarization we receive from one point on the sky. A polarization pattern is determined by re-garding the same intensity density around the electron, but seen from a different angle. By relation between Ylm’s, we know that only terms with

l = 2 can transform into a Y2,2 by rotation. Each of the terms m = 0,

m = ±1 and m = ±2 is affialited with a certain wave in the primordial universe (Hu & White 1997).

Y2,0 = C

q

2 3(3 cos

2_{(θ) − 1), where C = 4}q2π

15, is independent of the

az-imuthal angle and is thus generated by a scalar (density, temperature) wave heading in the z-direction, where the electron is positioned at one of the crests: it can be rotated over the φ angle without changing the physical situation. A sketch of the situation can be found in figure 8.

Figure 8: Sketch of a scalar wave and the associated density

Y2,±1 = ∓2C sin(θ) cos(θ)e±iφ is caused by a vector wave: in one of the

directions perpendicular to the direction of the wave, the material has a motion which sinusoidally changes sign. For an electron at the zero point of the wave, the Doppler effects of this movement creates a quadrupole. Changing the azimuthal angle by π, Y2,±1 changes sign, which coincides

with the velocity changing sign and thus the quadrupole. A sketch of the situation can be found in figure 9.

Y2,±2= C sin2(θ)e±2iφis caused by a tensor wave, or gravity wave. Since

it compresses and stretches space, density and therefore temperature fluc-tuate with it. It has the spin 2 property of a gravity wave that changing the azimuthal angle by 1

2π puts in an extra minus sign, for an expansion

in the x-direction means a contraction in the y-direction. A sketch of the situation can be found in figure 13.

Figure 10: Sketch of a tensor wave and the associated density

Note that these Y2m’s do not fully describe the physical situation: for

example, there is always a constant term added to the intensity, since it must be positive. However, by equation (63), only the l = 2 components matter when describing the polarization.

5.4

Rotating spherical harmonics by linear

rep-resentation

By using a representation of the group of spin-2 spherical harmonics as 3x3 traceless symmetric matrices, we can determine a polarization pattern for m = 2, m = 1 and m = 0. Noting that for a given point (θ, φ), the projections on the x, y and z axis are px= sin(θ) cos(φ), py= sin(θ) sin(φ)

and pz= cos(θ), we can regard matrices with matrix posts as

  pxpx pxpy pxpz pypx pypy pypz pzpx pzpy pzpz  . (64)

By virtue of this notation, we can write down our Ylm’s as matrices, for

example:   1 0 0 0 −1 0 0 0 0  ± i   0 1 0 1 0 0 0 0 0  = (pxpx− pypy) ± i(pxpy+ pypx)

= sin2(θ) cos(2φ) ± i sin2(θ) sin(2φ) = sin2(θ)e±2iφ= C Y2,±2(θ, φ). (65)

In the same way, the following equivalences hold: ∓1 C   0 0 1 0 0 ±i 1 ±i 0  = Y2,±1(θ, φ), (66) −1 C r 2 3   1 0 0 0 1 0 0 0 −2  = Y2,0(θ, φ). (67)

This representations gives us a natural way to regard an intensity from an different angle (β, α) on the sky, by transforming the basis using rotation matrices.

R(β, α) = 



cos(α) cos(β) − sin(α) − cos(α) sin(β) sin(α) cos(β) cos(α) − sin(α) sin(β)

sin(β) 0 cos(β)   (68) R−1(β, α) =  

cos(α) cos(β) sin(α) cos(β) sin(β)

− sin(α) cos(α) 0

− cos(α) sin(β) − sin(α) sin(β) cos(β) 

 (69)

Continuing in this fashion, we find that for the m = 0 case

−1 C r 2 3R −1 (β, α)   1 0 0 0 1 0 0 0 −2  R(β, α) = −1 C r 2 3  

cos2(β) − 2 sin2(β) 0 −3 cos(β) sin(β)

0 cos2_{(β) + sin}2_{(β) 0}

−3 cos(β) sin(β) 0 sin2(β) − 2 cos2(β) 

.

(70) All of the α dependence drops out, which is evident given the fact that Y20is independent of the azimuthal angle. For the m = ±1 case we get

∓1 CR −1 (β, α)   0 0 1 0 0 ±i 1 ±i 0  R(β, α) = ∓1 C  

2 cos(β) sin(β)e±iα ±i sin(β)e±iα

(cos2(β) − sin2(β))e±iα

±i sin(β)e±iα 0 ±i cos(β)e±iα

(cos2_{(β) − sin}2_(β)e±iα _{±i cos(β)e}±iα _{−2 cos(β) sin(β))e}±iα



.

(71) For the m = ±2 case:

1 CR −1 (β, α)   1 ±i 0 ±i −1 0 0 0 0  R(β, α) = 1 C  

cos2(β)e±2iα ±i cos(β)e±2iα _{− sin(β) cos(β)e}±2iα

±i cos(β)e±2iα −e±2iα ∓i sin(β)e±2iα − sin(β) cos(β)e±2iα _{∓i sin(β)e}±2iα

sin2(β)e±2iα 



(72) As one can easily check, all these matrices are still traceless and sym-metric. Since the Y2m’s span the 5-dimensional space of traceless and

symmetric 3x3 matrices, and change of basis is a linear operation, this statement holds for any linear combination, and we conclude that ro-tation is a well-defined operation on this group. So, after this change of basis, we can once again express the matrix in terms of the basis matrices, where we are interested in the Y2,2 component, for it gives an expression

for the polarization (equation 63).

We start off by using as straightforward linear algebra technique, by ex-pressing any traceless symmetric matrix in terms of its independent coor-dinates.   a c d c b e d e −a−b   (73)

We can use our basis matrices e2, e1, e0, e−1and e−2and find the matrix

which expresses the dependence of different coordinates on the basis:

      a b c d e       = 1 C         1 0 −q2 3 0 1 −1 0 −q2 3 0 −1 i 0 0 0 −i 0 −1 0 1 0 0 −i 0 −i 0               c2 c1 c0 c−1 c−2       (74)

Inverting this, we can determine an immediate relation between a matrix and its Y2,2 component:

      c2 c1 c0 c−1 c−2       =C 2        1 2 − 1 2 −i 0 0 0 0 0 −1 i −q3 2 − q 3 2 0 0 0 0 0 0 1 i 1 2 − 1 2 i 0 0              a b c d e       . (75)

From this expression, we read of the c2component. Combining expression

63 with 70, 71 and 72 respectively, writing the intensity density as a real cosine by taking for example 1

2(−Y2,1+ Y2,−1), we find that for the scalar

Q − iU = − 3σT 8πσB r π 5sin 2 (β), (76)

for the vector wave we have

Q − iU = − 3σT 8πσB

r 2π

15(cos(β) sin(β) cos(α) + i sin(β) sin(α)), (77) and for the tensor wave

Q − iU = − 3σT 16πσB r 2π 15((cos 2

(β) + 1) cos(2α) + 2i cos(β) sin(2α)). (78) Note that if we would have chosen to write the intensity density as a sine, the α terms would have changed from cosine to sine, from sine to -cosine. This is natural since it would have meant a rotation of the physical situ-ation. These equations match those given by (Hu & White 1997). A slightly more highbrow approach of looking at equation 75 is by re-garding the following inner product on the space of traceless symmetric matrices: hA, Bi =C 2 4 Tr(AB † ), (79)

where the Tr is the trace, defined as the sum over diagonal elements of a matrix. The factor preserves the orthonormality of the basis we have chosen earlier. By using the general expression for a traceless symmetric 3x3 matrix (equation 73), one can verify that taking this inner product yields exactly the same expression as (75).

5.5

Plot of polarization patterns

We can plot the outcome of equations 76, 77 and 78 using Mathematica. This is a little delicate, since we do not plot a common vector field but a tensor field. Nonetheless, we use the VectorPlot function of Mathematica, setting the size of the arrow heads to 0. We assign 4 functions, one for positive Q, which is 0 otherwise, and alike one for the absolute value of negative Q, one for positive U , one for the absolute value of negative U . In the vector plot the function for positive Q is multiplied by the arrow (0, 1), the one for negative Q by (0, −1) and the one for positive U by

1 √

2(−1, 1). For negative U , we can not simply multiply by 1 √

2(1, 1), for

in that case, negative Q and U can sum up to just positive Q. Therefore we multiply by 1₂(1, −1) in the case that Q is negative, and by √1

2(1, 1) is

Q is positive. The resulting plot shows the polarization seen by a viewer at the point (φ, θ) on the sphere. Note that φ runs around in the oppo-site direction compared to normal maps, which display the view from the outside.

Figure 11: Polarization from a scalar wave, seen from different sides

Figure 13: Polarization from a tensor wave, seen from different sides

These plots do not display a B-mode. If we examine the tensor polariza-tion, where a B-mode is expected (Cabella & Kamionkowski 2004), we see that it is symmetric around points that would lead to hot or cold spots on the sky, where the gravitational waves compresses or stretches the direction perpendicular to the line of sight. If we calculate 52PB =

ac∂b∂cPabaround for example (π₂,π₂), choosing it to be the origin, we get

52

PB = sin(2φ0) sin(θ0)(8 cos(θ0) + 6). Although this is not 0 around the

point, it does change sign 4 times, and so does the handedness, leaving no netto curl. Note that the sphere is regarded as a plane, which is possible for small angular scales.

Hu and White state that the B-modes arise when we modulate the waves across the surface (Hu & White 1997). The polarization pattern we look at now is a crest of a wave stretched over the surface of the sky, which should have been multiplied by ei~k·~x to actually regard the spectum of a wave. Since the wave vector must point up (figure 13), it would be only depedent on θ. This means the sign change caused by sin(2φ0) stays un-altered, preventing a handedness. A circular polarization does not seem to change this conclusion.

6

Conclusion and discussion

The behaviour of gravitational waves and quantum perturbations of the scalar field was derived, both from some basic formulas in general relativ-ity. Following a method presented by (Kosowsky 1999), the polarization patterns generated by temperature fluctuations through Thomson scat-tering were found. Altough they match the results found by (Hu & White 1997), no B-modes seem apparent. For their conclusion that gravitational waves produce B-modes, Hu and White do not present a derivation in their article, but other resources, like (Hu et al. 1998) and (Cabella & Kamionkowski 2004), use methods that are way more complicated. They have not been discussed here.

References

Ade, P. A. R., Aikin, R. W., Barkats, D., et al. 2014, Physical Review Letters, 112, 241101

Baumann, D. 2007, arXiv:astro-ph/0209273

Cabella, P. & Kamionkowski, M. 2004, arXiv:astro-ph/0403392 Carroll, S. 2004, Spacetime and Geometry (Addison Wesley)

Hu, W., Seljak, U., White, M., & Zaldarriaga, M. 1998, Phys. Rev. D, 57, 3290

Hu, W. & White, M. 1997, New Astronomy, 2, 323 Kosowsky, A. 1999, New Astronomy Reviews, 43, 157