Master Thesis

Master’s in Theoretical Physics

Cosmic Strings

Author:Evelyne Groen, 0310719

Supervisor:Dr. J.P. van der Schaar

University of Amsterdam

Institute for Theoretical Physics

August 13, 2009

To: C. Kelderman (1923 - 2008)

Thanks to: Dr. J.P. van der Schaar, O. Smits, I.S. Eliens, family and friends fortheir inspiration and support.

Summary

This thesis will give an introduction on the topic of cosmic strings. A cosmicstring is a topological defect that arises during phase transitions. In the earlyuniverse, when the temperature drops below a certain critical temperature, phasetransitions can occur, forcing the fields into a particular vacuum state. A cosmicstring is a one-dimensional defect, but defects of other dimensions like monopolesand domain walls can also occur. In the context of the Abelian-Higgs model,a simple formulation of a cosmic string will be derived. There are several waysto detect cosmic strings. In this thesis observational effects due to gravitationallensing and gravitational waves will be discussed. Using General Relativity wewill derive the Einstein equations in the presence of a cosmic string. In addition,the energy-momentum tensor will be derived from the effective string action. Wewill apply the Gauss-Bonnet theorem in order to calculate the wedge created inspace-time caused by the presence of the string. These wedges act as gravitationallenses that can be observed. The second type of observational effects is causedby gravitational waves that can be emitted either by contracting loops, or cusps.Loops are formed when strings intersect with themselves or others in the network.String networks have some remarkable properties implying that the networks arescale invariant. Finally we describe some possible observations.

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Contents

1 Introduction 81.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Symmetry breaking 112.1 Topological Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Abelian Higgs model . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 Cosmic strings in the universe (Kibble mechanism) . . . . . . . . . 172.4 Monopoles, Textures and Domain walls . . . . . . . . . . . . . . . . 18

3 A little bit of General Relativity 203.1 The metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 The geodesic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3 Einstein equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.4 Einstein gravity of a string . . . . . . . . . . . . . . . . . . . . . . . 23

4 Cosmic String as a Source 274.1 The action formalism of gravity . . . . . . . . . . . . . . . . . . . . 274.2 Effective Cosmic String Action . . . . . . . . . . . . . . . . . . . . . 294.3 Gravitational Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Cosmic String Dynamics 365.1 Solution of Cosmic String Wave Equation . . . . . . . . . . . . . . . 365.2 Oscillating loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.3 String interaction and network evolution . . . . . . . . . . . . . . . 42

6 Conclusion 47

A Higgs boson 50

B Derrick’s theorem 51

C String dynamics 52

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1 Introduction

Cosmic strings are a specific type of topological defects that arise during phasetransitions. Topological defects can occur when the field symmetries are broken.This means that the ground state does not exhibit the same symmetry as the fulltheory. Symmetry breaking happens when the universe cools down below somecritical temperature Tc and the field is forced to choose a vacuum state.

Due to symmetry breaking, energy can get trapped in specific regions of space.The topological structure of this trapped energy determines the nature of the de-fect. A line-like structure is called a cosmic string.

The universe, in its very early history, was an intense environment. Duringthe initial stages of the development of the universe, phase transitions occurredthat might have left traces that are still visible today. Phase transitions may haveled to symmetry breaking and thus the formation of topological defects. Besidesline-like defects, the cosmic string, other defects can occur, namely, domain wallsor monopoles. Some of these defects may have survived to present day. With everybroken symmetry there is a possibility for a topological defect, which could leadto the formation of cosmic strings.

As the universe cooled down it went through at least three phase-transitions:

1. The GUT transition occurs between 10−37s and 10−35s after the Big Bang.The Grand Unification Theory (GUT) predicts that at very high-energyscales the electroweak-nuclear and strong-nuclear forces are unified into oneforce. The GUT symmetries are broken by the rapid expansion that causeda cooling down of the universe.

2. Around 10−11s after the Big Bang the electroweak symmetry was broken.The electroweak symmetry unified electromagnetism and the weak interac-tion.

3. The quark-hadron transition at 10−5 s after the Big Bang caused the plasmaof free quarks and gluons to convert into hadrons (baryons and mesons, themore well-known baryons are protons and neutrons).

The Cosmological Principle states that on large scales the universe is homogeneousand isotropic: uniform in all directions. But on smaller scales this isn’t quite true.Galaxies and other clusters are not homogeneous and isotropic at all. A questionthat now immediately arises: where did the density fluctuations originate to formgalaxies?

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The cosmic microwave background (CMB) shows the afterglow of the Big Bang,with slight temperature differences indicating density fluctuations responsible forthe structure in our universe. When cosmic strings were first predicted, they werea candidate for the seeds of density perturbations that cause the formation ofstructure.Recent studies have indicated that although cosmic strings could stillbe partly responsible for the density perturbations, Cosmological Inflation theorygives a better explanation for the current data retrieved from the CMB radiation.Therefore most interest in cosmic strings was lost.

Recently the interest in cosmic strings has been shifted to a more theoreticalperspective. All Grand Unifying Theories predict the existence of cosmic strings.The observation of a cosmic string would validate one of the fundamental theoriesthat predict strings.

Cosmic strings have certain remarkable properties that could have caused themto survive to the present day. There are several ways to detect cosmic strings.When a cosmic string is formed during one of the phase transitions, it creates awedge in space-time. Light that passes a cosmic string is deformed due to thegravity exerted by the string. This effect is observable as the lensing of a cosmicstring. The lensing causes two exact similar objects to appear in the sky.

The second way of detecting a cosmic string is by measuring gravitationalwaves. Cosmic strings are capable of emitting gravitational radiation by formingcusps, kinks or loops. These configurations can be formed when strings intersectswith themselves or each other. Cusps emit radiation in a specific direction andwith a relatively narrow spectrum. Loops decay by emitting a rather broad spec-trum in a wide direction. Gravitational waves emitted by loops can therefore bemore easily detected. Both types of emitted gravitational waves can in principlebe detected, but that depends on the tension of the string. Cosmic strings have,until recently, not been found.

This thesis will give a review of past and current research on cosmic strings.

1.1 Outline

After a short introduction of symmetry breaking in chapter 2 this thesis will con-tinue with the Abelian-Higgs model to explain the behaviour of cosmic strings.With the use of a small recap of General Relativity, the concept of cosmic lensingwill be explained. Starting with the effective action of a cosmic string, the energy-momentum tensor will be derived in chapter 4. The Gauss-Bonnet theorem will beused to derive the wedge created in space-time that causes gravitational lensing.

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More will be explained about cusps and kinks, which are sources of gravitationalwaves. The evolution of a cosmic string network and its scaling behaviour ismentioned in the final section of chapter 5. The thesis concludes with the contem-porary status of observational evidence of the cosmic strings, including possiblediscoveries and recommended directions for future research.

1.2 Conventions

Throughout the thesis Einstein’s summation convention will be used:

3∑µ,ν=0

gµνxµxν = gµνx

µxν (1)

where the summation runs over µ, ν = 0, 1, 2, 3, where the zero indices is reservedfor the time-component and the other three indices are space-like components.The following abbreviation will be used:

g = det (gµν) (2)

In the case of a flat universe: gµν = ηµν , where

ηµν =

−1 0 0 00 1 0 00 0 1 00 0 0 1

is called the a mostly-plus metric. Furthermore Planck’s constant and the speedof light have been set to one:

h = c = 1 (3)

unless indicated otherwise.

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2 Symmetry breaking

In this chapter topological defects will be explained. The first model we will use todescribe a cosmic string, posesses a global symmetry which holds for all points inspace-time. Defects of different dimensions, like domain walls and monopoles, arebriefly mentioned and the Abelian-Higgs model is introduced. The Abelian-Higgsmodel exhibits a local symmetry. This means that the symmetry transforma-tions can be different in different points of space-time. This forms the basis forgauge theories that will help us to avoid some of the problems encountered whendiscussing cosmic string models possessing a global symmetry.

2.1 Topological Defects

A topological defect can be compared to a ball balancing on top of a mountain. Ata certain moment in time the ball will start to roll down and will eventually cometo a stop at the foot of the mountain. The valley can be compared to a certainvacuum state, this state is said to be degenerate because there is a nonzero falsevacuum state at x = 0 and (two) states equal to zero (see figure 1). To explorethis in more detail, let’s look at the simplest example of spontaneous symmetrybreaking. Consider a complex scalar field φ, sometimes called a Higgs field, whichhas the Lagrangian density [1]:

L = gµν∂νφ∗∂µφ− V (φ, φ∗) (4)

where the potential is given by:

V (φ, φ∗) =1

2λ(|φ|2 − 1

2η2)2 (5)

The self-interaction term is denoted by λ, it states how strongly two scalar particlesinteract and η is the mass term. This potential is also known as the Mexican hatpotential (fig. 1). The Lagrangian has a rotational symmetry. This means thatunder (circular) transformations in the φ, φ∗ plane the Lagrangian does not change.Just by looking at figure 1 it can be seen that the potential is invariant under U(1)transformations: the shape of the potential does not change, no matter in whatdirection you look at the x-y plane. The transformation

φ→ φeiα (6)

is unitary, since eiαe−iα = 1. When we apply this transformation to the Lagrangian(4):

L = gµν∂νφ∗e−iα∂µφe

iα − V (φ, φ∗) = gµν∂νφ∗∂µφ− V (φ, φ∗) (7)

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Figure 1: Mexican hat potential [2].

The dynamics of a real scalar field is described by the action:

S[φ] =∫dtL[φ] =

∫dt∫d3xL(φ, ∂µφ, φ) (8)

Hamilton’s extremal principle states that the configurations φ that are actuallyrealized are those that extremize the action, i.e. the action is stationary undersmall variations of the fields:

S[φ+ δφ] = S + δS (9)

The stationary action principle states that δS = 0. This means that for anysmooth curve:

limε→0

1

ε(S[φ+ εθ]− S[φ]) = 0 (10)

This becomes for the action:

δS =∫dmx[L(φ+ εθ, ∂µφ+ ε∂µθ)− L(φ, ∂µφ)]

=∫dmx[

∂L∂φ

θ +∂L

∂(∂µφ)∂µθ]ε+O(ε2)

=∫dmx[

∂L∂φ− ∂µ

∂L∂(∂µφ)

]θε+O(ε2) (11)

In order to set δS = 0 we need:

∂L∂φ− ∂µ

∂L∂(∂µφ)

= 0 (12)

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In our case, the Lagrangian contains a complex field that can be written as tworeal fields, so there should be an equation for each field [3]:

∂L∂φ− ∂µ

∂L∂(∂µφ)

= 0,∂L∂φ∗− ∂µ

∂L∂(∂µφ∗)

= 0 (13)

This is known as the Euler-Lagrange equation [4]. The first term of the Euler-Lagrange equation is given by:

∂L∂φ

= −∂V∂φ

= − ∂

∂φ(1

2λ(|φ|2 − 1

2η2)2)

= −λ(|φ|2 − 1

2η2)φ∗ (14)

And the second term is given by:

−∂µ∂L

∂(∂µφ)= −∂µ∂µφ = −∂2φ∗ (15)

The same equations are found for the real scalar field, which gives the followingfield equations:

∂2φ∗ + λ(|φ|2 − 1

2η2)φ∗ = 0, ∂2φ+ λ(|φ|2 − 1

2η2)φ = 0 (16)

The second equation is found by applying complex conjugation. The Higgs-bosoncan be derived in a somewhat similar expression, as can be seen in appendix A.When we look at the extrema of the potential, there is a solution obtained bysolving φ = 0, sometimes called the false vacuum. The real vacuum is at:

δV

δφ= λ(|φ|2 − 1

2η2)φ = 0 (17)

which gives for φ:

|φ|2 =1

2η2 (18)

So the minima of the potential lie on a circle (according to global U(1) symmetry).The field can be continuously changed by rotating through the complex field,staying always in a state of minimal potential energy:

φ(θ) =1√2ηeiθ (19)

where eiθ is an arbitrary phase constant. Any phase of the field describes a vacuumstate. But the ground states do not have the same symmetries as the Lagrangiananymore. Under a gauge transformation a state transforms into:

φ(θ)eiα =1√2ηeiθeiα (20)

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The symmetry is spontaneously broken. The string configuration arises from po-tential and gradient contributions to the energy density, which can be derived asfollows: a large circle can be drawn in space with the complex phase vectors alpointing outwards (this description is only valid for n = 1). All these points havethe same energy (the energy of the ground state). Somewhere in the middle thefield must pass through φ = 0, implying the existence of an excitation carryingenergy. This excitation is stable and called a cosmic string. The point inside thisloop has energy higher then the ground state. In polar coordinates:

φ(θ) =1√2ηeinθ (21)

where n is an integer. So what is the energy-density of such a configuration?We are interested in solutions matching specified boundary conditions at infinity(r →∞) [1]. The gradient in polar coordinates is given by [5]:

∇φ =∂φ

∂rr +

1

r

∂φ

∂θθ +

1

r sin θ

∂φ

∂ϕϕ (22)

Only the second term contributes, because φ only depends on θ:

∇φ(θ) =1

rin

η√2einθθ =

in

rφ(θ)θ (23)

Consider the energy density for a static configuration:

H = πφ− L (24)

where π = ∂L∂φ

. In a static configuration the energy density is equal to −L because

the time-dependent factors do not contribute:

H = −L = −gµν∂νφ∗∂µφ+ V (φ) = −|∇φ|2 + V (φ) (25)

For r → ∞, the potential term V (φ) goes to zero, but the squared kinetic termis proportional to 1

2r2. The total energy can be found by integrating over the

Lagrangian density [6]:

E = −∫d2x L =

∫ 1

r2rdrdθ (26)

Integrating over dθ will give a contribution of 2π. Integrating over dr will givean infinite contribution to the energy density. It is not possible to prevent thisoutcome from happening. This coincides with Derrick’s theorem (see AppendixB). But it can be evaded by adding an extra term to the Lagrangian.

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2.2 Abelian Higgs model

The symmetry described above is a global symmetry. Their symmetry transfor-mations involve rotating every point in the field by the same constant. A localsymmetry allows each point to vary by a different angle. This symmetry is presentin the Abelian Higgs model described below. The Lagrangian of the previousexample led to an infinite energy density contribution. In the case of the Abelian-Higgs model, a gauge field Aµ will cancel out the divergences, which will lead to anenergy density that will no longer be infinite, as happened in the previous section.In this case, the Lagrangian equals [7]:

L = −1

4FµνF

µν + |Dµφ|2 − V (φ, φ∗) (27)

where the potential is the same as the previous example, Dµ = ∂µ − ieAµ, andFµν = ∂µAν − ∂νAµ is the electromagnetic field strength. The local invariance isrealized by:

φ → eiα(x)φ(x) (28)

Aµ → Aµ +1

e∂µα(x) (29)

Deriving the field equations we again need the Euler-Lagrange equations [1] andthe complex conjugates:

∂L∂φ− ∂µ

∂L∂(∂µφ)

= 0,∂L∂φ∗− ∂µ

∂L∂(∂µφ∗)

= 0; (30)

∂L∂Aµ

− ∂ν∂L

∂(∂νAµ)= 0, (31)

Since Fµν does not depend on φ or φ∗, the first field equation (30) will be the sameas before:

D2φ+ λ(|φ|2 − 1

2η2)φ = 0, D2φ∗ + λ(|φ|2 − 1

2η2)φ∗ = 0 (32)

The first term of the second field equation (31) equals:

∂L∂Aµ

=∂

∂Aµ|Dµφ|2 =

∂

∂Aµ(∂µ − ieAµ)φ(∂µ + ieAµ)φ∗

=∂

∂Aµ(∂µφ∂

µφ∗ − ieAµφ∂µφ∗ + ∂µφieAµφ∗ − i2e2AµA

µφ∗φ)

= −ieφ∂µφ∗ + ∂µφieφ∗ − i2e2Aµφ∗φ+−i2e2Aµφ

∗φ

= ie(φ∗Dµφ− (Dµφ∗)φ) (33)

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since only |Dµφ|2 can be functional differentiated with respect to Aµ. For thesecond term only −1

4FµνF

µν depends on ∂νAµ:

∂ν∂L

∂(∂νAµ)= −∂ν

∂

∂(∂νAµ)

1

4(∂µAν − ∂νAµ)(∂µAν − ∂νAµ)

= −∂ν8

4(−∂µAν + ∂νAµ) = 2∂νF

µν (34)

The two terms combined give:

ie(φ∗Dµφ−Dµφ∗φ)− 2∂νFµν = 0 (35)

The gauge field is constructed in such a way that it cancels out the derivative ofthe gradient term from our previous example:

A =1

e∇(nθ) =

1

e∇(n)θ +

1

en∇(θ) = 0 +

n

e

1

r

∂θ

∂θ=

n

re(36)

as r →∞. The total contribution to the kinetic energy is now given by:

|Dµφ|2 = |(∇µ − ieAµ)φ|2 (37)

Keeping only static contributions and using (23):

Drφ = (∇r − ieA)φ =in

rφ− in

rφ = 0; (38)

Dθφ =1

r(∇r − ieAθ)φ = 0 (39)

The kinetic term no longer gives an infinite contribution to the energy density. Inthis case it gives a finite contribution to the energy density [1]:

H = −L =1

4FµνF

µν + V (φ, φ∗) (40)

Thus the kinetic term goes to zero in the Abelian Higgs model. The field tensorF µν is written in terms of the four-vector potential [5]:

F µν =∂Aν

∂xµ− ∂Aµ

∂xν(41)

where A = (Vc, Ax, Ay, Az). The Maxwell equations, defined up to a gauge trans-

formation, are:

B = ∇×A; (42)

E = −∇Φ− ∂A

∂t(43)

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The static contribution of the Maxwell equations is:

−1

4FµνF

µν = −1

4(∂iAj − ∂jAi)(∂iAj − ∂jAi)−

1

4(∂0Ai − ∂iA0)(∂0Ai − ∂iA0)

= −1

4(∇×A)2 − (

∂A

∂t−∇φ)2 = −1

4(B2 + E2) (44)

So the total contribution to the static energy density is:

H =1

4(B2 + E2) + V (φ) (45)

At large r, the energy density goes to zero, since the electric and magnetic fieldcomponents do not contribute. This solution carries a magnetic flux. This can beseen when considering the flux through an area around the cosmic string: Φ =∫

B · da. According to Stokes’ theorem [5] and using (36):

Φ =∫

B · da =∫∇×A · da =

∮A · dl =

∮ −nrerdθ =

2πn

e(46)

Thus the magnetic flux caused by the string is quantized [6]. In Appendix A isshown how the Higgs-model is related to the Abelian Higgs model.

2.3 Cosmic strings in the universe (Kibble mechanism)

The Kibble mechanism states that when a topological defect can form at a phasetransition, it will form. This statement is based on the argument that differentregions in space cannot communicate with each other. In the early universe causaleffects propagated with the speed of light. Different regions separated by morethan a distance d = ct do not know anything about each other. As stated before,c = 1 according to the conventions used throughout this thesis. This distance isalso called the causal horizon [6]. This can be written as:

ξ ≤ t (47)

where ξ is the correlation length, defined by the distance of possible interactionand t is the age of the universe. At some point there was a phase transition,that’s when different regions would pick different phases that leads to the cosmicstring. The strings that were formed grew as the universe expanded, giving rise toinfinite long strings [18] and were able to form a string network. During furtherdevelopment of the universe, the strings were able to interact and form loops. Moreabout string networks and string interaction will be discussed in chapter 5.

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2.4 Monopoles, Textures and Domain walls

Cosmic strings are two-dimensional topological defects. There can also appeartopological defects of other dimensions, causing monopoles and textures to arise.The type of defect that is produced depends on the symmetry that has been broken[8]. Although other topological defects like domain walls, monopoles and texturesare not the main focus of this thesis, I will give a short introduction. Where n arethe dimensions orthogonal to the defect:

• n = 1 Domain walls: different regions in space in one of the two phases areseparated by a domain wall. The universe is divided in discrete cells. Thishappens when a discrete symmetry is broken.

• n = 2 Strings: arise when the symmetry of a circle is broken, as the La-grangian of the beginning of this chapter.

• n = 3 Monopoles: are point-like defects, with a specific magnetic charge,either a north or a south pole. These types of defects arise when sphericalsymmetry has been broken.

• n = 4 Textures: are formed when more complex symmetries are broken.These types of defects are unstable.

When a discrete symmetry is broken at a phase transition, there is a possibilitythat two-dimensional domain walls appear. The domain walls will divide the uni-verse into different areas. An remarkable property of a domain wall is that thegravitational field is repulsive rather than attractive [6]. A universe that consistsof domain walls will look something like this:

Figure 2: Domain walls [9].

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Another topological defect is a zero-dimensional or point-like object called a monopole.This defect occurs when a spherical symmetry is broken. They are predicted bythe grand unified theories (GUTs) and are supposed to be super massive and carrymagnetic charge. They remain one of the problems of standard cosmology, as nonehave been observed. Cosmological Inflation resolves this problem by exponentialreduction in density due to the exponential expansion of the universe.

Figure 3: Magnetic monopole [9].

Textures will form when more complicated symmetries are broken. They are un-stable and able to unwind. Their configuration in one and two dimensions willlook something like this:

Figure 4: Texture in one dimension (a) and two dimensions (b) [9].

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3 A little bit of General Relativity

In this section I will introduce some basic details of General Relativity. The theoryof General Relativity, defined by Einstein in 1916, is a generalization of the theoryof Special Relativity, also introduced by Einstein in 1905. General Relativitydescribes a relativistic theory of gravity. The concept of a metric, the use ofgeodesics and applying the Einstein equations to the Friedmann-Robertson-Walkermetric are the main topics we will encounter hereafter. For more backgroundinformation on these subjects I refer to Spacetime and Geometry by Carroll [10],[11] and Modern Cosmology by Dodelson [12].

3.1 The metric

The metric turns coordinate distance into proper time. This proper time is in-variant: it does not depend on the observer. The metric includes the effects ofgravity so that particles can move freely in a curved space-time instead of regard-ing gravity as an external force. We can say that the curvature of space-time is amanifestation of gravity. In the next paragraph we will discover that the so-calledEinstein equations are the outcome of how space-time curvature responds to thepresence of matter and energy (and thus gravity). The line element, also calledthe proper time, is given by:

ds2 =∑µν

gµνdxµdxν (48)

The line element describes the infinitesimal length of the path of a particle movingthrough space-time. Special relativity is described by Minkowski space-time withthe metric: gµν = ηµν , with:

ηµν =

−1 0 0 00 1 0 00 0 1 00 0 0 1

In Cosmology, the metric is not of Minkowski type to allow for the expansion of theuniverse. There are several scenarios concerning the expansion of the universe: thespatial geometry can be closed, flat of open. These different types of geometries canbe visualized by considering two freely moving particles travelling parallel throughspace-time. When the universe is closed, the particles will gradually move towardseach other. When the universe is flat the particles will keep travelling parallel toeach other. The geometry of an open universe causes the particles to move furtheraway from each other. We will assume that the space-time geometry is flat, in

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line with observations. The metric in an expanding, flat universe (Friedmann-Robertson-Walker (FRW) metric) reads:

gµν =

−1 0 0 00 a2(t) 0 00 0 a2(t) 00 0 0 a2(t)

The scale factor a(t) is the co-moving distance between coordinates [12]: the dis-tance between the coordinates is proportional to the scale factor. At early times,during the beginning of the universe, the distance scaled according to:

a(t) ∝√t (49)

where t is the cosmic time.

3.2 The geodesic

The geodesic is the path followed by any particle in the absence of any forces. Toexpress this in equations, we must generalize Newton’s second law with no forces:F = d2x

dt2= 0 in the case of curved space-time. First I shall introduce the Principle

of Equivalence. This principle states that ”In small enough regions of space-time,the laws of physics reduce to those of special relativity; it is impossible to detectthe existence of a gravitational field by means of local experiments” [10]. Nowconsider a free-falling particle. According to the Einstein Equivalence Principle(EEP) there is no external force exerted on the particle, reducing Newton’s secondlaw to:

d2ξα(τ)

dτ 2= 0 (50)

The solution of this equation is given by: ξα(τ) = aατ + bβ, a straight line inspace-time. Now consider a different coordinate system: ξµ : ξα = ξα(xµ). Thesecoordinates could still be Cartesian, but can also rotate, be curved or acceleratedwith regard to our original coordinate system. The equation above becomes:

d

dτ(∂ξα

∂xµdxµ

dτ) =

∂ξα

∂xµd2xµ

dτ 2+

∂2ξα

∂xµ∂xνdxµ

dτ

dxν

dτ(51)

Multiplying both sides by ∂xλ

∂ξαgives:

δλµd2xµ

dτ 2+∂xλ

∂ξα∂2ξα

∂xµ∂xνdxµ

dτ

dxν

dτ(52)

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introducing the Christoffel symbol, defined as: Γλµν = ∂xλ

∂ξα∂2ξα

∂xµ∂xν, gives the equation

of motion:d2xλ(τ)

dτ 2+ Γλµν

dxµ(τ)

dτ

dxν(τ)

dτ= 0 (53)

This equation of motion can be used in a curved, accelerated or rotating coordinatesystem. In a flat geometry, non-accelerating or non-rotating system, the Christoffelsymbol is equal to zero and the equation above is reduced to (50).

3.3 Einstein equations

The Einstein equation relates components of the Einstein tensor to the energy-momentum tensor. It ”governs” how the metric responds to energy and momen-tum. This field equation must be postulated, but can be derived from some plau-sible arguments. We would like to find a relativistic generalization of the Poissonequation of the Newtonian potential:

∇2Φ = 4πGρ (54)

Where ∇2 is the Laplacian and ρ the mass density. To transform this equation intoa relativistic form, let’s look at the current characteristics first. On the left-handside a Laplacian is acting on the gravitational potential, on the right-hand sidethere is the mass distribution. The generalization of the mass distribution is theenergy-momentum tensor Tµν . The gravitational potential should be replaced bya metric tensor. Unfortunately, it’s not just the metric tensor. Instead we needto construct the appropriate tensor from the metric and the derivatives of themetric: the Riemann tensor. The Ricci tensor in terms of the Christoffel symbolis described as:

Rµν = Γαµν,α − Γαµα,ν + ΓαβαΓβµν − ΓαβνΓβµα (55)

where the commas are the derivatives with respect to x. The Christoffel symbolin terms of the metric is described as:

Γµαβ =gµν

2[∂gαν∂xβ

+∂gβν∂xα

− ∂gαβ∂xν

] (56)

where gµν is the inverse of gµν . The Ricci tensor contains the curvature of space-time. A first guess could be to generalize the Newtonian potential by plugging inthe Ricci-tensor. If we want to preserve conservation of energy ∇µTµν = 0, thefield equations would imply:

∇µRµν = 0 (57)

This is equation is not true. We do know a symmetric tensor, constructed from theRicci tensor, which is automatically conserved: the Einstein Tensor. The Einsteintensor is automatically conserved due to the Bianchi identity:

Rλµκν;η +Rλµηκ;ν +Rλµνη;κ = 0 (58)

22

we have Rµν = gλκRλµκν , Rµκ = Rλµλκ and Rλµκν = −Rλµνκ. Multiplying (58)

first by gλκ and secondly with gµν gives the desired result:

gµν(gλκRλµκν;η + gλκRλµηκ;ν + gλκRλµνη;κ) = 0 (59)

gµνRµν;η − gµνRµη;ν + gµνRκµνη;κ = 0 (60)

R;η −Rνη;ν −Rκ

η;κ = 0 (61)

This is known in a more well known form as [13]:

(Rµν − 1

2gµνR);µ = 0 (62)

Our field equation is now of the form:

Gµν = Rµν −1

2Rgµν ∝ 4πGTµν (63)

where G corresponds to Newton’s gravity constant. The Einstein equation is equalto:

Gµν = Rµν −1

2gµνR = 8πGTµν (64)

where Gµν is the Einstein tensor, Rµν is the Ricci Tensor, and R is the Ricciscalar: the contraction of the Ricci tensor (R ≡ gµνRµν), G is Newton’s constantand Tµν the energy-momentum tensor. Einstein’s field equation describes howthe metric responds to energy and momentum. The field equation is postulated.The energy-momentum tensor contains the energy and momentum of matter; theEinstein equation relates energy to curvature.

3.4 Einstein gravity of a string

In this section the Einstein equations of a cosmic string will be derived. A stringin Minkowski space lying along the z-axis (fig. 5) is invariant under certain trans-lations and rotations: time translations, spatial translations in the z-direction,rotations around the z-axis and Lorentz boost in the z-direction.

The reasonable assumption is made that the space-time of a gravitating stringhas the same symmetries. The metric in terms of cylindrical coordinates can nowbe written as [6]:

ds2 = dt2 − dr2 − dz2 − C2(r)dθ2 (65)

in matrix form:

ds2 = gµνdxµdxν =

1 0 0 00 −1 0 00 0 −1 00 0 0 −C2(r)

dxµdxν

23

- x

6z

����

����

��1 y

Figure 5: A cosmic string along the z-axis

Only the gθθ = C(r)2 depends on r, and: θ ⊂ [0, 2π]. When C2 = r2 the metricdescribes flat space-time, but when it is proportional to Ar2, a slice is taken outof space-time. We assume that the energy-momentum tensor is reduced to:

Tµν = µδ(x)δ(y)

−1 0 0 00 0 0 00 0 0 00 0 0 1

(66)

The energy of the energy-momentum tensor is aligned along the z-direction. Lateron we will more or less derive this expression from the effecive action S. There willbe only a contribution to the Einstein equations (64) of the Ttt and Tzz components,so they will reduce in this case to:

Rtt −1

2gttR = 8πGTtt (67)

Rzz −1

2gzzR = 8πGTzz (68)

The Einstein equations are given in terms of the Ricci tensor and the Ricci tensorcan be expressed in terms of the Christoffel symbol (55). As we can easily seefrom expression (56), the only contribution to the Christoffel symbol are comingfrom the θθ-components. The Christoffel symbol can be described in terms of themetric (56). This component is the only term depending on r. The other terms do

24

not depend on r, or on any other variable, and will give a zero contribution whendifferentiated. There are three terms contributing to the Christoffel symbol:

Γθθr =gθθ

2(∂gθθ∂r

) =+1

2C2

∂C2

∂r=

1

2C22CC ′ =

C ′

C; (69)

Γθrθ =gθθ

2(∂gθθ∂r

) =C ′

C; (70)

Γrθθ = −gtt

2(∂gθθ∂r

) = −1

22CC ′ = −C ′C (71)

where the primes denote differentiation with respect to r. There are only threecontributions to the Christoffel symbol that are unequal to zero. The two contri-butions to Rrr are given by:

Rrr = Γαrr,α − Γαrα,r + ΓαβαΓβrr − ΓαβrΓβrα = −Γθrθ,r − ΓθθrΓ

θrθ

= − ∂

∂r

C ′

C− C ′

C

C ′

C= −C

′′C − C ′C ′

C2− C ′2

C2= −C

′′C

C2= −C

′′

C(72)

Where again the first and third term do not contribute. The second contributionto the Ricci-tensor is given by:

Rθθ = Γαθθ,α − Γαθα,θ + ΓαβαΓβθθ − ΓαβθΓβθα = Γrθθ,r + ΓθrθΓ

rθθ − ΓθrθΓ

rθθ − ΓrθθΓ

θθr

=∂

∂r(−C ′C)− (−C ′C)

C ′

C= −C ′′C − C ′C ′ + C ′2 = −C ′′C (73)

The second term in the first line cancels, and the fourth term has two contributionsleading to the cancellation of the second and third term against each other in thesecond line. The Ricci scalar is given by:

R = gµνRµν = gttRtt + grrRrr + gzzRzz + gθθRθθ

= grrRrr + gθθRθθ (74)

At the second equality we used the fact that the components of the Ricci tensorthat do not depend on r or θ are zero. The Ricci scalar is equal to:

R = grrRrr + gθθRθθ =C ′′

C+−1

C2(−C ′′C) =

C ′′

C+C ′′

C= 2

C ′′

C(75)

We already know that the Rtt and Rzz components are equal to zero because theydo not contain any r-dependence. The Einstein equations simple reduce to:

Rtt −1

2gttR = −1

2(2C ′′

C) = −8πGµ; (76)

Rzz −1

2gzzR = −1

2(−1)(2

C ′′

C) = 8πGµ (77)

25

We can conclude that the Einstein equations, with the use of (66), have beenreduced to [6]:

C ′′

C= 8πGµ (78)

In the final paragraph of the next chapter this result will be derived in a differentmanner. The result will be used to derive the wedge that has been removed fromspace-time by the presence of the cosmic string.

26

4 Cosmic String as a Source

During this chapter and the next we will go into more detail about possible ob-servations or detection of cosmic strings. There are several ways to perceive acosmic string. The final paragraph of this chapter will give more insight in grav-itational lensing by a cosmic string. In order to fully comprehend the theoremnecessary to derive the lensing angle, the effective cosmic string action and theenergy-momentum tensor of a cosmic string will be derived.

4.1 The action formalism of gravity

Following Lecture Notes on General Relativity by Carrol [11] we will derive theEinstein equation from the action principle. Re-deriving the Einstein equationsfrom the effective action will show that the Lagrangian L used in chapter 2 is asuitable choice. In the case of the Lagrangian formulation the Einstein equationscan be derived from:

SH =∫dnxLH (79)

The Lagrangian density is written as: L =√−gR, where R = gµνRµν is the Ricci

scalar and g = det(gµν). The action reads:

SH =∫dnx√−ggµνRµν (80)

This is the simplest choice for the Lagrangian. This is the only non-trivial scalarconstructed in terms of the metric, which has no higher than second order deriva-tives. As we have seen before (10) the equations of motion come from varying theaction with respect to the metric:

δS =∫dnx

[√−ggµνδRµν +

√−gRµνδg

µν +Rδ√−g]

(81)

= (δS)1 + (δS)2 + (δS)3 (82)

The second term is already of a form that can be used directly. The first andthird term have to be rewritten. Starting with (δS)1, for arbitrary variations ofthe Ricci tensor, which can be expressed in terms of the Christoffel symbol (55):

Γρνµ → Γρνµ + δΓρνµ (83)

When taking the covariant derivative:

∇λ(δΓρνµ) = ∂λ(δΓ

ρνµ) + ΓρλσδΓ

σνµ − ΓσλνδΓ

ρσµ − ΓσλµδΓ

ρνσ (84)

∇ν(δΓρλµ) = ∂ν(δΓ

ρλµ) + ΓρνσδΓ

σλµ − ΓσνλδΓ

ρσµ − ΓσνµδΓ

ρλσ (85)

27

It can be shown that for (δS)1:

δRµν = ∂λ(δΓρνµ) + δΓρλσΓσνµ + ΓρλσδΓ

σνµ − ∂ν(δΓ

ρλµ)− δΓρνσΓσλµ − ΓρνσδΓ

σλµ

= ∇λ(δΓρνµ)−∇ν(δΓ

ρλµ) (86)

Because the Christoffel symbol is symmetric in the two lower indices, the thirdterm in (84) and (85) will disapear when subtracted. When some dummy-indicesare relabelled the first contribution can be written as:

(δS)1 =∫dnx√−ggµν [∇λ(δΓ

λνµ)−∇ν(δΓ

λλµ)]

=∫dnx√−g[gµν∇λ(δΓ

λνµ)− gµν∇ν(δΓ

λλµ)] (87)

=∫dnx√−g∇σ[gµσ(δΓλλµ)− gµν(δΓσµν)] (88)

The equivalance principle demands that: ∇σgµσ = 0, now (87) can be written

as (88). The result is an integral with respect to the natural volume element ofthe covariant divergence of a vector. According to Stokes’ theorem this is equalto the boundary contribution at infinity. We can set this to zero by making thevariation vanish at infinity. Therefore the first term does not contribute to thetotal variation. Looking at the (δS)3, the following identity can be applied for anymatrix M:

Tr(lnM) = ln(detM) (89)

The variation of this identity gives:

Tr(M−1δM) =1

detMδ(detM) (90)

In this case: M = gµν , inserting M = gµν in the equation above, then detM = g−1

and:

δ(g−1) =1

ggµνδg

µν (91)

Now with the use of the equation above:

δ(√−g) = δ[(−g−1)−1/2] = −1

2(−g−1)−3/2δ(−g−1)

= −1

2(−g−1)−3/2 1

−ggµνδg

µν = −1

2

√−ggµνδgµν (92)

And so (δS)3 = −∫dnx√−g 1

2Rgµνδg

µν . Since the first term (δS)1 does not con-tribute, we are left with the second and third term:

δS =∫dnx√−g[Rµν −

1

2Rgµν ]δg

µν (93)

28

Since this must be true for arbitrary variations:

1√−g

δS

δgµν= Rµν −

1

2Rgµν = 0 (94)

these are the vacuum field equations. In the presence of matter, the total actionshould look like:

S =1

8πGSH + SM (95)

If we apply the variational principle again this will lead to:

1√−g

δS

δgµν=

1

8πG(Rµν −

1

2Rgµν) +

1√−g

δSmδgµν

= 0 (96)

when we look at the Einstein equations, we can identify:

Tµν = − 1√−g

δSMδgµν

(97)

So by choosing the simplest form of the Lagrangian we have re-derived the Ein-stein equations. We know that the energy momentum tensor is symmetric andconserved. We can check what this formulation gives for varying the Lagrangian(4) and S =

∫ √−gL with respect to the inverse metric:

δSMδgµν

=∫dnx[√−gδgµν(∂µφ∂νφ∗) + δ

√−g(gµν∂µφ∂νφ

∗ − V (φ, φ∗))] (98)

As we have seen before: δ√−g = −1

2

√−ggµνδgµν . Inserting this equality into (98)

gives:

δSMδgµν

=∫ √−gδgµν [∂µφ∂νφ∗ + (−1

2gµν)(g

ρσ∂ρφ∂σφ∗ − V (φ, φ∗))] (99)

= −√−gTµν (100)

So the energy-momentum tensor equals:

Tµν = −∂µφ∂νφ∗ +1

2gµν [g

ρσ∂ρφ∂σφ∗ − V (φ, φ∗)] (101)

This result coincides with what we expected.

4.2 Effective Cosmic String Action

Consider a straight cosmic string in the z-direction. On physical grounds thestring has a tension and its thickness can be neglected. The stress energy tensor

29

can therefore be approximated by (66). We now construct an effective action ofa cosmic string, which will give us the expected energy-momentum tensor. TheNambu-action of a single particle in one dimension will be used to derive theaction of a string in two dimensions, leading to the energy-momentum tensorin flat space-time of a cosmic string. The ”physics” of a single particle movingthrough space-time can be described by the simple action [6], [14]:

Sparticle =∫ds (102)

This will look like:

r -

�

x(t)

Figure 6: A particle moving through space-time.

where ds2 is defined as usual by: ds2 = c2dτ 2 = c2dt2 − dx2. So (102) can bewritten as:

Sparticle =∫ √

1− gµν xµxν (103)

The equations of motions are derived by taking the variation of the action withrespect to xµ: δS

δxµ= 0, as will be done in chapter 5. The simplest form of the action

of a string is slightly more complicated. Since a string moves in two dimensions, itcreates a world sheet in space-time (fig. 7). In this case our action will look like:

Sstring = −µ∫d2ζ√−γ (104)

where µ has the dimension of the tension of the cosmic string. The action hasnow become a dimensionless quantity. In this case there is a different metric γ,depending on the choice of or coordinate system. The function xµ maps everycoordinate from flat space-time to curved space-time:

xµ → xµ(ζa), a = 0, 1 (105)

30

-

-

-

-

-

��7ζa

- ζb

Figure 7: A moving string creating a world sheet in space-time.

q

xµ(ζa)

Figure 8: Going to curved space-time.

31

Going from flat to curved space-time is visualized in fig. 8. The new metricdefined in terms of the old metric is now given by:

γab = gµν∂xµ

∂ζa∂xν

∂ζb(106)

This is called the induced metric. We have now found the two dimensional worldsheet metric. The string energy-momentum tensor can be found by varying theeffective action with respect to the metric gµν :

δS

δgµν= −µ

∫d2ζδ(

√−γ)xµ,ax

ν,b (107)

As we have seen before in (92): δ(√−γ) = −1

2

√−γγabδγab and δgµν = xµ,ax

ν,bδγ

ab.In order to derive the energy-momentum tensor the variation of the action shouldbe set to −

√−gTµν in agreements with (97):

δS

δgµν= µ

∫d2ζ

1

2

√−γγab

δγab

δgµνxµ,ax

ν,b = −

√−gTµνxµ,axν,b (108)

This can be rewritten as:

T µν√−g = µ

∫d2ζ√−γγabxµ,axν,bδ4(xσ − xσ(ζa)) (109)

For a straight string in the z-direction in flat space-time directed in the z-direction,the equation above reduces to:

T µν = µδ(x)δ(y)

1 0 0 00 0 0 00 0 0 00 0 0 1

(110)

because all derivatives are equal to one in flat space-time, and is, as was antici-pated, equal to (66). Only the delta-function for the x- and y-components are left,due to the fact that we integrated over the t- and z-components. This is exactlythe energy-momentum tensor we have been looking for.

4.3 Gravitational Lensing

The Gauss-Bonnet theorem connects the geometry to the topology of a surface[15]. This theory is very useful in our case: knowing the (Gaussian) curvature ofthe surface and the Euler Characteristic we can derive the deficit angle. Suppose Σ

32

is a two-dimensional Riemannian manifold, with boundary ∂Σ, then the theoremis given by [1]: ∫

ΣK · dS = 2πχ(M)−

∫∂Σkg · dl (111)

where K is the Gaussian curvature, in our case given by the Ricci tensor as K =Rx

x. The element dS is the surface area, and dl the line element along the boundaryof Σ, given by: dl = rdθ. The geodesic curvature is denoted by kg, and is given by1r

for a circle. The Euler characteristic (EC) is given by χ(M). A complete proofof this theorem can be found in [15]. The Euler characteristic of a surface is givenby:

χ(M) = F − E + V (112)

where F are the faces, E are the edges and V are the vertices (corners) of atirangulation of the surface. The Euler characteristics of a disc in two dimensionshas F = 1, E = 1, V = 4, giving χ(M) = 1. Inserting the Euler characteristicsand the geodesic curvature: ∫

RxxdS = 2π −

∫∂Σ

1

rrdθ (113)

The second term on the right hand side will give:∫∂Σ

1rrdθ = 2πC

r, where A is a

constant in the function C(r) = Ar. An alternative formulation of the Einsteinequation emerges when the Einstein equations are contracted with gµν :

gµνRµν − gµνgµν1

2R = gµν8πGTµν (114)

R− 2R = 8πGT ρρ (115)

giving R = −8πGT ρρ. Inserting this into the original Einstein equation (64), it isfound that:

Rµν = 8πGTµν +1

2gµνR = 8πG(Tµν −

1

2gµνT

ρρ) (116)

where T ρρ equals the trace of the energy-momentum tensor. For our two-dimensionalsurface we only need to know the x-components of the Riemann tensor. The x-components of the energy-momentum tensor does not contribute, since the ten-sor is aligned along the z-axis. The trace of the energy-momentum tensor gives:T = 2µδ(x)δ(y)µ. So R x

x is given by:

R xx = 8πG(T x

x −1

2T ) = −8πGδ(x)δ(y)µ (117)

The Gauss-Bonnet equation (113) becomes:∫8πGµδ(x)δ(y)µ dxdy = 8πGµ = 2π − 2πA (118)

33

it follows that: A = 1− 4Gµ. The cosmic string metric becomes:

ds2 = −dt2 + dr2 + dz2 + (1− 4Gµ)r2dθ2 (119)

An angular wedge of δ = 8πGµ is removed from flat space-time (fig. 9). Whenlooking at (fig. 10), it can be seen that this structure will cause gravitationallensing. When the angular wedge δ is removed it will appear that light (the

Figure 9: Gravitational field of a cosmic string [16].

dashed line) travelling from a distant light source, crossing a cosmic string, to anobserver, it will appear to the observer that there are two light sources in the sky.This can be visualized as a circle from which a piece has been removed and thesides have been glued together to create a cone. To calculate the lensing angle,the distance of b needs to be calculated. According to geometry, we know:

tan δ′ =b

y= tan

1

28πGµ (120)

where δ′ = 12δ. It follows that b = y tan 1

28πGµ = y4πGµ for Gµ � 1. The angle

α is given by:

tanα =y4πGµ

x+ y(121)

34

(( (( (( (( (( (( (( (( (( (( (( (( (( (( (( (( ((

hh hh hh hh hh hh hh hh hh hh hh hh hh hh hh hh hh

����

����

����

���

PPPPPPPPPPPPPPP

Source

String

x y

α δ

b

Figure 10: The angular wedge removed from space-time by a cosmic string.

����

����

����

���

PPPPPPPPPPPPPPP

δ Identify

I

∗

∗∗

∗

∗

∗∗

∗∗

Figure 11: Accretion disk in the cosmic string wake.

This gives for the lensing angle 2α:

2α = 8πGµ(1 +x

y)−1 (122)

Looking at fig. 10, it is also understood why a cosmic string can cause densityperturbations that eventually cause galaxy formation. A string moving througha region of dust will leave an over-dense accretion region behind it [1]. Considera string moving through a dusty region (see fig. 11). Because there is anarea cut out of space-time caused by the string the dust particles will eventuallycollide, resulting in an accretion disk in the cosmic string wake. This could give acontribution to primordial density fluctuations [6].

35

5 Cosmic String Dynamics

In this chapter we will first solve the wave equation of a cosmic string, and usea rewritten constraint to the wave equation to derive a cusp. A cusp is a certainkind of singular string configuration that can form when strings form a loop or canbe caused by the gravity of a black hole. Finally some comments will be made onthe properties of a cosmic string network in the last section.

5.1 Solution of Cosmic String Wave Equation

Having just derived the effective action of a string (104) we can now derive theequations of motion of the string. A two-dimensional surface is represented by:xµ = xµ(ζa), and the line element is defined according to (106) as [6]:

ds2 = gµν∂xµ

∂ζa∂xν

∂ζbdζadζb = γabdζ

adζb (123)

Variation of the effective cosmic string action with respect to xµ gives:

δS

δxµ(ζa)= 0 (124)

For arbitrary coordinates, according to (53) the equations of motion become:

∂2xµ

∂ζ2+ Γµνσγ

ab∂xnu

∂ζa∂xσ

∂ζb= 0 (125)

In Minkowski space (or flat spacetime) gµν = ηµν and we may set Γµνσ = 0, sincethere is no curvature in flat space time. The equations of motions of a string takethe form

∂2xµ

∂ζ2=

1√γ∂a(√−γγab∂x

µ

∂ζb) = 0 (126)

this is equal to

∂a(√−γγab∂x

µ

∂ζb) = 0 (127)

The parameterization of the world-sheet can be chosen freely, for example in thelight-cone gauge:

ζ0 = t, ζ1 = z − t (128)

A convenient choice of the gauge of a string in flat space-time is [6]:

γ01 = 0; (129)

γ00 + γ11 = 0 (130)

36

with this information we can rewrite the metric as:

γ01 = ηµν∂xµ

∂ζ0

∂xν

∂ζ1= x · x′ = 0; (131)

γ00 + γ11 = ηµν∂xµ

∂ζ0

∂xν

∂ζ0+ ηµν

∂xµ

∂ζ1

∂xµ

∂ζ1= x2 + x′2 = 0 (132)

The dots and primes are derivatives with respect to ζ0 = t and ζ1 = z − t respec-tively. The metric has become conformally flat:

γab =√−γηab (133)

γab =1√−γ

ηab (134)

The equation of motion of the string becomes;

∂α(√−γγab∂x

µ

∂ζb) =√−γ 1√

−γηab∂a(

∂xµ

∂ζb) = xµ − x′′µ = 0 (135)

When ζ0 is set equal to t, ζ0 = x0 = t, and ζ1 = ζ, the string trajectory can bewritten as a as a three-vector

x(ζ, t) (136)

The string equation of motion becomes the two-dimensional wave equation:

x− x′′ = 0 (137)

where x depends on a space-coordinate ζ and a time-component t. The solutionof this two dimensional wave equation is known as d’Alembert’s solution [17] andis derived in appendix C. A general solution of the equation of motion is given by:

x(ζ, t) =1

2[a(ζ − t) + b(ζ + t)] (138)

representing a left and a right moving wave. The constraints (131) and (132) canbe summarized as:

x · x′ = 0; (139)

x2 + x′2 = 1; (140)

x− x′′ = 0 (141)

With (139) and (140) and the solution of the wave-equation (138), we can summa-rize the constraint in a stronger statement. We transform the variables on whicha and b are depending back to:

u = ζ − t, v = ζ + t (142)

37

Now we are going to reformulate the differentiation of a and b with respect to ζand t. The reason for this reformulation will become clear during the derivationof the constraint. The differentiation with respect to ζ and t in terms of the newcoordinates reads:

a′ =∂a

∂ζ=∂a

∂u

∂u

∂ζ=∂a

∂u; (143)

a =∂a

∂t=∂a

∂u

∂u

∂t= −∂a

∂u(144)

Where we have used (209) and (212). We can conclude that:

a′ = −a (145)

Along the same line of reasoning, we can take both derivatives of b:

b′ =∂b

∂ζ=∂b

∂v

∂v

∂ζ=∂b

∂v; (146)

b =∂b

∂t=∂b

∂v

∂v

∂t=∂b

∂v(147)

Now we can conclude that:b′ = b (148)

With this knowledge we can write the first constraint as:

x · x′ =1

4[a + b][a′ + b′] =

1

4[−a′ + b′][a′ + b′]

=1

4[−a′

2 − a′b′ + a′b′ + b′2] = 0 (149)

From this equation it follows that: −a′2 + b′2 = 0, or:

a′2 = b′2 (150)

When we insert the solution (221) in the second constraint we find together with(150) that:

x2 + x′2 =1

4[a + b]2 +

1

4[a′ + b′]2 =

1

4[−a′ + b′]2 +

1

4[a′ + b′]2

=1

4[a′2 + b′2 − 2a′b′ + a′2 + 2a′b′ + b′2]

=1

4[2a′2 + 2b′2] =

1

4[4a′2] = 1 (151)

We can conclude that the constraints (139) and (140) can be reformulated as:

a′2 = 1 = b′2 (152)

38

The second constraint can be rewritten as:

x2 = 1− x′2 = 1− 1

4[a′2 + 2a′b′ + b′2]

= 1− 1

4a′2 − 1

2a′b′ − 1

4b′2 (153)

When we use a′2 = b′2 = 1, (153) becomes:

x2 = 1− 1

4− 1

2a′b′ − 1

4=

1

2− 1

2a′b′ =

1

4− 1

2a′b′ +

1

4=

1

4a′2 − 1

2a′b′ +

1

4b′2

=1

4[a′2 − 2a′b′ + b′2] =

1

4[a′ − b′]2 (154)

This result (x2 = 14[a′ − b′]2) will be a useful identity in the next paragraph.

Summarizing the results above, the wave equation of a moving string is equal to:

x− x′′ = 0 (155)

The solution of the wave equation of an oscillating string is equal to:

x(ζ, t) =1

2[a(ζ − t) + b(ζ + t)] (156)

representing a right and a left moving wave. From the two-dimensional metric twoconstraints can be derived:

x · x′ = 0 (157)

x2 + x′2 = 1 (158)

and be rewritten into: a′2 = b′2 = 1.

5.2 Oscillating loops

The motion of a closed loop is described by (156) and (152) with ζ varying in therange [6]:

0 ≤ ζ ≤ L (159)

where L = ε/µ is the invariant length of the loop, µ the tension of the string andε the total energy, since tension is defined as an energy density (energy per unitof length). For a closed loop, we want the solution to be periodic:

x(ζ + L, t) = x(ζ, t) (160)

In terms of a and b this is written as:

b(ζ + t+ L)− b(ζ + t) = −a(ζ − t+ L) + a(ζ − t) = 0 (161)

39

It is also demanded that the function a and b are to be periodic:

a(ζ + L) = a(ζ) (162)

b(ζ + L) = b(ζ) (163)

The motion of the loop must also be periodic in time:

x(ζ +L

2, t+

L

2) =

1

2[a(ζ +

L

2− t− l

2) + b(ζ +

L

2+ t+

L

2)]

=1

2[a(ζ − t) + b(ζ + t+ L)] = x(ζ, t) (164)

where we used that the function b is periodic in L. The period is L2. Because the

timescale of the oscillation is comparable to the loop length L, the loop motionmust be relativistic. An interesting property of these solutions is that the stringcan reach the velocity of light during each period. To describe the behaviour ofthe string near luminal motion (i.e. moving with the speed of light), we choosethe space-time coordinates so that the luminal point is at ζ = t = 0 and x = 0.For a continuously differentiable function we can apply a Taylor expansion:

f(x) = f(a) +f ′(a)

1!(x− a) +

f ′′(a)

2!(x− a)2 + ... (165)

Expanding the functions a and b near ζ = 0:

a(ζ) = a′

0ζ +1

2!a

′′

0ζ2 +

1

3!a

′′′

0 ζ3 + ...; (166)

b(ζ) = b′

0ζ +1

2!b

′′

0ζ2 +

1

3!b

′′′

0 ζ3 + ... (167)

Where the abbreviaton a′(ζ = 0) = a

′0 has been used. When we use the constraint

a′2 = b

′2 = 1 we can write

a′2 = (a

′

0 + a′′

0ζ +1

2a

′′′

0 ζ2 + ...)2; (168)

b′2 = (b′

0 + b′′

0ζ +1

2b

′′′

0 ζ2 + ...)2 (169)

these can only be equal if:

a′

0 = −b′

0; (170)

|a′

0| = |b′

0| = 1; (171)

a′0 · a′′0 = b′0 · b′′0 = 0; (172)

a′′20 + a′0 · a′′′0 = b′′20 + b′0 · b′′′0 = 0, ... (173)

40

From the equation (154) above we find that:

x2(ζ, t) =1

4[a

′ − b′]2 =

1

4[(a′0 + a′′0ζ +

1

2!a′′′0 ζ

2 + ...)− (b′0 + b′′0ζ +1

2b′′′0 ζ

2 + ...)]2

=1

4(a′20 − a′0b

′0 − b′0a

′0 + b′20 )− 1

4(a′′0ζ + b′′0ζ)2 + ... (174)

= 1− 1

4[(ζ − t)a′′0 + (ζ + t)b′′0]2 + ... (175)

At the second line (170) - (173) have been used. There exists a point a long thiscurve for which the speed of the string is equal to the speed of light: |x| = 1.

The shape of the string at t = 0, with the use of (166), (167) and (170), isgiven by:

x(ζ, 0) =1

2[a + b]

=1

2[(a

′

0 +1

2a

′′

0ζ2 +

1

3!a

′′′

0 ζ3 + ...) + (b

′

0ζ +1

2!b

′′

0ζ2 +

1

3!b

′′′

0 ζ3 + ...)]

=1

2(a

′

0 + b′

0)ζ +1

2(

1

2!a

′′

0 +1

2!b

′′

0)ζ2 +1

2(

1

3!a

′′′

0 +1

3!b

′′′

0 )ζ3 + ...

=1

4(a′′0 + b′′0)ζ2 +

1

12(a′′′0 + b′′′0 )ζ3 + ... (176)

For (a′′0 + b′′0) 6= 0 the string develops a ’cusp’, i.e. a certain kind of singularity,at the origin: ζ = t = 0. The shape of the cusp at the origin is visualised in fig. 12.

The velocity at the cusp approaches the speed of light, as we have seen in (175).With a tension of the string not so far below Planck scale it emits an intense beamof gravitational waves in the direction of its motion [6].

When the string reaches the speed of light, the cusps can emit gravitationalwaves. There are two specific case when gravitational waves are emitted by acosmic string, both cases could be detected on earth, but are relatively difficult todiscover:

1. Cusps: The cusps beam the radiation in a specific direction, perpendicularto the formed cusp. These gravitational beams have a spectrum that can bedistinguished from gravitational waves from other sources. But due to theirspecificly aimed direction and the number of sources, the chances are slimthat they are aimed at the earth, and therefore difficult to detect.

41

-

6

Figure 12: The shape of a string near a cusp.

2. Kinks: are formed when contracting loops form a cusp. Just like the ’open’string it will emit a specific spectrum of gravitational radiation, but areaimed at various directions. Loops are created during intersection of stringsand will be discussed in more detail in the next section. The amplitude ofthe emitted waves is directly connected to the tension of the string. Due tothe directionless emission and decrease of the tension, the amplitude of thegravitation waves will decrease fast, making them also difficult to detect.

5.3 String interaction and network evolution

A string network is web of strings that during the expansion of our universe havebeen intersecting with each other. It turns out that the behaviour of a stringnetwork is more involved than one would naively expect. First we will describethe most naive expectation. The Friedmann equations can be used to determinethe scaling of the energy density of the cosmic string network. The Friedmannequations assume that the universe is uniform and isotropic. The metric is of theform:

gµν =

(−1 00 a2(t)

)(177)

When we consider an ideal fluid, this can be described by [12]:

T µν =

−ρ 0 0 00 P 0 00 0 P 00 0 0 P

(178)

42

where ρ is the density of the fluid and P is the pressure of the fluid. The conser-vation of the energy-momentum tensor in an expanding universe is equal to thecovariant derivative of the energy-momentum tensor being zero:

T µν;µ ≡∂T µν∂xµ

+ ΓµαµTαν − ΓανµT

µα = 0 (179)

The conversation law for the expanding universe is equal to:

∂ρ

∂t+a

a[3ρ+ 3P ] = 0 (180)

is known as the continuity equation. The equation of state is given by:

ρ = ωP (181)

When we insert the equation of state into (180)

∂ρ

∂t+a

a[3ρ+ 3ωρ] = 0 (182)

∂ρ

∂t+a

a3ρ[1 + ω] = 0 (183)

da

dt

dρ

da+a

a3ρ[1 + ω] = 0 (184)

this can be written as:a

a[dρ

d ln a+ 3ρ(1 + ω)] = 0 (185)

and

dρ

d ln a= −3(1 + ω)ρ (186)

d ln ρ

d ln a= −3(1 + ω) (187)

This results in:ln ρ = −3(1 + ω) ln a+ constant (188)

When we exponentiate both sides, we find the following solution:

ρ = ρ0a−3(1+ω) (189)

Inserting this equation into the second Friedmann equation we get:

(a

a)2 =

8πG

3ρ0a−3(1+ω) (190)

43

where we indicate: ρ0 = 8πG3ρ. To get a solution of a in terms of the variable t, we

try the following ansatz:a ∝ tα (191)

The scale factor is now equal to:

tα−1

tα

2

= t−2 = ρ0t−α3(1+ω) (192)

Then α equals 23(ω+1)

. And this results in the following solution of the equation ofstate:

a(t) = a0t2

3(ω+1) (193)

where a0 = 1ρ0

.The equation of state is different for every type of matter. The equation of statefor respectively radiation, matter and string equals:

ω =1

3, Pr =

1

3ρr (194)

ω = 0, Pm = 0ρm = 0 (195)

ω = −1

3, Ps = −1

3ρs, (196)

Where the equation of state for strings can be obtained from (110) and (178).When we plug these equations into (189), we find that the density for these typeof matter scale respectively as:

ρr ∝ a−3(1+1/3) =1

a4(t); (197)

ρm ∝ a−3(1+0) =1

a3(t); (198)

ρs ∝ a−3(1−1/3) =1

a2(t)(199)

The density of cosmic string networks scales as 1a2(t)

[18], [19].But as it turns out to be, matter and radiation give the most important contri-

bution, and therefore it is not possible that the strings scale as 1a2 . Cosmic strings

are not a large contributor to today’s spectrum, indicating that their contribu-tion is diminished by some events that can not be accounted for in the expressionabove. There are some events that have not been incorporated in this rather naivecalculation. Strings can for example intersect with each other or with themselves,thereby creating loops that gradually decay because the loops emit gravitationalwaves. So there are two loop formation mechanisms that eventually reduce thestring energy density [18]:

44

Figure 13: Scale-invariant evolution of a cosmic string network [9].

1. Collision of two strings: two strings intercommute with each other. Duringthis intersection a loop is created that gradually decays by the means ofgravitational radiation.

2. Self-intersection: a single string intersects with itself creating a loop1.

Besides those two loop formation mechamisms, there is also the phenomenom ofwiggly strings that has to be considered when looking at options that can reducethe string density. The density of wiggly strings, cosmic strings with kinks orcusps, grow more slowly compared to strings that are stretched [1]. In the firsttwo cases metioned above, the loops begin to oscillate and decay by emitting grav-itational waves.

Self-intersection and the collision of two strings is visualised in figure 14.Effectively, a cosmic string network will therefore not scale as found in the (naive)

Figure 14: Collision of two strings (a) and self-intersection (b) [9].

1When two strings intersect they can also entangle or simply cross each other without forminga loop, it is not always the case that loops are created during intersection.

45

calculation above, but will scale with a density that does not dominate over matteror radiation. Simulations have shown that effectively the density of strings scalesas

ρs ∝1

a4(t)(200)

during the radiation dominated area [1]. When the density is now multiplied bythe four-volume it will remain constant, thus scale invariant, as can be seen in fig.13. This means that the cosmic strings scale just like radiation and do not dodominate over them [1]. The density of the string decreases fast enough to be inagreement with present cosmological data.

Gravitational waves emitted by loops will have a relatively broad spectrum andare beamed into several directions, unlike gravitational waves emitted by cusps.Gravitational waves emitted by loops will therefore be more easy to find. Anotherreason for this is that cusps might only have been formed during the initial stages ofthe universe, while loops can be created throughout the lifespan of the universe. Itwill be more likely to detect gravitational radiation emitted by loops than by cusps.

An interesting remark that can be made of the properties of one-dimensional de-fects (strings) versus two-dimensional defects (domain walls) and zero-dimensionaldefects (monopoles), concerns their energy density. Between a network of domainwalls, the typical spacing will be of order t. The energy density in domain wallswill then scale as 1

t, and would dominate over the matter and radiation densities

[1]. This network is therefore excluded from the scenario. Point like defects cannotfind each other like strings do. The spacing between them scales as 1

a(t)and the

density as 1a(t)3

, just like other massive matter. Point-like defects would come todominate the radiation-dominated era and can also be excluded.

46

6 Conclusion

Cosmic strings are cosmological phenomena that came into being during the initialstages of the development of the universe. Three different types of phase transi-tions were mentioned that could cause the cosmic strings to arise: GUT transition,electroweak transition and the quark-hadron transition. Cosmic strings are a spe-cific type of topological defects that arises during phase transitions. Topologicaldefects can occur when the field symmetries are broken. This happened when theuniverse cools down below some critical temperature Tc and the field is forced tochoose a vacuum phase.

There are several ways to detect cosmic strings. When a cosmic string is formedduring one of the phase transitions, it creates a wedge in space-time. Light comingfrom a distant object that passes a cosmic string before it is observed on earth,will be deformed by the gravity exerted by the string. The lensing causes twoexact similar objects to appear in the sky. There have not yet been identified anyobjects appearing in the sky that could have been lensed by a cosmic string, butif they were found, they would probably look something like fig 15.

To calculate the angular wedge, some derivations were necessary. First of al alittle bit of General Relativity has been used to calculate the Einstein gravity ofa string. An approximation of the energy-momentum tensor combined with theRiemann tensor were necessary to apply the Gauss-Bonnet theorem. In this casean angular wedge of δ = 8πGµ is removed from the metric of flat space-time. Thiscan be imagined as a circle from which an angle δ is removed and the ends areglued together, creating a cone.

Another way of detecting cosmic strings is by the gravitational waves emittedby the cusps or kinks that originate on the strings. Cusps emit gravitational wavesin a very particular direction with a specific spectrum. Due to the the specificlyaimed directions of these waves they are difficult to detect on earh. Kinks behavelike cusps but are formed on contracting loops. The contraction of the loop causesthe amplitude of the emitted beams to decrease fast, making them also hard to find.

Multiple strings together form a sting network, that is web of strings that dur-ing the expansion of our universe have been intersecting with each other. TheFriedmann equations can be used to determine the scaling of the energy density ofthe cosmic string network. Naively this would scale as 1

a2(t), but that’s ruled out

because it would imply strings to dominate the energy budget. There are two loopformation mechanisms that eventually reduce the string energy density beyondthat of the initial approximation. The final scaling of the mass density of a string

47

behaves as ρs = 1a4(t)

, and therefore does not dominate over matter or radiation.

Even thought no detection of a of cosmic string has been reported, there weresome observations that were first believed to be explained by cosmic strings. Thefirst possible observation of a cosmic string was done by Sazhin. On January 12,2006 the Hubble Space telescope observed his double extragalactic object [20].They reported the observation of a lensing candidate called CSL-1 (Capodimonte-Sternberg Lens Candidate number one), see fig. 15. There are also three othercandidates, going by the not so startling names CSL-2 through CSL-4. The twoimages of CSL-1 separated by 2” look almost identical, both have the same red-shift and magnitudes. If the images were indeed of the same galaxy, they had tohave been created due to a cosmic lens, as we have just seen in paragraph 4.3 [19].

Figure 15: Image of the region surrounding CSL-1 [20].

Unfortunately, the high-resolution image of CSL-1 showed that the object is infact a pair of giant interacting elliptical galaxies [20]. Despite of the similaritiesin energy and light distribution and radial velocities, CSL-1 is not the lensing ofan cosmic string. Another argument supporting t that CSL-1 is not created by acosmic lens, is that all objects falling inside the narrow strip defined by the deficitangle computed along the string should be affected, and more lensed images areto be expected. Examination of the area around CSL-1 revealed no more lensedobjects, leading to the rejection of the cosmic string hypothesis.

Except for observing of the effect of lensing there is a different way of detectingcosmic strings. A recent article of Bevis and Hindmarsh et al. [21] compared mod-els of the cosmic microwave background (CMB) power spectra with CMB models

48

including cosmic strings. The CMB data give a moderate preference to the modelincluding cosmic strings. The wideley accepted inflation model also fits the CMBdata. Including cosmic strings in this scenario might better explain the sourcing ofadditional anisotropies in the CMB radiation. In the combined inflation plus stringscenario, inflation creates primordial perturbations still visible today, and the cos-mic strings cause additional perturbations. The observed CMB anisotropies areconsidered to be small, therefore the coupling between inflation and cosmic stringsperturbations are ignored. The string and inflation perturbations are calculatedseparately and are simply added together in order to give the total power spec-trum. So including the cosmic string in the inflation model gives better resultswhen compared with the actual CMB data.

To summarize: cosmics strings are an interesting topic to study from a theoret-ical point of view. Finding cosmic strings would clearly have serious implicationsfor physics and allowing for a good chance to distinguish between different candi-dates for a fundamental theory [22].

49

A Higgs boson

According to Goldstone’s theorem, models showing spontaneous symmetry break-ing lead to massless scalar particles. According to the standard model, at tempera-tures above a critical temperature, when symmetry is unbroken and all elementaryparticles are massless, with the exception of the Higgs particle. A popular state-ment about the Higgs boson is that it gives mass to massless particles, in particularthe particles described by the electro-weak nuclear force. Below this critical tem-perature, the Higgs field spontaneously breaks into a vacuum expectation value forthe Higgs field. To explain the mechanism of the Higgs-boson we need a slightlydifferent version of our previously encountered Abelian-Higgs model. To explainthe Higgs mechanism, we start with the following Lagrangian:

L = DµφDµφ∗ − V (φ, φ∗)− 1

4FµνF

µν (201)

where φ is a complex scalar field and the anti-symmetric tensor Fµν = ∂µAν−∂νAµ.The partial derivatieve is now replaced by the covariant derivative and is given by:Dµ = ∂µ − ieAµ, with e the gauge coupling and Aµ the gauge vector field andV (φ, φ∗) our familiar Mexican hat potential (5). This model is invariant under thefollowing transformations:

φ(x) → eiα(x)φ(x); (202)

Aµ(x) → Aµ(x) +1

e∂µα(x) (203)

This symmetry is sponteneausly broken and the Higgs field acquires an expectationvalue not equal to zero. To study the properties of the different particles, representφ as φ = η + φ1√

2. The Lagrangian becomes:

L =1

2(∂µφ1)2 − 1

2λη2|φ1|2 −

1

4FµνF

µν + e2η2AµAµ + Lint (204)

where Lint includes the second and higher order terms in φ1 and Aµ. This La-grangian also includes a massive scalar particle: the Higgs-boson with mass mH =√λη. As of yet, the Higgs boson has not been found.

50

B Derrick’s theorem

Derrick’s theorem states that there are no stable time-independent, localized so-lutions for any scalar models in more than one dimension. This theorem uses anscaling argument proposing their non-existence. A brief outline of the proof issketched below.Consider the energy of a localized solution φ(x) in n-dimensions [6], [23]:

E =∫dnx[(∇φ)2 + V (φ)] = I1 + I2 (205)

where I1 represents the gradient term and I2 for the potential term. Suppose werescale φ(x) by: x→ αx. The rescaled energy becomes:

Eα = α2−n∫dnx[(∇φ)2 + α−n

∫dnxV (φ) = α2−nI2 + α−nI2 (206)

When n ≥ 2 the equation above will collapse. Fortunately, there are some waysin which the unfortunate outcome of Derrick’s theorem can be avoided. This caneither be done by adding higher derivative terms, or by allowing time-dependency.

51

C String dynamics

The solution to the two-dimensional wave-equation:

x− x′′ = 0 (207)

First two new variables are introduced, let’s call them u and v:

u = ζ + t, v = ζ − t (208)

The three-vector now depends on: x(ζ, t)→ x(u, v). When u and v are differenti-ated with respect to ζ then:

∂u

∂ζ= 1,

∂v

∂ζ= 1 (209)

When differentiating x(u, v) once with respect to ζ:

∂x

∂ζ=∂x

∂u

∂u

∂ζ+∂x

∂v

∂v

∂ζ=∂x

∂u+∂x

∂v(210)

At the second equality (209) has been used. When differentiating again, the righthand side of (137)becomes:

∂2x

∂ζ2=

∂

∂ζ(∂x

∂u+∂x

∂v) =

∂

∂u(∂x

∂u+∂x

∂v)∂u

∂ζ+

∂

∂v(∂x

∂u+∂x

∂v)∂v

∂ζ

=∂2x

∂u2+ 2

∂2x

∂u∂v+∂2x

∂v2(211)

This can also be done for the left-hand-side of (137). When we differentiate thenew variables with respect to t:

∂u

∂t= 1,

∂v

∂t= −1 (212)

Differentiating x(u, v) once with respect to t gives :

∂x

∂t=∂x

∂u

∂u

∂t+∂x

∂v

∂v

∂t=∂x

∂u− ∂x

∂v(213)

At the second equality (212) has been used. When differentiating again, the lefthand side of (137)becomes:

∂2x

∂t2=

∂

∂t(∂x

∂u− ∂x

∂v) =

∂

∂u(∂x

∂u− ∂x

∂v)∂u

∂t+

∂

∂v(∂x

∂u− ∂x

∂v)∂v

∂t

=∂2x

∂u2− 2

∂2x

∂u∂v+∂2x

∂v2(214)

52

When we insert (211) and (214) into (137):

∂2x

∂u2− 2

∂2x

∂u∂v+∂2x

∂v2− ∂2x

∂u2− 2

∂2x

∂u∂v− ∂2x

∂v2= 0 (215)

This results in:

4∂2x

∂u∂v= 0 (216)

The wave equation becomes:∂2x

∂v∂u= 0 (217)

By integrating this equation twice, the general solution can be derived. Whenintegrating (217) once:

∂x

∂u= c(u) (218)

Performing the integration again gives:

x(u, v) =∫

c(u)du+ a(v) (219)

The integral is a function of u, let’s assume that the integral is equal to some b(u),then the solution becomes:

x(u, v) = b(u) + a(v) = b(ζ + t) + a(ζ − t) (220)

A general solution of the equation of motion is given by:

x(ζ, t) =1

2[a(ζ − t) + b(ζ + t)] (221)

representing a left and a right moving wave.

53

References

[1] S. Poletti, Topological Defects, Department of Physics and MathematicalPhysics, University of Adelaide

[2] Figure adapted from: ireswww.in2p3.fr/ires/obernai/greiner/greiner16.pdf

[3] M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory,Westview Press (1995)

[4] A. Altland and B. Simons, Condensed Matter Field Theory, CambridgeUniversity Press (2007)

[5] D.J. Griffiths, Introduction to Electrodynamics, Prentice Hall International,Inc. (1999)

[6] A. Vilenkin and E.P.S. Shellard, Cosmic Strings and other TopologicalDefects, Cambridge University Press (1994)

[7] M.B. Hindmarsh and T.W.B. Kibble, Cosmic Strings, Rep. Prog. Phys (2008)

[8] J.A. Peacock, Cosmological Physics, Cambridge University Press (1999)

[9] Figure adapted from the Cambridge Cosmology website:www.damtp.cam.ac.uk/user/gr/public.html

[10] S.M. Carroll, An Introduction to General Relativity: Spacetime and Geome-try, Addison Wesley (2004)

[11] S.M. Carroll, Lecture notes on General Relativity, december 1997

[12] S. Dodelson, Modern Cosmology, Elsevier (2003)

54

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[15] E.P. van den Ban and E. Looijenga, Riemann Geometry - An introductorycourse: notes to a course by E.P. van den Ban and E. Looijenga, (2008)

[16] This figure is drawn by R. Penrose, from his book The Road to Reality: AComplete Guide to the Laws of the Universe (2004)

[17] E. Kreyszig, Advanced Engineering Mathematics, 8th edition, John Wiley &Sons, inc

[18] J. Polchinski, Introduction to Cosmic f- and d- strings, arXiv: hep-th/0412244v1 (21 Dec 2004)

[19] T.W. B. Kibble, Cosmic strings reborn? arXiv: astro-ph/0410073v2 (20 Oct2004)

[20] M.V. Sazhin, M. Capaccioli, G. Longo, M. Paolillo, O.S. Khavanskaya, N.A.Grogin, E.J. Schreier, G. Covone The true nature of CSL-1 Mon. Not R.Astron Soc. 1-6 (2006)

[21] N. Bevis, M. Hindmarsh, M. Kunz, J. Urrestilla, Fitting CMB data withcosmic strings and inflation arXiv: astro-ph/0702223v3 (25 Jan 2008)

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55

[24] V.F. Mukhanov and S. Winitzki, Introduction to Quantum Fields in ClassicalBackgrounds, draft version (2004)

56