Andrew Stewart- Constraining cosmological parameters with the cosmic microwave background

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Constraining cosmological parameters with

the cosmic microwave background

Andrew Stewart

Master of Science

Department of Physics

McGill University

Montreal, Quebec

November 3, 2008

A thesis submitted to McGill University in partialfulfilment of the requirements of the degree

of Master of Science

Andrew Stewart 2008



In fine, he gave himself up so wholly to the reading of romances, that

a-nights he would pore on until it was day, and a-days he would read

on until it was night; and thus, by sleeping little and reading much, the

moisture of his brain was exhausted to that degree, that at last he lost the

use of his reason. A world of disorderly notions, picked out of his books,

crowded into his imagination; and now his head was full of nothing but

enchantments, quarrels, battles, challenges, wounds, complaints, amours,

torments, and abundance of stuff and impossibilities; insomuch, that all

the fables and fantastical tales which he read seemed to him now as true

as the most authentic histories.

Miguel de Cervantes Saavedra, Don Quixote

ii



ACKNOWLEDGEMENTS

I would like to thank Robert Brandenberger for supervising all of the work pre-

sented in this thesis, for numerous enlightening conversations, and, in particular,

for his patience. Thanks to Joshua Berger and, especially, Stephen Amsel for mak-

ing their code available and answering many questions regarding the edge detection

method. A big thanks to Eric Thewalt for debugging some parts of the code and mak-

ing many helpful suggestions. I would like to thank the WMAP Science Team for the

use of the image shown in Figure 1–1 and I acknowledge the use of the Legacy Archivefor Microwave Background Data Analysis (LAMBDA). Support for LAMBDA is pro-

vided by the NASA Office of Space Science. Thanks, also, to my family for all of their

support during my time at McGill. Last, but most certainly not least, I would like

to thank Rachel Faust, Guillaume Giroux, Martin Auger, Francois Aubin, Razvan

Gornea, John Idarraga, Amelie Bouchat, Marie-Cecile Piro, Francis-Yan Cyr-Racine,

Aaron Vincent, Nima Lashkari, Jean Lachapelle, Anke Knauf, Jamie Sully and Paul

Franche for useful distractions.

iii



ABSTRACT

We investigate the constraints which can by applied on two different cosmological

parameters using observations of the cosmic microwave background (CMB). First,

we develop a method of constraining the cosmic string tension, Gµ, which uses the

Canny edge detection algorithm as a means of searching CMB temperature maps

for the signature of the Kaiser-Stebbins effect. We test the potential of this method

using high resolution, simulated CMB temperature maps. By imitating the future

output from the South Pole Telescope project, we find that a bound Gµ < 5.5×10−8

could potentially be imposed. Second, motivated by the string gas cosmological

model, we examine the constraint levied by the CMB on a blue tilted gravitational

wave spectrum. We find that the CMB cannot provide a tighter bound than those

coming from other observations, the most stringent of which is nT 0.15 from

nucleosynthesis.

iv



ABREGE

Nous etudions les contraintes qui peuvent etre appliquees sur deux parametres

cosmologiques en utilisant les observations du rayonnement de fond cosmologique

(CMB). Premierement, nous developpons une technique de contrainte de la tension

des cordes cosmiques, Gµ, en utilisant l’algorithme de detection des contours Canny

afin de detecter la signature de l’effet Kaiser-Stebbins dans les cartes de temperature

du CMB. Nous testons le potentiel de cette methode avec des cartes de la temperature

du CMB simulees a haute resolution . En imitant les futures donnees du projet

South Pole Telescope, nous trouvons qu’une limite de Gµ < 5.5 × 10−8 pourrait

etre imposee. Deuxiemement, dans le cadre du modele cosmologique d’un gaz de

cordes, nous examinons la limite imposee par le CMB sur un spectre des ondes

gravitationnelles incline vers le bleu. Nous trouvons que le CMB ne peut fournir

de contrainte plus stricte que celles provenant d’autres observations, nommement la

nucleosynthese, qui impose deja une limite de nT 0.15.

v



TABLE OF CONTENTS

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

ABREGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Constraint on the Cosmic String Tension . . . . . . . . . . . . . . . . . . 6

2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Map Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 The Gaussian Component . . . . . . . . . . . . . . . . . . . 18

2.2.2 The String Component . . . . . . . . . . . . . . . . . . . . 212.3 The Canny Edge Detection Algorithm . . . . . . . . . . . . . . . . 27

2.3.1 Non-maximum Suppression . . . . . . . . . . . . . . . . . . 282.3.2 Thresholding with Hysteresis . . . . . . . . . . . . . . . . . 32

2.4 Edge Length Counting . . . . . . . . . . . . . . . . . . . . . . . . 392.5 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 432.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3 Constraint on the Blue Tilt of Tensor Modes . . . . . . . . . . . . . . . . 60

3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.2 Current Bounds from Other Observations . . . . . . . . . . . . . . 63

3.2.1 Pulsar Timing . . . . . . . . . . . . . . . . . . . . . . . . . 64

vi



3.2.2 Interferometers . . . . . . . . . . . . . . . . . . . . . . . . . 663.2.3 Nucleosynthesis . . . . . . . . . . . . . . . . . . . . . . . . 69

3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

vii



LIST OF TABLES

Table page

2–1 Definition of the approximate gradient directions used in the edge de-tection algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2–2 Summary of the ability of the Canny algorithm to make a significant

detection of a cosmic string signal for SPT specific simulations . . . 51

2–3 Summary of the ability of the Canny algorithm to make a significantdetection of a cosmic string signal for simulations corresponding toa hypothetical CMB survey . . . . . . . . . . . . . . . . . . . . . . . 54

viii



LIST OF FIGURES

Figure page

1–1 The WMAP 5-year TT power spectrum along with recent observa-tional results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2–1 The geometry of the space-time near a cosmic string . . . . . . . . . . 11

2–2 Components of a simulated temperature anisotropy map . . . . . . . 26

2–3 Maps produced by the Canny edge detection algorithm . . . . . . . . 38

2–4 A histogram of edge lengths . . . . . . . . . . . . . . . . . . . . . . . 42

2–5 An averaged histogram of edge lengths . . . . . . . . . . . . . . . . . 45

2–6 Comparison of CMB maps with and without a component of cosmicstring induced temperature fluctuations . . . . . . . . . . . . . . . 49

2–7 Comparison of histograms for maps with and without a component of

cosmic string induced temperature fluctuations . . . . . . . . . . . 50

2–8 Comparison of CMB maps with and without a component of instru-mental noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3–1 Magnitude of the difference between the angular power spectra of amodel with a blue tensor spectral index and a model with a standardtensor spectral index . . . . . . . . . . . . . . . . . . . . . . . . . . 72

ix



CHAPTER 1

Introduction

The cosmic microwave background (CMB), famously first reported by Penzias

and Wilson as a universal background radio noise [1], has since become a cornerstone

of modern cosmology and the measure against which many physical predictions are

tested. Over the past few decades there have been a multitude of experiments dedi-

cated to measuring and characterizing the signatures of the background radiation, the

two most famous of which are arguably the Cosmic Background Explorer (COBE)

[2] and Wilkinson Microwave Anisotropy Probe (WMAP) [3] satellites. The initial

measurement by the Far-Infrared Absolute Spectrophotometer (FIRAS) instrument

on COBE of the near perfect black-body spectrum of the background radiation, char-

acterized by a temperature of T = 2.726 ± 0.010 K [4], supported the theory of a hot

and dense early universe. The subsequent discovery of anisotropies in the background

temperature by the Differential Microwave Radiometer (DMR) instrument [5], also

on COBE, matching those predicted by theory, established the CMB as the best win-

dow onto the high-energy physics of the early universe. For example, the anisotropies

1



of the CMB hold information about the formation of large scale structure. In par-

ticular, each potential seed of structure formation will leave a different imprint of

anisotropies on the CMB, providing a powerful tool to discriminate against various

cosmological models. The anisotropies in the CMB also probe other early processes,

including inflation, quantum gravity and topological defects related to symmetry

breaking [6]. In the modern era of precision cosmology, the CMB has been measured

to unprecedented accuracy by a multitude of different surveys (see [7] for some recent

results from WMAP). The mission of WMAP is to determine the geometry, content

and evolution of the universe via a full sky map of the anisotropies, and it has pro-

vided excellent measurements of the first few acoustic peaks in the angular power

spectrum [8]. These peaks match those predicted by inflationary cosmological mod-

els and provides strong evidence in favour of that paradigm. The position, height

and shape of the acoustic peaks provide enough information to determine the values

of the key parameters in the inflationary scenario. This has led to the adoption of

a minimal six parameter ΛCDM model as a “standard” cosmology, with deviations

from that template highly constrained [7]. Despite this, there is still much to learn

from the CMB and observations will continue into the foreseeable future with a gen-

eral focus on higher sensitivity at smaller angular resolution. This includes precision

measurements of the CMB polarization, which is part of the search for a signature of

a primordial gravitational wave background. Arguably, the most significant experi-

ment of the foreseeable future is the Planck satellite [9], which is designed to provide

unparalleled measurements of the CMB, including the polarization.

2



The background radiation appears to arrive in all directions from a spherical

surface corresponding to the surface of last-scattering. Thus, the CMB temperature

anisotropy field is commonly decomposed in terms of spherical harmonics, Y lm, as

[6]:

δT

T =lm

almY lm(θ, φ) , (1.1)

where alm are the spherical harmonic coefficients and θ and φ are the polar and

azimuthal coordinates, respectively. The index l is commonly referred to as the

multipole moment. One can then define the angular power spectrum of anisotropy

C l ≡ |alm|2 , (1.2)

where the angled brackets indicate the average over m, that is, over all observers in

the universe. It then follows that the spectrum depends solely on l, since there is no

preferred direction in the universe [6]. If the primordial fluctuations responsible for

the anisotropies are described by a Gaussian random field, as is the case with infla-

tion, then the angular power spectrum alone is sufficient to characterize the results

[6]. Typically, cosmological models do not predict the exact temperature pattern in

the sky but rather a statistical distribution of anisotropies and their angular power

spectrum. Therefore, the CMB temperature observed in the sky is considered a sin-

gle realization of the model. This leads to a sample variance uncertainty, usually

referred to as cosmic variance [6]. Figure 1–1 shows the angular power spectrum of

anisotropy as predicted by the standard inflationary paradigm along with the mea-

sured values from multiple surveys. The remarkable agreement between theory and

3



Figure 1–1: The WMAP 5-year TT power spectrum along with recent results fromthe ACBAR [10] (purple), Boomerang [11] (green), and CBI [12] (red) experiments.The pink curve is the best-fit ΛCDM model to the WMAP data. [Figure from [8]]

experiment is clear and illustrates the power of the CMB to constrain cosmological

theories.

Efficient computer codes have been written for calculating the CMB anisotropy

spectra up to an arbitrary multipole moment based on a given set of cosmological

parameters and desired physical effects. The most popular of these codes are cmb-

fast [13] and camb [14], the latter of which is based on the former, with both in

wide use in the physics community. The details of how these codes are implemented

are outside the scope of this work, though we note that these programs can compute

4



the angular power spectrum in a matter of minutes on a typical desktop computer

to very high accuracy (∼ 0.1% for camb).

In this thesis, we investigate the constraints which can by applied on two different

cosmological parameters using the CMB. In the first part of this work, we develop an

edge detection method of searching CMB temperature anisotropy maps for the effects

of cosmic strings, and we examine how it could be used to constrain the cosmic string

tension. In the second part of this work, using the alternate cosmological model string

gas cosmology as a motivation, we study how well the angular power spectrum of the

CMB can constrain the possible blue tilt of the power spectrum of tensor fluctuations,and how it compares to the constraints applied by other observations. We end with a

review and discussion of the main results emerging from each of these investigations.

5



CHAPTER 2

Constraint on the Cosmic String Tension

2.1 Overview

At very early times it is believed that the universe underwent a series of symme-

try breaking phase transitions which led to the formation of different types of topo-

logical defects. Among them are linear topological defects know as cosmic strings

[15, 16]. In some scenarios, these defects are stable and can survive until later times.

Topologically stable cosmic strings do not have endpoints and must either extend to

infinity or form closed loops. The quantity which characterizes these cosmic strings

is their tension, µ, which is equivalent to the mass per unit length of the string.

The tension of the cosmic strings is directly determined by the energy scale of the

symmetry breaking during which they were formed, meaning they carry enormous

amounts of energy. It is possible that cosmic strings could have been formed at a

grand unification transition, or later during an electroweak transition, or at any point

in between. Thus, the tension of the cosmic strings can take a wide variety of values.

6



After creation, the cosmic strings form a random network of infinite strings

and closed string loops. Initially the strings are in a very dense environment and

their motion is damped, but as the universe cools this damping diminishes and the

strings move independently of the contents of the universe. Curved string segments

experience an accelerating force that goes like the string tension, and they quickly

develop relativistic speeds. The arrangement of the cosmic string network then

evolves over time through string interactions. In the event that two cosmic strings

cross there are two possible outcomes: they can simply pass through one another,

or they can break at the point where they come into contact and exchange partners.

The latter case is referred to as intercommutation . Numerical calculations indicate

that intercommutation occurs in almost all cases of string crossing [17]. Of course,

strings can also have self-interactions in which they cross themselves somewhere

along their length. In this case, intercommutation causes the formation of a closed

loop that breaks off of the longer segment. When formed, cosmic string loops are

essentially free from the rest of the network and they continue to oscillate, losing

energy via gravitational radiation, until eventually decaying. Infinitely long strings,

on the other hand, cannot decay into gravitational radiation and survive indefinitely.

Without a mechanism for the string network to lose energy density, cosmic

strings would rapidly come to dominate the universe because they would scale as

non-relativistic matter [15]. Therefore, this transfer of energy density into gravity

waves through the formation of loops is crucial. In conjunction with this energy loss

mechanism, it is expected that the string network eventually approaches a scaling

regime in which the number of strings crossing a given horizon volume is fixed in

7



such a way that the energy density in cosmic strings scales like radiation [15]. In this

regime, the strings then contribute some fraction of the total energy. The existence

of a scaling solution is supported by numerical simulations of the evolution of the

cosmic string network [18, 19, 20].

When discussing cosmic strings it is common to work with the dimensionless

parameter Gµ, where G is Newton’s constant. The quantity Gµ is of interest be-

cause it characterizes the strength of the gravitational interaction of the strings.

The gravitational perturbations produced by the strings, and thus the density per-

turbations and the induced fluctuations in the CMB, are all of the order of Gµ [15].Until the late 1990’s, cosmic strings were studied as potential seeds for structure

formation [21], fuelled in part by the realization that the density fluctuation from

a string formed around the grand unified epoch would be of the same order as the

temperature anisotropy discovered by COBE [5]. As mentioned in the Introduction,

acoustic peaks were eventually discovered in the CMB angular power spectrum and

subsequently measured with great accuracy. This lead to cosmic strings being ruled

out as the main origin of structure in favour of the inflationary paradigm, since the

angular power spectrum predicted by cosmic strings consists of only a single broad

peak. Despite this, cosmic strings can still contribute partially to the CMB angular

power spectrum (less than 10% [22, 23]). Therefore, there still exists a great deal of

interest in cosmic strings since there are many cosmological models in which their

formation is generically predicted (see [24, 25, 26] for just a few possibilities).

The observational signatures of cosmic strings are distinct and lie within ob-

servational reach. The current bounds on the string tension come from a variety of

8



measurements. For instance, the gravitational waves emanating from many string

loops at different times produce a stochastic background which is the focus of current

interferometer and pulsar timing experiments. Pulsar timing, specifically, places a

bound Gµ < 10−7−10−8 on the cosmic string tension [27, 28]. However, we note that

in order to place a bound on Gµ using gravitational wave constraints one must make

assumptions about the size of the loops which are formed in the string network, the

probability that strings will intercommute when crossing, and even the string model

under consideration. Therefore, the strength of these bounds can be questioned. As

well as gravitational wave constraints, there is also a bound on the tension coming

from the angular power spectrum of the CMB. As mentioned above, the shape of

the spectrum from cosmic strings alone does not match the observed acoustic peak

structure, meaning they can only contribute a fraction of the cosmological fluctua-

tions. Some work has been devoted to finding the size of this string contribution,

the results of which translate directly into a bound Gµ < 5 × 10−7 [29, 30].

While the above phenomena can be used to place indirect constraints on the

cosmic string tension there is another observational signature unique to cosmic strings

which could be directly detected, namely, linear discontinuities in the temperature

of the CMB. This signature was first studied by Kaiser and Stebbins [31], and is

commonly referred to as the KS-effect. This effect occurs because the space-time

around a straight cosmic string is flat, but with a wedge, whose vertex lies along

the length of the string, removed. The angle subtended by the missing wedge, φ, is

9



determined by the tension of the cosmic string as [32]

φ = 8πGµ. (2.1)

Figure 2–1 shows the geometry of the space-time near a cosmic string. For an

observer looking at a source while a cosmic string is moving transversely through the

line of sight between the two, the photons passing from the source to the observer

along one side of the string will appear to be Doppler shifted relative to those passing

along the other side. If the source that the observer is viewing happens to be the

CMB, this effect will manifest itself as discontinuities in the microwave background

temperature along curves in the sky where strings are located. The magnitude of

the step in temperature across a cosmic string is

δT

T = 8πGµγ svs |k · (vs × es)| , (2.2)

where vs is the speed with which the cosmic string is moving, γ s is the relativistic

gamma factor corresponding to the speed vs, vs is the direction of the string move-

ment, es is the orientation of the string and k is the direction of observation [33].

This temperature jump is independent of the redshift at which the photons pass

by the cosmic string. Some work has already been dedicated to searching for the

KS-effect in current CMB data [34, 35], but a cosmic string signal was not found,

leading to a constraint on the tension Gµ 4 × 10−6.

The goal of the first part of this work is to develop a method of detecting the

temperature discontinuities in the CMB produced by cosmic strings via the KS-

effect using an edge detection algorithm commonly employed in image analysis. The

10



Source Observer

String

Figure 2–1: The geometry of the space-time near a cosmic string. Shown here is aslice of the space-time perpendicular to the orientation of the string. The coloured

area represents a missing wedge with deficit angle φ, while the dashed lines representthe paths of photons travelling from a source to an observer, and the arrow showsthe direction of motion of the string. The photons passing on one side of the cosmicstring will appear to be Doppler shifted with respect to those passing on the otherside due to this non-trivial geometry.

motivation behind this choice is clear since the cosmic strings literally appear as

edges in the CMB temperature. Depending on the sensitivity of the edge detection

algorithm to these temperature edges, we can then place a bound on the cosmic string

tension through Equation (2.2). We expect the bounds arising from this method to

be more robust than those coming from gravitational waves since we do not need to

make assumptions about the nature of the cosmic string network. This work is a

continuation of the study presented in [36].

We are interested in the cosmic strings in the network that survive until later

times, specifically, the times relevant to the production of an edge signature in the

CMB, that is, the time of last scattering until the present time. Based on the

evolution of the network, cosmic strings are more numerous around the time of last

11



scattering than later times. On today’s sky, those strings correspond to an angular

scale of approximately 1 . Therefore, an observation of the CMB with an angular

resolution substantially less than 1

is necessary in order to be able to detect the

edges related to these strings. With this in mind, we also focus on the application

of the edge detection method to high resolution surveys of the CMB, in particular

the future data from the South Pole Telescope project.

The South Pole Telescope (SPT) [37] is a 10m diameter telescope being deployed

at the South Pole research station. The telescope is designed to perform large area,

high resolution surveys of the CMB to map the anisotropies. The telescope is de-signed to provide 1′ resolution in the maps of the CMB. This makes the SPT ideal

to search for the KS-effect (even better than Planck), and we believe that with such

high resolution data our method could provide bounds on the cosmic string tension

competitive with those of pulsar timing.

The remainder of this chapter is arranged as follows: In section 2.2, we discuss

the simulated CMB maps used in our analysis with a focus on the anisotropies

coming from Gaussian fluctuations and cosmic strings. In Section 2.3, we outline

the edge detection algorithm we are using, highlighting the details of our particular

implementation. In Section 2.4, we discuss how we quantify the edge maps that are

output by the edge detection algorithm. In Section 2.5, we explain the statistical

analysis used to determine if a significant difference has been detected. In Section

2.6, we present the results of running the edge detection algorithm on simulated

CMB maps and the possible constraints on the cosmic string tension that could be

applied. Finally, in Section 2.7, we discuss our results.

12



2.2 Map Making

To use an edge detection algorithm to search for a cosmic string signal we first

need an image, or map, to run it on. For this initial investigation of edge detection

as a method for constraining or even detecting cosmic strings, we generate CMB

temperature anisotropy maps by means of numerical simulations and use these as the

input for the edge detection algorithm. By utilizing numerical simulations, we know

exactly the parameters used to generate each temperature anisotropy map. That way,

when we compare the output of the edge detection algorithm for different input maps,

we can conclude if the method is able to make a significant discrimination or not. If

edge detection proves to be a feasible method for detecting a string signal, the goal is

to eventually examine real CMB data for the presence of cosmic strings. To do so, we

would first need to compute an idealized set of data corresponding to the appropriate

cosmological theory and compare it to what is found in the real microwave sky. This

presents us with another reason to develop a method of generating simulated CMB

maps.

The simulated maps are constructed through the superposition of different tem-

perature anisotropy components based on the type of effects being reproduced. We

are interested in the simulation of small angular scale patches of the microwave sky,

so we employ the flat-sky approximation [38]. In this approximation, the geometry

of a small patch on the sky can be considered to be essentially flat. Thus, each

map component, as well as the final map itself, is a two dimensional square image

characterized by an angular size and an angular resolution. Specifically, we work

13



with a square grid that has a size corresponding to the angular size being simulated,

and a pixel size corresponding to the angular resolution being simulated. The pixels

in the grid are indexed by two dimensional Cartesian coordinates (x, y) and we take

the upper left corner of the grid to be the origin.

The common component in every simulated CMB map is a set of temperature

anisotropies produced by Gaussian inflationary fluctuations. The normal distribution

of these fluctuations is predicted by various cosmological models and is supported

by current observations [7]. These fluctuations must be included because they cor-

respond to an angular power spectrum like that measured in the real microwave sky[7]. In fact, we simulate the Gaussian fluctuations such that they account for all of

the observed power in the CMB. Thus, in the absence of any other effects the final

simulated map is simply equivalent to the Gaussian component and is consistent

with observations. That is, we define

T (x, y) ≡ T G(x, y) , (2.3)

where T (x, y) represents the the final temperature anisotropy map and T G(x, y) rep-

resents the Gaussian component. We signify the maps by T simply as a choice of

notation, but we note that the value of each pixel is actually that of the temperature

anisotropy δT/T .

To make a CMB map including the effects of cosmic strings, we simulate a

separate component of string induced temperature fluctuations produced via the

KS-effect. The final temperature anisotropy map is then given by a combination of

the string and Gaussian components. Denoting the string component by T S (x, y),

14



we define the final temperature as

T (x, y)

≡α T G(x, y) + T S (x, y) , (2.4)

where α is a scaling factor which depends on the tension of the cosmic strings in

T S (x, y). We must scale the amplitude of the Gaussian component to compensate for

the excess power we introduce by adding a component of string-induced fluctuations.

In this way the strings can contribute a fraction of the total power, while the final

map is still in agreement with current CMB survey results.

Let us comment in more detail on the nature of this scaling. We demand that theangular power of the final combined temperature map match the observed angular

power for multipole values up to the first acoustic peak, i.e. l 220. We choose

this multipole range because it is tightly constrained by current observations [7].

Then again, as mentioned above, the Gaussian component alone accounts for all

of the observed angular power in the CMB. Thus, this demand is equivalent to

requiring that the angular power of the combined map match that of a pure Gaussian

component. Working in the flat-sky approximation allows us to replace the usual

spherical harmonic analysis of the CMB fluctuations by a Fourier analysis [38]. We

can then express our condition as

|T G(k < k p)|2 = α2|T G(k < k p)|2 + |T S (k < k p)|2 , (2.5)

where k p is the wavenumber corresponding to the first acoustic peak of the angular

power spectrum of the CMB, |T S (k < k p)|2 is the average of the Fourier tempera-

ture anisotropy values from the string component for wavenumbers less than k p and

15



|T G(k < k p)|2 is the equivalent object for the Gaussian component. From Equation

(2.2) one can see that the average of the temperature anisotropy values in the string

component should go as the cosmic string tension squared. Therefore, if we define a

reference cosmic string tension, Gµ0, we have

|T S (k < k p)|2 = |T S (k < k p)|20

Gµ

Gµ0

2

, (2.6)

where |T S (k < k p)|20 is the average for a string component corresponding to the

reference tension and Gµ is the cosmic string tension corresponding to the string

component on the left-hand side of the equation. Substituting this into (2.5) we can

solve for the final form of the scaling factor:

α2 = 1 − |T S (k < k p)|20|T G(k < k p)|2

Gµ

Gµ0

2

. (2.7)

The benefit of having α in this form is we need only calculate the ratio of averages

once using the reference tension. After this we can calculate the value of the scaling

factor with only the cosmic string tension used in the given simulation, Gµ. More

detailed studies of combining string anisotropies and Gaussian anisotropies have

concluded that, in general, a cosmic string contribution of less than 10% of the

observed CMB power on large scales cannot be ruled out [22, 23].

A third component which we can include in the final map is a simulation of

instrumental noise. Any real CMB survey has some amount of noise associated with

its observation and we add this component to examine the effect that noise has on

the ability of an edge detection algorithm to detect a cosmic string signal. As a crude

approximation, we simulate an instrumental noise component that is simply white

16



noise with some given maximum amplitude in the temperature difference δT N,max.

If an instrumental noise component is included we do not need to perform any addi-

tional scaling since it is an unphysical effect, and it is simply summed directly to the

other components. Therefore, denoting the noise component by T N (x, y), we have

T (x, y) ≡ T G(x, y) + T N (x, y) (2.8)

for a simulation without cosmic strings, or

T (x, y) ≡ α T G(x, y) + T S (x, y) + T N (x, y) (2.9)

for a simulation including cosmic strings.

The dominant portion of the final simulated map is the Gaussian temperature

fluctuations. As such, these Gaussian fluctuations represent the most significant

“noise” when trying to directly detect the effect of cosmic strings with the edge

detection algorithm. The significance of the instrumental noise component in the

final map is determined by the maximum amplitude of the noise, which should in

general be small compared to the amplitude of the Gaussian fluctuations. The size

of the temperature anisotropies in the string component depends directly on the

tension of the cosmic strings which are being simulated, as described by Equation

(2.2). For sensible values of the string tension, the amplitude of the string-induced

anisotropies will lie from a factor of a few up to orders of magnitude below the

amplitude of the Gaussian temperature anisotropies, thus presenting the difficulty

in directly detecting them.

17



The simulation of an entire CMB temperature anisotropy map is not particularly

resource intensive and typically takes of the order of a few minutes on a standard

desktop computer, depending on the angular resolution and angular size of the sim-

ulated survey and on which components are being included.

Before moving on to discuss the edge detection algorithm itself, we first review

our methods for generating the Gaussian and string components since they contain

all of the interesting physics.

2.2.1 The Gaussian Component

In this section we discuss in more detail the actual numerical simulation of the

Gaussian temperature fluctuations. The free parameters in the simulation of the

Gaussian component are the angular resolution and the angular scale.

As touched on above, the spherical harmonic expansion of the CMB temperature

anisotropies, as given in equation (1.1), can be replaced by a Fourier expansion when

using the flat-sky approximation [38]. Therefore, when generating the component of

Gaussian fluctuations, we choose to work on a grid in Fourier space. In this case, each

pixel in the grid is indexed by the coordinates (kx, ky), which are the components

of the wavevector pointing to that pixel. The size and resolution of the grid still

correspond to the two angular scales in the simulation. The advantage of being able

to use a Fourier analysis is that it greatly simplifies the calculations, and the value

of the temperature anisotropy at a particular pixel on the the grid is then given by

18



the relation

δT GT

(kx, ky) = g(kx, ky) a(kx, ky) , (2.10)

where g(kx, ky) is a random number taken from a normal probability distribution

with mean zero and variance one [39]. The quantity a(kx, ky) is the Fourier space

equivalent of alm in (1.1) and is related to the angular power spectrum of the tem-

perature anisotropy in the same way,

< |a(kx, ky)|2 >= C l . (2.11)

In the flat-sky approximation the multipole moment is related to pixel position inthe grid by

l =2π

θ

k2x + k2

y , (2.12)

where θ is the angular size of the survey area [39].

When performing the numerical simulation, we first compute the COBE normal-

ized angular power spectrum up to the required multipole moment using the camb

software. One can see from Equation (2.12) that, in general, the largest multipole

moment required for a simulation increases as the resolution increases. Therefore,

since we are interested in simulating high resolution CMB maps, we require the

values of the angular power spectrum at very large multipole moments. There are

currently no CMB surveys taking measurement up to the l-values needed in our

simulated maps, so we run camb with input cosmological parameters determined

by surveys at lower angular resolution. To be precise, when computing the angular

power spectrum with camb, we choose our input parameters to be those derived

19



using the CMBall data set, which combines the results from multiple surveys [10].

We stress, however, that we are free to choose any particular set of parameters we

like.

The next step is to compute the temperature fluctuations pixel by pixel using

the angular power spectrum output by camb. For each pixel (kx, ky) we compute

the corresponding multipole moment using Equation (2.12). Clearly though, l can

take non-integer values, whereas the angular power spectrum is computed for only

integer values. Therefore, we approximate the value of the angular power spectrum

at l using the linear interpolation

C l = C lb + (l − lb)(C la − C lb) , (2.13)

where la is the integer value lying above l and lb is the integer value lying below.

With the value of the angular power spectrum at (kx, ky), it is then straightforward

to calculate the Fourier temperature anisotropy value using Equations (2.11) and

(2.10). Once we have computed the value of each pixel in the grid we take the inverse

Fourier transform of the array using a fast Fourier transform (FFT) algorithm. This

produces a temperature anisotropy map in position space.

By choosing the origin of the grid to be at the top left corner, we have introduced

a preferred direction into the simulation of the Gaussian fluctuations. To compensate

for this asymmetry we construct the final Gaussian component, T G(x, y), by superim-

posing four separate sub-components, which we label as T 1...T 4, each computed sep-

arately using the method described above. When combining these sub-components

we reflect each along one of the four axes on the grid. In this way, we eliminate any

20



irregularity in the final map. Therefore, the Gaussian component is defined as

T G(x, y)

≡

1

2T 1(x, y) + T 2(xmax

−x, y)

+T 3(x, ymax − y) + T 4(xmax − x, ymax − y)

, (2.14)

where xmax and ymax are the maximal x and y values based on the simulation pa-

rameters. The factor of 1/2 in front of the sum is required to maintain the original

standard deviation.

Figure 2–2 shows a Gaussian component produced using the method described

above. One can see the familiar smooth regions of positive and negative deviations

from the background temperature, present in all maps of the CMB. The amplitude

of the fluctuations are of the order of 10−5 matching those measured in the real

microwave sky [5]. We also note that there is no evidence of a preferred direction in

the final map component.

2.2.2 The String Component

As with the Gaussian component, in this section we discuss the details of the

numerical simulation of the string component. The free parameters in the simulation

of the string component are the angular resolution, the angular scale, the cosmic

string tension and the number of cosmic strings per Hubble volume.

Detailed numerical simulations of cosmic string networks have been performed

for a number of years and as a whole share the trait of being very computationally

intensive [18, 19, 20, 40]. Since the focus of this work is on testing the edge detection

21



method, not the details of the cosmic string network evolution, we utilize a toy

model of the network for simplicity. We then examine the resulting temperature

anisotropies caused by the strings which photons encounter between the time of last

scattering and the present day. We choose to use the toy model originally presented

by Perivolaropoulos in [41]. In this model the time from last scattering to the present

time is divided into multiple Hubble times. At each Hubble time, a network of

straight strings with a length equal to two times the size of the Hubble volume at

that time, each with random position, orientation and velocity, is laid down according

to a scaling solution in which there are a fixed number of string segments crossing

each Hubble volume. Each individual string in the simulated network generates

a temperature discontinuity in the microwave sky via the KS-effect. The network

of strings produced at each Hubble time is assumed to be uncorrelated with that

of the previous Hubble time. This is justified since the cosmic strings move with

relativistic speeds, meaning that between Hubble times there will be multiple string

interactions causing the network to enter into a completely different configuration.

The final string-induced anisotropy map is then given by the superposition of the

effects of all of the strings in all of the Hubble slices. We note that in this toy model

cosmic string loops and their subsequent effects are not included.

When performing the numerical simulation, we first separate the period between

the present time, t0, and the time of last scattering, tls, into N Hubble time steps

such that ti+1 = 2ti. For a redshift of last scattering zls = 1000 we then have [33]

N = log2

t0tls

≃ 15 . (2.15)

22



For large redshifts and assuming Ω0 = 1, the angular size of the Hubble volume

at a given Hubble time is approximated by θH i ∼ z−1/2i ∼ t

1/3i . Therefore, we have

θH ls ≃ z−1/2

ls ≃ 1.8

for the Hubble volume corresponding to the time of last scattering

and θH i+1 ≃ 21/3θH i for all subsequent Hubble time steps [33]. We calculate the

contribution to the final temperature anisotropy map from each of the Hubble slices

separately. For a specific Hubble time step ti we start with an extended region that

has a total angular size equivalent to the angular size of the string component being

simulated plus two times the angular size of the Hubble volume at that particular

Hubble time. The number of strings ni that should exist in that particular region is

then given by the scaling solution

ni = M (θ + 2θH i)

2

θ2H i

, (2.16)

where M is the number of cosmic strings crossing each Hubble volume and θ is the

angular size of the string component being simulated [33]. As usual, we work on a

square grid, this time placed over the entire extended region, with pixel size given

by the angular resolution being considered.

Pixels within the entire extended area are then chosen at random to be the

midpoints of strings, with a probability such that the average number of strings in a

single Hubble volume is in agreement with the number M of the scaling solution. If

a pixel is chosen to be a midpoint, we choose a random orientation about that pixel

and we place a straight string of length 2θH i . We then simulate the temperature

23



fluctuation produced by that string by adding a temperature anisotropy

δT S

T

= 4πGµγ svsr (2.17)

to a rectangular region on one side of the string, and subtracting the same amount

from a rectangular region on the other side. This temperature anisotropy corresponds

to the KS-effect as given by Equation (2.2), where r = |k ·(vs× es)| takes into account

the projection effects. The direction of observation k is approximately constant over

the entire field of view while the quantity vs × es is a random unit vector since both

the string orientation and velocity are random. Thus, the value of r is uniformly

distributed over the interval [0,1] [33]. In Equation (2.17), we take the RMS speed

of the strings to be vs = 0.15 [33], so the amplitude of the fluctuation is determined

entirely by the string’s tension and its orientation. Each rectangular region affected

by the temperature fluctuation has a length 2θH i along the direction of the string and

extends a distance θH i in the direction perpendicular to the string. Thus, each cosmic

string gives rise to five separate temperature discontinuities: one at its position, two

parallel to it at a distance θH i and two perpendicular to the string at the endpoints.

After placing all of the cosmic strings and calculating the temperature fluctuation

for each, we have finished simulating the cosmic string network for the given Hubble

time step.

Since we began with a region which is larger than the string component we

wanted to simulate in the first place, we must crop the larger area to the correct size.

We choose to discard pixels equally from all four sides of the extended area, so that

we retain only those from the central region of the larger area. By identifying the

24



correctly sized simulated area with the centre of the extended area, one can see that

what we essentially did when first defining the extended region was to enlarge the

actual simulation area by a Hubble volume in each direction. The reason that we

expand our simulated area in this way is because any string whose midpoint is within

a distance θH i of the actual area we want to simulate could enter into it. Thus, we

must also account for these strings which lie around the edges of the area of interest,

not only those centred within it.

Finally, when we have simulated the string network for each Hubble time step,

we sum together these fifteen sub-components pixel by pixel. This superpositionapproximates what the contribution from the entire, more complex cosmic string

network would be, and gives the final cosmic string component, T S (x, y).

In the model described above, we have fixed values for the the speed of the

strings, the length of the strings and the depth of the temperature fluctuation region

around the string. These values were obtained from particular numerical simulations

[33], however, these parameters can vary significantly for different models of the

string network (see [16] for a review) and should not be considered as established.

Figure 2–2 shows a cosmic string component simulated using the model described

above. Clearly visible are the sharp temperature discontinuities caused by individual

straight strings as well as the cumulative effect of the entire cosmic string network.

The amplitude of the fluctuations in the string component are small compared to

those appearing due to Gaussian fluctuations. The random way in which the cosmic

strings are positioned and oriented in the network is also apparent.

25



4.01e-5 -4.46e-5 1.12e-6 -2.78e-6

Figure 2–2: Components of a simulated temperature anisotropy map. On the leftis an example of a component of Gaussian temperature fluctuations. On the right

is an example of a component of cosmic string induced temperature fluctuations.In both components, the angular size of the simulated region is 2.5 ×2.5 and theangular resolution is 1′ per pixel (22,500 pixels). In the string component the tensionof the cosmic strings was taken to be Gµ = 6 × 10−8 and the number of strings perHubble volume in the scaling solution was taken to be M = 10. The colour of a pixelrepresents the value of the temperature anisotropy at that pixel, as described by thescale below each image.

26



2.3 The Canny Edge Detection Algorithm

At this point we are ready to discuss how to run the edge detection algorithm on

one of the simulated CMB anisotropy maps. When looking for edges in an image we

are looking for curves across which there is a strong intensity contrast. The strength

of an edge can then be quantified by the magnitude of the contrast from one side

of the edge to the other, or equivalently, the magnitude of the gradient across the

edge. For CMB temperature anisotropy maps, the intensity that we are dealing with

is simply the amplitude of the fluctuations. Thus, we define the edges in the CMB

maps as lines across which the temperature difference is large.

To search for edges in CMB temperature anisotropy maps we use the Canny edge

detection algorithm. The Canny algorithm is a multi-stage edge detection algorithm

first developed in 1986 by John F. Canny [42]. Despite its age, this algorithm remains

one of the most commonly used edge detection methods in image analysis. Canny’s

goal was to develop an optimal edge detector which combined good detection and

localization of edges without being prone to false detection. He found that, given

his criteria, the ideal edge filter was well approximated by first-order derivatives of

a Gaussian [42]. The benefit of using the Canny algorithm is that it has a relatively

simple and straightforward implementation which also offers a certain amount of

flexibility, allowing us to optimize the procedure for our purpose.

The process of detecting the edges in a temperature anisotropy map using the

Canny algorithm takes of the order of one minute, depending on the angular size and

resolution.

27



In the following sections we review in detail how we apply the Canny algorithm to

CMB maps to search for edges. The free parameters in the edge detection algorithm

are the size of the gradient filter (the filter length) and the value of the three edge

thresholds.

2.3.1 Non-maximum Suppression

Since we are interested in temperature gradients, the first step of the Canny edge

detection algorithm is to simply compute the gradient of the temperature anisotropy

map and use it to determine which pixels could be part of an edge.

For consistency with the above sections, we denote the input temperature ani-

sotropy map by T (x, y). To compute the gradient of T (x, y) we first construct two

square filters, F x(x, y) and F y(x, y), which are first-order derivatives of a two dimen-

sional Gaussian function along each of the two map coordinates (x, y), respectively.

These filters have the form:

F x(x, y) = − x

2πσ4e−(x2+y2)

2σ (2.18)

F y(x, y) = − y

2πσ4e−(x2+y2)

2σ . (2.19)

We then apply each filter separately at every pixel in the temperature anisotropy map

in order to find the gradient magnitude along each coordinate axis. The components

of the gradient magnitude along the x-direction and y-direction, denoted by Gx and

28



Gy respectively, are given by

Gx(x, y) = i,j

F x(i, j)T (x + i, y + j) (2.20)

Gy(x, y) =i,j

F y(i, j)T (x + i, y + j) , (2.21)

where the maximum and minimum values of i and j are determined by the filter

length. In practise, we actually compute Gx(x, y) and Gy(x, y) by a convolution of

the temperature map with the filter using a FFT for the sake of increased speed.

With the component of the gradient in each direction known at every pixel, we can

construct a new map

G(x, y) =

G2x(x, y) + G2

y(x, y) , (2.22)

which is the map of the gradient magnitude, or edge strength, corresponding to the

original temperature anisotropy map. We can also construct a second map

θG(x, y) = arctan

Gy(x, y)

Gx(x, y)

, (2.23)

which is the map of the gradient angle, or gradient direction. In the above equation

the sign of both components is taken into account so that the angle is placed in the

correct quadrant. Therefore, the arctangent has a range of (−180 , 180 ].

In the Canny algorithm, part of the definition of a pixel that is considered to be

on an edge is that it must be a local maximum in the gradient magnitude. By local

maximum we mean that the gradient magnitude at a given pixel is larger than that of

both pixels which neighbour it along the axis defined by the gradient direction at thatpixel. Using the gradient magnitude and direction maps, it is straightforward then

29



to check the local maximum condition pixel by pixel and determine which could be a

part of an edge and which could not be part of an edge. Since we are only interested

in constructing a final map of edges, if a pixel does not satisfy the local maximum

condition we immediately discard that pixel. Therefore, this process is referred to as

non-maximum suppression .

On a square grid there are only eight distinguished directions which form four

axes, namely the two directions along each coordinate axis and the two directions

along each diagonal axis. For the sake of simplicity, when referring to the eight

directions on the grid we make an analogy with the eight directions on the face of acompass (i.e the positive x-direction is equivalent to east, etc.). However, as already

mentioned, the gradient direction as calculated in Equation (2.23) can take any value

(−180 , 180 ]. Thus, in order to relate the gradient direction, or equivalently the edge

direction, to one that we can trace on the grid, we must approximate the value of

θG(x, y) at each pixel to lie along one of the eight grid directions. The definition of

the approximated gradient directions is given in Table 2–1.

For the purpose of performing the non-maximum suppression we first check the

approximated gradient direction at a given pixel to determine which of the four grid

axes corresponds to the gradient axis at that pixel. We then record the gradient

magnitude of the two pixels which neighbour the original pixel along that gradient

axis. For example, if the gradient direction is approximated as north-west then we

record the gradient magnitude of the pixel to the north-west and the pixel to the

south-east. Lastly, we compare the gradient magnitude of the original pixel to the

gradient magnitudes of the two neighbours. Only if the gradient magnitude is larger

30



Table 2–1: Definition of the approximate gradient directions used in the edge detec-tion algorithm. In the left column are the different ranges of values that the gradientdirection can take. In the right column are the approximated gradient directionsmatching each of the eight directions on the grid. Depending on which range a given

pixel falls into in the left column, the gradient direction at that pixel will then bereplaced by the corresponding approximation in the right column.

Actual Gradient Direction Approximated Gradient Direction−22.5 ≤ θG(x, y) < 22.5 θG(x, y) ≃ 0 (east)

22.5 ≤ θG(x, y) < 67.5 θG(x, y) ≃ 45 (north-east)

67.5 ≤ θG(x, y) < 112.5 θG(x, y) ≃ 90 (north)

112.5 ≤ θG(x, y) < 157.5 θG(x, y) ≃ 135 (north-west)

157.5 ≤ θG(x, y) < −157.5 θG(x, y) ≃ 180 (west)

−157.5

≤θG(x, y) <

−112.5 θG(x, y)

≃ −135 (south-west)

−112.5 ≤ θG(x, y) < −67.5 θG(x, y) ≃ −90 (south)

−67.5 ≤ θG(x, y) < −22.5 θG(x, y) ≃ −45 (south-east)

than both neighbours is the pixel considered a local maximum, or possibly part of

an edge.

Figure 2–3 shows a gradient magnitude map after non-maximum suppression

has been performed. To clearly illustrate the result of performing non-maximum

suppression, we present the map of local maxima corresponding to the same cosmic

string component shown in Figure 2–2 with no other components added to it, yet

this does not represent a legitimate final simulated CMB map. Many of the orig-

inal pixels have been discarded, as expected, and we are left with a rough map of

edges. Although curves corresponding to certain edges in the original temperature

anisotropy component can be seen, there are many other pixels marked as local max-

ima corresponding to extremely weak edges, making the signal from stronger edges

difficult to detect.

31



2.3.2 Thresholding with Hysteresis

The map produced after performing non-maximum suppression represents pixels

which could perhaps be on an edge. Thus, the next step of the Canny algorithm is

to produce the final map of genuine edge pixels from the map of local maxima.

When performing non-maximum suppression we only compared a single pixel

with two of its neighbours to determine if it could be part of an edge. Pixels with

a small gradient magnitude may have still been marked as a local maxima if the

gradient magnitudes of their neighbours were also small. As discussed above, Figure

2–3 shows that this is indeed the case. The magnitude at such pixels can in fact be

so small that we do not want to consider them as edge pixels, since they can dilute

the more significant signal coming from stronger edges. In addition, because we want

to detect edges which appear due to cosmic strings via the KS-effect, we expect the

gradient direction to be consistent across the length of the string induced edge. This

directionality needs to be taken into account to determine which local maxima pixels

belong to the same string edge. Taking these two points into consideration, we must

further expand our definition of exactly what constitutes an edge pixel.

The Canny algorithm outlines a process of applying multiple thresholds to define

the edges in an image, known as thresholding with hysteresis. First, we choose an

upper gradient threshold, tu < 1, such that we can then define a pixel which is

definitely part of an edge, which we name a true-edge pixel , as one which is not only

a local maximum but also satisfies

G(x, y) ≥ tuGm . (2.24)

32



Here Gm is the mean maximum gradient magnitude computed from simulated tem-

perature maps which contain only strings. The value of Gm depends on the param-

eters of the simulation being performed, most notably the string tension, and must

be computed separately for each parameter set using a selected number of simulated

string maps. One can think of Gm as representing the strongest possible edge that

could be formed by cosmic strings alone. Therefore, with this threshold, we are sim-

ply stating that if the gradient magnitude at a given pixel is some chosen fraction of

the maximum possible, then it must be a true-edge pixel.

It is not sufficient, however, to define the edges using only one threshold becausethe gradient magnitude can fluctuate at each pixel along the length of an edge. This

variation can be caused by both instrumental noise and the random nature of the

Gaussian anisotropies. If we applied only an upper threshold, we would reject the

pixels at which the gradient magnitude fluctuates below that threshold, but should in

fact still be considered as a part of a given edge. This would lead to edges being cut

into smaller segments, making them look like dashed lines, rather than continuous

curves on the map. To avoid this, we also choose a lower gradient threshold, tl < tu,

and define a pixel which is possibly part of an edge, which we name a semi-edge

pixel , as a local maximum pixel satisfying

tlGm ≤ G(x, y) < tuGm . (2.25)

We then further assert that any semi-edge pixel which is in contact with a true-

edge pixel and has the appropriate gradient directionality is also a true-edge pixel

sharing the same edge (see the later discussion in this section for a full explanation

33



of these conditions). This allows us to fill the gaps between true-edge pixels and

avoid incorrect breaking up of the edges. If a semi-edge pixel is not in contact with

a true-edge pixel, then it is rejected. If a local maximum pixel still falls below the

lower threshold then it is also rejected. The latter case is the requirement that an

edge pixel have some minimum strength, and cures the problem of a local maxima

with extremely small gradient magnitudes being included in the final edge map.

Since we are interested in edges appearing due of the presence of cosmic strings,

we also apply a “cutoff” threshold such that we reject all pixels for which

G(x, y) > tcGm , (2.26)

where tc ≥ 1. We apply this third threshold because the Gaussian temperature

fluctuations in the CMB map dominate those coming from the cosmic strings. As

such, they lead to edges with much stronger gradient magnitudes, that is, greater

than Gm. If we only applied the upper bound tu, these edges would overwhelm the

edge detection algorithm, washing out the cosmic string signal. By setting a cutoff

threshold, we can discard the pixels with a gradient magnitude which we consider to

be too strong to have been caused by cosmic strings, and keep only those representing

the cosmic string signature. We choose tc ≥ 1 because we also consider the slight

enhancement of weak edges corresponding to Gaussian fluctuations, as a result of

the underlying cosmic string edges, to be part of the cosmic string signal.

To perform the final edge detection on the map of local maxima we first apply the

thresholds as described above. After applying the thresholds we no longer need the

information about the gradient magnitude. Thus, we introduce a simplified notation

34



in which we mark true-edge pixels as 1, semi-edge pixels as 1/2 and all rejected pixels

as 0.

We then check which semi-edge pixels are actually true-edge pixels. We begin

by searching the map for a pixel which is a 1 and has not already been examined

during the tracing of a different edge. If we find one we then check the gradient

direction at that pixel to determine the axis along which the gradient lies. Given the

gradient axis, we inspect each of the six neighbouring pixels which do not lie along

that axis for ones which are non-zero. For example, if the gradient lies along the

north-south axis then we would check the pixels to the north-west, west, south-west,south-east, east and north-east. The two directions perpendicular to the gradient

axis represent the edge axis while the other four directions represent the two axes

which are next to parallel to the edge axis. The reason that we look at the neighbours

along six directions, rather than only the two directions along the edge, is because

we are working on a grid with finite resolution. As such, a wiggle in a real string,

which occurs on a scale below the grid resolution, may manifest itself in the map

as an abrupt jump in the edge position from one pixel to the next. Even a straight

string, depending on its orientation, may appear to have one or more “steps” when

it is viewed at the resolution of the grid. Thus, we cannot expect an edge to be a

continuous chain with the next edge pixel always lying along the edge axis defined

by previous pixel. If we did not account for this, it could lead to the tracing of edges

being prematurely terminated, causing an overabundance of short edges.

If any of the six neighbouring pixels is marked as 1/2 we check the gradient

direction at that pixel. If the gradient direction is parallel or next to parallel to

35



the gradient direction at the original pixel, we immediately change the neighbouring

pixel from a 1/2 to a 1, that is, we change it from a semi-edge pixel to a true-edge

pixel. If the gradient direction is not parallel or next to parallel, then we do not

consider the neighbour as part of the same edge and we ignore it. The comparison

of the gradient directions represents our demand that the temperature gradient be

consistent along an entire edge. If any neighbouring pixel is already marked as a 1

and has not been examined during the tracing of another edge, then we check the

gradient direction at that pixel. If it is in agreement, in the above sense, with that

of the original pixel, we consider it part of the same edge. Once we have checked all

six neighbours of the original pixel we mark it as having been examined.

If we did find a neighbour which is considered to be an edge pixel on the same

edge, regardless of whether it was originally a 1/2 or a 1, we then move to that pixel

and repeat the process of checking the neighbouring pixels. If that pixel then has

another neighbour sharing the same edge that neighbour will be marked as a 1 (if

necessary) and we move to that pixel, and so on. The process of moving pixel by

pixel continues until we reach a pixel that has no neighbours considered to be sharing

the same edge. In this way we will eventually trace the entire edge.

We note two additional points related to tracing the edges in the map. Clearly

when we move to a neighbouring pixel the original pixel will then be a neighbour

of that pixel. By keeping track of which pixels have been examined at each step we

know not to move back to the original pixel again and we do not repeat the process

for the same pixels over and over. Secondly, if the original edge pixel is not the

endpoint of an edge, then it should have two neighbours which share the same edge.

36



If this is the case we mark both as true-edge pixels (if necessary) and then check the

neighbours of each of those pixels separately. In this way we trace the edge along two

separate paths simultaneously, but the end result is still a single continuous curve.

The entire process described above traces a single edge in the map. When we

have finished with a particular edge, we then search for the next pixel in the map

which is a 1 and has not already been examined. If we find one, we then start from

that pixel and trace the corresponding edge until its end. When we can no longer

find a pixel which is a 1 and has not been examined, we consider all of the edges in

the map to have been traced. If there are any remaining pixels which are still markedas 1/2, we consider them not to be in contact with a true-edge pixel and we mark

them as 0. The edge detection process is then finished, and the end result is the final

map of true-edge pixels corresponding to the original temperature anisotropy map.

Figure 2–3 shows a final edge map after thresholding with hysteresis has been

performed. Once again, we show the edge map corresponding to the same cosmic

string component shown in Figure 2–2 and the same map of local maxima shown

in Figure 2–3. Many of the pixels appearing in the map of local maxima have now

been rejected, especially those with very small gradient magnitudes, and the stronger

edges are now much better defined. This is a direct result of applying the thresholds

and directionality conditions. Comparing the original temperature anisotropy map

to the final edge map, it is clear that not only is the Canny algorithm good at

locating the edges which are clearly visible, but that it is also sensitive to the faint

edges which are not easily detectable by eye.

37



2.68e-7 5.34e-9

Figure 2–3: Maps produced by the Canny edge detection algorithm. On the left is anexample of a map of local maxima generated after non-maximum suppression. Thesize of the gradient filters used was 5 × 5 pixels. The colour of a pixel represents themagnitude of the gradient at that pixel, as described by the scale below the image.On the right is an example of a final map of edges generated after thresholding withhysteresis. The values of the thresholds used were tu = 0.25, tl = 0.10 and tc = 3.5.The value of Gm was calculated using a cosmic string tension of Gµ < 6 × 10−8. Theyellow pixels represent pixels which were determined to be on an edge. Together,these pixels show the the position, length and shape of the edges occurring in theoriginal temperature anisotropy map. In both maps, the grey pixels represent pixelswhich were discarded from the image. The above images correspond to the samecosmic string component shown in Figure 2–2.

38



2.4 Edge Length Counting

After applying the Canny algorithm we have created an edge map corresponding

to some initial CMB temperature anisotropy map. To facilitate a comparison with

edge maps generated from different input temperature anisotropy maps, we need a

way to quantify each individual edge map. Since we are considering cosmic strings

as a source of edges in CMB temperature anisotropy maps, one might intuitively

expect that in the presence of strings one would observe a larger number of edges of

all lengths, or at least a larger number in some finite range of lengths. With this in

mind, we employ a simple method of quantifying the edge maps, which is to record

the length of each edge appearing in the edge map. We can then use this data to

construct a histogram of edge lengths which describes each map. The method we

use to count the length of the edges in the edge map is almost identical to that used

to perform the edge tracing during the edge detection. This time, though, every

pixel in the map is already in one of two categories, true-edge pixels (or simply edge

pixels) and non-edge pixels.

Beginning in the same way, we search the map for an edge pixel which has not

already been counted as part of another edge. When one is found we initialize a

counter corresponding to the number of pixels on the edge and set it equal to one.

We then check the gradient direction at that pixel and look at the six neighbours not

along the gradient axis for ones which are also edge pixels and which have a gradient

direction that is parallel or next to parallel to that of the original pixel. We choose

to look at six neighbours rather than just the two perpendicular to the gradient

39



axis for the same reasons discussed in Section 2.3.2. One may wonder why we again

check the gradient direction of the neighbouring pixels when we have already done

so while tracing the edges. The reason is the same as before: to be consistent with

the assumption that the edge was created by a cosmic string, we must confirm that

pixels which share the same edge have similar gradient directions. For example,

two local maxima which appear as neighbours may have been immediately marked

as edge pixels if their gradient magnitudes were greater than the upper threshold.

These pixels would survive to the point where the length counting takes place. If we

did not compare their gradient directions now, rather we imposed only the condition

that they be neighbours, we may incorrectly count them as part of the same edge.

Moreover, two separate edges may happen to occur next to each other in a map. By

checking the gradient directions we ensure the two edges are counted as two shorter

edges rather than being counted as one long edge.

If any of the neighbouring pixels is also an edge pixel sharing the same edge, then

we increase the value of the counter by one and mark the original pixel as counted.

By keeping track of which pixels have already been counted we make sure not to

include the same pixels twice, which would lead to overestimating the length of the

edge. We then step to the neighbouring pixel and repeat the process of checking its

neighbours. The process of moving pixel by pixel continues until we reach a pixel

that has no neighbours sharing the same edge. Each time we step to a new pixel we

increase the value of the counter by one. As was the case when tracing the edges,

the original pixel could have two neighbours which share the same edge. If this is

the case we again step along the edge in two different directions simultaneously. In

40



spite of this, we record all of the steps in both directions with the same counter so

that we still obtain the correct value for the number of pixels on the edge.

When we reach the endpoint(s) of the edge we are finished measuring the length

and we record the value of the counter. This value is exactly the length of the edge

in units of pixels. Therefore, we increase the value of the total number of strings of

that length by one in the histogram of edge lengths. We do not consider a single

pixel to represent an edge, therefore, the minimum edge length that we include in our

histograms is two pixels long. If the counter returns a value of one then we simply

ignore that pass.

After counting the length of an edge and recording it in the histogram, we then

search for the next edge pixel in the map that has not already been counted. If we

find one, we start counting the length of the corresponding edge. When we can no

longer find an edge pixel that has not been counted, then we are finished counting

the length of all of the edges in the map, and the histogram corresponding to the

edge map is complete.

Figure 2–4 shows a typical histogram of edge lengths for a simulated CMB

temperature anisotropy map. There is an abundance of edges with short lengths and

the distribution of the total number of edges decays rapidly as the length increases.

This means that we are unlikely to see very long edges with a gradient magnitude

less than Gm, which makes sense, since the Gaussian fluctuations lead to gradients

which are much stronger. The histograms corresponding to simulations including

different combinations of the map components share this behaviour in general.

41



T o t a l N u m

b e r o f E d g e s

Edge Length [pixels]

Figure 2–4: A histogram of edge lengths. This histogram corresponds to a simulated

CMB map without cosmic strings and without instrumental noise. The angular sizeof the map was 10 × 10 and the angular resolution was 1′ (360,000 pixels). In theedge detection algorithm the gradient filter length was 5 pixels and the thresholdswere tu = 0.25, tl = 0.10 and tc = 3.5. The value of Gm was calculated using acosmic string tension of Gµ < 6 × 10−8. The height of each bar corresponds to thetotal number of edges at that edge length. The inset plot shows a closeup of the tailof the larger plot.

42



2.5 Statistical Analysis

Now that we have a numerical description of our edge maps we need to develop

a way to compare them and look for differences. Specifically, we are looking for

a change in the distribution of the total number of edges between an edge map

corresponding to a simulation without cosmic strings and an edge map corresponding

to a simulation with cosmic strings.

Both the Gaussian and string components in the simulated CMB temperature

anisotropy maps are generated using random processes. If we were to compare two

histograms generated from only one simulated temperature anisotropy map each, we

would not be able to draw a very meaningful conclusion. Therefore, to make our

comparison more robust, we simulate many temperature anisotropy maps with the

same input parameters and perform the edge detection and length counting on each

one separately. This provides a set of histograms from which we can then compute

the mean number of edges of each length occurring over all the runs. We also compute

the standard deviation from each mean value. In the end this provides us with a

new averaged histogram of edge lengths that has statistical error bars. Comparing

two of these averaged histograms then allows us to assign a statistical significance to

the difference in the distributions. Figure 2–5 shows a typical averaged histogram.

The distribution of edges is very similar to that for a single simulated map, as shown

in Figure 2–4. The statistical error bars computed over all of the maps are small

compared to the mean value for short lengths. However, they become of the order

of the mean for longer lengths, since those edges are rare. From this point on,

43



whenever we mention a histogram we mean an averaged histogram computed using

many simulations.

When comparing two histograms, we compare the mean value for each specific

length separately, rather than perform a single general test based on the overall

shapes of the distributions. We prefer to treat each bin separately because each has

a separate standard deviation associated with it. Furthermore, we assume that the

underlying values used to compute each mean are normally distributed. That way

we can use Student’s t-statistic to determine the significance of the difference at each

length.

For two samples of equal size n, Student’s t-statistic is defined as

t =

N 1 − N 2 n

(σ1)2 + (σ2)2, (2.27)

where N 1 and σ1 are the mean and sample standard deviation of the first sample

and N 2 and σ2 are the mean and sample standard deviation of the second sample.

Given two histograms, we compute t for each length occurring in the two histograms

for which N i ≥ 3σi, where i = 1, 2. This constraint on the lengths we consider stems

from our assumption that the underlying distribution of each mean value is normal.

Since it would be inconsistent to consider negative values for the total number of

strings at any given length, we choose only lengths for which the total number of

strings is positive definite at the 3σ level. The p-value corresponding to each t is

then computed from a t-distribution with 2n − 2 degrees of freedom.

44



M e a n N

u m b e r o f E d g e s


Figure 2–5: An averaged histogram of edge lengths. This averaged histogram corre-sponds to 40 simulated CMB maps without cosmic strings and without instrumental

noise. The angular size of each map was 10 × 10 and the angular resolution of each was 1′ per pixel (360,000 pixels). In the edge detection algorithm the gradientfilter length was 5 pixels and the thresholds were tu = 0.25, tl = 0.10 and tc = 3.5.The value of Gm was calculated using a cosmic string tension of Gµ < 6 × 10−8. Theheight of each bar corresponds to the mean number of edges at that edge length. Theerror bars represent a spread of 3σ from the mean value, where σ is the standarddeviation of the mean, which is calculated separately for each length. Shown hereare only the lengths in the histogram for which the mean is greater than 3σ.

45



We then combine the probabilities calculated for each length, denoted by pL,

into a single statistic which characterizes the difference between the two histograms.

Using Fisher’s combined probability test, we can define the new statistic χ2

as

χ2 = −2LmL=2

ln( pL) , (2.28)

where Lm is the maximum length at which a p-value was computed. The final p-value

corresponding to the statistic χ2 is then determined from a chi-square distribution

with 2Lm − 2 degrees of freedom.

The final step is to compare this single p-value to a significance level ǫ to conclude

whether or not the difference in the two histograms is significant. We choose to work

with the customary significance level ǫ = 0.0027 corresponding to 3σ of a normal

distribution. If our p-value is less than ǫ we state that the difference in the two edge

maps is statistically significant.

2.6 Results

We present the results of running the Canny edge detection algorithm on simu-

lated CMB anisotropy maps in two parts. First, we report the results for simulations

which are designed to mimic the expected output from the SPT. Using these re-

sults, we determine what kind of bound on the cosmic string tension one could hope

to achieve using the edge detection method on data from that survey. Second, we

present the results for simulations corresponding to a hypothetical survey that has

different specifications than those of the SPT. We use these results to investigate

46



how the potential constraint on the tension changes with respect to the design of the

survey.

The SPT is capable of producing a 4,000 square degree survey of the anisotropies

in the CMB [37]. To replicate the same amount of sky coverage, we simulate 40 sep-

arate 10 × 10 maps, where the angular resolution of each of these maps is 1 ′ per

pixel, again matching that specified for the SPT. To test the edge detection method,

we simulate two separate sets of 40 maps, the first set including the effect of cosmic

strings, and the second set excluding the effect of cosmic strings. Each set of maps

gives rise to a histogram of edge lengths via the edge detection and edge lengthcounting algorithms. We then compare these two histograms using the statistical

analysis described in Section 2.5 to determine if the difference in the distributions

is significant. We repeat this process for many different values of the cosmic string

tension, until we can no longer identify a statistically significant difference in the two

histograms. Figure 2–6 shows a side by side comparison of a simulated CMB map

without a cosmic string component and a simulated CMB map which does include

a cosmic string component. The effect of the cosmic strings in the final temperature

anisotropy map is not apparent and any difference in the typical structure between

the two maps is unnoticeable by eye. Figure 2–7, on the other hand, shows a his-

togram corresponding to a set of maps without a cosmic string component and a

histogram corresponding to a set of maps with a cosmic string component. The two

histograms show that the edge detection method is in fact able to detect a difference

which is not evident by eye, with maps including strings having slightly higher mean

47



values for certain lengths. Although the difference in histograms may not seem large,

this particular example would generate a significant result.

Although the angular size and resolution of the simulation are determined by

the specifications of the survey in question, the values of the other free parameters

in each step of process must also be fixed. We take the number of cosmic strings

per Hubble volume in all of the string component simulations to be M = 10 [33],

regardless of the cosmic string tension. In every run of the edge detection algorithm,

we choose the gradient filter length to be 5 pixels, the value of the upper threshold

to be tu = 0.25 and the value of the lower threshold to be tl = 0.10. These values forthe thresholds may appear small, but as one can see from the scale in Figure 2–3,

the gradient magnitude in the string component can take a large range of values.

Therefore, Gm can be quite a bit larger than the average gradient magnitude on a

string induced edge, so we must choose low values for the thresholds in order to not

throw away the entire string signal. We have not mentioned the value of the scaling

factor in the map addition, α, nor the value of the cutoff threshold, tc. The reason is,

we do not fix the value of these two parameters for all of the runs. In the case of the

scaling factor, its value must change for each given cosmic string tension, as described

by Equation (2.7). The value of the cutoff threshold, on the other hand, is chosen

deliberately based on the value of the tension, such that we get the best results from

our edge detection method. We note the value of both of these parameters when

presenting our findings.

For the SPT specific simulations, the capability of the edge detection method

to make a significant detection of the cosmic string signal for different choices of the

48



3.55e-5 -3.82e-5 2.76e-5 -2.53e-5

Figure 2–6: Comparison of CMB maps with and without a component of cosmic

string induced fluctuations. On the left is the map without a cosmic string componentand on the right is the map with a cosmic string component. Both maps show a 2.5

× 2.5 patch of sky at 1′ resolution (22,500 pixels). The values of the free parametersin the cosmic string simulation were Gµ = 6 × 10−8 and M = 10. The scaling factorin the map component addition that produced the map on the right was α = 0.987.

49



M e a n N u m b e r o f E d g e s


Without cosmic stringsWith cosmic strings

Figure 2–7: Comparison of histograms for maps with and without a componentof cosmic string induced fluctuations. Each histogram corresponds to a set of 40simulated CMB maps. The angular size of each map was 10

×10 and the angular

resolution of each was 1′ per pixel (360,000 pixels). In the maps including a cosmicstring component, the string free parameters were taken to be Gµ < 6 × 10−8 andM = 10 while the scaling factor in the map component addition was α = 0.987. Inthe edge detection algorithm the gradient filter length was 5 pixels and the thresholdswere tu = 0.25, tl = 0.10 and tc = 3.5. The value of Gm was calculated using thesame cosmic string tension given above. The height of each bar corresponds to themean number of edges at that edge length. The error bars represent a spread of 3σfrom the mean value, where σ is the standard deviation of the mean. Shown hereare only the lengths for which the mean is greater than 3σ in both histograms.

50



Table 2–2: Summary of the ability of the Canny algorithm to make a significantdetection of a cosmic string signal for SPT specific simulations. Shown here arethe results corresponding to simulated CMB maps excluding instrumental noise aswell as simulated CMB maps including instrumental noise. In the first column are

different choices for the tension of the cosmic strings. In the second, third and fourthcolumns are the values of the scaling factor, cutoff threshold and p-value respectively,corresponding to each of the tensions. A p-value of less than 2.7 × 10−3 indicatesthat the simulations including cosmic strings produced significantly different resultsfrom those without cosmic strings.

String Tension (Gµ) Scaling Factor (α) Cutoff Threshold (tc) p-valueWithout Instrumental Noise

6.0 × 10−8 0.987 3.5 7.19 × 10−12

5.5 × 10−8 0.989 4.2 6.99 × 10−4

5.0 × 10

−8

0.991 5.5 2.39 × 10

−3

4.5 × 10−8 0.993 6.0 9.95 × 10−3

With Instrumental Noise6.0 × 10−8 0.987 3.5 2.92 × 10−10

5.5 × 10−8 0.989 4.2 1.45 × 10−3

5.0 × 10−8 0.991 5.5 1.36 × 10−2

4.5 × 10−8 0.993 6.0 1.96 × 10−2

cosmic string tension is summarized in Table 2–2. We find that our edge detection

method can distinguish a signal arising from cosmic strings down to a tension of

Gµ = 5 × 10−8. Therefore, if the edge detection method was used on ideal data

from the SPT, but was unable to distinguish a difference from a theoretical data set

without the effect cosmic strings, we could then impose a constraint on the cosmic

string tension of Gµ < 5 × 10−8.

The above mentioned results were determined from simulated maps which did

not contain a component of instrumental noise. To examine the effect that detector

noise will have on the ability of the edge detection method to constrain the cosmic

string tension, we repeat the same process described above, with the same choices

51



for all of the parameters, but this time with instrumental noise included in the sim-

ulation of the CMB maps. As mentioned earlier, we simulate a component of white

noise with a given maximum temperature change. Here, we choose the maximum

temperature change caused by the instrumental noise to be δT N,max = 10µK, roughly

corresponding to that planned for the SPT [37]. Figure 2–8 shows a side by side com-

parison of a simulated CMB map which includes instrumental noise and one which

does not. The effect that the noise has on the map is clear, making it appear pix-

elated and non-Gaussian, yet the overall structure of the image is still visible since

the temperature fluctuations caused by the noise are sub-dominant compared to the

Gaussian fluctuations.

For the SPT specific simulations including instrumental noise, the results of

using the edge detection method to detect a cosmic string signal are also presented

in table 2–2. We find that detector noise does not have a substantial effect, and it

weakens the possible constraint that the edge detection method could place on the

cosmic string tension only slightly to Gµ < 5.5

×10−8.

Along with the results specific to the SPT, we explore how the constraint which

could be applied by the edge detection method changes based on the specifications

of the survey. For this purpose, we imagine a theoretical observatory which has the

same specifications as the SPT but could map five times the amount of sky with the

same resolution, that is, produce a 20,000 square degree survey of the anisotropies in

the CMB. To replicate the output of a survey with this design, we instead simulate

200 separate 10 × 10 maps at 1′ resolution. In this hypothetical case we again

choose the maximum temperature change caused by the instrumental noise to be

52



3.55e-5 -3.82e-5 3.91e-5 -3.64e-5

Figure 2–8: Comparison of CMB maps with and without a component of instrumen-tal noise. On the left is a simulated CMB map excluding instrumental noise. Onthe right is a simulated CMB map including a component of instrumental noise withmaximum temperature fluctuation δT N,max = 10 µK. Both maps show a 2.5 × 2.5

patch of sky at 1′ resolution (22,500 pixels). Neither map includes a component of cosmic string induced temperature fluctuations.

53



Table 2–3: Summary of the ability of the Canny algorithm to make a significantdetection of a cosmic string signal for simulations corresponding to a hypotheticalCMB survey. Shown here are the results corresponding to simulated CMB mapsexcluding instrumental noise and simulated CMB maps including instrumental noise.

See the caption of Table 2–2 for a description of the columns.

String Tension (Gµ) Scaling Factor (α) Cutoff Threshold (tc) p-valueWithout Instrumental Noise

3.5 × 10−8 0.995 8.0 1.95 × 10−5

3.0 × 10−8 0.997 8.8 8.16 × 10−4

2.5 × 10−8 0.998 9.6 7.87 × 10−3

With Instrumental Noise3.5 × 10−8 0.995 8.0 2.37 × 10−5

3.0 × 10−8 0.997 8.8 1.46 × 10−3

2.5 × 10−8

0.998 9.6 2.80 × 10−1

δT N,max = 10 µK. The analysis follows the same procedure as outlined above, and

we keep the same values for all of the free parameters.

For the larger survey size, the results of using the edge detection method to

detect a cosmic string signal are summarized in Table 2–3. By increasing the survey

size from that of the SPT by a factor of five, while keeping all other specifications the

same, the ideal output from such an observatory could have the potential to improve

the constraint on the cosmic string tension to Gµ < 3.0 × 10−8. When instrumental

noise is included, we find that the effect on the edge detection method is in this case

negligible and the possible bound remains the same as that found using simulations

without instrumental noise.

54



2.7 Discussion

There is renewed interest in cosmic strings formed during phase transitions in

the early universe despite the fact that they were ruled out as being the single source

of structure formation. The evolution of the random network of cosmic strings over

time has been well studied and accurately modelled. The observational signatures

arising from this cosmic string network lie within observational reach, and many

works have been devoted to finding a string signal in order to constrain the value of

the cosmic string tension, Gµ. The existence of cosmic strings has yet to be verified

or ruled out by observations and the continued search for strings is motivated by

cosmological models in which their formation is generally predicted.

We have developed a method of searching for linear discontinuities in the mi-

crowave background temperature caused by the presence of cosmic strings along our

line of sight to the surface of last scattering. The method which we have developed

involves applying an edge detection algorithm to CMB temperature anisotropy maps

in order to identify the effect of cosmic strings. We have applied our edge detection

method to simulated CMB maps both including cosmic strings, and without cosmic

strings, to test its ability to discriminate between the two. This then translates di-

rectly into a possible constraint on the cosmic string tension. In particular, we have

focused on two different sets of simulations, one which mimics the future output com-

ing from the SPT and one which corresponds to a theoretical survey which covers five

times as much sky as the SPT with the same angular resolution. We find that the

edge detection method could potentially place a bound on the cosmic string tension

55



of Gµ < 5 × 10−8 for a perfect CMB observation from the SPT and that this could

be lowered to Gµ < 3 × 10−8 for the larger survey. For more realistic simulations

which include instrumental noise, we find that the potential bound corresponding

to the SPT weakens by only a small amount to Gµ < 5.5 × 10−8 while the possible

bound corresponding to the theoretical survey does not change at all, and is still

Gµ < 3 × 10−8. We consider the constraint corresponding to the SPT specific simu-

lations which include a component of instrumental noise to be the main conclusion of

this work. This possible bound is approximately an order of magnitude better than

those arising from other methods which use CMB observations and approximately

two orders of magnitude better than those arising from other methods which search

for the KS-effect. Therefore, we believe that using the output from the SPT along

with the edge detection method has the potential to greatly improve the constraint

on the cosmic string tension. This bound is not tighter than the constraints arising

from current pulsar timing data, although it is competitive, falling directly within

the range of values reported by different observations. Nevertheless, as mentioned

in the Overview, we consider our method of constraining the tension to be more ro-

bust since we make less assumptions about some of the unknown parameters which

describe the cosmic string network and its evolution. Therefore, we believe that the

possible bound on Gµ given above would in fact represent a stronger constraint.

We conclude that instrumental noise does not have a very major effect on the

ability of the edge detection method to identify the cosmic string signal. We believe

that this is an indication that the thresholding with hysteresis performs as it should,

since noisy pixels could destroy the edge signal by causing large fluctuations in the

56



gradient magnitude. Furthermore, as one can see from Figure 2–7, the largest dif-

ference between histograms occurs at short lengths rather than longer lengths. The

instrumental noise leaves this difference in the short edge signal between maps with

and without strings relatively unchanged, since the probability of a particularly noisy

pixel falling on a short edge, resulting in it being incorrectly detected by the Canny

algorithm, is small compared to that for longer edges. We also found that increasing

the simulated survey size increases the statistical significance of the deviations be-

tween the histograms for similar values of the cosmic string tension. This behaviour

is expected though, since more edge maps were used to compute the mean values

in each of the histograms and one can see from Equation (2.27) that the value of t

scales as√

n. While the p-values are smaller for similar tensions, the final constraint

which can be levied by the larger survey is not drastically different from that corre-

sponding to the SPT specific simulations. Increasing the survey size by 5 times only

lowered the possible constraint by a factor of roughly√

5. Based on this result, we

conclude that the survey size does not have a major influence on the ability of the

edge detection method.

While we have chosen to focus on the SPT in this work, the edge detection

method is quite versatile and could be used with virtually any high resolution CMB

survey. We conclude that this method presents a powerful and unique way of con-

straining the cosmic string tension which has the potential to perform better than

current methods, or, at the very least, to provide a complimentary technique to those

already in use. In the following, we note some improvements which could be made

to the method and also the way in which it is tested.

57



When generating the simulated CMB maps, we employed a toy model of the

cosmic string network which includes only straight strings and no cosmic string

loops. More detailed models of the network and its evolution have been developed

in other works [18, 19, 20, 40] and can be implemented numerically. Therefore,

one obvious way to improve the testing method we have outlined here would be to

implement one of these more complex models which would in turn produce a more

realistic map of the temperature anisotropies induced via the KS-effect. On the other

hand, we stress that a change of this nature would come at a large computational

expense. On a similar note, it may also be useful to develop a more robust method

of combining the string induced temperature anisotropies with those coming from

Gaussian fluctuations, to make sure that the final simulated map agrees with other

observations.

While on the topic of the simulated CMB maps, we reiterate that we have

included only a simplified white noise component as instrumental noise. A more

complex simulation of instrumental noise would include a low frequency piece which

results in stripes appearing in the final map of the CMB. Based on the method

described in this thesis, it is clear that this striping would be crucial, since it would

result in maps with more edges than that predicted by the cosmological theory and

this could be confused with the effect of cosmic strings. One redeeming feature of this

type of low frequency noise is that the stripes which are introduced would lie along

the scanning direction, thus, when dealing with actual SPT data, it may be possible

to subtract this effect out of the final map or to simply ignore edges lying along the

known scanning direction in the edge detection algorithm itself. In future work it

58



would be useful to investigate this type of noise as well as the removal strategies in

more detail to determine if they would change the behavior of the edge detection

method and whether they could weaken the limits quoted here.

Lastly, after applying the Canny edge detection algorithm to the CMB temper-

ature anisotropy maps, we quantify the corresponding edge map by recording the

length of every edge appearing in it. As mentioned in Section 2.4, this is one of the

simplest ways of describing the edge map, and it may be beneficial to investigate

an alternative method of image comparison which provides a more powerful way of

discriminating between the two edge maps.

59



CHAPTER 3

Constraint on the Blue Tilt of Tensor Modes1

3.1 Overview

A stochastic background of primordial gravitational waves represents valuable

information about the very early universe as well as a way to discriminate between

the myriad of cosmological models currently proposed. Although a stochastic back-

ground of gravitational waves has yet to be directly detected, efforts are being un-

dertaken to do so by many current and future experiments, some of the largest being

LIGO, GEO600, TAMA, and LISA [44]. As mentioned in the introduction, CMB

polarization also offers a way to detect a signature generated by inflationary grav-

itational waves. The CMB polarization observations could provide one of the two

detections needed in order to determine the slope of the gravitational wave spectrum.

1 The results of this portion of the thesis led to the publication [43]. The work was co-authored

by R. H. Brandenberger who made some slight revisions to the final text.

60



Furthermore, they allow the value of the tensor to scalar ratio to be determined at

large scales.

A stochastic background of gravitational waves is characterized by the gravita-

tional wave spectrum

Ωgw =1

ρc

dρgwd ln f

, (3.1)

where ρc is the critical density of the universe, ρgw is the energy density of the

background gravitational waves and f is frequency [44]. In the above, the energy

density in gravitational waves is written as an integral over ln f , and the derivative

picks out the integrand. In practise, the gravitational wave spectrum is commonly

assumed to depend on frequency as a power of f . The value of this power is denoted

by nT , and is known as the tensor spectral index.

Within the framework of the inflationary universe scenario the primordial grav-

itational wave spectrum is predicted to be nearly scale invariant with a slight red tilt

[45], i.e. more power at large scales. The reason for the red tilt is that the amplitude

of the gravitational wave spectrum on a fixed scale k is set by the Hubble constant

H at the time ti(k) when the scale k exits the Hubble radius during the period of

inflation. Smaller scales exit the Hubble radius later when the Hubble constant is

smaller, leading to the red tilt. However, there do exist alternative cosmological

models to the standard inflationary scenario which predict a blue tilt of the gravita-

tional wave spectrum, i.e. more power at small scales, which have not yet been ruled

out by observations. One cosmological model that predicts such a tilt is string gas

cosmology [46].

61



String gas cosmology is an approach to string cosmology which starts from the

new degrees of freedom and symmetries which string theory contains, but particle

physics-based models lack, namely string winding modes, string oscillatory modes

and T-duality symmetry, and uses them to develop a new cosmological model [46].

The claim is that by making these crucial stringy additions, one obtains a new

cosmological model which is singularity free [46], generates nearly scale-invariant

scalar metric perturbations from initial string thermodynamic fluctuations [47, 48]

and provides a natural explanation for the observed dimensionality of space [46] (see

[49] for an overview of the string gas cosmology structure formation scenario). A

key result which emerges from string gas cosmology is that the gravitational wave

spectrum has a slight blue tilt, giving rise to a testable prediction different from that

of the inflationary universe paradigm [50].

The goal of the second part of this work is to investigate how a tensor spectrum

with a blue tilt can be constrained by the measurements of the CMB, and how this

compares to similar constraints derived from other astrophysical observations. The

motivation for this is provided by string gas cosmology, yet these constraints apply to

all cosmological models. We believe that precise numerical constraints on the tensor

spectral index have not been presented in the literature and this analysis represents

an important result.

The remainder of this chapter is arranged as follows: In Section 3.2, we use

pulsar timing, laser interferometer and big bang nucleosynthesis constraints on the

gravitational wave spectrum to calculate the bounds on the required blue tilt. We

also investigate the prospects for improved constraints from some future experiments.

62



In Section 3.3, we present the results of our analysis into whether the angular power

spectrum of the CMB temperature anisotropy is compatible with the bounds calcu-

lated from the other observations, or if it will offer an even tighter constraint. Lastly,

in Section 3.4, we discuss our findings and other issues related to the method used.

3.2 Current Bounds from Other Observations

The starting point of our analysis will be an expression for the primordial grav-

itational wave spectrum normalized by CMB observations. Chongchitnan and Efs-

tathiou [51] have derived just such an expression with a pivot scale k0 = 0.002Mpc−1

using the combined results from multiple surveys. Using a value P S (k0) ≃ 2.21×10−9

for the amplitude of the scalar power spectrum evaluated at the pivot scale, they

found Ωgw(f ) can be written as

h2Ωgw(f ) ≃ 4.36 × 10−15r

f

f 0

nT

, (3.2)

where f 0 = 3.10 × 10−18

Hz. Solving this result for the tilt of the gravitational wave

spectrum we get the explicit expression

nT ≃ 1

ln(f ) − ln(f 0)ln

2.29 × 1014

h2Ωgw(f )

r

. (3.3)

In the above equations r is the tensor-to-scalar ratio evaluated at the pivot scale,

r =P T (k0)

P S (k0)

. (3.4)

63



The calculation of the tensor-to-scalar ratio depends quite sensitively on the pa-

rameters of the cosmological model under consideration. For that reason we choose

to leave r as a free parameter in the main expressions calculated in this work. Nev-

ertheless, for the sake of examining some numerical values of the constraints derived

here, we will insert a value of r corresponding to the current upper bound into our

results. Note, however, that the bounds we derive depend only logarithmically on

r. In each case we choose to use the value of the tensor-to-scalar ratio given by

the combined three-year WMAP and lensing normalized Sloan Digital Sky Survey

(SDSS) data2 applied to the standard ΛCDM model, but including tensors [45].

3.2.1 Pulsar Timing

High precision measurements of millisecond pulsars provide a natural way to

study low frequency gravitational waves. A gravitational wave passing between the

earth and a pulsar will cause a slight change in the time of arrival of the pulse

leading to a detectable signal. In the case of gravitational waves, the fluctuating

time of arrivals will be correlated between widely spaced pulsars, producing a unique

signature. Therefore, it can be discriminated from other effects which can cause a

varying time of arrival for a single pulsar. Jenet et al. [53] have developed a technique

to make a definitive detection of a stochastic gravitational wave background which

involves cross-correlating the time derivative of the timing residuals for multiple

2 We note that this portion of the work was completed before the release of the five-year WMAPresults [7], but this new data would not have a significant effect on any part of this study.

64



pulsars. The value of Ωgw is then determined directly from the power spectrum of

the timing residuals [53]. The significance of a detection using the method of Jenet

et al. depends on the number of pulsars observed, the rms timing noise, the number

of observations and the power spectrum of the measured timing residuals.

The Parkes Pulsar Timing Array (PPTA) project [52] is a pulsar timing exper-

iment using the Parkes 64m radio telescope located in Australia with the ultimate

goal of reaching the required sensitivity to make a direct detection of gravitational

waves. The PPTA project hopes to make timing observations of a sample of twenty

millisecond pulsars, ten or more of which with a precision of less than approximately100 ns.

Jenet et al. have applied their method to data from seven pulsars observed

by the PPTA project combined with an earlier data set to find a constraint on the

amplitude of the characteristic strain spectrum. They then used this result to place

a bound on the primordial gravitational wave spectrum [28]

h2Ωgw(1/8 yr) ≤ 2.0 × 10−8 . (3.5)

Plugging this bound into Equation (3.3) at the frequency f = 1/8yr ≃ 3.96×10−9Hz,

we obtain a constraint on the tensor spectral index

nT 0.0477ln

4.59 × 106

r

. (3.6)

The WMAP+SDSS data places a bound r < 0.30 on the tensor-to-scalar ratio [45].

Inserting r = 0.30 into the above equation we find that the current pulsar timing

observations constrain the blue tilt of the tensor spectrum to nT 0.79.

65



Jenet et al. have also used simulated data to determine the upper bound on the

primordial gravitational wave spectrum expected from future pulsar observations.

Using a simulated data-set of twenty pulsars timed with an RMS timing residual of

100 ns over 5 years they calculated [28]

h2Ωgw(1/8 yr) ≤ 9.1 × 10−11 . (3.7)

Plugging this improved constraint into Equation (3.3) at the frequency f = 1/8 yr

we calculate a bound

nT 0.0477ln2.09 × 104

r , (3.8)

and again using the value r = 0.30, we find that, in the absence of a detection, future

pulsar timing observations could tighten the constraint on the blue tilt to nT 0.53.

3.2.2 Interferometers

Interferometer experiments offer a way to directly measure the gravitational

wave strain spectrum with many observatories currently running or planned for the

future. Interferometers in different locations form a network that will search for a

correlated signal between detectors beneath uncorrelated detector noise, in order to

improve sensitivity.

The Laser Interferometer Gravitational Wave Observatory (LIGO) [54] is a

ground based interferometer project operating in the frequency range of 10 Hz -

a few kHz. LIGO consists of two collocated Michelson interferometers in Hanford,

66



Washington, H1 with 4 km long arms, and H2 with 2 km long arms, along with a

third interferometer in Livingston Parish, Louisiana, L1 with 4 km long arms.

Most recently LIGO has performed its fourth science run, S4, with improved

interferometer sensitivity. Abbott et al. [54] have used the S4 data to calculate

a limit on the amplitude of a frequency independent gravitational wave spectrum.

They computed a bound

Ωgw < 6.5 × 10−5 (3.9)

in the frequency range 51-150 Hz. Inserting this value into Equation (3.3) at the

frequency f = 100 Hz, we get a constraint

nT 0.0223 ln

1.49 × 1010h2

r

. (3.10)

The WMAP+SDSS data also provides a value of h = 0.716 for the Hubble param-

eter3 . Inserting this along with r = 0.30 into the above bound, we find that the

current LIGO results place a constraint nT 0.53 on the blue tilt of the tensor

spectrum.

The final phase of LIGO, named Advanced LIGO, hopes to reach a detection

sensitivity of [54]

Ωgw ∼ 10−9 . (3.11)

3 The complete parameter table is available on-line at: http://lambda.gsfc.nasa.gov/

product/map/dr2/params/lcdm_tens_wmap_sdss.cfm

67



Plugging this value into Equation (3.3) at f = 100 Hz, we find that if Advanced

LIGO does not make a positive detection of a gravitational wave background, then

it will place a bound on the blue tilt of the tensor spectrum

nT 0.0223 ln

2.29 × 105h2

r

. (3.12)

Using r = 0.30 and h = 0.716 in this expression, Advanced LIGO would then

constrain the blue tilt of the tensor spectrum to nT 0.29.

The Laser Interferometer Space Antenna (LISA) [55] is a planned space-based

interferometer experiment operating in the mHz range. LISA will consist of threedrag-free spacecraft each at the corner of an equilateral triangle with sides of length

5 × 109 m. Each spacecraft has two optical assemblies pointed towards the other

two spacecraft forming three Michelson interferometers. This triangle formation will

orbit the sun in an Earth-like orbit separated from us by approximately fifty million

kilometres. The goal of LISA is to reach a sensitivity of [56]

h2

Ωgw(1 mHz) ≃ 1 × 10−12

. (3.13)

At the LISA sensitivity level one would expect gravitational wave signals from super-

massive black hole binaries, other binary systems and super-massive black hole for-

mation to be present. Assuming these predicted signals could somehow be removed

and LISA does not detect any primordial signal, we can plug this predicted bound

into Equation (3.3) at f = 1 mHz to obtain a limit on the blue tilt of the primordial

68



gravitational wave spectrum

nT 0.0299ln2.29 × 102

r . (3.14)

Inserting r = 0.30 into this equation we find that LISA could potentially place a

constraint nT 0.20 on the blue tilt.

3.2.3 Nucleosynthesis

The theory of big-bang nucleosynthesis (BBN) successfully predicts the observed

abundances of several light elements in the universe. In doing so, BBN places con-

straints on a number of cosmological parameters. This in turn results in an indirect

constraint on the energy density in a gravitational wave background as follows: the

presence of a significant amount of gravitational radiation at the time of nucleosyn-

thesis will change the total energy density of the universe, which affects the rate of

expansion in that era, leading to an over-abundance of helium and thus spoiling the

predictions of BBN [44].

Assuming N ν = 4.4, where N ν is the effective number of neutrino species at the

time of nucleosynthesis, the BBN bound is [54]

f 2f 1

Ωgw(f ) d(ln f ) < 1.5 × 10−5 . (3.15)

Plugging Equation (3.2) into the left-hand side and performing the integration we

obtain the inequalityf nT 2 − f nT 1

nT < 3.4 × 109

h2f nT 0

r. (3.16)

69



In order to apply the above result, we must first discuss the two integration limits

f 1 and f 2. The lower cutoff frequency f 1 corresponds to the Hubble radius at the

time of BBN and takes the value f 1 ∼ 10−10

Hz. For wavelengths larger than the

Hubble radius, the gravitational waves are frozen out [57] (see [58] for a review) and

thus do not act like radiation. The upper cutoff frequency f 2 is the ultraviolet cutoff.

We will take it to be given by the Planck frequency, i.e. f 2 = f Pl = 1.86 × 1043 Hz.

Substituting these two limits into Equation (3.16) along with the WMAP+SDSS

values r = 0.30 and h = 0.716 then solving numerically for nT , we find the bound

on the blue tilt of the tensor spectrum from BBN to be

nT 0.15 . (3.17)

Had we instead inserted for f 2 the scale of grand unification, 1016GeV, or the Hubble

rate during a simple large field inflation model, which is 1013 GeV, the bound would

be slightly relaxed to 0.16 or 0.17, respectively. Thus, the dependence of the bound

on the uncertain ultraviolet cutoff scale f 2 is quite mild.

We do not want Ωgw > 1 at any scale within the integration bounds. Since

we are working with such large frequencies we should check to be sure that this

condition is satisfied. Substituting the value of the tensor spectral index determined

by BBN back into Equation (3.2) we find that Ωgw(f Pl) = 3.93 × 10−6, meaning our

requirement is indeed satisfied for all frequencies within the interval of integration.

70



3.3 Results

The three-year WMAP results are an improvement upon previous observations.

A reduction in instrument noise produced spectra which are three times more sensi-

tive in the noise limited region, independent years of data allow for cross-checks, the

instrument calibration and response have been better characterized and a thorough

analysis of the polarization data has improved the understanding of the data [59].

Using the three-year WMAP data, the derived angular power spectrum of the tem-

perature anisotropy, C l, is cosmic variance limited to l = 400 and the signal to noise

ratio exceeds unity to l = 1000 [59]. Thus, this high precision cosmological data may

provide another method of constraining the value of the tensor spectral index.

Using camb, we can simulate how a blue tilt of the primordial gravitational

wave background would effect the observed angular power spectrum of the CMB. To

examine possible constraints, we employ the following method: First, we calculate

C l using camb for each of the three current bounds on nT calculated in the previous

sections. Second, we calculate C l using camb, but this time with the standard

inflationary relation nT = −r/8 [45]. Finally, we compare the output C l values for

the models with a blue tilt against the output for the model with the standard value

of the tensor spectral index (i.e. the above relation from inflationary cosmology).

For consistency with the previous sections, when running camb we choose our input

cosmological parameters to be those calculated using the WMAP+SDSS data for a

ΛCDM model with tensors.

71



10 102

103

0

100

200

300

400

500

600

700

800

900

l

D i ff e r e n c e µ K 2

Figure 3–1: Magnitude of the difference between the angular power spectra of amodel with a blue tensor spectral index and a model with a standard tensor spectralindex. Shown here are the cases nT = 0.15 (green), nT = 0.53 (yellow) and nT = 0.79(orange) respectively. The dashed line represents the cosmic variance error at eachl.

From Figure 3.3 we can clearly see that the power spectrum of the temperature

anisotropy for models with a blue tensor spectral index does not vary much from

that calculated using a standard inflationary definition of the tensor spectral index.

In fact, the difference is within the cosmic variance error at all l ≤ 1000 for each of

the three bounds calculated using the PPTA observations, LIGO observations and

the theory of BBN. Thus, we conclude that the CMB does not offer any tighter

constraints on the blue tilt of the gravitational wave spectrum than those already

calculated.

72



3.4 Discussion

A stochastic background of primordial gravitational waves is a prediction of

many cosmological models. Assuming that the gravitational wave spectrum depends

as a power on frequency, then this spectrum can be characterized by its tilt and

amplitude. Most models predict the tilt to be nearly scale-invariant but slightly red,

while some models, like string gas cosmology, predict a slight blue tilt. Although a

gravitational wave background has yet to be directly detected, observational results

can already be used to constrain it.

Using the current results from pulsar timing observations, direct detection ob-

servations and the theory of nucleosynthesis, we have placed bounds on the possible

blue tilt of the gravitational wave background. By far the tightest constraint on

the tilt comes from big-bang nucleosynthesis. If we take the tensor to scalar ratio

on CMB scales to be given by the current observational upper bound, and if we

take the ultraviolet cutoff scale in the spectrum of gravitational radiation to be the

Planck scale, then the bound is nT 0.15, tighter than even Advanced LIGO, the

future PPTA, and LISA can hope to achieve. It is not surprising that nucleosynthesis

provides the tightest bounds on a blue tilt of the gravitational wave spectrum since

nucleosynthesis probes physics on scales much smaller than the other experiments we

analyzed, and spectra with blue tilts have more power on the smallest scales. That

is, BBN gives us the largest “lever arm” to probe gravitational waves in conjunction

with CMB observations, a point also made in [60]. From our results we can clearly

73



see the trend that the bound on the blue tilt of the tensor spectral index tightens as

the length scale probed by the given experiment decreases.

Simulations of the angular power spectrum of the temperature anisotropies in

the CMB did not offer any tighter constraints on the tensor spectral index, with each

of the constraints calculated in this paper producing a temperature power spectrum

that was within the cosmic variance error of one calculated for a standard ΛCDM

model with tensor modes included.

After completion of this work we became aware of [61] in which a master equa-

tion was derived which relates the short wavelength observable Ωgw(f ) to the tensor

to scalar ratio measured with the CMB. The goal of that work was to develop a

formulation which is as general as possible. In particular, the equation of state pa-

rameter and the tensor spectral index are taken to be arbitrary functions of the scale

factor and wavenumber respectively, not constants as is often assumed. Their master

equation is thus a more general version of our “master” equation (3.2) and we could

have just as easily used it as our starting point. In fact, we have confirmed that by

choosing a constant value w = 1/3 for the equation of state parameter (which is de-

scribed as the most logical value in [61]) and numerical values for other cosmological

parameters that match our choices above, we can indeed re-derive all of our results

from the master equation of [61]. We consider this a good consistency check for

the constraints derived in this work. The authors of [61] do include a discussion of

some of the same types of observations mentioned here, namely laser interferometer

and pulsar timing, however, we stress that they do not use the current numerical

constraint from any particular observatories to compute actual upper bounds on nT ,

74



which was the purpose of this work. The authors of [61] also discuss a constraint

on nT coming from BBN, but they take the constraint on Ωgw from BBN to be a

constant across all frequencies rather than integrating their master equation as was

done in this study. In the end they find a weaker bound on the tilt, nT 0.36, than

the one obtained here.

As mentioned in Section 3.2 , Equation (3.2) has been normalized at the scale

of cosmic microwave background observations. However, those experiments probe

scales that are approximately ten orders of magnitude larger than those probed by

the PPTA and approximately nineteen orders of magnitude larger than those probedby LIGO, with LISA probing between the two. Extrapolating between such a large

difference in scales is not straightforward and we note that in [51] the authors con-

clude from their analysis that even within the framework of the inflationary universe

paradigm the formula for the primordial gravitational wave spectrum (3.2) is too re-

strictive, and they believe it is indeed not possible to extrapolate reliably over such a

large difference in scales. Whether or not this is the case in the string gas cosmology

model should perhaps be examined more carefully in future work.

Continuing with string gas cosmology, we conclude that the current bounds on

the tilt of the gravitational wave spectrum are weak. The predicted magnitude of the

blue tilt of the gravity wave spectrum is thought to be comparable to the magnitude

of the red tilt of the spectrum of scalar metric fluctuations [50]. If the latter is taken

to agree with the current bounds, we predict a blue tilt of less than nT = 0.1 which

will not be easy to detect. There may, on the other hand, be models similar to string

75



gas cosmology in which the scalar and tensor tilts are not related, and for which

planned experiments could set valuable constraints on the model parameter space.

76



CHAPTER 4

Conclusions

The cosmic microwave background has had, has, and will continue to have im-

portant implications for cosmology. It represents the premier tool with which to gain

insight into the physics of the early universe, as well as a number of other phenomena.

In this thesis, we have explored the ability of the CMB to constrain two particular

parameters related to alternative cosmological models.

We have developed a method which makes use of the Canny edge detection

algorithm as a means of searching maps of the CMB temperature for the signature

of the KS-effect. Since this effect is caused by the presence of cosmic strings, the edge

detection method represents a way of directly constraining the cosmic string tension.

By testing this method on simulated CMB maps, we found that, using the future

output from the South Pole Telescope project, a bound on the cosmic string tension

of Gµ < 5.5 × 10−8 could potentially be imposed. This bound is approximately an

order of magnitude lower than the best current constraint imposed by the CMB and

is competitive with those reported by from pulsar timing experiments. Nevertheless,

we believe that the edge detection method provides a more robust result than those

77



from pulsars, since less assumptions about the nature of the cosmic strings have to

be made. We also found that using the edge detection method with data from a

much larger survey observing with the same angular resolution as the SPT would

not drastically reduce the possible constraint that could be levied, reducing it by

only a factor of a few.

We have also investigated whether the angular power spectrum of the CMB

could provide a stronger constraint on the possible blue tilt of the gravitational

wave background than those imposed by other means, namely, pulsar timing and

laser interferometer observations and the theory of nucleosynthesis. As a first step,we calculated the specific constraint on the blue tilt related to each of these three

observations, which we believe represents in itself an important result. We discovered

that the tightest current bound on the blue tilt of the tensor spectrum comes from

BBN at nT 0.15, tighter than even some future gravitational wave observatories

could hope to achieve. In the end, we found that the CMB could not impose a tighter

bound than any of these three methods.

In closing, we believe that in this thesis we have shown two important results as

well as some unique ways of extracting information from the CMB. Considering the

topics focused on in this work, we have highlighted the influence that the CMB has

on a wide range of cosmological issues. Then again, by the same token, we have also

shown that alternative models are still viable, and cannot be entirely ruled out by the

precise measurements which have already been taken. On that note, looking ahead

to the future of CMB physics, we believe that there are still many more exciting

discoveries to be made.

78



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Documents

Andrew Stewart- Constraining cosmological parameters with the cosmic microwave background