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 partial fulfilment of the requirements of the degree ' of Master of Science ©Andrew Stewart 2008 1*1 Library and Archives Canada Bibliotheque et Archives Canada Published Heritage Branch Direction du Patrimoine de I'edition 395 Wellington Street OttawaONK1A0N4 Canada 395, rue Wellington OttawaONK1A0N4 Canada Your file Votre reference ISBN: 978-0-494-53778-7 Our file Notre reference ISBN: 978-0-494-53778-7 NOTICE: AVIS: The author has granted a nonexclusive license allowing Library and Archives Canada to reproduce, publish, archive, preserve, conserve, communicate to the public by telecommunication or on the Internet, loan, distribute and sell theses worldwide, for commercial or noncommercial purposes, in microform, paper, electronic and/or any other formats. L'auteur a accorde une licence non exclusive permettant a la Bibliotheque et Archives Canada de reproduire, publier, archiver, sauvegarder, conserver, transmettre au public par telecommunication ou par I'lnternet, prefer, distribuer et vendre des theses partout dans le monde, a des fins commerciales ou autres, sur support microforme, papier, electronique et/ou autres formats. The author retains copyright ownership and moral rights in this thesis. Neither the thesis nor substantial extracts from it may be printed or otherwise reproduced without the author's permission. L'auteur conserve la propriete du droit d'auteur et des droits moraux qui protege cette these. Ni la these ni des extraits substantiels de celle-ci ne doivent etre imprimes ou autrement reproduits sans son autorisation. In compliance with the Canadian Privacy Act some supporting forms may have been removed from this thesis. Conformement a la loi canadienne sur la protection de la vie privee, quelques formulaires secondaires ont ete enleves de cette these. While these forms may be included in the document page count, their removal does not represent any loss of content from the thesis. Bien que ces formulaires aient inclus dans la pagination, il n'y aura aucun contenu manquant. 1+1 Canada 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 n ACKNOWLEDGEMENTS I would like to thank Robert Brandenberger for supervising all of the work presented in this thesis, for numerous enlightening conversations, and, in particular, for his patience. Thanks to Joshua Berger and, especially, Stephen Amsel for making 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 making 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 Archive for Microwave Background Data Analysis (LAMBDA). Support for LAMBDA is provided 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, Frangois 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. m 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, G/i, 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 G\i < 5.5 x 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 rix < 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, G\i, en utilisant ralgorithme 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 pro jet South Pole Telescope, nous trouvons qu'une limite de G\i < 5.5 x 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 T%T ^ 0.15. 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 2.2 2.3 2.4 2.5 2.6 2.7 3 Overview Map Making . . . . 2.2.1 The Gaussian Component 2.2.2 The String Component The Canny Edge Detection Algorithm 2.3.1 Non-maximum Suppression 2.3.2 Thresholding with Hysteresis Edge Length Counting Statistical Analysis Results Discussion . „. . ' 6 13 18 21 27 28 32 39 43 46 55 Constraint on the Blue Tilt of Tensor Modes 60 3.1 3.2 60 63 64 Overview Current Bounds from Other Observations 3.2.1 Pulsar Timing vi . 3.3 3.4 4 3.2.2 Interferometers 3.2.3 Nucleosynthesis . Results . Discussion . 66 69 71 ' . . . 73 Conclusions 77 References 79 vn LIST OF TABLES Table page 2-1 Definition of the approximate gradient directions used in the edge detection algorithm 2-2 31 Summary of the ability of the Canny algorithm to make a significant detection of a cosmic string signal for S P T specific simulations . . . 51 2-3 Summary of the ability of the Canny algorithm to make a significant detection of a cosmic string signal for simulations corresponding to a hypothetical CMB survey 54 vin LIST OF FIGURES Figure , page 1-1 The WMAP 5-year TT power spectrum along with recent observational 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 cosmic string 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 instrumental noise 53 3-1 Magnitude of the difference between the angular power spectra of a model with a blue tensor spectral index and a model with a standard tensor 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 dedicated 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, characterized 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 window 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 particular, 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 provided excellent measurements of the first few acoustic peaks in the angular power spectrum [8]. These peaks match those predicted by inflationary cosmological models 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 ACDM 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 general 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 experiment 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, Yim, as [6]: ^ = X>m*Zm(M) , (1-1) Im where a/m are the spherical harmonic coefficients and 9 and <fi are the polar and azimuthal coordinates, respectively. The index I is commonly referred to as the multipole moment. One can then define the angular power spectrum of anisotropy G = (|a,m|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 I, since there is no preferred direction in the universe [6]. If the primordial fluctuations responsible for the anisotropics are described by a Gaussian random field, as is the case with inflation, 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 anisotropics and their angular power spectrum. Therefore, the CMB temperature observed in the sky is considered a single 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 measured values from multiple surveys. The remarkable agreement between theory and 3 6000 $i> WMAP 5yr Acbar Boomerang CBI ?• 5000 h 1 cvT ^ r 4000 h o o o o / C\J f - 3000 o ^ « 2000 - # \ J 4 \ 5 ®$ 1000 V* 0 10 100 500 1000 1500 Multipole moment / Figure 1-1: The WMAP 5-year TT power spectrum along with recent results from the ACBAR [10] (purple), Boomerang [11] (green), and CBI [12] (red) experiments. The pink curve is the best-fit ACDM 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 CMBFAST [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 topological 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, p, 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 Gfi, where G is Newton's constant. The quantity G/.J, is of interest because it characterizes the strength of the gravitational interaction of the strings. The gravitational perturbations produced by the strings, and thus the density perturbations and the induced fluctuations in the CMB, are all of the order of G/u, [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 observational 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 G[i < 10~7 —10~8 on the cosmic string tension [27, 28]. However, we note that in order to place a bound on G[i, 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 fluctuations. Some work has been devoted to finding the size of this string contribution, the results of which translate directly into a bound G/i < 5 x 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, (j>, is 9 determined by the tension of the cosmic string as [32] <J) = 8 T T G H . Figure 2-1 shows the geometry of the space-time near a cosmic string. (2.1) 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 S.rp — = 8TTGHJSVS \k • (vs x e s ) [ , (2.2) where vs is the speed with which the cosmic string is moving, 7S is the relativistic gamma factor corresponding to the speed vs, vs is the direction of the string movement, 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 Gfi < 4 x 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 KSeffect using an edge detection algorithm commonly employed in image analysis. The 10 Figure 2-1: The geometry of the space-time near a cosmic string. Shown here is a slice of the space-time perpendicular to the orientation of the string. The coloured area represents a missing wedge with deficit angle 0, while the dashed lines represent the paths of photons travelling from a source to an observer, and the arrow shows the direction of motion of the string. The photons passing on one side of the cosmic string will appear to be Doppler shifted with respect to those passing on the other side 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 anisotropics. The telescope is designed to provide V 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 temperature 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 t o 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 anisotropics 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 correspond 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 t h e Gaussian component and is consistent with observations. That is, we define T(x,y) = TG(x,y), (2.3) where T(x, y) represents the the final temperature anisotropy map and TG(X, y) represents 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 ST/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 14 Ts{x,y), we define the final temperature as T(x, y) = a TG{x, y) + Ts(x,y), (2.4) where a is a scaling factor which depends on the tension of the cosmic strings in Ts(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 the angular power of the final combined temperature map match the observed angular power for multipole values up to the first acoustic peak, i.e. / < 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 (\TG(k<kp)\2) = a2{\TG(k<kp)\2) + (\Ts(k<kp)\2), (2.5) where kp is the wavenumber corresponding to the first acoustic peak of the angular power spectrum of the CMB, (\Ts(k < kp)\2) is the average of the Fourier temperature anisotropy values from the string component for wavenumbers less than kp and 15 (\Ta{k < kp)\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, G/J,O, we have (\Ts(k < kp)\2) = (\Ts(k < kp)\\ (^^j , (2.6) where (\Ts(k < kp)\2)0 is the average for a string component corresponding to the reference tension and G/x 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: , (\Ts(k<kp)\\(Gy\2 (\TG(k < kPW) \Gfio) • . { •' The benefit of having a 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, G\x. 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 <5Tjv,maxIf an instrumental noise component is included we do not need to perform any additional scaling since it is an unphysical effect, and it is simply summed directly to the other components. Therefore, denoting the noise component by TN(x,y), T(x,y) = TG(x,y) + TN(x,y) we have (2.8) for a simulation without cosmic strings, or T(x,y) = aTG(x,y) + Ts(x,y) + TN(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 t o 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 t o the amplitude of the Gaussian fluctuations. The size of the temperature anisotropies in the string component depends directly on t h e 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 t o orders of magnitude below t h e 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 simulated 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 anisotropics, as given in equation (11), 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 5TG \Kx,Ky) where g(kx,ky) = g{kXi Ky) a\^xi ky) j (2-1'J) 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 a/TO in (1.1) and is related to the angular power spectrum of the temperature anisotropy in the same way, <\a(kx,ky)\2>=Q. (2.11) In the flat-sky approximation the multipole moment is related to pixel position in the grid by / = y V / fc2 + fc2) (2.12) where 6 is the angular size of the survey area [39]. When performing the numerical simulation, we first compute the COBE normalized 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 ^-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, / 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 I using the linear interpolation Cl = Clh+(l-lb)(Cla-Clb), (2.13) where la is the integer value lying above I and Z& 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, TG{X, y), by superimposing four separate sub-components, which we label as T1...T4, each computed separately 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 TG(X, y) = - [Ti(x, y) +. T2(xmax - x, y) •E, y-max where xmax and ymax -y) + TA(xma:!. - x, ymax - y)] , (2.14) 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 1CT5 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 anisotropics 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 uncorrected 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, tis, into N Hubble time steps such that ti+i = 2t{. For a redshift of last scattering z\s = 1000 we then have [33] iV = l o g 2 ( ^ ) ~ 1 5 . 22 (2.15) For large redshifts and assuming fi0 — 1> the angular size of the Hubble volume — 1/2 at a given Hubble time is approximated by 9Hi ~ zi 1/3 ~ V • Therefore, we have 8Hls ~ zz~ ' ~ 1.8° for the Hubble volume corresponding to the time of last scattering and 0ffi+1 — 21/30Hi 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 U 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 rii that should exist in that particular region is then given by the scaling solution n, = M{e\2y\ (2.16) where M is the number of cosmic strings crossing each Hubble volume and 6 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 29HZ- We then simulate the temperature 23 fluctuation produced by that string by adding a temperature anisotropy -^- = AirGfi%Vsr (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-(vsx 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 x 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 20ni along the direction of the string and extends a distance 6H1 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 9^ 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 9^ 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 superposition approximates what the contribution from the entire, more complex cosmic string network would be, and gives the final cosmic string component, Ts(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-5MHHHBBI^^^H^^Hi^^^H-4.46e-5 12e-6 • • ^ ^ ^ ^ • • • • • • • • • • • • • H - 2 . 7 8 e - 6 Figure 2-2: Components of a simulated temperature anisotropy map. On the left is 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° x 2.5° and the angular resolution is V per pixel (22,500 pixels). In the string component the tension of the cosmic strings was taken to be Gfi = 6 x 10~8 and the number of strings per Hubble volume in the scaling solution was taken to be M = 10. The colour of a pixel represents the value of the temperature anisotropy at that pixel, as described by the scale 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 anisotropy map by T(x,y). To compute the gradient of T(x,y) we first construct two square filters, Fx(x,y) and Fy(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: • e = S F*(*,V) = - ^ Fy(x,y) = -^M- e h ( 2 - 18 ) -^. .(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) = J^Fx(i,j)T(x Gy(x,y) = Y^Fy(i,j)T{x.+ + i,y + j) i,y + j),- (2.20) (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) = y/(Px(x,y) + Gl(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 .^-•"""(SfeS- (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 that pixel. 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 a compass (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 9Q(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 detection algorithm. In the left column are the different ranges of values that the gradient direction can take. In the right column are the approximated gradient directions matching 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 be replaced by the corresponding approximation in the right column. Actual Gradient -22.5° < 6G(x,y) 22.5° < 0G(x,y) 67.5° < 0G(x,y) 112.5° < eG(x,y) 157.5° < 9G{x,y) -157.5° < 6G{x,y) -112.5° < 6G{x,y) -67.5° < 9G(x,y) Direction <22.5° <67.5° < 112.5° < 157.5° < -157.5° < -112.5° <-67.5° < -22.5° Approximated Gradient Direction 0G{x,y)~O° (east) 6G(x,y) ~ 45° (north-east) 9G(x,y) ~ 90° (north) 9G(X,V) — 135° (north-west) 9G(x,y) ~ 180° (west) 8G(x,y) ~ —135° (south-west) 6G(x,y) ~ -90° (south) 6G(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 original 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 maxima 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. 32 (2.24) Here Gm is the mean maximum gradient magnitude computed from simulated temperature maps which contain only strings. The value of Gm depends on the parameters 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 simply 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 because the 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, ti < 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 tiGm<G{x,y)<tuGm. (2.25) We then further assert that any semi-edge pixel which is in contact with a trueedge 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 a/2 we check the gradient direction at that pixel. If the gradient direction is parallel or next td parallel to 35 the gradient direction at the original pixel, we immediately change the neighbouring pixel from a xji 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 J/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 marked as Y2, 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 ,fcW*5^r -. ' i- v "fj hm Figure 2 3: Maps produced by the Canny edge detection algorithm. On the left is an example of a map of local maxima generated after non-maximum suppression. The size of the gradient filters used was 5 x 5 pixels. The colour of a pixel represents the magnitude 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 with hysteresis. The values of the thresholds used were tu = 0.25, t\ = 0.10 and tc = 3.5. The value of Gm was calculated using a cosmic string tension of G[i < 6 x 10~8. The yellow 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 the original temperature anisotropy map. In both maps, the grey pixels represent pixels which were discarded from the image. The above images correspond to the same cosmic 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 td 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 1000 r 900 - Edges 800 - (4-1 Total Number 0 700 600 500 400 300 200 100 0 I 0 1 1 1 I 5 I I 10 I h-^-l I , 15 , I 20 25 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 size of the map was 10° x 10° and the angular resolution was V (360,000 pixels). In the edge detection algorithm the gradient filter 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 Gft < 6 x 10~8. The height of each bar corresponds to the total number of edges at that edge length. The inset plot shows a closeup of the tail of 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 '-m)\fj^T^' (227) where Nj and <TI are the mean and sample standard deviation of the first sample and N2 and 02 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 ./V, > 3<7j, 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 3cr level. The p-value corresponding to each t is then computed from a t-distribution with 2n — 2 degrees of freedom. 44 c oc oo Number of Edges i nnn 4U0 c3 CD 200 I 1 I1 T 1 0 T 1 8 T 1 T ^T^^i 10 I 12 14 ]S—r-E—^ 16 Edge Length [pixels] Figure 2-5: An averaged histogram of edge lengths. This averaged histogram corresponds to 40 simulated CMB maps without cosmic strings and without instrumental noise. The angular size of each map was 10° x 10° and the angular resolution of each was 1' per pixel (360,000 pixels). In the edge detection algorithm the gradient filter length was 5 pixels and the thresholds were tu = 0.25, ti = 0.10 and tc = 3.5. The value of Gm was calculated using a cosmic string tension of G{i < 6 x 10"8. The height of each bar corresponds to the mean number of edges at that edge length. The error bars represent a spread of 3cr from the mean value, where a is the standard deviation of the mean, which is calculated separately for each length. Shown here are only the lengths in the histogram for which the mean is greater than 3a. 45 We then combine the probabilities calculated for each length, denoted by pi, into a single statistic which characterizes the difference between the two histograms. Using Fisher's combined probability test, we can define the new statistic \2 as 2 X = -2J2HPL), (2.28) L=2 where Lm is the maximum length at which a p-value was computed. The final p-value corresponding to the statistic x2 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 e to conclude whether or not the difference in the two histograms is significant. We choose to work with the customary significance level e = 0.0027 corresponding to 3cr of a normal distribution. If our p-value is less than e 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 results, 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 separate 10° x 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 length counting 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 histogram 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 t\ = 0.10. These values for the 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, a, 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 M H B B B B I B B a H H l H I ^ ^ ^ H -3.82e-5 2.76e-5 • • • • • M M B a a i M ^ ^ ^ H -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 component and on the right is the map with a cosmic string component. Both maps show a 2.5° x 2.5° patch of sky at V resolution (22,500 pixels). The values of the free parameters in the cosmic string simulation were G\i = 6 x 10~8 and M = 10. The scaling factor in the map component addition that produced the map on the right was a = 0.987. 49 Without cosmic strings With cosmic strings moo 0) hO H 800 0 S-i <U J3 S •z g 000 400 CO 0) T 200 „ _J_ ._ 1 L JL i T L fi 8 1 4 10 i i h^~r^ 12 T ,-^_14 16 Edge Length [pixels] Figure 2-7: Comparison of histograms for maps with and without a component of cosmic string induced fluctuations. Each histogram corresponds to a set of 40 simulated CMB maps. The angular size of each map was 10° x 10° and the angular resolution of each was 1' per pixel (360,000 pixels). In the maps including a cosmic string component, the string free parameters were taken to be Gfi < 6 x 10~8 and M = 10 while the scaling factor in the map component addition was a = 0.987. In the edge detection algorithm the gradient filter length was 5 pixels and the thresholds were tu = 0.25, U = 0.10 and tc = 3.5. The value of Gm was calculated using the same cosmic string tension given above. The height of each bar corresponds to the mean number of edges at that edge length. The error bars represent a spread of 3cr from the mean value, where a is the standard deviation of the mean. Shown here are only the lengths for which the mean is greater than 3cr in both histograms. 50 Table 2-2: Summary of the ability of the Ganny algorithm to make a significant detection of a cosmic string signal for SPT specific simulations. Shown here are the results corresponding to simulated CMB maps excluding instrumental noise as well 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 fourth columns 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 x 10~3 indicates that the simulations including cosmic strings produced significantly different results from those without cosmic strings. String Tension (Gii) 6.0 x icr 8 5.5 x lCT8 5.0 x 10" 8 4.5 x 10" 8 6.0 x 10" 5.5 x .10' 5.0 x 10" 4.5 x 10" Scaling Factor (a) Cutoff Threshold (tc) Without Instrumental Noise 0.987 3.5 4.2 0.989 0.991 5.5 0.993 6.0 With Instrumental Noise p-value 7.19 x 10" 12 6.99 x 10"4 2.39 x 10" 3 9.95 x 10" 3 2.92 x 10~ 10 1.45 x 10" 3 1.36 x 10" 2 1.96 x 10" 2 0.987 0.989 0.991 0.993 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 Gfi — 5 x 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 Gp, < 5 x 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 simulation 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 8T^^max = 10/^K, roughly corresponding to that planned for the SPT [37]. Figure 2-8 shows a side by side comparison 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 pixelated 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 G[i, < 5.5 x 1CT8. 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° x 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 i ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ M -3.82e-5 1 e-5 ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ M H B -3.64e-5 Figure 2-8: Comparison of CMB maps with and without a component of instrumental noise. On the left is a simulated CMB map excluding instrumental noise. On the right is a simulated CMB map including a component of instrumental noise with maximum temperature fluctuation 5TN,max = 10 /xK. Both maps show a 2.5° x 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 significant detection of a cosmic string signal for simulations corresponding to a hypothetical CMB survey. Shown here are the results corresponding to simulated CMB maps excluding instrumental noise and simulated CMB maps including instrumental noise. See the caption of Table 2-2 for a description of the columns. String Tension (G/i) 3.5 x 10"8 3.0 x 1CT8 2.5 x 1CT8 3.5 x lfn 8 3.0 x 10"8 2.5 x 10"8 Scaling Factor (a) Cutoff Threshold (tc) Without Instrumental Noise 0.995 8.0 0.997 8.8 0.998 9.6 With Instrumental Noise 0.995 8.0 0.997 8.8 0.998 9.6 p-value 1.95 x 10" 5 8.16 x 10"4 7.87 x 10" 3 2.37 x 10" 5 1.46 x lO- 3 2.80 x 10- 1 STN}max = 10 /iK. 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 G/i < 3.0 x 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, G\i. 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 microwave 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 directly 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 coming 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 Gfi < 5 x 10 - 8 for a perfect CMB observation from the SPT and that this could be lowered to G/J, < 3 x 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 G/i < 5.5 x 10~8 while the possible bound corresponding to the theoretical survey does not change at all, and is still Gfj, < 3 x 10~8. We consider the constraint corresponding to the SPT specific simulations 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 robust 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 G/i 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 difference 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 between 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 y/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 corresponding to the SPT specific simulations. Increasing the survey size by 5 times only lowered the possible constraint by a factor of roughly y/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 constraining 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 temperature 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 Modes 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 background of gravitational waves has yet to be directly detected, efforts are being undertaken 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 gravitational 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 gravitational wave spectrum pcd\nf where pc is the critical density of the universe, pgw is the energy density of the background gravitational waves and / is frequency [44]. In the above, the energy density in gravitational waves is written as an integral over In / , and the derivative picks out the integrand. In practise, the gravitational wave spectrum is commonly assumed to depend on frequency as a power of / . The value of this power is denoted by TIT, and is known as the tensor spectral index. . Within the framework of the inflationary universe scenario the primordial gravitational 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 U(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 gravitational 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 calculated 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 Efstathiou [51] have derived just such an expression with a pivot scale k0 — 0.002 Mpc - 1 using the combined results from multiple surveys. Using a value Vs{k0) ~ 2.21 x 10~9 for the amplitude of the scalar power spectrum evaluated at the pivot scale, they found flgW(f) can be written as h2Qgw(f) ; - 4-36 x 10" 15 r (£) Jo, T , (3.2) where /o = 3.10 x 10~18 Hz. Solving .this result for the tilt of the gravitational wave spectrum we get the explicit expression "^R7PMA)H2-29xl°^^J- (3 3) ' In the above equations r is the tensor-to-scalar ratio evaluated at the pivot scale, 63 The calculation of the tensor-to-scalar ratio depends quite sensitively on the parameters 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. Nevertheless, 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) data 2 applied to the standard ACDM 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 WMAP results [7], but this new data would not have a significant effect on any part of this study. 64 pulsars. The value of Qgw 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 experiment 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 P P T A project hopes to make timing observations of a sample of twenty millisecond pulsars, ten or more of which with a precision of less than approximately 100 ns. Jenet et al. have applied their method to data from seven pulsars observed by the P P T A 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] h2Qgw(l/8yv) < 2.0 x 10" 8 . (3.5) Plugging this bound into Equation (3.3) at the frequency / = l / 8 y r ~ 3.96 x 10~ 9 Hz, we obtain a constraint on the tensor spectral index n,<0.04771n(4-59rXl°6). (3.6) The W M A P + S D S S 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 n-r < 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] h2Qgw(l/8yr) < 9.1 x 10" 1 1 . (3.7) Plugging this improved constraint into Equation (3.3) at the frequency / = 1/8 yr we calculate a bound nT /2.09xl04\ < 0.0477 In ( J , , , (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, HI with 4 km long arms, and H2 with 2 km long arms, along with a third interferometer in Livingston Parish, Louisiana, LI 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 ngw < 6.5 x 1(T5 (3.9) in the frequency range 51-150 Hz. Inserting this value into Equation (3.3) at the frequency / = 100 Hz, we get a constraint n T < 0.0223In ( 1 - 4 9 X r 1 0 ' 0 ' ' 2 ) . (3.10) The WMAP+SDSS data also provides a value of h = 0.716 for the Hubble parameter 3 . Inserting this along with r = 0.30 into the above bound, we find that the current LIGO results place a constraint n-r < 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] ngw ~ l o - 9 . 3 The complete parameter table is available on-line at: product/map/dr2/params/lcdm_tens_wmap_sdss.cfm 67 (3.ii) http://lambda.gsfc.nasa.gov/ Plugging this value into Equation (3.3) at / = 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 „ T < 0.02231„ ( 2 - 2 9 X 1CW 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 n-r < 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 three drag-free spacecraft each at the corner of an equilateral triangle with sides of length 5 x 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] h2ngw(l mHz) ~ 1 x 10" 12 . (3.13) At the LISA sensitivity level one would expect gravitational wave signals from supermassive black hole binaries, other binary systems and super-massive black hole formation 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 / = 1 mHz to obtain a limit on the blue tilt of the primordial 68 gravitational wave spectrum nT < 0.0299m r 2 - 2 9 r X l 0 2 " ) . (3.14) Inserting r = 0.30 into this equation we find that LISA could potentially place a constraint nr < 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 constraints 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 nucleosynthesis 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 Nu = 4.4, where Nv is the effective number of neutrino species at the time of nucleosynthesis, the BBN bound is [54] h < 1.5 x l O - 5 . ngw(f)d(\nf) (3.15) / Plugging Equation (3.2) into the left-hand side and performing the integration we obtain the inequality fTlT fir h.—a_ L2 flT < 3.4xi09^A_. nr r 69 (3.16) In order to apply the above result, we must first discuss the two integration limits / i and /2- The lower cutoff frequency / i corresponds to the Hubble radius at the time of BBN and takes the value f\ ~ 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 / 2 is the ultraviolet cutoff. We will take it to be given by the Planck frequency, i.e. f2 = fpi = 1.86 x 10 43 Hz. Substituting these two limits into Equation (3.16) along with the W M A P + S D S S values r = 0.30 and h = 0.716 then solving numerically for n^, 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 / 2 the scale of grand unification, 1016 GeV, or the Hubble rate during a simple large field inflation model, which is 10 13 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 / 2 is quite mild. We do not want Qgw > 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 figw{fpi) = 3.93 x 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 sensitive 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 temperature anisotropy, C;, is cosmic variance limited to I = 400 and the signal to noise ratio exceeds unity to I = 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 Ci using CAMB for each of the three current bounds on nr calculated in the previous sections. Second, we calculate C\ using CAMB, but this time with the standard inflationary relation rir = —r/8 [45]. Finally, we compare the output C\ 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 ACDM model with tensors. 71 900 | 800 - '•. 700 - 600 - 3L 500 - a; a I 400 ft! Q 300 - 200 - 100 - 102 10 103 I Figure 3-1: Magnitude of the difference between the angular power spectra of a model with a blue tensor spectral index and a model with a standard tensor spectral index. Shown here are the cases rtr = 0.15 (green), nr = 0.53 (yellow) and rix = 0.79 (orange) respectively. The dashed line represents the cosmic variance error at each I. 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 I < 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 observations and the theory of nucleosynthesis, we have placed bounds op 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 ACDM model with tensor modes included. After completion of this work we became aware of [61] in which a master equation was derived which relates the short wavelength observable flgw(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 parameter 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 described 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 nr, 74 which was the purpose of this work. The authors of [61] also discuss a constraint on rir coming from BBN, but they take the constraint on flgw 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, nr < 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 probed by 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 conclude 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 restrictive, 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 nr = 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 important 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 G/J, < 5.5 x 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 S P T 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 7%T < 0.15, tighter than even some future gravitational wave observatories could hope to achieve. 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