Wiggly Cosmic String Evolution

Wiggly Cosmic String Evolution José Pedro Pinto Vieira Mestrado em Física Departamento de Física e Astronomia 2013/2014 Orientador Carlos José Amaro Parente Martins, Investigador Coordenador, Centro de Astrofísica da Universidade do Porto Todas as correções determinadas pelo júri, e só essas, foram efetuadas. O Presidente do Júri, Porto, ______/______/_________ Acknowledgements First of all, my sincerest thanks to Dr. Carlos J. A. P. Martins, the most supportive supervisor a student can have, for all his patience and invaluable physical insight. If I ever do manage to become a cosmologist, I owe it to you. Thanks also to Dr. Tasos Avgoustidis, who opened the door to extra dimensions. Without him, this thesis would be one chapter shorter. In addition, a word of gratitude towards those who have taught me Physics and Mathematics for the last five years. It has been a pleasure to learn from (almost) all of you. Furthermore, thanks to all the companions who have walked this path beside me, for all the pain and laughter we have shared, and for all the stimulating discussions since the Great Debate About The Nature Of Fire of 2009. In particular, special thanks to Artur Sousa, Catarina Cosme, Daniel Passos, Diogo Lopes, and Ester Simões. And finally, a very special word of appreciation to the staunch supporters who have backed me up from behind the scenes. For your wholehearted dedication, thank you, mother, father, sister, grandparents, and Ana. God knows what you have to put up with. I acknowledge the Gulbenkian Foundation for the significant financial support provided through Programa de Estímulo à Investigação 2013, grant number 132590. Needless to say, this thesis is much richer thanks to it. 1 Resumo Cordas cósmicas são uma classe especial de defeitos topológicos que se podem formar num grande leque de cenários cosmológicos, incluindo extensões naturais do modelo padrão. Compreender a sua dinâmica e evolução é portanto uma questão de grande importância para aprender acerca dos mecanismos físicos desconhecidos que têm desempenhado papeis determinantes na História do Universo - especialmente agora que temos finalmente à nossa disposição um conjunto de dados experimentais que podem ajudar a constringir estes modelos (notavelmente de Planck [2] e BICEP2 [1, 17, 26]). A enorme complexidade e não-linearidade destes sistemas, no entanto, faz com que mesmo os exemplos mais simples de redes de defeitos sejam impossíveis de descrever completamente por meios analíticos. Além disso, a grande gama de escalas de tempo e comprimento envolvidos limitam seriamente a nossa capacidade de retirar informação de simulações computacionais. Assim, a abordagem típica a este imbróglio resulta numa simbiose entre estas duas filosofias, usando-se dados de simulações para calibrar simplificações analíticas do problema. O desafio é então a construção de um modelo resolúvel que ainda assim capture a essência dos fenómenos mais importantes. O modelo de Uma Escala dependente da Velocidade - “Velocity-dependent One-Scale model” (VOS), no original [19, 20, 21] - é o mais bem sucedido modelo deste género no que toca a modelar o comportamento de grande escala de uma rede simples de cordas cósmicas [25]. Em contrapartida, peca por ser fundamentalmente limitado em relação à informação que pode fornecer acerca da importante estrutura de pequena escala que se vai acumulando ao longo da História de uma rede. Nesta dissertação, apresenta-se uma generalização natural e original do VOS que tem em conta a evolução da estrutura de pequena escala na rede [24]. Após uma discussão detalhada do formalismo matemático em causa, considera-se a sua aplicação a alguns limites fisicamente relevantes - em particular, deduzimos condições concretas sob as quais a estrutura de pequena escala na rede deve evoluir para um regime de “scaling”. Finalmente, é dado algum foco a um trabalho em desenvolvimento que explora a relação entre este modelo e uma versão do VOS com dimensões extra. 2 Abstract Cosmic strings are a special class of topological defects which can be formed in a broad range of cosmological scenarios, including natural extensions of the standard model. Understanding their dynamics and evolution is thus of great importance to learn about the unknown physical mechanisms that have played significant roles in the History of the Universe - especially now that valuable data which may be used to constrain these models has finally become available (notably from Planck [2] and BICEP2 [1, 17, 26]). The highly nonlinear complexity of these objects, however, makes even the simplest examples of these networks impossible to completely describe analytically. Moreover, the wide range of time and length scales involved severely limit how much can be learned from computational simulations. In the end, a typical approach is a compromise between these two philosophies in which simulation data is used to callibrate analytical simplifications of the problem. The challenge, then, is building a solvable model which still captures most of the important phenomena. The Velocity-dependent One-Scale model (VOS) [19, 20, 21] is the most successful such model when it comes to modeling the large-scale behaviour of a “vanilla” network [25]. However, it is fundamentally limited in how much it can tell us about the important small-scale structure that builds up during the History of the network. In this dissertation, we present an original natural generalization of the VOS which takes into account the evolution of small-scale structure in the network [24]. After a detailed discussion of the mathematical formalism involved, its application to a few physically relevant limits is considered - in particular, we deduce concrete conditions under which the small-scale structure in the network should evolve towards a scaling regime. Finally, we also focus on on-going work which explores the relationship between this model and an extra-dimensional version of the VOS. 3 Contents 1. Introduction to Cosmic Strings 8 1.1. Topological Defects and Spontaneous Symmetry Breaking in Phase Transitions ..................................... 8 1.2. The Abelian-Higgs: A Simple Model . . . . . . . . . . . . . . . . . . . . 10 1.3. Basics of String Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4. StringIntersections.............................. 14 2. Cosmic String Evolution 16 2.1. Scaling Solutions: A Simple Picture . . . . . . . . . . . . . . . . . . . . . 16 2.2. TheOne-ScaleModel............................. 17 2.3. The Velocity-dependent One-Scale Model . . . . . . . . . . . . . . . . . . 18 3. Wiggly Cosmic String Evolution 21 3.1. Elastic String Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2. AveragedEvolution.............................. 23 3.3. NetworkDynamics .............................. 27 3.4. PerturbativeLimits.............................. 28 3.4.1. The Tensionless Limit . . . . . . . . . . . . . . . . . . . . . . . . 29 3.4.2. TheLinearLimit........................... 30 3.5. Scaling In The Full Model . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4. Bonus Chapter: Extra-dimensional Analogies! 42 4.1. Strings in Extra Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2. Extra Dimensions and Wiggles . . . . . . . . . . . . . . . . . . . . . . . . 44 5. Conclusions and Further Work 47 A. Multifractal Analysis 49 B. Guessing Energy Loss Terms 52 4 Bibliography 53 5 List of Figures 1.4.1. (figure borrowed from [30]) A loop is produced due to exchange of pairs in (a) a crossing between two colliding strings (b) self-intersection in a singlestring. ................................. 15 3.5.1. Possible scaling regimes in the matter era for increasing values of ηand s= 0. The Abelian-Higgs case corresponds to the intersection with the dashed line (for c= 0.23). .......................... 37 3.5.2. Possible scaling regimes for η= 15 and s=D(1 −X)in the matter era. The Abelian-Higgs case corresponds to the intersection with the dashed line (for c= 0.23)................................ 38 3.5.3. Possible scaling regimes in the radiation epoch for varying values of η and D. The Abelian-Higgs case corresponds to the intersection with the dashed line (for c= 0.23). .......................... 39 3.5.4. Results for η= 17 and D= 0.2in the matter era. The Abelian-Higgs case corresponds to the intersection with the dashed line (for c= 0.23). . 40 3.5.5. Results for η= 17 and D= 0.2in the radiation era. The AbelianHiggs case corresponds to the intersection between the blue points and the dashed line (for c= 0.23). ........................ 41 A.0.1. Schematic picture of a renormalization procedure. `is the renormalization scale below which the details of the original shape (in black) are being smoothed while `0is the length of the renormalized shape with finite curvature (in red). In numeric simulations, the renormalization procedure being used is usually the version which does not assign curvature to the renormalized segments (in green). . . . . . . . . . . . . . . . . . . . . . . 50 6 List of Abbreviations EDVOS Extra-Dimensional Velocity-dependent One-Scale model FRW Friedmann-Robertson-Walker GUT Grand Unification Theories rms root mean square VOS Velocity-dependent One-Scale model 7 E=µ0aˆdσ (1.3.9) The zero-component of eq. 1.3.3 then becomes ˙+ 2˙a a˙x2= 0 (1.3.10) while its i-components give ¨x + 2 ˙a a˙x 1−˙x2=1  x0 !0(1.3.11) Note also that averages in a string can be conveniently defined at the expense of by hQi ≡ ´Qdσ ´dσ (1.3.12) 1.4. String Intersections As the configuration of a string network evolves according to eq. 1.3.3 it often happens that two string segments (either from different strings or from separate locations of the same string) travel towards each other and get very close. Then, clearly, the zerowidth approximation which enables us to write the effective action 1.3.1 breaks down and we momentarily need to consider the full microscopic model in order to know what happens. Fortunately, when this kind of crossing takes place, there are generaly only three possible outcomes (the probability of each happening depends on the particular model we are working with, and can usually only be estimated by numerical means due to the nonlinear complexity of these phenomena) [30]: 1. The segments exchange partners or intercommute, i.e., their ends detach from one segment and get stuck to another at the intersection. This kind of process can often lead to loop production as illustrated in figure 1.4.1. A generic consequence of this phenomenon is the appearance of discontinuities in the direction of the string tangent (called kinks). This is typically the most likely outcome and GotoNambu simulations usually just assume this happens every time segments cross (as is the case for the Abelian-Higgs model). 2. The strings become connected by a bridge which can be point-like (junction) or string-like (zipper), depending on the specific model. This kind of consequence is 14 especially relevant in the study of cosmic superstrings. 3. The segments do not interact and come out of the “collision” intact - by far the least interesting and least likely result. Figure 1.4.1.: (figure borrowed from [30]) A loop is produced due to exchange of pairs in (a) a crossing between two colliding strings (b) self-intersection in a single string. 15 2. Cosmic String Evolution In section 1.3 we discussed the motion of a single Goto-Nambu string. In this chapter, we shall review standard ways to make use of these results in the context of network evolution. 2.1. Scaling Solutions: A Simple Picture The first assumption that is canonically made about cosmological string networks, and which can be seen to hold approximately in high-resolution simulations [23], is that they look like random walks on large scales (i.e., they are Brownian networks). In particular, their large-scale properties are assumed to be described by a single characteristic length Lwhich determines the typical distance between neighbouring strings (moreover, Lis usually also identified with the radius of curvature of strings - yet another assumption which can be tested numerically [25]). On average, there is about one segment of length Lper volume L3, meaning that the energy density of the network can be written as ρ=µ0 L2(2.1.1) As a first approximation, one might be tempted to assume that cosmological expansion would stretch Las L∝a(which essentially corresponds to focusing a limit in which the strings are “frozen” and thus do not interact). That scenario, however, is not cosmologically viable as that would mean ρ∝a−2, implying that the string energy density would decay slower than both radiation and matter (which decay as a−4and a−3, respectively) and thus soon dominate the Universe. Clearly then, the interactions between strings and consequent loss of energy into loops (which are expected to decay both gravitationally and into smaller loops) are of paramount importance when studying the fate of a network. Interestingly, when these losses into loops are taken into account, they are expected to balance the evolution equation for ρin such a way that the network approaches a 16 scaling regime with Lproportional to the horizon size dH∝t. This kind of asymptotic behaviour is observed in simulations as well as in the standard models for network evolution which will be discussed in subsequent sections. This is because a feature which is typically taken into account in these models (and which, once again, is in reasonable agreement with numerical simulations) is that the energy loss into loops goes as dρ dt loops ∼ −µ0 L3=−ρ L(which roughly corresponds to having a segment travel a distance of Lbefore encountering another segment and producing a loop of size L). This means that the evolution equation for ρcan be written approximately as the sum of a contribution from eq. 1.3.10 with this term, becoming dρ dt ≈ −2Hρ −ρ L(2.1.2) which can be rewritten in terms of γ≡L/t and λ=Ht ≡t a da dt (meaning λ= 1/2in the radiation era and λ= 2/3in the matter era) as 1 γ dγ dt =1 2t2 (λ−1) + γ−1(2.1.3) which has the scaling solution γ=1 2 (1 −λ)(2.1.4) as an attractor. 2.2. The One-Scale Model The One-Scale model, due to works by Kibble [16] and Bennet [6] in the mid-eighties, is among the first efforts to describe the cosmological evolution of a system of cosmic strings and make some quantitative predictions about the attractor scaling solution. Even though it is not the most accurate model available, it is worth some consideration since its basic assumptions are the starting point of the standard more reliable alternatives (such as the velocity-dependent one-scale model, which will be discussed in section 2.3). This model is essentially a more rigorous version of the picture in section 2.1 - one in which both the average rms velocity of strings and energy losses into loops are taken into account. The main simplification in this model (and the one in the origin of its name) is that the only relevant length scale in a string network is the characteristic length L. In particular, the fraction of energy lost into loops of size between land l+dl per correlation volume is determined by some loop production function f(l/L)(which 17 we write as an undefined function we expect to peak around l=L, but whose exact shape can be sought numerically) so that the energy loss into loops becomes dρ dt loops =−µ0 L3 ∞ ˆ 0 f(l/L)dl L≡ −cρ L(2.2.1) If to this we add the contribution from eq. 1.3.10 (this time not neglecting the velocity) the analogue of eq. 2.1.2 becomes dρ dt =−2Hρ 1 + v2−cρ L(2.2.2) where v2is just the rms velocity of the strings defined as v2≡D˙x2E≡´˙x2dσ ´dσ (2.2.3) which is treated as a constant in this equation (depending only on whether the Universe is dominated by radiation or matter). As expected, there is an attractor scaling solution given by (once again making γ≡L/t and H=λ/t) γ=c/2 1−λ(1 + v2)(2.2.4) (note that cand v2should both depend on λ). 2.3. The Velocity-dependent One-Scale Model As has been mentioned before, the One-Scale model is not the most accurate of models. That is because it fails to capture some important physical phenomena due to two main limitations: the fact that the rms velocity is being considered a constant, which is not expected to hold but in a scaling regime, and the fact that it is completely “blind” to the presence of small-scale structure in the network (i.e., kinks and wiggles formed by intercommutation). The latter is a more fundamental difficulty in the sense that it is an unavoidable consequence of the assumption that there is only one relevant length scale L, so we will postpone its discussion until chapter 3, where we focus on original work on how to get around this problem. The former, however, is basically due to a purely mathematical simplification which can be dropped if the resulting additional terms can all be written in terms of vand L. This can be achieved by noting the evolution of vin 18 eq. 2.2.2 depends on ¨x ·˙x and using the identities (see [19]) 1 (1 −˙x2) x0 !0·˙x =−x0·˙x0 x02=a R(˙x ·ˆu)(2.3.1) where ˆu is the unit vector parallel to the curvature radius vector. The resulting system of equations is then (after identifying L=Rand substituting c for cv to take into account that the rate at which intersections occur is proportional to the velocity of the string segments involved)      2dL dt = 2HL (1 + v2) + cv dv dt = (1 −v2)hk(v) L−2Hvi(2.3.2) where k, called the momentum parameter, is defined as k=h(1 −˙x2) ( ˙x ·ˆu)i v(1 −v2)(2.3.3) and in most relevant regimes (both relativistic and non-relativistic) can be written as (see [22]) k(v) = 2√2 π 1−8v6 1+8v6(2.3.4) In doing so, one ends up with a model that retains most of the simplistic features of the one-scale model but is still valid in regimes with varying v(notably in frictiondominated regimes) and can thus be used to make predictions across the whole history of a network (i.e., there is no need to change parameters as we change cosmological eras). In particular, we find the attractor scaling solution      γ2=k(k+c) 4λ(1−λ) v2=k(1−λ) λ(k+c) (2.3.5) when a∝tλ(note that the second equation is an implicit equation for the velocity which fixes k). Nevertheless, scaling solutions do not seem to exist when the scale factor is not a power law - as during the transition between the radiation and the matter epoch or after the onset of dark energy domination (around the present time). During these transition epochs, eqs. 2.3.2 have to be treated numerically - and it is found that realistic cosmic string networks should not have enough time to reach a scaling regime during the matter-dominated era [22]. 19 It is this improved version of the one-scale picture, first developed by Martins and Shellard [19, 20, 21], which is commonly known as the Velocity-dependent One-Scale model (VOS). Despite its simplicity, it performs remarkably well when tested against high-resolution simulations [25], which makes its use the most reliable method when it comes to making quantitative predictions about the evolution of the large-scale properties (i.e., Land v) of a network. It is also interesting to note that, even though Lis still the only length scale playing a role in this model, the momentum parameter, as defined in eq. 2.3.3, clearly depends on the shape of small-scale structure. 20 3. Wiggly Cosmic String Evolution As was briefly mentioned in section 2.3, one of the most serious problems with models assuming a one-scale approximation is that they are intrinsically limited in how much they can tell us about what happens in length scales below the characteristic length L. This problem is especially relevant since it has been realised that realistic networks should build up a significant amount of structure on these scales - mainly as a consequence of kinks formed by intercommutations [3, 23]. There have been previous attempts to tackle this problem. A significant step was taken by Kibble and Copeland in 1991 with the introduction of a two-scale model [15] which successfully shed some light into the fate of wiggly networks in spite of a few quantitative shortcomings (briefly addressed in [30]). Subsequently, the authors, together with Austin, suggested a three-scale model [4] to overcome these difficulties - however, the additional degree of freedom this implied seriously compromised the predictive power of the model. With this in mind, in this chapter we present a generalisation of the VOS model which explicitly takes into account the evolution of small-scale structure in the network while still preserving the main victories of the VOS. The basic formalism introduced herein is the result of original work done in collaboration with C. J. A. P. Martins (the supervisor of this thesis) and E. P. S. Shellard which can be found in [24]. 3.1. Elastic String Dynamics In the zero-width approximation, a generic string model can be defined by an action S=−ˆL√−γd2ζ(3.1.1) where the Lagrangian density can depend on the background metric gµν, background fields such as Maxwell-type gauge potentials Aµor a Kalb-Ramond gauge field Bµν (but not their gradients) and relevant internal fields encoded in a function Λ(usually called 21 the master function) [10, 9] L= Λ + JµAµ+1 2WµνBµν +... (3.1.2) It is easy to see that the action for the Goto-Nambu strings we have focused on can be obtained by simply setting Λ = −µ0and making all else null. Superconducting strings, for example, correspond to the case in which the Maxwell field is important while the term with the Kalb-Ramond field is ideal for describing global strings (i.e., strings arising from the breaking of a global symmetry, which typically have long-range interactions mediated by the massless Goldstone boson associated with this breaking). A well known feature arising from the Goto-Nambu action is that the string tension and energy density coincide U=T=µ0(3.1.3) An elastic string model is one for which Uand Tdo not necessarily coincide (and can even vary). These models can be described [9, 10] by a function Λ = Λ (χ)where for a simple case in which external fields are not relevant (such as the one we are interested in) χ≡γabφ,aφ,b (3.1.4) φbeing a scalar field identified with a stream function defined on the worldsheet which is constant along the flow lines of a conserved current. At this point, the attentive reader might be wondering about the relevance of this little detour. If we are only interested in Goto-Nambu strings, why should we look at these more general models? The answer is related to the property 3.1.3 of Goto-Nambu strings: if we look at a very wiggly Goto-Nambu string from afar, we are not able to make out the small-scale details of its configuration - instead, it seems as though we are looking at a much smoother string which happens to have a much higher energy density and a much lower tension! This means that a macroscopic description of a wiggly Goto-Nambu string is equivalent to the microscopic description of an elastic string. In particular, it makes sense to think of wiggly strings as carrying a mass current which “renormalizes” the bare energy per unit length µ0. This interpretation motivates the use of the Lagrangian density L=−µ0q1−γabφ,aφ,b (3.1.5) 22 which has the equation of state UT =µ2 0(3.1.6) which has been shown to be the exact equation of state for a macroscopic (averaged) wiggly Goto-Nambu string [8, 18]. Consistently with this physical interpretation, φis treated as a mesoscopic quantity which depends only on the worldsheet time and some “renormalization scale” `which must be large enough for the spatial dependences of U to be negligible but still much smaller than the correlation length. It is useful to introduce the dimensionless parameter w≡√1−χ(3.1.7) in terms of which the local string tension and energy density can be written: T=µ0w, U =µ0 w(3.1.8) Note that wmust be between 0and 1(w= 1 corresponding to the Goto-Nambu case). The equations of motion now become ˙  w+ w˙a ah2w2˙x2+1 + ˙x21−w2i= 0 (3.1.9) ¨x +˙x 1−˙x2˙a a1 + w2=w2  x0 !0(3.1.10) ˙w w=1−w2 ˙a a+x0·˙x0 x02!(3.1.11) where 3.1.9 is the general (wiggly) form of 1.3.10, 3.1.10 is the general form of 1.3.11, and 3.1.11 comes simply from varying the action with respect to φand using the definition of w. 3.2. Averaged Evolution There are now two independent measures of energy one can work with. These are the total energy in a string segment E=aˆUdσ =µ0aˆ wdσ (3.2.1) 23 In the general case with a varying renormalization scale ˙ E0 E0∼ − ˙µ µ∼˙a a−[dm(`)−1] ˙ ` `(3.4.5) which generically makes µdecay slower than a−1. In fact, if we were to assume a simple fractal network (i.e., with a constant dm) we could find some suggestive solutions for a few natural choices of `. Following a scale proportional to the scale factor, for instance, would result in µ∝adm−2, ξ ∝a(1+dm)/2(3.4.6) while following a scale proportional to the correlation length would lead to µ∝a4/(3−dm)−3, ξ ∝a2/(3−dm)(3.4.7) It is pertinent to note that 3.4.6 implies that µ(and thus E0) is constant for a large-scale Brownian network (dm= 2) while 3.4.7 means this only happens when dm= 5/3(which, interestingly, is the fractal dimension of a self-avoiding random walk [29]). 3.4.2. The Linear Limit The most useful perturbative limit is without a doubt the linear limit, when there is only a little small-scale structure and therefore both wand µare very close to 1which simulations predict should be the case [23]. At the mesoscopic level we just define w= 1 −y(3.4.8) where y1; macroscopically this corresponds to µ∼1 + hyi ≡ 1 + Y(3.4.9) where Yis also very small and positive. Curiously, there is now a more intuitive relation between the two averaging procedures hQi ∼ ´Q(1 + y)dσ (1 + y)dσ ∼ hQi0+corr0(y, Q)(3.4.10) (where the correlator is just corr0(z, w) = hzwi0−hzi0hwi0) and in particular they are 30 equivalent for quantities independent of w. Assuming that is the case for ˙x2then Dwα1˙x2α2E∼(1 −α1Y)v2α2(3.4.11) which greatly simplifies our evolution equations. In what follows we assume k(v)maintains its standard form (eq. 2.3.4). This assumption is related to the ubiquitous expectation that our undefined renormalization procedure should transform our complex strings in strings for which the VOS assumptions apply. Still, since we know that the VOS assumptions are not always obeyed everywhere in realistic networks, a generalization of eq. 2.3.4 might be needed to study relativistic scenarios with very high wiggliness. Linearising the averaged evolution equations (3.2.13, 3.2.18, 3.2.19, and 3.2.20) one finds ˙ E E=h1−2v2−2Y1−v2i ˙a a(3.4.12) ˙ E0 E0 =h1−2v2+ 2Y v2i˙a a−2kaY v R−[dm(`)−1] ˙ ` `(3.4.13) ˙ (v2)=2v1−v2"ka R(1 −2Y)−2v(1 −Y)˙a a−[dm(`)−1] 2v ˙ ` `#(3.4.14) ˙ Y= 2Y kav R−˙a a!+ [dm(`)−1] ˙ ` `(3.4.15) Finally, switching from conformal to physical time and introducing the energy loss terms discussed in section 3.3, we end up with the following generalised (linear wiggly) VOS model evolution equations 2dL dt = 2 h1 + v2+Y1−v2iHL +cfv 1−1 2Y(3.4.16) 2dξ dt = 2 h1 + (1 −Y)v2iHξ + [2kY +c(f0+s)] v+ [dm(`)−1] ξ ` d` dt (3.4.17) dv dt =1−v2"k ξ(1 −2Y)−2Hv (1 −Y)−[dm(`)−1] 2v` d` dt#(3.4.18) dY dt = [2kY +c(f0+s−f)] v ξ−2HY +[dm(`)−1] ` d` dt (3.4.19) which naturally reduce to the VOS equations for Y= 0 (when necessarily dm= 1). Recall that f0,f, and sare in principle functions of Ysuch that f0+s−fis linear in Y(just like dm−1). In order to demonstrate the utility of this simplified model, an analysis of its scaling 31 solutions can be carried out. Given the range of natural choices of `we may resort to, it makes sense to consider `∝tin addition to ξ=γξt,v=const, and Y=const. Moreover, we assume dm∼1 + 2Y, which numerical simulations [23] suggest may be a reasonable approximation in expanding space. As expected the resulting solution for (γξ, v)is only a linear perturbation in Yaround the VOS solution (γGN , vGN )which can be found by solving the system 2.3.5. It can be obtained (for c6= 0) by solving the algebraic system              v2 v2 GN ∼1 + βY γ2 ξ γ2 GN ∼1 + β+ 2B−A+2λ(1+v2 GN )(β−1) 1−λ(1+v2 GN )Y Y∼2(λ−1)[(2A+1)k+(A+1)c] [1−2A+2D+(A−2D)λ]k+[2D(1−λ)−A(2−λ)]c (3.4.20) where β=cλ(2−A−2B)−2k(1−2λ) 2λ(k+c),a∝tλas before, and the energy loss terms have been conveniently rewritten as            f0+s−f∼AY f0+s+f∼2 (1 + BY ) f0+s∼1 + DY (3.4.21) Focusing on the third equation in the system 3.4.20, we can use the physical requirement that Ybe positive to impose constraints on the linear term in the expansion of s(Y)and even on Aand B. For example, if we assume eqs. 3.3.6 and 3.3.7 (meaning A∼ −1/2 + D,B∼1/4 + D/2, and DY ∼s), then Y∼2 (λ−1) [4kD + (1 + 2D)c] (4 −[1 + 2D]λ)k+ (2 −λ(1 + 2D)) c(3.4.22) and if k > 0(which is favoured by simulations [23]) requiring Yto be positive is equivalent to imposing (recall that Dmust also be positive) D > (4 −λ)k c+ 2 −λ 2λk c+ 1(3.4.23) while demanding that Y < 1implies D < 3 + k cλ−4k c+ 1 4+8k c−10k c+ 2λ(3.4.24) 32 and surprisingly these two conditions are incompatible in both the matter and the radiation epoch (considering c= 0.23, as found in numerical simulations of Abelian-Higgs strings [25]), which shows that it is not necessarily trivial to find natural energy loss parameters which enable small-scale scaling - and also that eqs. 3.3.6 and 3.3.7 may not be valid, since simulations suggest small-scale scaling can happen in the linear limit [23]. A more complete discussion of the consistency relation the energy loss functions must obey is left for section 3.5. 3.5. Scaling In The Full Model First of all, let us write the full evolution equations for a network with uniform wiggliness at a scale `(t). 2dL dt =HL "3 + v2−(1 −v2) µ2#+cfv √µ(3.5.1) 2dξ dt =Hξ "2 + 1 + 1 µ2!v2#+v"k 1−1 µ2!+c(f0+s)#+ [dm(`)−1] ξ ` d` dt (3.5.2) dv dt =1−v2"k ξµ2−Hv 1 + 1 µ2!−1 1 + µ2 [dm(`)−1] v` d` dt#(3.5.3) 1 µ dµ dt =v ξ"k 1−1 µ2!−c(f−f0−s)#−H 1−1 µ2!+[dm(`)−1] ` d` dt (3.5.4) As before, we assume the multifractal dimension can be written as a function of µ. For expanding space, simulations [23] suggest dm(`)=2−1 µ2(3.5.5) which, as expected, approaches the Brownian case (dm∼2) at large scales. Once again the scaling solutions2can be found by solving an algebraic system (for µ6= 1)              v2=[4X2−2λX(1+X)](k/c)−X(1−X)(f0+s) λ(1+X)2[(k/c)+f0+s] γξ=vk(1−X)+c(f0+s) 1+X−λ[2+(1+X)v2] v γξ[k(1 −X)−c(f−f0−s)] + (1 −λ) (1 −X)=0 (3.5.6) where we have used the useful quantity X≡1/µ2. 2Consistently with our previous approach, we are assuming that `is also scaling. 33 Now we are interested in answering the general question: what conditions must f, f0,s, and cbe subjected to so that scaling solutions exist? If we are only interested in mathematical existence (i.e., if we are not worried about non-physical results such as speeds above the speed of light) then there is an unexpected result: for any specific shape of f,f0, and s, given any X, there exist at most two values of c,cX, such that there is a scaling solution with that constant value of X. It is easy to find such a cX, if it exists: first just compute v2=[4X2−2λX (1 + X)] ϕX−X(1 −X) (f0+s) λ(1 + X)2[ϕX+f0+s](3.5.7) where ϕXis a real solution of the quadratic equation Aϕ2 X+BϕX+C= 0 (3.5.8) where A= (1 −λ) (1 −X)1−X2−(1 −X)h4X2−2λ(1 + X)Xi+1−X2[1 + X−2λ] (3.5.9) B= (1 −λ) (1 −X2) (2 −X) (f0+s)+(f−f0−s) (4X2−2λ(1 + X)X) + + (f0+s)X(1 −X)2+ [(f0+s) (1 −X)−f+f0+s]h(1 + X)2−2λ(1 + X)i (3.5.10) C= (f0+s)2(1 −λ)1−X2−(f0+s) (f−f0−s)hX(1 −X) + (1 + X)2−2λ(1 + X)i (3.5.11) (of course, if there are no real solutions to 3.5.8 that just means that scaling is impossible for that X), then compute k(v)using eq. 2.3.4 and the cXwe are after is simply cX=k(v) ϕX (3.5.12) if it is positive and less than 1(otherwise there is no scaling). In order to understand what this purely mathematical consistency condition implies, let us focus on a specific kind of shape for the loop-chopping functions. In the absence of solid evidence in favour of a specific shape3(and given that the discussion in the previous section shows that the “guesses” 3.3.6 and 3.3.7 may not be the most convenient to use if we are interested in studying scaling), we illustrate the kind of analysis that can be carried out using the following family of loop-chopping functions (see appendix B for a 3Which, as has been mentioned, should in principle be obtainable from high-resolution simulations. 34 heuristic derivation)      f0= 1 f= 1 + η1−1 √µ(3.5.13) where ηis some positive number which is related to how much energy is lost by the string to small-scale loops. Since the equations we are solving are simple but rather lenghty, we choose to do our analysis simply by computationally solving them for a uniform sampling of the interval X∈(0,1) (testing about 1000 points for each η). Firstly, let us consider only the matter era (which is when simulations seem to suggest it is the easiest to achieve scaling [23]), with λ= 2/3. For the sake of simplicity let us assume that s= 0 for the time being. The corresponding results are summarised in figure 3.5.1. In this era only one type of solution seems to be acceptable (i.e., results in physical values of c): the solution associated with the greater root of eq. 3.5.8 (called cX+in the plots). Even this kind of solution does not seem to be physical for η.7 (note that η= 1 makes eqs. 3.5.13 equivalent to eqs. 3.3.6 and 3.3.7). However, as we increase η, there is a clear tendency: the cXnecessary to have scaling with a “fixed” X decreases (which makes sense if we interpret an increase in µas an increase in energy lost to small-scale loops) and scaling becomes possible for increasing values of µ(smaller X). Based on available simulations [23], which predict an increasing µduring the radiation era, one would expect this kind of tendency to remain in this epoch - since for scaling to occur in these simulations it would be necessary for the small-scale component of the network to lose more energy in the form of small-scale loops. Interestingly, it does not seem to be possible to have scaling for µ&2.2(X. 0.2), suggesting that networks in the matter era cannot stabilise in the tensionless limit regardless of how much energy they lose to loops (although we shall see that accounting for kink formation by intercommutation, via s6= 0, can change this picture). The effect of adding s > 0is not so straightforward. Supposing (for simplicity) a simple linear dependence s=D(1 −X)(with D > 0) and fixed η= 15 (which seems to give reasonable results for the Abelian-Higgs model, when c∼0.23 [25]) we find that at first an increase in Dseems to have the same effect as a decrease in ηin the previous discussion, until scaling becomes impossible after D∼2. After that, however, for D∼5, scaling becomes possible again and increasing Dnow seems to have an effect similar to increasing ηin the case with D= 0. This behaviour is captured in figure 3.5.2. It is also worthwhile to note that we can now have scaling closer to the tensionless limit. Looking at the radiation era (λ= 1/2) we find the same kind of qualitative behaviour 35 with a few important differences. Firstly, there is now a second kind of solution, which is rather insensitive to ηand is consistently suppressed by an increase in D(and even for D= 0 only applies to values of cbelow 0.2). Secondly, the solution we were following earlier does not allow scaling near the Goto-Nambu limit unless Dis high enough4 (D∼1.7). And finally, there does not seem to exist any kind of mechanism stopping us from getting scaling in the tensionless limit. These results can be seen in figure 3.5.3. If we are interested in physical solutions then we must also impose 0< v2<1and 0< γξ<1 1−λ(where the last inequality just expresses the causal requirement that the correlation length must be below the horizon length). These requirements naturally discard some of the situations permitted above. In particular, γξseems to behave in a rather chaotic manner for the cX−solution5. Since the specifics of the dynamics of each of these quantities make this analysis much more complex, and keeping in mind that our goal in this section is simply to illustrate the new paths this formalism uncovers and provide some intuition regarding the physical processes taking place, we will not pursue a complete analysis. Instead, we just show the results for a specific choice of energy loss functions which seems to give plausible results for c∼0.23 (and which was “guessed” using the intuition from the discussion so far): η= 17,D= 0.2. Figure 3.5.4 relates to the matter era while figure 3.5.5 is about the radiation epoch. Naturally, future work will have to focus on determining conditions of stability for these scaling regimes. One interesting feature which may also warrant further study is the fact that some values of cseem to admit more than one solution (we have seen some examples with three possible solutions in the radiation era, but there is even one example of two possible solutions in the matter epoch - see figure 3.5.2 for D= 5). It may be that the existence of more than one solution to a given model gives rise to “cross-over” phenomena which would be interesting to study - in particular, it would be interesting to determine whether a clever choice of parameters could relate a momentary stabilisation of µin radiation era simulations [23] to this kind of behaviour. 4Note that this kind of feature is especially welcome since simulations seem to suggest small-scale scaling in the matter era but not in the radiation era. Bearing this in mind, one would expect Dto be small enough to retain it. 5This is probably at least partly due to numerical errors. Since both the numerator and the denominator of the second equation in 3.5.6 are rapidly oscillating about zero with very small amplitudes, there is not a simple way to present these results free of these errors. 36 Figure 3.5.1.: Possible scaling regimes in the matter era for increasing values of ηand s= 0. The Abelian-Higgs case corresponds to the intersection with the dashed line (for c= 0.23). 37 Figure 3.5.2.: Possible scaling regimes for η= 15 and s=D(1 −X)in the matter era. The Abelian-Higgs case corresponds to the intersection with the dashed line (for c= 0.23). 38 Figure 3.5.3.: Possible scaling regimes in the radiation epoch for varying values of η and D. The Abelian-Higgs case corresponds to the intersection with the dashed line (for c= 0.23). 39 strings can also be translated into a correspondence between wiggly strings in the linear limit and superconducting strings with zero charge in a limit with φ022˙ φ2. These kinds of analogies are a good way to gain insight about particular limits of very complex models. In this case, they might be particularly useful since simulations with superconducting or wiggly strings are much easier to run than simulations with strings in extra dimensions. Further work should clarify how much this connection allows us to learn about extra-dimensional strings from the study of wiggly (or realistic Goto-Nambu) strings and vice-versa. 46 5. Conclusions and Further Work In this thesis, we have successfully developed a novel formalism to model cosmic string network evolution. Building upon solid work already established in the literature (notably the foundations of the VOS laid down by Martins and Shellard [19, 20, 21, 22] and the rigorous work on elastic string models carried out by Carter and Martin [8, 9, 10, 18]), we have managed to devise a relatively simple model which retains the main advantages of the VOS while enabling us to describe the evolution of small-scale structure in the network. Naturally, bringing out the full potential of this new approach will take further work. In particular, it is important to use simulations to gain a more quantitative understanding of the way energy is lost by the small-scale component of the network - as we have shown how the details of this process determine which kinds, if any, of small-scale scaling regimes are possible. In the end we also point at an unexpected connection between elastic and extradimensional strings as a potential way of using our model to learn more about different kinds of string models. Exploring exactly how that can be done is the subject of work in progress. 47 Appendices 48 A. Multifractal Analysis Given an arbitrary one-dimensional shape and some undefined renormalization procedure (that smoothens it so that the renormalized shape has no structure below some predefined renormalization length scale), its multifractal dimension at a scale `,dm(`), is defined as [29] dm(`) = dlog M(`) dlog `(A.0.1) where M(`)is the length of the renormalized figure for a renormalization length `. This quantity generalizes the usual notion of fractal dimension in the sense that it allows us to assign scale-dependent fractal dimensions to physical objects which cannot have a “normal” fractal dimension because physical constraints limit the minimum size of structure - dm(`)can be thought of as the fractal dimension a shape seems to have when we cannot make out details whose scale is below `. The definition of fractal dimension that is usually used in these kinds of problems can be simply recovered by D= lim `→0+dm(`)(A.0.2) which is trivially unity for any physical one-dimensional shape. As `increases, however, complex shapes which approximate fractal structures should see an increase in this value. In particular, for the case of cosmic strings, the fact that they look Brownian on large scales implies that at large scales dm(`)should approach 2. For now, let us keep in mind the simple renormalization procedures illustrated in figure A.0.1 when thinking about cosmic strings. The algorithm is simple: pick a point on the string, find the first point at a distance `(first in the sense that you will reach it first if you are traveling on the string starting from the point you have picked), and substitute the segment uniting them for a simpler smoother one. In our case, since it is convenient for the network to have a characteristic finite curvature R, we should choose a segment with constant curvature R(as shown in red) - the straight (green) case is shown because numeric simulations usually use that kind of simpler renormalized shape to calculate µ, but we will see that our results are not significantly affected by that choice provided 49 that `is much smaller than R(as it should in order for us to be modeling small-scale structure). Figure A.0.1.: Schematic picture of a renormalization procedure. `is the renormalization scale below which the details of the original shape (in black) are being smoothed while `0is the length of the renormalized shape with finite curvature (in red). In numeric simulations, the renormalization procedure being used is usually the version which does not assign curvature to the renormalized segments (in green). We can define the “computational” energy renormalization factor ¯µ≡M(`) `=E/µ0 `(A.0.3) (where Eobviously refers to the segment, not the network) and it is trivial to verify that ` ¯µ ∂¯µ ∂` =dm(`)−1(A.0.4) Using µ= ¯µ` `0(A.0.5) and the relation between `and `0(which can be found using the fact that the red segment in figure A.0.1 is an arch with radius R) sin `0 2R=` 2R(A.0.6) 50 we can write µ= ¯µ 1 + O   ` R!2   (A.0.7) which then implies ` µ ∂µ ∂` =` ¯µ ∂¯µ ∂` +O   ` R!2   =dm(`)−1 + O   ` R!2   (A.0.8) which is basically eq. 3.2.17. 51 B. Guessing Energy Loss Terms A discussion of the full wiggly model requires some assumption regarding the form of the energy loss functions. As has been said, it should be possible to investigate the dependence of these functions on µby computational means. For the purpose of this discussion, though, we propose a plausible ansatz for the loop-chopping terms which relies on the same kind of logic as 2.2.1. Firstly, we argue that the argument used in the writing of 2.2.1 should also apply to the smooth-looking renormalised string. Therefore the loop terms in the equation of ρ0should be the same as their VOS counterparts, with the correlation length being identified with ξ. This means1 dρ0 dt loops =−µ0v ξ3 ∞ ˆ 0 g(l/ξ)dl ξ≡ −cvρ0 ξ(B.0.1) and thus f0(µ)=1 (B.0.2) If the total energy lost to loops were merely the energy contained in the large loops (with size ∼ξ) considered to carry the energy lost by the bare string, then the analogue of eq. B.0.1 for ρcould be written simply multiplying everything by µand in the end we would have f(µ) = 1 as well. However, when a loop is formed by intercommutation in a wiggly string, there is usually the possibility that a class of much smaller loops will be formed - for that reason, for µ > 1, one would expect f(µ)>1. The typical length of these smaller loops, we conjecture, can be related to a characteristic length scale which can be written as a combination of Land ξsuch that it is zero when L=ξand clearly the simplest scale with these properties is simply ξ?=ξ−L. Bearing this in mind, we 1Notation warning: we are now noting the loop production function as g(l/ξ)instead of f(l/ξ)to avoid confusion with the energy loss function f(µ). 52 try the substitution g(l/ξ)→g(l/ξ) + g?(l/ξ?)and get dρ dt loops =−µ0µv ξ3 ∞ ˆ 0 g(l/ξ)dl ξ−µ0µv ξ3 ∞ ˆ 0 g?(l/ξ?)dl ξ≡ −cvρ ξ"1 + ηξ? ξ#(B.0.3) (where ηis a positive number defined as η=c−1´g?(x)dx), which is clearly only acceptable in the Goto-Nambu limit if ξ?→0+as µgoes to unity. Using the simple ξ?=ξ−Lthis leads to f(µ) = 1 + η 1−1 √µ!(B.0.4) which by eq. 3.3.4 implies f1(µ) = 1 −η√µ √µ+ 1 (B.0.5) Incidentally, in the linear limit f(Y)∼1 + η 2Yand if η= 1 this ansatz becomes indistinguishable from eqs. 3.3.6 and 3.3.7. This derivation, we emphasise, is everything but rigorous. However, faced with the lack of a rigorously obtained set of energy loss functions, we use these to illustrate the kind of useful calculations that can be carried out with our wiggly formalism. If further study unveils a more reliable set of functions, all we have to do is redo the calculations in section 3.5 with the corresponding substitutions. 53 Bibliography [1] P.A.R. Ade, W. Aikin, R. D. Barkats, J. Benton, S. A. Bischoff, C. J. Bock, J. A. Brevik, J. I. Buder, E. Bullock, D. Dowell, C. L. Duband, P. Filippini, J. S. Fliescher, R. Golwala, S. M. Halpern, M. Hasselfield, R. Hildebrandt, S. C. Hilton, G. V. Hristov, V. D. Irwin, K. S. Karkare, K. P. Kaufman, J. G. Keating, B. A. Kernasovskiy, S. M. Kovac, J. L. Kuo, C. M. Leitch, E. M. Lueker, P. Mason, B. Netterfield, C. T. Nguyen, H. R. O’Brient, W. Ogburn, R. A. Orlando, C. Pryke, D. Reintsema, C. S. Richter, R. Schwarz, D. Sheehy, C. K. Staniszewski, Z. V. Sudiwala, R. 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