Topological Defects Formation
According to the Hot Big Bang Model, which describes the history of the Uni-verse from the initial instant (“explosion”) to the present days, the Universe started about 14 billion years ago and since then it has expanded and cooled down. During the earliest times of evolution the existing expansion rate sce-nario had been interrupted by an inflation era, that is a period of very rapid (exponential) expansion where the Universe was dominated by a vacuum en-ergy density. To understand how the Universe has expanded one may study the thermodynamics of the CMB. In the past, the CMB radiation was much hotter and then dominated the gravity of the Universe. This period is known as radiation era. However, as time went on, the CMB cooled down and decreased energy. At some point, the radiated energy became equal to the energy of the ordinary matter (matter-radiation equality). At present, we live in the matter era, where the gravity of all matter in the Universe is much greater than the radiation gravity. Therefore, by measuring the content of the energy of the matter and radiation of a large region of space, we are able to understand how this region expands with the Universe [HH05]. Incidentally, observations of the anisotropy spectrum of the CMB have been mentioned as evidence for sup-porting the idea of an accelerating Universe [Spe+03]. With the expansion of the early Universe it has undergone a succession of phase transitions involving spontaneous symmetry breaking mechanisms which may have led to formation of cosmic topological defects [KT90]. To better understand the general idea of a SSB phase transition let us to have a look at a very common example, the freezing of water to ice5. When the water is still a liquid at some tempera-ture it is constituted of translational and rotational symmetry (high symmetry). However, once the water temperature cools down it gets frozen and such symme-tries are broken. That is, the symmetry of the system is reduced to the discrete symmetry (low symmetry) of the ice crystal. During such process, the crys-talline orientations of the ice are led to be diﬀerent in diﬀerent parts, so that some “defects” are then formed. At this point, we recognize a strong connec-tion between particle physics and cosmology, since the Standard Model (SM) of particle physics is mostly based on the concept of symmetry breaking. Indeed, the early universe has been often studied as a manner of testing the ideas of the SM at non-accessible energy scales to terrestrial accelerators. By the way, the SM has been tested to a very high precision and shows up that below an energy scale MGUT ∼ 1016 GeV [Wei08], the electroweak interactions (weak and electromagnetic forces), represented by the symmetry group SU(2) ×U(1) and the strong interactions associated with the gauge group SU(3) can be unified, forming what is called Grand Unified Theory. The idea here is that in the early times (close to the singularity) of the Hot Big Bang model, when the universe was at the highest temperature, all interactions were merged into a single one. Actually, gravity is the only force that does not make part of the GUT; besides being a geometric theory, at the nuclear scale, such interaction is insignificantly weak as compared to the others fundamental forces. However, at the Planck energy scale (MPl ∼ 1019 GeV ), the gravitational interaction becomes as strong as the others forces [Kib97]. Above MGUT, strong and electroweak interactions unify within a larger gauge symmetry group G, where grand unified theories involving Supersymmetry (SUSY) have been considered as suitable description for such energy scales [Baj+04; Fuk+05; Rab11]. For such reason, SUSY GUT arises as a possible way to unify all the four fundamental forces of nature in a single one (this unification is often called as Theory of Everything) [Kib97].
As we have mentioned before, the ideas of the GUT are generally speak-ing based on the notion of SSB phase transitions: a system represented by a high symmetric group G is spontaneously broken to a subgroup H with less symmetry, G → H → ···SU(3) ×SU(2) ×U(1) → SU(3) ×U(1)em.
Topological Classification of Defects
In order to determine what kind of topological defect emerges for a given SSB transition G → H, one may study the content of homotopy groups πk(G/H) of the vacuum manifold M = G/H, since the defect to arise is strictly determined by the topology of M. When the vacuum manifold M has a non-trivial topol-ogy, is multiply connected, πk(G/H) 6= 1 (1 corresponds to the trivial topology), stable topological defects of dimension6 2 − k will appear with a characteristic length scale of the size of the correlation length ξ [VS94; JRS03; Ken06].
Below, we present further details about domain walls, monopoles and tex-tures, where the type of non-trivial mapping of the vacuum manifold M in each defect is specified. Cosmic strings being the main topic of this manuscript will be left for a more extensive presentation in Section 2.3 and Chapter 3.
Observational Evidences of Cosmic Strings
There are many theoretical justifications for the existence of cosmic strings, but the observational evidences are still feeble and mostly indirect. Nevertheless, if cosmic strings do exist, important cosmological eﬀects should be observed as consequence. Among the possible ways of observing their presence in the Universe are: the gravitational lensing signatures (see Section 3.2.1) and grav-itational waves left behind by such objects. The presence of a cosmic string somewhere in the Universe aﬀects light trajectories, forming double images of objects behind the string. Such eﬀect is known as planar gravitational lensing and has not been detected yet. Curiously, the observation of a pair of giant elliptical galaxies was erroneously reported by Sazhin et al. [Saz+03], as de-tection of lensing signatures induced by a cosmic string. However, high quality data from the Hubble space telescope showed that was not a lensing eﬀect (two images of the same galaxy), but a pair of similar galaxies [AHP06; Saz+06]. Another possibility for detecting cosmic strings is through observation of gravi-tational waves, since the oscillating string loops created as a mechanism for loss of energy by strings emit gravitational radiation. Infinite cosmic strings also emit gravitational radiation as the small-scale structures (wiggles) on the string are source of gravitational waves as well [VV85; VS94; HK95; BPOS18]. Cosmic strings can also be observed by formation of a wake of matter behind moving strings. Actually, this is an important fact that may corroborate the density inhomogeneity in the Universe. However, maybe the most suitable manner to observe strings is through the measurements of the induced anisotropies in the cosmic microwave background radiation [Kib97; CPV11]. For example, the con-ical geometry of a cosmic string may cause discontinuity in the temperature of the cosmic microwave background as a string moves through the space. Such phenomenon is known as Kaiser-Stebbins eﬀect [KS84].
Data on the CMB collected from Planck Satellite have not confirmed the existence of these objects yet, but they have set upper boundaries on their mass-energy density Gµ0 < 10−7 (c = 1) [Ade+14]. As warned by Copeland and Kibble [CK10], “Both cosmic strings and superstrings are still purely hypo-thetical objects. There is no direct empirical evidence for their existence, though there have been some intriguing observations that were initially thought to pro-vide such evidence, but are now generally believed to have been false alarms. Nevertheless, there are good theoretical reasons for believing that these exotic objects do exist, and reasonable prospects of detecting their existence within the next few years.” Indeed, the search for cosmic strings is currently still very active and it happens in all the fronts mentioned above: CMB radiation mea-surements [Her+17] and gravitational wave bursts [SES17].
It is worth mentioning here that topological defects found in CMP systems have been explored as a laboratory for better understanding of the ideas of cos-mology. For example, string-like defects as line disclinations in liquid crystals, vortex-line in superfluid helium and magnetic flux lines in type II superconduc-tors have been used for creating analogies with cosmic strings. On the other hand, such analogies have helped out the CMP defects be better comprehended by themselves.
Topological Defects in Liquid Crystals
In 1888, the botanist Friedrich Reinitzer reported the physicist Otto Lehmann about the existence of certain organic substance (cholesteryl benzoate) that curi-ously exhibited properties of both liquid and solid. Besides having been assigned to Lehmann the invention of the term liquid crystal to describe such substance, he was the first to detect the birefringence (anisotropy) and main structural properties of nematic8 substances. But it was just in 1922 that Georges Friedel in collaboration with Grandjean rigorously described the mesophases (nematic, smectic and cholesteric) according a molecular order criterion [Fri22]. Surpris-ingly, even after these interesting findings, the study of LCs gradually lost inter-est. The interest for physics of liquid crystal was intentionally revived by Frank [Fra58], who introduces the word “disinclination” later renamed disclination for line singularities in LCs. However, maybe the most important fact about the study of LCs was the contribution of Pierre-Gilles de Gennes, rewarded by the Nobel Prize in 1991 for his studies on Soft Matter.
The structure of a liquid crystal can be understood as a fluid made of rod-like (cigar-shaped) molecules being symmetric by rotation about its own axis. At high temperatures no order is established, as thermal agitation eﬀects do-main dipole ordering eﬀects all molecules are equally oriented, and the phase is isotropic (an ordinary liquid). However, when the temperature cools down interactions (dipole-dipole interactions) between the molecules lead them to be nearly aligned (approximately parallel to each other on average). Such ordered configuration is known as nematic phase, which is symmetric under rotations around a parallel axis to the molecules (average order). The orientation of the molecules in the LCs is measured by introduction of an order parameter, which is zero in the isotropic phase and non-zero in the nematic phase. Such order pa-rameter is related to a unit three vector ~n named director field, which describes the average direction of alignment relative to the axes ~a (unit vector) of the in-dividual molecules, see Fig. 2.4. If we chose ~n along the axis z, we find that the degree of orientational order is given by s = (1/2)h3 cos2 θ −1i [Tsv42]. For the most ordered phase, we would have θ = 0 or θ = π, with the molecules ~a parallel to the optical axis ~n. On the other hand, when the sample is entirely random we should get s = 0. Because the molecules are rod-like the director vector has no preferred polarity, that is ~n and −~n are equivalent. The orientation of the director field leads to formation of defects, and it is given by an angle that changes as 2πm, where the topological factor m gives the strength (winding number) of the defect. For example, when the director field ~n goes around π along a closed loop as we circle around the defect, m = ±1/2 disclinations arise. On the other hand, when the director points outwards or inwards everywhere, defects with strength m = ±1 appear. More precisely, for disclination in the z-const plane (~n confined to planes perpendicular to the defect axis), the director field components are ~n = cos(mφ+ φ0)ˆx+ sin(mφ+ φ0)ˆy, where m measures the director field rotation as one goes around the defect, φ is the angular coordinate and φ0 is a constant parameter. The two-dimensional cross-section for m = 1 disclinations is well represented in Fig. 2.3 [KL07; GPP95].
Table of contents :
2 Topological Defects
2.1 Topological Defects Formation
2.2 Topological Classification of Defects
2.2.1 Domain Walls
2.3 Cosmic Strings
2.3.1 Observational Evidences of Cosmic Strings
2.4 Topological Defects in Liquid Crystals
2.4.1 Isotropic-Nematic Phase Transition
2.4.2 Nematic Liquid Crystals as a Laboratory for Cosmology
3 Geometric Aspects of Linear Defects
3.1 Geometric Theory of Gravitation
3.1.1 Geodesic Equation
3.2 Weak Field Approximation for Regular Cosmic String
3.2.1 Gravitational Lensing
3.3 Geometric Method for Disclination
4 Wiggly String as a Waveguide
4.1 The Wiggly Cosmic String
4.2 Propagation of Massless Fields
4.3 Propagation of Massive Fields
4.4 Optical Analogues of Spacetimes
4.4.1 An Optical Waveguide Analogue of the Wiggly String
5 Optical Concentrator from a Hyperbolic Metamaterial
5.1 Hyperbolic Liquid Crystal Metamaterial
5.2 Light Trajectories in the HLCM Media
5.3 Propagating Wave Modes
6 Conclusions and Perspectives
A.1 Finslerian Method
A.2 Numerical Analysis
A.2.1 Finite Difference Method
A.2.2 Applying the Finite Difference Method
A.2.3 Discretization Codes
A.3 Eikonal Approximation