Dynamics of Domain Wall Systems

Msci Physics Research Project

Authors: Supervisor: Ciaran O’Hare Dr. Adam Moss James Stevenson

May 31, 2013

Abstract

Domain walls are cosmic topological defects that arise when the potential of a real scalar ﬁeld has a discrete symmetry broken at a phase transition. We present the results of numerical simulations of domain walls in Minkowski and FRW cosmological backgrounds in box sizes up to 20482 and 1283. We have obtained scaling solutions for wall properties in 2+1 and 3+1 dimensions and in radiation and matter dominated epochs, identifying scale invariant evolution as the asymptotic scaling law. We also study the eﬀects of a conserved charge condensed inside the domain walls in a hybrid domain wall model, a 2+1 dimensional analogue to cosmic vortons. We simulate the formation of domain wall systems in this model and obtain analytic solutions known as “kinky vortons”. We also probe the stability of these solutions under axial and non- axial perturbations and classify the outcomes of collisions between two vortons. Contents

1 Theory of Domain Walls 2 1.1 Introduction ...... 2 1.2 Domain wall formation ...... 3 1.3 Domain walls in FRW ...... 6

2 Numerical Simulations 8 2.1 The PRS algorithm ...... 8 2.2 Evolution algorithm ...... 8 2.3 Tests of the code ...... 10 2.4 Initial conditions ...... 11 2.5 Visualisation ...... 12

3 Cosmological scaling 13 3.1 Motivation ...... 13 3.2 Wall calculations ...... 14 3.3 Parameter choice ...... 16 3.4 Scaling results ...... 18 3.5 Discussion ...... 19

4 Theory of Charged Domain Walls 22 4.1 Introduction ...... 22 4.2 The model ...... 22 4.3 Symmetry breaking conditions ...... 24 4.4 Condensate formation conditions ...... 25

5 Evolution of Charged Domain Walls 27 5.1 Numerical implementation ...... 27 5.2 Dynamical properties and scaling ...... 28 5.2.1 Charged domain walls - Minkowski ...... 28 5.2.2 Charged domain walls - FRW ...... 30 5.2.3 Discussion ...... 31

6 Analytic Solutions and the Construction of Vortons 32 6.1 Inﬁnite planar wall revisited ...... 32 6.2 Kinky vortons ...... 34 6.3 Energy minimising radius ...... 36

7 Vorton Stability 38 7.1 Stability analysis ...... 38 7.2 Vorton candidates via energy minimisation ...... 40 7.3 Vorton collisions ...... 41

1 Chapter 1

Theory of Domain Walls

1.1 Introduction

Phase transitions are ubiquitous in nature. A common feature associated with many phase transitions is the notion of symmetry breaking, a familiar and often quoted example being the symmetry breaking that occurs during the freezing of liquid water to ice. When the rotationally symmetric liquid state has degrees of freedom removed at a phase transition it crystallises into a regular lattice structure. In a macroscopic system of water, the phase transition of freezing will occur independently at many different sites at once each forming a separate crystal structure. As freezing progresses, these “domains” of ice will be forced to meet each other, at which point they are unlikely to be able to mesh together precisely, the result being a crack in the ice - a topological defect. The restoration of broken symmetries at high temperatures is a well known phenomenon, hence topological defects are familiar in many fields of science. A similar domain-domain wall configuration is present in ferromagnets when regions of the solid with aligned magnetic spins lock together. After the superfluid phase transition in Helium, point-like defects known as vortices appear [1] and similar defects have also been observed in superconductors [2] and protein folding [3]. Early Universe phase transitions, too, predict an array of topological defects (see [4] for review) as symmetry breaking plays a central role in grand unified theories (GUTs) which attempt to unify three of the four fundamental interactions of nature [5]. Here the symmetry breaking is associated with quantum fields where the vacuum states in their potentials are not simply connected. The breaking of different symmetry groups can lead to the formation of a range of defects including domain walls, cosmic strings, monopoles, textures and any number of hybrids. Defects have been a fascination in cosmology and particle theory since the description of domain walls and cosmic strings in the 1970s [6, 7]. Offering a unique avenue into the study of early Universe phase transitions, defects can provide valuable constraints on GUTs. They are also of importance in cosmology; for instance, monopoles were famously used in an argument that lead to the development of the theory of inflation [8, 9] and domain walls have also been considered as candidates for dark energy [10] and dark matter [11, 12]. The defects known as cosmic strings have been awarded much of the attention of the literature and have been linked with many areas including gamma-ray bursts [13] and time travel [14], and have recently been shown to provide a potential link to observables in string theory [15, 16]. Defects operate on a principle of “if they can form, they will form” [7]. If symmetry breaking occurs at GUT phase transitions then the formation of the associated defect is inevitable. Al- though there is currently no evidence for any class of defect, the observational search is ongoing. If defects are massive enough they should be evidenced by a number of sources, for example their signatures in the cosmic microwave background [18], primordial density fluctuations [17] or relic gravitational waves emitted under their collapse [19].

2 Classical field theory simulation of defect producing models has been an active topic of research since the late 1980s. Domain walls present the simplest case and therefore make a useful testbed to compare numerical results with analytic predictions. The most intriguing aspect of domain walls and one which is important for understanding their cosmology is the dynamical scaling they exhibit. Systems of domain walls can be characterised by a scaling law for their energy density and the initial effort of this study was to find this for two domain wall models. There has been some speculation in the literature [23, 24] as to the artificial effect of the simulation size on the scaling law and this is something that has been investigated here. This report is made up of two distinct parts which cover the work we have carried out on two domain wall models. The first part extends some previous research of the Z2 domain wall model (e.g. [20–22]). We describe the analytic steps taken in developing a code to simulate and analyse domain walls in 2+1 and 3+1 dimensions and the measures employed to ensure accuracy. The second domain wall model that has been investigated is a hybrid Z2 U(1) model [25] which × involves introducing a conserved charge condensing inside the walls. We fully outline the analytic approach taken in creating a simulation of this model. We then present results of work done on this model which extends the research carried out by Battye et al. (2009, [26]) to wall formation in a Friedmann-Robertson-Walker (FRW) cosmological background (expanding frame). Finally we describe the search for analytic solutions and the tests performed on their stability under perturbations and collisions. This report makes use of standard conventions used in the field. Firstly relativistic indexing, where positions in D+1 dimensional spacetime are written with an index µ running from 0 to µ ∂ D, x = (ct, x). Their respective partial derivatives are written as a µ ∂µ, note that the ∂x ≡ Einstein summation convention places an implicit summation sign wherever there are repeated spacetime indices. Finally, all equations are written in a system of natural units in which ~ = c = kB = 1.

1.2 Domain wall formation

Domain walls occur at the breaking of a discrete symmetry associated with the potential of a real scalar field φ(t, x). The temperature dependent symmetry breaking potential considered here has the form, λ 2 (φ, T ) = φ2 η(T )2 . (1.1) V 4 − (shown in Figure 1.1) where η(T ) and λ are model dependent parameters. The Lagrangian density is written as the difference of kinetic and potential terms and packages up the necessary dynamics of the field, 1 µν = η ∂µφ∂νφ (φ, T ). (1.2) L −2 − V Where ηµν = diag( 1, 1, 1, 1) is the Minkowski metric which describes the geometry of flat, − non-expanding space. The symmetry group that is broken in this potential is known in group theory jargon as Z2, or symmetry under reflection φ φ. Above some critical phase transition temperature, Tc, → − the potential possesses a single minimum at φ = 0, known as vacuum state of the potential. With this form, the potential is symmetric under reflections about its vacuum state. As the temperature cools and the phase transition progresses, the potential acquires two separated minima at φ = η, and the previous minimum acquires a non-zero energy value. There are now ± two new vacuum states at the two minima and the state at φ = 0 which is now a maximum in potential becomes known as the “false” vacuum state. The field will then naturally tend to fall down into one of the two minima with equal probability. The set of vacuum states that the field may occupy will be referred to as the vacuum manifold. The vacuum manifold before the phase transition possesses a Z2 symmetry because fluctuations in the field about the vacuum

3 (φ = 0 + δφ) are invariant under reﬂections φ φ. However expansions around the vacua → − that appear after the phase transition φ = η + δφ do not possess this symmetry. The process ± of changing the vacuum state of the ﬁeld from one which possesses a symmetry to one which does not is known as spontaneous symmetry breaking.

V (φ) T > Tc T < Tc

Figure 1.1: Z2 symmetry breaking potential, (φ) above and below the phase transition tem- V perature, Tc. The initial vacuum state at φ = 0 rises and becomes an unstable maximum after the phase transition. The ﬁeld must then choose to fall down into one of the two new vacuum states, spontaneously breaking the Z2 symmetry.

For the purposes of this study the dynamics prior to the phase transition are not needed, so from here on the temperature dependence in the potential will be turned off and only temperatures T < Tc will be considered. This restricts the parameter η to be constant and fixes the locations of the vacuum states. To describe the appearance of domain walls it is necessary to obtain the equation of motion for φ. To find this one must vary the action, Z S = d4x (1.3) L under small perturbations in the field. This eventually leads to the Euler-Lagrange equation for φ, ∂2φ ∂ = 2φ V . (1.4) ∂t2 5 − ∂φ Where t is cosmic (physical) time, and is the Laplacian operator in physical co-ordinates. 5 This equation of motion when placed under the fluctuations present after the phase transition describes the subsequent evolution of the field. The tendency of the field to fall randomly into one of the minima and the appearance of the Laplacian operator which acts to correlate the field locally, leads to the partitioning of space into “domains” in which the field is aligned into one of the two minima, (Figure 1.2). Regions that have had enough time to talk to one another, i.e. are causally correlated, will tend to arrange themselves into the same vacuum state, however regions that are not correlated will be randomly aligned in either state. Walls appear when oppositely aligned domains meet and the field is forced to leave the vacuum manifold and traverse the potential barrier separating the two vacuum states.

4 Figure 1.2: Partitioning of the Universe into domains soon after the phase transition. Blue/white colours correspond to each vacuum state. The domain walls that appear here in 2D are shown as black lines (in 3D they appear as 2 dimensional surfaces).

Domain walls are described by a mathematical function known as the kink solution [6], displayed in Figure 1.3. In the simple one dimensional case this solution describes how the field interpolates between the vacua, φ(x ) = η, but is a general form that also profiles walls → ±∞ ± in 2 and 3 dimensions. This solution can be obtained by considering the energy functional which must be minimised for a stable field configuration, Z E = dx . (1.5) H Where is the Hamiltonian density for the field (the sum of kinetic and potential terms). H Minimising this functional under the boundary conditions for a kink one can obtain the exact solution (see [19, 27] for full derivation), r λ φ(x) = ηtanh( ηx). (1.6) 2 The important feature of this solution is that it is stable. It cannot be simply smoothed out to a single field value, as taking one half of the field that resides in one domain to the other would require an infinite amount of energy. For a system of these kinks, the only possible evolution that can take place is for the domains to organise themselves so as to minimise their total surface area. In the description of domain walls we will frequently refer to them as ‘objects’ moving around in space, but this is perhaps a somewhat misleading interpretation. For cosmological purposes they can be interpreted in this way, as physical surfaces that possess mass which contributes to the energy density of the Universe and evolve according to relativistic equations for elastic surfaces. However another equally valid interpretation, and one which is more appropriate when thinking about their appearance in simulations, is that they are the necessary solutions to the non-linear field equation (1.4) and that domain walls themselves are simply regions of the field with particularly high spatial gradients.

5 φ

+η

η −

Figure 1.3: The kink solution which interpolates between the two vacua, η, and describes the ± proﬁle of a domain wall. The width is determined by the parameters, λ and η.

1.3 Domain walls in FRW

For the purposes of understanding the impact of walls in a physical context the domain wall forming ﬁeld equation must now be derived in a Friedmann-Robertson-Walker (FRW) cosmological background, essentially an expanding Universe. This involves using the line element,

ds2 = dt2 + a2(t) dx2 + dy2 + dz2 . (1.7) − In which a(t) is the dimensionless scale factor, parametrising the expansion of space. The co- ordinates x, y, z are referred to as “co-moving” as they remain ﬁxed as the Universe expands. The metric and inverse metric for this line element are,

2 2 2 gµν = diag 1, a (t), a (t), a (t) (1.8) − and, gµν = diag 1, a−2(t), a−2(t), a−2(t) (1.9) − respectively. The inverse FRW metric must now replace the Minkowski metric in the Lagrangian (1.2) to become, 1 µν λ 2 2 2 = g ∂µφ∂νφ (φ η ) . (1.10) L −2 − 4 − Following the same procedure as before, this time varying the action, Z Z S = dt d3x√ g (1.11) − L (where g is the determinant of the metric) under small perturbations in the ﬁeld δφ, one may arrive at the FRW equation of motion,

∂2φ ∂φ 1 ∂ = 3H(t) 2 φ V (1.12) ∂t2 − ∂t − a2(t) 5 − ∂φ

6 where is now the Laplacian operator in co-moving co-ordinates. The function H(t) = 1 ∂a is 5 a ∂t the Hubble expansion parameter and acts as a damping term. It is now convenient to perform a dt co-ordinate transformation to conformal time: dτ = a(t) . Conformal time is a measure of time that incorporates the scale factor and greatly simpliﬁes equations in cosmology. The equation of motion under this transformation is,

∂2φ ∂ ln a(τ) ∂φ ∂ = 3 2φ a2(τ) V . (1.13) ∂τ 2 − ∂τ ∂τ − 5 − ∂φ Note that in 2 dimensions, this equation has a factor of 2 instead of 3 in the damping term. To solve this equation of motion, a form for a(τ) is required, which in turn requires a choice of cosmological era. Eras that may be in place during the formation of domain walls are those in which the energy density of the Universe is dominated by single component, either radiation or matter (including dark matter) which are modelled as fluids 1. The form of the scale factor can be found by solving the Friedmann and fluid equations in the presence of a radiation or matter fluid of density ρ and pressure P . In a flat Universe without a cosmological constant, the Friedmann equation is (in physical time),

1 ∂a2 8πG = ρ(t), (1.14) a ∂t 3 which is derived from the Einstein field equations. The fluid equation is ∂ρ 1 ∂a + 3 (ρ + P ) = 0, (1.15) ∂t a ∂t which is a consequence of conservation of energy and momentum in an expanding frame. The density and pressure terms for a fluid can be related by an equation of state,

P = wρ (1.16)

1 where the numerical factor is w = 0 for matter and w = 3 for radiation. With an equation of state chosen, the Friedmann and ﬂuid equations can be solved together for either radiation or matter domination.

1/2 arad t (1.17) ∝ 2/3 amat t . (1.18) ∝ Which can then be transformed to conformal time to obtain a form for a(τ),

arad τ (1.19) ∝ 2 amat τ . (1.20) ∝ ∂ ln a Hence the term ∂τ in the equation of motion is simply, 1/τ for radiation and 2/τ for matter. With this choice made, the equation of motion is suﬃciently prepared to be implemented in a numerical integration scheme.

1Radiation and matter domination eras are also known as Tolman and Einstein de Sitter universes respectively

7 Chapter 2

Numerical Simulations

2.1 The PRS algorithm

Field theory simulations of topological defects are carried out in a discretised box of fixed co- moving volume, i.e. an expanding volume with a minimum spatial separation that appears fixed in the coordinates. This allows spatial gradients to be approximated over the full length of the simulation without having to adjust the spatial discretisation. Unfortunately this presents an issue for domain wall formation, as the width of the walls (which is constant in physical co-ordinates) will appear to shrink with time. The method by which this effect is overcome is known as the PRS algorithm [23] and involves a modification to the equation of motion in the form of two variables α and β,

∂2φ ∂ ln a ∂φ ∂ + α 2φ = aβ V . (2.1) ∂τ 2 ∂τ ∂τ − 5 − ∂φ Maintaining constant co-moving width, ensuring the walls are resolvable for the full dynamical range, requires the choice of β = 0. Although this adjustment was initially rather ad hoc, the validity of this approach in producing physical results has been proven by Sousa et al. (2010, [28]) if the condition β α + = D (2.2) 2 is met, where D is the number of spatial dimensions. This condition enforces momentum conservation so the wall expansion is physical. With the choice β = 0, the variable α is set to be 2 or 3 (as it appears in the unmodiﬁed equation of motion).

2.2 Evolution algorithm

The equation of motion is a second order partial differential equation which must be solved using a finite difference scheme. A grid of φ values, φi,j,k = φ(i∆x, j∆x, k∆x), is set up with a constant spacing ∆x, up to a maximum of N grid points along each dimension. The grid is then evolved in time by the modified equation of motion (2.1) along discrete timesteps of duration ∆τ. The Laplacian operator 2 must be approximated to some order. To obtain good estimates 5 to the large spatial gradients present at the walls, a fourth order approximation centred on φi,j,k is found to be sufficient, for example 2nd-order x derivatives,

2 ∂ φ φi+2,j,k + 16φi+1,j,k 30φi,j,k + 16φi−1,j,k φi−2,j,k = − − − + (∆x4). (2.3) ∂x2 12∆x2 O The choice of ∆x and ∆τ is limited by the Courant-Friedrichs-Lewy (CFL) condition which must be met if a finite difference scheme for solving partial differential equations is to be conver- gent [29]. This condition places a lower bound on the spatial separation for a given timestep size

8 and originates from the tracking of disturbances travelling over the grid. Disturbances in the field move at light speed (c = 1), so if a finite difference scheme is to calculate the amplitudes of these disturbances along a discrete spatial grid at discrete intervals, then for the series to not diverge the timestep must be shorter than the time taken for a disturbance to travel across adja- cent grid points. If the equation of motion was a pure wave equation (i.e. without the potential or damping term) the CFL condition would be ∆x √D∆τ (in D spatial dimensions). For the ≤ equation of motion used in the simulation there will be a slightly larger minimum separation depending upon the size of the damping term (this can be found by trial and error if needed). However, it is only necessary to choose a value of ∆x to be sufficiently small to resolve the walls. For this reason it must be balanced with the wall width, which is set by the model parameters λ and η. For now we will fix the separations at ∆x = 1 and ∆τ = 0.1 to ensure the CFL condition is always met, the wall width will be tuned accordingly when parameter choice is discussed. Approximating spatial gradients at each grid point using equation (2.3) requires knowledge of four neighbouring points in each dimension. This presents an issue for finite simulation boxes as gradients cannot be calculated at the edges of the box. So to be able to consistently evaluate 2φ at all points in the simulation box, periodic boundary conditions are employed, identifying 5 each side of the box with the opposing side thus giving the illusion of infinite space. However, this artificially changes the topology of the simulation to that of a torus, and as such imposes 1 a light crossing time of 2 N∆τ; the time taken for signals emitted in opposite directions from the same point to cross the boundary and interfere. It is therefore roughly how long it takes for the field to recognise it is in a finite space. Close to this limit, the field will have correlated itself where on average there will be only two domains left in the system (see examples in Figure 2.1). This means that the evolution close to and past the light crossing time will deviate from a network scaling solution as the wall sample size is made up of only one or two walls. For this reason, the simulations presented here are only run up to the light crossing time, unless there is any reason why there may be structures existing past it.

Figure 2.1: Typical appearance of systems after the light crossing time. Often there will remain on average only one or two walls in the system, the change in total length will therefore be unrepresentative of network scaling.

The equation of motion (2.1) is solved using a second order leapfrog method. This requires re-writing the second order equation of motion as two ﬁrst order equations: the rate of change of the ﬁeld and the right hand side of the equation of motion, ∂φ u(τ) = (2.4) ∂τ ∂u ∂ ln a ∂φ ∂ f(τ) = = 2φ α V (2.5) ∂τ 5 − ∂τ ∂τ − ∂φ

9 These are solved using a Taylor expansion to second order in ∆τ, 1 φ(τ + ∆τ) = φ(τ) + u(τ)∆τ + f(τ)∆τ 2 (2.6) 2 1 u(τ + ∆τ) = u(τ) + [f(τ) + f(τ + ∆τ)] ∆τ 2. (2.7) 2 This process makes up the evolution algorithm; the field is first evaluated at each point using predefined initial conditions for φ(0) and v(0). The evolved field is then taken into the second line in which u(τ) is prepared for the next timestep, and the process repeats.

2.3 Tests of the code

The code can be checked if it is functioning correctly by input of special initial conditions for which the subsequent evolution is known analytically. The simplest of these is a kink-antikink configuration, two oppositely oriented kinks placed end to end (Figure 2.2, left). If this solution ∂φ is entered into the equation of motion with ∂τ = 0 then it should remain fixed and not decay, if it does this then the evolution algorithm is working correctly. The code can be further tested by giving the configuration a Lorentz boost in the x-direction by performing the transformation, r ! λ φ = η tanh ηX (2.8) 2 X = γ(x vτ) (2.9) − Where v is the velocity of the full kink solution and γ = 1/√1 v2 is its Lorentz factor. It − follows that the rate of change of φ is (Figure 2.2 right), r r ! ∂φ λ λ = γv η2sech2 ηX (2.10) ∂τ − 2 2 By implementing these as initial conditions the kink solution can be tracked along the x-direction with a given velocity v. This will be used later when considering colliding analytic solutions in the Kinky Vorton model.

∂φ φ ∂τ

y y x x

Figure 2.2: Profile of the field and rate of change of the field in the kink-antikink solution. The kink here is moving with velocity v in the positive x-direction. Recall the presence of periodic boundary conditions in x and y.

A test of the evolution algorithm one can perform in 3 dimensions is to input a sphere of domain wall given by a radial version of the kink solution, r ! λ φ = η tanh η(r R) (2.11) 2 −

10 where r2 = x2 + y2 + z2 and R is the radius of the sphere. This solution describes a sphere of domain wall with one domain inside the sphere and the other outside the sphere. The code is working correctly when this bubble collapses due to the imbalance of forces and releases its energy radially as waves in the ﬁeld which are then damped away by the expansion term. Spheres of domain walls are also useful in testing the accuracy of wall area calculations which will be covered in the next chapter.

2.4 Initial conditions

The final piece in setting up a simulation is the initial conditions that lead to the formation of domain walls. There appear to be two opposing choices of initial conditions used in simulations of defects. Both choices involve setting the initial rate of change in the field everywhere to be 0, this sets the total energy of the system to a minimum and reduces the effect of excessive fluctuations caused by the field oscillating around the potential minima. However the choice in initial field values may either be a random number chosen from a uniform distribution between the vacuum states (in this case the range, η φ +η) [20–22], or precisely at the vacuum − ≤ ≤ states ( η) with equal probability [23, 25, 26]. The latter choice has faster domain formation ± but produces very high initial gradients, and if the damping term is small (for example in a 2D radiation dominated era) the field may not be able to settle at all. Studies that have chosen initial conditions of this form have often resorted to periods of artificially high damping to force the field to settle [26, 35]. Initial field values used in this study however, were selected from a Gaussian distribution centred on φ = 0, with 5 standard deviations lying between the vacua, η. This allows manageable initial gradients and is a more physical choice than uniformly ± distributed values, better reflecting the quantum fluctuations left following a phase transition. A comparison of initial conditions is shown in Figure 2.3.

1.2

0.8

φ 0.6

0.4

0.2

−0.2 0 10 20 30 40 50 60 70 Conformal time

Figure 2.3: Evolution of a single grid point φ value settling, into an η = 1 domain under diﬀerent initial conditions. Gaussian initial conditions are shown in the black curve, uniform in the blue curve and η in the red curve. ±

11 2.5 Visualisation

It is both useful and interesting to visualise the formation and evolution of domain walls. For instance it is by far the easiest way to check that the code is working as desired and if changes have had a qualitative eﬀect on the evolution. Often changes can be more obvious visually, even when measurable properties of the network may be very similar. The visualisation presented in Figures 2.4 and 2.5 make use of imaging tools built into the program MATLAB1.

τ = 150 τ = 400 τ = 1000 τ = 2560

Radiation

Matter

Figure 2.4: Evolution of a 2D domain wall network in a box size of 5122 for radiation and matter eras for times up to the light crossing time.

Figure 2.5: Evolution in 3D, box size 643 in radiation and matter eras. The images show φ = 0 isosurfaces.

To pre-empt a later result, it is already clear that the cosmological era appears to have little impact on the qualitative nature of the network evolution.

1Visualisation becomes extremely computationally taxing for large box sizes, so for this reason only mid-sized boxes have been shown here

12 Chapter 3

Cosmological scaling

3.1 Motivation

The crucial question regarding domain walls is their compatibility with present day cosmology. Establishing this requires knowledge of how the energy density of domain wall systems forming in the early Universe scales with time. This scaling can then be compared to other components of the energy budget of the Universe like matter and radiation. For instance if domain walls were found to scale slowly enough such that they are not sufficiently diluted by the expansion, they may come to dominate the Universe with multiple walls existing inside the horizon [22]. A prediction of this kind is in clear conflict with present day observations (we don’t see any domain walls out there). Therefore models that predict domain walls to scale in this way would have to impose some measure of dealing with them such as inflation1, or be adjusted so that they do not form. It may also be the case that even if domain walls are expanded out of the horizon their energy density may be much larger than the observationally confirmed value which is very 2 close to the critical energy density required for a flat Universe, ρc = 3H0 /8πG [31], where H0 is the Hubble parameter at the present day.. The scaling solution that we are able to extract from our simulations is how the total area A of the domain wall network varies with time. This is a slightly unphysical quantity but is the most directly measurable one in simulations and can be related to other properties of domain walls in general. For example, sizes of domains are characterised by a correlation length ξ V/A, where A is the average wall area and V is the volume in which the wall is contained, ∼ which may be measured in either physical or co-moving co-ordinates. The correlation length is the length scale over which the field has become aligned into a single domain. The upper limit set by causality on the size of this length at time, t is of order ct (in physical co-ordinates). Because the energy contained in the walls is proportional to the wall area, the energy density of the domain wall system ρDW (τ) will scale inversely proportional to the co-moving correlation length. The evolution of components of the energy density of the Universe, for example radiation and matter typically follow power law evolution, so the scaling solution is therefore expected to be some power law in conformal time, A τ q. (3.1) ∼ In terms of the correlation length this is, ξ τ −q. This power law can be converted into physical ∼ time t (in 3D) by, 1 (2−q) ξ t 3 . (3.2) ∼ The solution that is of particular interest is that of scale invariance, q = 1, this is when lengths − scale with the same exponent in both cosmic time and conformal time, i.e. the dynamics are unaffected by the scale factor a(t). There have been some analytic arguments for the presence

1However, it is entirely possible that domain walls form after inﬂation

13 of scale invariance [32, 33], but all numerical simulations thus far have observed deviations from −2 this solution. Note that the critical energy density of the Universe scales as ρc t so a domain −1 ∼ wall system with scale invariant dynamics (ρDW t ) would very quickly come to dominate ∼ energy density of the Universe.

3.2 Wall calculations

Scaling solutions are extracted from simulations by calculating the total area (or length in 2D) of the wall network at each timestep. Domain walls described by the kink solution (1.6) can be identified between points where the field crosses φ = 0. A crude estimate to the wall area can be obtained by finding and counting all neighbouring grid points which have opposite signs. These zero-crossing points are found by evaluating a condition at every grid point along each dimension of the box, for example in the x-direction,

φi,j,k + φi+1,j,k < φi,j,k + φi+1,j,k , (3.3) | | | | | | This condition is evaluated for each i, j, k = 1, ..., N. If the condition is true at any two neighbouring grid points (i, i+1) then a wall is deemed to exist and a count is increased. This routine is performed along each dimension of the simulation and the count multiplied by ∆x2 in 3D or ∆x in 2D. A grid-point by grid-point check gives a crude approximation of the wall length/area but will inevitably lead to an over-estimate. For example, in a box size of 10242 the calculation described gives a persistent error of 25% in the circumferences of randomly placed circular domain walls. ∼ The over counting is caused by the discrete grid inaccurately representing smooth surfaces. For example if there existed a wall oriented such that at a single grid point (i, j, k) the wall checking condition read true in all 3 directions, that single grid point would contribute a total of 3∆x2 to the total area, but depending on the orientation of the wall the “true” area contained in the grid point volume would be some factor smaller. This problem is related to a classic geometric fallacy illustrated in Figure 3.1. In this example a square is sequentially “smoothed” out by turning in its corners. The shape in this example may appear to approach the smooth circle but even in the limit of inﬁnite iterations the shape does not reach the continuum limit.

Figure 3.1: As the number of steps in this iterative procedure increases the length of the perimeter of the resulting jagged shape remains the same at 8. It appears as though it will gradually approach the circle as more iterations are performed, thus creating a circle with a perimeter of 8 and radius of 1. This is a fallacy because even in the limit of inﬁnite iterations the shape does not approach a circle.

The representation of smooth domain walls on a cubic lattice is directly related to this problem; even in the limit of an infinitesimally fine grid the lattice calculated wall area will always be an overestimate to the area one would expect in the continuum limit. So in addition to finding the zero-crossing points where a wall exists, we must also divide by a weighting factor

14 which relates lattice calculated areas to their continuum value. Considerations of this kind are commonplace in cosmic string simulations [30] but appear to be absent from the domain wall literature so we will explain the calculation in the simple 2 dimensional case here. Consider a square lattice which is used to approximate an arbitrarily oriented line (Figure 3.2) and imagine trying to calculate the length of the hypotenuse of a right-angled triangle with sides of length x and y on a square lattice. The lattice calculated value for the length of this p 2 2 line is Llat = x + y, whereas the continuum length is L = x + y .

Figure 3.2: Representing a diagonal line on a square lattice of increasing resolution. The lattice calculated value for the length of the diagonal line is always Llat = x + y, even though the resulting shape appears to converge on a straight line.

The true length of an arbitrary line in the continuum limit can be found in terms of the lattice calculated length by averaging over all possible orientations of the line with θ in the range (0, π/2). p L x2 + y2 = (3.4) Llat x + y | | | | 1 π = = . (3.5) sin θ + cos θ 4 | | | | The modulus of the lengths has been used so that all orientations contribute positively to the wall count. By multiplying the lattice calculated length by the average weighting π/4, a better estimate to the wall length can be calculated. In the 3D case this weighting factor is 2/3, and is derived in a similar manner. The accuracy of the wall calculations is displayed in Figure 3.3. These plots show that the error in calculating diﬀerent sized objects in a single grid is roughly constant but increasing the resolution leads to a more accurate calculation for a given sized object. The error is always less than 3% for the range of box sizes used in this study and a roughly constant error will be removed with the use of logarithms when calculating the scaling exponent. To make an even better estimate the quantity ( cos θ + sin θ ) should be calculated at | | | | every zero-crossing point, however this involves a much more computationally intensive wall calculation algorithm which requires the orientations of the walls at each point (this is related to gradient terms). It was found that doing this does not improve the error by an amount that justiﬁes the increase in computing time.

15 3 1.4

1.2 2.5

2 0.8

0.6 1.5

% error0.4 in length % error in wall area

1 0.2

0.5 0 20 25 30 35 40 45 50 200 400 600 800 1000 1200 1400 1600 1800 2000 Radius of sphere N

Figure 3.3: On the left is the error in the calculation of the areas of spheres of increasing radius, in a box size of 1283; the error is persistent at around 1.5%. On the right is the accuracy of the 2 dimensional calculation, showing the error in the circumference of a circle of domain wall with a radius of N∆x/4.

3.3 Parameter choice

There remain two free parameters of the theory that are yet to be chosen: λ and η. The parameter η controls the locations of the potential minima and hence the value φ takes in the vacuum. The field can always be transformed so that these minima in the potential are at φ = 1, so η may be set to unity without changing the dynamics. With η fixed, the parameter ± λ now controls the width of the walls, which given the finite resolution of any one simulation is choice that needs to be made carefully. To discuss the choice of this parameter it is convenient to change variables from λ to w, 2π2 λ = , (3.6) w2 where w has dimensions of length [6]. This allows one to interpret the parameter choice in a more physical sense as the width of the wall. It also allows comparison to previous studies where this variable is often used. The kink solution in terms of this variable and η = 1 now reads, πx φ(x) = tanh (3.7) w Thinner walls are desirable as they are observed to form faster, leading to faster onset of dynamical scaling and hence a simulation with a longer dynamical range. Unfortunately a wall width that is too thin may not be resolvable for a given ∆x. When this occurs the walls become discontinuities in the field and the network evolution ceases. On the other hand, if the walls are too thick the evolution will start to become dominated by the thickness of the wall, meaning the number of domains in the network is smaller and hence the sample size with which to obtain the area is reduced. A balance must be found with a choice of wall width that can both produce accurate scaling and be resolvable to the simulation. This balance can be considered more precisely by calculating the quantity F which is proportional to the ratio of the kinetic and potential energy of the system of walls, 2 1 X ∂φi,j,k F = (3.8) A ∂τ i,j,k

This quantity is useful as it can be used to determine when a scaling solution begins to take hold. The regime of interest is the period after the field has rolled down into the vacua and has finished oscillations about the potential minima. At this point the dynamics of the field are no

16 longer dominated by motion left over from the phase transition and is now solely made up of the dynamics and collapse of the wall system. Hence domain wall evolution that is described by a scaling law will take place when there is no longer potential energy being converted into kinetic energy, i.e. the period in which F is constant (Figure 3.4). Provided the walls can be resolved by the simulation a scaling solution will be found eventually, so the choice of wall width becomes an issue obtaining network dynamics over the longest period possible. In Figure 3.5 one can see that between the cases w = 4 and w = 6 the scaling over- or under-shoots the eventual evolution that settles. For the 2D simulations the optimum value was w = 5. In 3D a value of w = 3 was chosen for similar reasons.

0 0 10 10

−1 −1 10 10

w = 2

−2 −2 F 10 F 10 w = 3 w = 4

w = 6 w = 5 −3 −3 10 w = 8 10 w = 10 w = 7 w = 9

−4 −4 10 10 0 1 2 3 0 1 2 10 10 10 10 10 10 10 Conformal time Conformal time

Figure 3.4: Ratio of kinetic and potential energies in 2D and 3D (left to right), with different values of w. Thinner walls settle into a scaling regime faster. However in the 2D case, w = 2 does not enter a period of constant F, but instead slowly decreases; in this case the walls are not sufficiently resolvable for the system to evolve correctly, as demonstrated in the next figure.

6 10 w = 2 w = 4 w = 6 w = 8 w = 10

5 10

Wall length 4 10

3 10 0 1 2 3 10 10 10 10 Conformal time

Figure 3.5: Evolution of the total length of the wall network for the same widths as in the previous ﬁgure, where it can be can seen that the w > 2 cases possess power law scaling whereas w = 2 does not.

17 The ﬁnal choice that must be made is the box size, set by N. To obtain the best estimate to a scaling law the largest box size possible is desired, this both increases the total sample number of walls and extends the light crossing time meaning a scaling solution can be found over a longer range. However practically there are constraints from both memory and computing time. The largest box sizes used for this study were 1283 and 20482 (although simulations in smaller boxes have been performed to test for a dependency on simulation size). Because the algorithm performs calculations on a matrix of side length N the computing time per timestep scales as N 2 in 2D and N 3 in 3D, but can be reduced slightly with eﬃcient memory usage, ∼ ∼ for example using matrices with 2n elements, for some integer n. The software MATLAB is also highly optimised for matrix operations, so the code took advantage of this. Instead of using ‘for’ loops to perform calculations over the matrix, the entire matrix was evolved at once using the in built element-by-element matrix operations.

3.4 Scaling results

The total area of the domain wall network is observed to follow a power law in conformal time,

A τ q (3.9) ∼ where q is a measurable scaling exponent (recall that q = 1 is the case of scale invariant − dynamics). This area is proportional to the energy density of the domain wall network ρDW and inversely proportional to the co-moving correlation length ξ. The same is true of the total length of the wall network in 2D. Figures 3.6 and 3.7 show a set of results for the largest box sizes in 2 and 3 dimensions in matter and radiation eras. The scaling exponent was calculated by performing a linear fit to a log-log plot of wall length/area against conformal time. The dynamical scaling regime takes hold after an initial settling period, which can be identified as mentioned earlier by considering the ratio of kinetic and potential energies, F . The fit is only taken after F takes on a region of constancy. In addition to this the scaling exponent has been calculated in successive bins in conformal time, so the eventual approach to an asymptotic scaling solution can be seen. The errors presented are are standard deviation in the exponents, this is because the errors provided by the fitting tool are negligible and the roughly constant small error that remains in the wall calculation is removed when taking the logarithm of the wall length/area. A summary of the largest simulations performed is displayed in following table.

Dimension N Era # runs Fit range (timesteps) w Scaling exponent (q) 2D 1024 Mat. 1000 3400 5 0.95917 0.06405 − ± 2D 1024 Rad. 1000 3400 5 0.94792 0.06704 − ± 2D 2048 Mat. 1000 6800 5 0.98910 0.03102 − ± 2D 2048 Rad. 1000 6800 5 0.97931 0.04512 − ± 3D 128 Mat. 1000 330 3 0.93472 0.10052 − ± 3D 128 Rad. 1000 330 3 0.92695 0.11866 − ±

Table 3.1: Results of the scaling solutions of Z2 domain walls in an FRW cosmological background.

18 7 10 −0.6

6 −0.7 10

5 −0.8 10

−0.9 4 10 −1

3 Wall length 10 −1.1 Scaling Exponent

2 10 2 2048 −1.2 Radiation era Matter era 1 10 −1.3 1 2 3 10 10 10 0 200 400 600 800 1000 Conformal time Conformal time

Figure 3.6: Scaling of wall length in 2D for the box size 20482 over a total of 1000 simulations. The dashed lines show the region in which the scaling exponent was calculated. The ﬁgure on the right shows how the scaling exponent calculated in bins varies over the dynamical range.

7 10 0

−0.2

6 10 −0.4

−0.6 5 10 −0.8 Wall area

4 −1 10 Scaling Exponent 3 128 −1.2 Radiation era Matter era 3 10 −1.4 0 1 10 10 0 10 20 30 40 50 60 Conformal time Conformal time

Figure 3.7: The same plot, for the scaling of wall area in 3D, for the box size 1283.

3.5 Discussion

The scaling exponents found in all box sizes show both consistency with results of previous studies [20–24, 34–37] and consistency with the analytically predicted scale invariant evolution (q = 1) [32, 33]. Moreover, deviations from scale invariance found in these studies could − be attributed to the limited dynamical range of small simulations, as it appears that there is convergence on scale invariance for larger box sizes and longer dynamical ranges, Figure 3.8. There is also good agreement with the large-scale simulations performed most recently by Leite et al. (2013, [21]) using an alternative formulation known as the “one scale model”, which transfers ignorance into diﬀerent physical characteristics of the network at the cost of a more involved analysis routine.

19 −0.65 −0.65 Matter D Radiation 2D 3 −0.7 −0.7

−0.75 −0.75 −0.8 −0.8 −0.85 −0.85 −0.9 −0.9 −0.95 Scaling exponent Scaling exponent−0.95 −1

−1 −1.05

−1.05 −1.1 6 7 8 9 10 11 12 4 5 6 7 8 N N log2 log2

Figure 3.8: Results obtained for the full range of N, with 1000 simulations carried out for each box size. We observe some convergence on the scale invariant solution q = 1. − Although scale invariant evolution appears to be the asymptotic scaling law, the relaxation of the system into this regime is very slow. If possible even larger box sizes would be preferred to further confirm that this solution was approached at late times. Unfortunately the light crossing time is present in all simulations of this type and will be the limiting factor to every study. The only way to combat this issue is to have a box size large enough to allow the system to unambiguously settle into a dynamical scaling regime within the light crossing time. This of course would require greater computer power than was used in this investigation. It is possible however, that the system could be allowed to relax faster by further consideration of initial conditions, which would require a full treatment of the type of a phase transition that could lead to domain wall formation. Furthermore, the results also have a small but present dependence on cosmological era; the exponent in the radiation era is slightly smaller (less negative) for every test performed. This can be understood by considering the form that a(τ) takes during each era, the Universe expands faster during matter domination so the energy density of domain walls should drop faster. However this result is simply a facet of the idealised cosmological era placed in the equation of motion. In a true physical scenario, the domain wall forming field is expected to couple at least weakly, to matter and radiation. So the form of a(τ) would not be as trivial as has been used in these simulations, and would change once the domain walls began to dominate. Nevertheless the fact that there is very little sensitivity to cosmological era here suggests that scale invariant dynamics are to be expected to be found for domain walls forming in most non-accelerating cosmological eras. To further investigate the dynamics of domain walls it may be necessary to consider simulating a more realistic fluid of matter and radiation coupled with the domain wall forming field, and using a more complicated form for a(τ) that includes the influence of all three components. It may also be interesting to consider domain walls interacting with a form of cosmological constant. Deviations from scale invariant evolution are disastrous for the resulting cosmology; for walls forming as early as possible around T 0.1 MeV, deviations as small as +0.1 in the ∼ exponent would lead to a present-day Universe that could contain as many as 10s of domain walls within the observable Universe [20]. Whilst scale invariant evolution allows domain walls to be expanded out of the horizon it would still lead to a Universe with an energy density far greater than what is observed. The results obtained in this study confirm domain walls as a catastrophic cosmological force and one that is to be avoided, at least in its simple form. In hybrid or constrained models however there is still scope for a domain wall type defect to have

20 existed at some point in time, for example a domain wall network that suﬀers a more extreme energy loss mechanism thereby making them unstable [24] or is biased to form larger domains [34, 38]. The presence of dark energy in the local Universe, too, has sparked an idea of a “frozen- in” domain wall network in a so-called elastic dark energy model [39, 40]. These models involve introducing a conserved charge to the walls, which adds a great deal more depth to model and has the prospect of interesting scaling results.

21 Chapter 4

Theory of Charged Domain Walls

4.1 Introduction

The discussion of charged topological defects traces back to a pioneering paper published by Edward Witten in 1985 on superconducting cosmic strings [41]. The status of being superconducting arises from the theory facilitating the emergence of non-dissipative currents confined to the interior of the string. Localised configurations of this nature are typically referred to as condensate fields. Associated with this internal structure are objects are known as cosmic vortons, string loops stabilised by the presence of a circulating current. In the cosmological context, these vorton strings are potential sources for a number of astrophysical phenomena, such as cosmic rays and gravitational waves [42, 43]. Historically, progressions in their study have often focused on the global version of Witten’s model in which the symmetry transformations do not possess a space-time dependence, eliminating the need for gauge fields. This allows analysis of general features of these systems whilst making their investigation significantly less demanding from both theoretical and computational perspectives. Adapting the recipe for superconducting cosmic strings, it is possible construct an analogous theory of domain walls. If the same stabilising behaviour were to manifest in the domain wall scenario, one would anticipate an effect to their dynamical evolution. This could correspond to departures from the documented scaling properties and thus leave implications regarding their compatibly with cosmology. It is also known that the vorton phenomena appearing in domain wall models represent a (2+1) dimensional analogue of cosmic vortons, with many of their qualitative features expected to be the same [25][44]. Hence, with the benefit of being able to identify exact analytic solutions instead of having to resort to approximations, the motivation to study charged domain walls is not only as an independent system but also as a truncation of a higher dimensional problem. The structure of this chapter is as follows; beginning by examining the mathematical formulation of the model, the conditions necessary for the existence of charged domain walls will be derived. Following this, an account will be presented of the numerical results regarding their scaling behaviour, and the natural occurrence of stable vortons will be discussed. The remainder will then focus on the dynamics of such objects, where analytic solutions will be isolated. This will include the conditions for their existence, stability analysis and the investigation of their interaction properties.

4.2 The model

The study of charged domain walls may be approached through the extension of the Z2 model with the inclusion of an additional complex scalar ﬁeld and an appropriate coupling term between the two constituent ﬁelds. The Lagrangian for such a system, is essentially a reduced version of that appearing in superconducting cosmic strings - the possible emergence of walls instead of

22 strings hinges on restricting one of the ﬁelds to be real [42],

1 µ µ = ∂µφ∂ φ ∂µσ∂¯ σ (φ, σ), (4.1) L −2 − − V with φ and σ corresponding to real and complex fields respectively. The potential in consideration contains symmetry breaking contributions for both fields accompanied by a φ, σ interaction term, λφ 2 λσ 2 (φ, σ) = φ2 η2 + σ 2 η2 + βφ2 σ 2 (4.2) V 4 − φ 4 | | − σ | | where λσ,λφ, ησ and ηφ are real non-negative constants. The theory possesses a Z2 U(1) × symmetry, associated with the respective field transformations of reflection, φ φ and global → − phase rotations σ eiΛσ (with Λ a constant). A general result from Noether’s Theorem is that → the existence of a global U(1) symmetry entails a conserved 4-current density denoted jµ:

jµ = Im(¯σ∂µσ) (4.3)

By analogy with electromagnetism, the 0-th component j0 is typically referred to as the charge density and the spatial components, current densities [45]. Accordingly, it is beneficial to define a Noether charge, Q as, Z 1 Z Q = dDx j0(x) = dDx(σ ˙ σ¯ σσ¯˙ ) (4.4) 2i − with D corresponding to the number of spatial dimensions. It will later become evident that value of this charge has a critical impact on the subsequent evolution of the system. Evaluating the Euler-Lagrange equations for each field leads to two coupled equations of motion,

µ λφ 2 2 2 ∂µ∂ φ φ η φ + βφ σ = 0, (4.5) − 2 − φ | |

µ λσ 2 2 2 ∂µ∂ σ σ η σ + βφ σ = 0. (4.6) − 2 | | − σ The roles played by each field in the theory depend intricately on the theory parameters. The presence of the coupling term opens the possibility of preventing both symmetries from being broken simultaneously within the vacuum, found to be achievable with sufficiently high β. The knowledge of the accessible vacuum states of the theory forms a platform in which to constrain the parameter space. However, different approaches to such have been presented in the literature, leading to a fair amount of ambiguity on the subject. Therefore the process relevant to this model will be covered in a reasonable amount of detail here in an attempt to eliminate any scope for confusion. In the conventional situation, the ground states are specified subject to minimising the potential . In a model containing a complex field, this approach turns out to be a little too V restrictive, failing to reveal the entire spectrum of degenerate minimum energy solutions accessible to the system1. The alternative perspective essentially shifts the task of energy minimisation to that of finding the global minima of an effective potential. Considering the general form of a complex scalar field σ = σ(x) eiα(x), the effective potential is defined as; | | 2 µ λφ 2 22 λσ 2 22 2 2 eff = σ ∂µα∂ α + φ η + σ η + βφ σ . (4.7) V | | 4 − φ 4 | | − σ | | Now, it is further supposed that the derivative terms appearing in the effective potential equate to a constant2. This contrasts with the restrictive nature of the approach mentioned initially, 1At the very least, such “true” vacuum solutions are ensured to appear at spatial infinity where derivatives must vanish to ensure normalisability. However, if for example the states at ±∞ are different, the field configuration will be forced out of the true vacuum as in the kink solution. 2This may be motivated by requiring such to be stable solutions of the field equation(s)

23 φ |σ| x x

y y

Figure 4.1: Three dimensional visualisation of the infinite planar wall (left) and that for a prototype condensate profile (right). which corresponds to the particular case of setting this constant to zero. To make the vacuum solutions more explicit, it is possible to reformulate equations (4.5) and (4.6), by shifting by an appropriate constant, as this will not impact the dynamics of the system. In doing this, the effective potential reduces to,

2 λφ 2 2 λσ 2 2 2 µ 2 2 eff = φ ηφ + σ ησ + ( ∂µα∂ α) + βφ σ . (4.8) V 4 − 4 | | − λσ − | | Note that the minimum energy is consistent with the previous method, instead now the ﬁeld values leading to these extrema vary according to the form of α, demonstrating a much broader spectrum of vacuum solutions than initially anticipated.

4.3 Symmetry breaking conditions

The scenario of interest is when a localised condensate field forms, confined to the locale of a domain wall (Figure 4.1). The essential question to address is: what values of the theory parameters are compatible with the formation of such a condensate? The first requirement is that the vacuum state reflects the Z2 symmetry broken, and the U(1) unbroken. That is, φ2 = η2 and σ = 0. The former of these conditions ensures that the topological criteria for the φ | | formation of domain walls is met. An effective potential consistent with this symmetry breaking pattern is presented in Figure 4.2. Supposing the case of an infinite planar domain wall in 2 + 1 dimensions lying along the y-axis, a condensate field flowing along the wall may be represented by the ansatz, σ = σ(x) ei(ky+ωt). (4.9) | | Relating to the conserved quantities of the theory, (4.3) and (4.4), it is easily extracted that the spatial and temporal frequencies k and ω are representative of the current and charge respectively. Substituting this ansatz into the effective potential, it becomes apparent that the presence of charge and current alters the energy and necessarily the vacuum solutions of σ to compensate,

2 λφ 2 2 λσ 2 2 2 2 2 2 2 Veff = φ ηφ + σ ησ + (ω k ) + βφ σ . (4.10) 4 − 4 | | − λσ − | | Comparing candidate vacuum states of the system, the aforementioned symmetry breaking situation can be achieved subject to satisfying following parameter constraint, 2 4 2 2 2 2 λφηφ > λσ ησ + (ω k ) . (4.11) λσ −

24 However, the above restriction neglects the possibility of both symmetries being spontaneously broken in the vacuum, which to be valid, requires the stability of the σ = 0 state. Returning | | to the Lagrangian formalism, such stability is verified by utilising the σ field equation (4.6) to consider the behaviour of the system under small pertubations from σ = 0. Seeking departures | | from the vacuum of the form σ = ϕ(t)ei(ky+ωt), ϕ 1, with φ2 = η2, an eigenvalue equation φ may be formulated in terms of the dynamical field ϕ, 2 2 1 2 2 2 ∂ ϕ = βη λση + ω k ϕ, (4.12) t − φ − 2 σ − where the spatial derivative terms vanish due to the consideration of correlated regions in the same degenerate energy state. Drawing parallels to the familiar oscillator problem, requiring stability forces the term within the parenthesis to be positive. Thus the limit of small fluctuations about σ = 0 recovers stable oscillatory motion by imposing the second restriction on the | | parameter space λσ 2 2 2 2 ησ + ω k β > 2 − . (4.13) ηφ

This substantiates the previous claim that the spontaneous breaking of both the Z2 and U(1) symmetries in the vacuum can be escaped with suﬃciently high β. At this point it is convenient to deﬁne the composite parameter χ,

χ = ω2 k2. (4.14) − Emphasising that it is only the collective contributions of charge and current that influence the dynamics of the system. The prescription outlined in the literature of cosmic vortons [42] is to classify these solutions as, χ = 0 : Chiral, • χ > 0 : Electric, • χ < 0 : Magnetic. • Furthermore, the mass terms for the fields (the coefficients of their quadratic terms) can now be written as, λφ λσ m2 = η2, m2 = η2 + χ. (4.15) φ 2 φ σ 2 σ which aid in providing a compact form for the constraints (4.11) and (4.13),

4 4 mφ m > σ , λφ λσ 2 (4.16) λφmσ β > 2 . 2mφ

4.4 Condensate formation conditions

The conditions derived so far are alone inadequate for the formation of a condensate inside a wall. Further insight into the problem can be gained by turning focus to the wall proﬁle. With parameters controlled such that the U(1) symmetry remains unbroken within the vacuum ( σ = 0), it is reasonable to assume that the proﬁle of the domain wall will retain the previously | | introduced form of the kink solution, p ηφ λφx φ = η tanh (4.17) φ 2

25 V V V |σ| |σ| φ φ

Figure 4.2: Illustration of an eﬀective potential in accordance with the spontaneous breaking of the Z2 symmetry only.

Towards the centre of the kink, φ is forced further from the true vacuum and towards φ = 0 state. It may therefore become energetically favourable that this “trapped” energy be compensated for with the formation of a bound state solution with σ = 0. This possibility is explored through a | | 6 similar perturbative approach to before, which first involves linearising the field equation (4.6) around the kink profile (4.17), 3 p µ λσ 2 2 2 2 ηφ λφx ∂µ∂ σ σ η + βη tanh σ = 0. (4.18) − 2 | | − σ φ 2

The emergence of a condensate solution, corresponding to a non-zero expectation value for the σ field insists instability in the σ = 0 state. Following the analytic approach of [19, 25, 42] the | | stability of the trivial σ = 0 solution is studied with departures of the form σ = ζ(x)eiνtei(ky+ωt) | | with ζ(x) 1.4 Implementing such into (4.18) generates a Schrödinger-like eigenvalue equation: p ηφ λφx ∂2ζ + βη2tanh2 m2 ζ = ν2ζ (4.19) − x φ 2 − σ A little thought on the supposed form of σ = ζ(x)eiνtei(ky+ωt) exposes that negative eigenvalues ν2 < 0 lead to subsequent growth of the magnitude of the field, thus representing instability. The method of identifying the bound state solutions and the associated spectra for the problem of (4.19) is outlined in [27]. It follows that, seeking a negative mode ν2 < 0 amounts to satisfying the condition 2 2 2 λφmσ(2mσ + mφ) β < 4 . (4.20) 2mφ Together with (4.11) and (4.13), this condition establishes how to confine the parameter space of the model to accommodate for the existence of superconducting domain wall networks. As some concluding remarks on this section; by introducing the effective potential it is possible to recover the full set of vacuum states necessary to consider the formation of condensate fields. A total of three restrictions were obtained on the model parameters, the first two of these ensured that the true vacuum state corresponded to a spontaneously broken Z2 symmetry, with the U(1) remaining unbroken. The last of these, builds on the former by forcing instability of the true vacuum of the σ field within the confines of the wall, generating a condensate field.

3The process of “linearising” essentially refers to the removal of the coupling between the two ﬁeld equations by inserting an explicit solution to the other obtained in absence of an interaction, in this case corresponding to the kink solution. 4More generally, using the perturbative degrees of freedom in this way corresponds to deviations from the ansatz (4.9) of the form |σ(x)| → |σ(x)| + ζ(x) and ω → ω + ν.

26 Chapter 5

Evolution of Charged Domain Walls

5.1 Numerical implementation

Having established the conditions for their formation, it is natural to question the dynamical impact of a localised condensate field. Remarked upon during the discussion of the vanilla domain wall model, the aspect of a network most relevant to cosmological compatibility is the associated scaling law. It proves useful to recall the formal definition of the conserved charge of the theory, supplemented by the most general ansatz describing a condensate field σ = σ(x, y) ei(k·x+ωt), this reads, | | Z Q = ω σ 2d2x (5.1) | | This implies that the magnitude of an emergent condensate network is controlled by the value of the conserved charge Q. The impact of condensate structures on dynamical scaling may therefore be probed by formulating test scenarios over a suitable range of Q. The framework for numerical implementation of the model follows the prescription detailed in chapter 2. The scheme is adapted to accommodate for the additional complex field by evolving the two component fields in parallel. The focus was to first replicate and then extend previous work on the study of charged domain wall networks [25, 44]. Therefore, to maintain consistency, the simulations presented here were assigned the same parameter values,

ηφ = 1, , ησ = √3/2, , λφ = 2, λσ = 2, β = 1 (5.2)

Modelling the initial conditions around a Z2 symmetry breaking phase transition, the earlier convention (each lattice point acquiring an arbitrary value from a Gaussian distribution) was applied to the φ field. To reflect the nature of random fluctuations around the σ = 0 vacuum | | state, the initial conditions for the σ field were composed accordingly. However, to accommodate for charge in the model the inclusion of a time varying component was necessary, leading to the i(ωt−θ ) projected form σ = Ae 0 defined at t = 0. The value θ0 specifies the initial phase value of the complex field which was extracted from a uniform distribution between 0 and 2π for each lattice point. To allocate the two remaining unknowns, A and ω, it is valuable to relate them to the conserved charge Q; Z Z 2 2 2 Q = A ω d x = ρQ(0) d x (5.3) where it is evident that they collectively define the charge density of the system, ρQ. It is beneficial from a practical perspective to absorb this dependence into a single parameter. Hence, from here on we set ω = 1. This enables the extent of the subsequent condensate structure to p be controlled exclusively by the initial field amplitude A = ρQ(0). It was noted in previous work [26] that the respective treatments of these parameters may be reversed (that is to say, fix

27 A and vary ω) without significantly altering the results. Unfortunately these initial conditions generate unphysical initial energy gradients between neighbouring points. In an FRW cosmological background these gradients are dealt with by a damping term in the equations of motion. However, there is no such term in the Minkowski equation, leading to erratic behavior and field unable to approach a steady configuration. The effects may be suppressed by introducing an artificial damping term in operation for a brief period during the initial phase of the network evolution.

5.2 Dynamical properties and scaling

Numerical simulations were performed on a 2 dimensional lattice with N = 2048 with spatial and temporal separations of ∆x = 1 and ∆t = 0.1 as before. Investigations into the scaling properties of charged domain walls were conducted for both Minkowski and FRW backgrounds. The former of these scenarios was performed with the intent of replicating the results of previous work [26]. In both scenarios, the dynamical range was extended to three times the light-crossing time with the interest of identifying structures existing in spite of the ﬁnite simulation box.

5.2.1 Charged domain walls - Minkowski Numerical simulations in a Minkowski spacetime were performed by introducing an artificial constant damping term for the first 200 timesteps of the simulation. The variation in the scaling behaviour with the initial charge density ρQ is displayed in Figure 5.1. Following an initial transient period the differently charged cases rapidly diverge into distinct scaling regimes. It is evident that the effect of the charge is to suppress the collapse of the domain structure. Furthermore the results suggest that as the wall length decreases, the network eventually reaches a point at which the walls are saturated by the charged condensate. At this point, the magnitude of the condensate becomes sufficient to prevent further collapse. However, this only develops prior to the light-crossing time for the example of ρQ = 0.4. The occurrence of charge saturation may also be appreciated from the visualisations of network evolution presented in Figure 5.2. It is apparent, that the quantity of charge possessed by the system reaches such a value that the condensate field may no longer be confined to the wall. One might suspect that it is naive to immediately attribute deviations from a scaling regime as being due to the periodic environment. This becomes particularly relevant for the curve ρQ = 0.27 which does not reach a halt until past the light-crossing time. The possibility was challenged by restricting scaling calculations to a subspace of a much larger simulation box. This construction would give the illusion of a significantly larger light crossing time for the considered sample box, allowing for comparison with some of the results required adopting the conventional method. This led to the observation that network freezing phenomena appeared to manifest regardless of numerical limitations. Of course, having to simulate a mostly discarded lattice is very inefficient and thus was not the primary source of results but merely served to eliminate the lattice geometry as the origin of certain static configurations.

28 5 10 1.5

0.5

4 10 −0.5

−1 Wall length

Scaling exponent−1.5 ρQ(0) = 0.01 ρQ(0) = 0.14 ρQ(0) = 0.27 −2 ρQ(0) = 0.4 3 10 −2.5 2 10 100 200 300 400 500 600 Conformal time Conformal time

Figure 5.1: Scaling results obtained for the range of initial charge densities ρQ = 0.01, 0.14, 0.27, 0.4 simulated in Minkowski spacetime. The vertical dashed line indicates the light-crossing time. The case, ρQ = 0.01 can be seen to approach scale invariance shown by the lighter dashed line.

τ = 40 τ = 100 τ = 500 τ = 700

ρQ = 0.01

ρQ = 0.14

ρQ = 0.27

ρQ = 0.4

Figure 5.2: Snapshots of the network evolution for the 4 initial charge densities considered.

29 5.2.2 Charged domain walls - FRW As for previous considerations of an FRW universe, the PRS algorithm was invoked to ensure the walls remain resolvable over the dynamical range. In contrast to the outcome in Minkowski spacetime, we have discovered that the presence of charge is found to be inconsequential on the scaling behaviour of the system, illustrated by Figure 5.3. Establishing that scale-invariance is approached regardless of the initial charge density of the system. The appearance of a condensate structure in FRW is captured by Figure 5.4.

5 10 −0.5

−1

4 10 −1.5 Wall length

Scaling exponent −2 ρQ(0) = 0.01 ρQ(0) = 0.14 ρQ(0) = 0.27 ρQ(0) = 0.4 3 10 −2.5 0 1 2 10 10 10 50 100 150 200 250 300 350 400 Conformal time Conformal time

Figure 5.3: Scaling results obtained for the same range of initial charge densities, simulated in an FRW cosmological background.

Figure 5.4: Snapshots depicting the development of a condensate network at four instants of the systems evolution.

5.2.3 Discussion On physical grounds the relevant scenario here is the case of an FRW background. However the example in Minkowski oﬀered signiﬁcant insight into the exact nature of the condensate

30 phenomena, not being distorted by the effects of cosmological expansion. It was concluded that it may transpire that the condensate becomes so intense that it acts to balance collapse entirely. Although less pronounced, such behaviour also occasionally appeared in the numerical simulations of an FRW universe. To elaborate, visualisations of the field configurations frequently saw the emergence of circular objects demonstrating the ability to stabilise from collapse. However, with substantial levels of structure still present at the light crossing time, these candidate stable solutions would only produce noticeable effects on the scaling dynamics at late times (far beyond the light-crossing time) when they would dominate the wall budget. Reaching this epoch, the scaling profile would be expected to be reminiscent the freeze-out phenomena demonstrated for Minkowski. A final detail is with regard to the conserved charge of the model. The simulations presented here were also performed for a neutral universe scenario, in which the lattice was populated evenly with regions of positive and negative charge, allowing for the emergence of localised regions of net charge. Interestingly, regions of opposing charge develop an opposing flow of current. However this adjustment did not lead to an observable change in scaling behaviour so to prevent redundancy in information the results were omitted.

31 Chapter 6

Analytic Solutions and the Construction of Vortons

It was noted in the previous section that numerical simulations occasionally saw the emergence of objects demonstrating stability, at least over the dynamical range. Resembling the behaviour of cosmic vortons, the stability of these loop structures may be attributed to circulating currents inhibiting their collapse. With the underlying topological defect corresponding to a kink variation as opposed to a cosmic string, these analogous phenomenon are referred to as kinky vortons [25]. Hereafter, the labels vorton and kinky vorton will be used interchangeably. In order to explore further the possibility of stable vortons, it is ideal to reveal the solutions of the model from a fully analytic treatment. This will allow for the issue of stability to be tackled directly by subjecting the results to numerical study.

6.1 Inﬁnite planar wall revisited

The analysis will proceed by first considering the setup of an infinite planar domain wall com- plemented with a condensate field of the form σ = σ(x) ei(ky+ωt). Following this format, the | | non-linear coupled field equations of 4.5 and 4.6 reduce to, λφ ∂2φ = (φ2 η2) + β σ 2 φ x 2 − φ | | (6.1) 2 λσ 2 2 2 2 ∂x σ = σ (ησ + χ) + βφ σ . | | 2 | | − λσ | |

Coupled equations possessing this level of complexity have been the subject of previous study [46] where numerous exact solutions have been documented. The ﬁeld solutions relevant to the discussion of localised condensates and thus kinky vortons are found to exist if and only if, λσ 2 2χ λφ 2β 2 2 ησ + = + 2 ηφ. (6.2) − β λσ 2β λσ −

Solutions of the above may be sought by forcing both sides to equate to zero. This enables to one to compute the following relationship between several parameters of the theory if the condition of (6.2) is to be met, λφ = λσ λ = 2β. (6.3) ≡ Given their equivalence, the parameters λφ and λσ will be referred to as a single λ in future discussion. Returning to previous constraints made on the parameter space of the theory, en- forcing (6.3) also makes the problem of identifying compatible parameters far more tractable. 2 2 The conditions presented in (4.16) both reduce to the constraint mφ > mσ and that of (4.20)

32 2 2 becomes mφ < 2mσ. The collective requirements of these may therefore be written as a single limitation on the mass terms, 1 m2 < m2 < m2 . (6.4) 2 φ σ φ 1 As a reminder, the mass components restated in terms of the parameter λ are mφ = 2 ληφ and 1 mσ = 2 λησ + χ. Having secured their existence by satisfying the criteria of (6.3), the solutions of (6.1) are now presented, s q 2(2m2 m2 ) q 2 2 σ φ 2 2 φ = ηφtanh x m m , σ = − sech x m m . (6.5) φ − σ | | λ φ − σ As noted in the previous chapter, the specific parameter values adopted in our numerical investigations were chosen to be consistent with those considered in previous work on the subject. However, with the appropriate tools now at hand, it is worth at least briefly motivating their election in the first place. With (6.3) removing the independence of two elements within the parameter space, three degrees of parameter freedom remain, these are ηφ, ησ and λ. The important point is that, upon assigning some arbitrary values to these constants, the equation (6.4) reduces to a formal statement of the values of χ accessible to the system. Aiming to study the behaviour of these objects in the chiral, electric and magnetic regimes, a set of values which accommodate for such are,

ηφ = 1, ησ = √3/2, λφ = λσ = λ = 2, β = 1. (6.6)

Confining χ to lie within range, 1 1 < χ < . (6.7) −4 4 The desirability of this particular choice derives from them offering a generic route into the study of kinky vortons, offering solutions in all regimes without being preferential to those either magnetic or electric. Returning to the condensate solutions of an infinite planar wall (6.5), with the injection of the selected parameter set they reduce to r x√1 4χ 1 + 4χ x√1 4χ φ = tanh − , σ = sech − . (6.8) 2 | | 2 2 The above solutions demonstrate the impact of the presence of charge and current on the overall features of the condensate. Focusing on the argument appearing in both the kink and condensate profiles, it can be seen that an increase in charge, or decrease in current leads to an increase in the characteristic width of both structures. Additionally, the coefficient of the condensate suggests that introducing additional charge leads to an enhancement of the amplitude whereas current acts to extinguish it. Combining these behaviours, it is evident that the addition of charge fuels the condensate, increasing both its width and magnitude. As illustrated in Figure 6.1, approaching the upper limit of the allowed range χ 1/4, the condensate becomes delocalised. → Conversely, current competes against charge acting to extinguish the condensate, diminishing both its width and amplitude until eventually driving it to vanish in the limit that χ 1/4, → − shown in Figure 6.2.

33 χ =0.1 χ =0.2 χ =0.24

|σ|

Figure 6.1: Proﬁles of σ demonstrating that increasing χ fuels the condensate structure. | |

χ =-0.1 χ =-0.2 χ =-0.24

|σ|

Figure 6.2: Proﬁles of σ demonstrating that decreasing χ extinguishes the condensate structure. | |

6.2 Kinky vortons

The previous discussion focused on identifying the exact analytic solutions of condensate phenomena appearing in the infinite planar wall scenario. The task now is to attempt to generalise these solutions to describe kinky vortons: circular loops of condensate forming on a radial kink profile. The first necessary adjustment to the theory reverts back to its foundations where an appropriately modified ansatz is utilised,

σ = σ(r) ei(Nθ+ωt), φ = φ(r), (6.9) | | subject to the boundary condition ∂r(φ(0)) = ∂r σ(0) to prevent any discontinuities at the | | origin. The notion of spatial frequency k has been adapted into a winding number, N. Recalling that the current component k was previously associated with variation perpendicular the kink, the equivalent in polar co-ordinates is given by the combination rθ. Therefore, for some loop of radius r and winding number N, the two formulations may be related via k = N/r. Furthermore, requiring σ to be single valued imposes the condition that N takes an integer value, making it a topologically conserved quantity. The convention to follow will refer to the radius of the vorton as the distance from the origin to the centre of the kink, denoted R. Finally, simple theoretic considerations allow one to justify that the solutions of the form (6.8) should be valid to good approximation if this radius of curvature R is substantially larger than the characteristic kink and condensate widths. This may be demonstrated by inserting the ansatz of (6.9) into the ﬁeld equations and accounting for the symmetry of the problem by transforming into cylindrical co-ordinates, 1 λφ 2 2 2 ∂rr∂rφ = (φ η + β σ φ r 2 − φ | | 2 (6.10) 1 λσ 2 2 2 2 N 2 ∂rr∂r σ = σ (ησ + (ω 2 ) + βφ σ . r | | 2 | | − λσ − r | |

34 If variations in the component fields φ, σ are localised to a scale small compared to the loop radius, the radial entry in the above coupled equations remains at roughly the constant R in the region where the spatial derivative terms become non-zero, i.e. the region proximate to the condensate core. Acknowledging such, it is reasonable to assume from the resulting field equations that the profile of kinky vortons may be described by suitably modifying the those presented in (6.8) r √1 4χ 1 + 4χ √1 4χ φ = tanh − r R , σ = sech − r R . (6.11) 2 − | | 2 2 − with χ = ω2 N 2/R2. In the vorton literature this considered region of solution space is com- − monly referred to as the “thin ring limit”. The appearance of the fields in a vorton configuration is displayed in Figure 6.3.

φ Re(σ) |σ|

Figure 6.3: Depictions of the kink and condensate ﬁeld conﬁgurations assigned to kinky vortons. The central illustration conveys the presence of current by explicitly resolving the winding number N.

Aiming to study the behaviour of these objects, it is much more desirable to be able to construct and quantify them entirely using only parameters with assigned physical significance, such as Q, N and R. This may be achieved by first restating the integral expression for the conserved charge Q, which reads: 1 Z ∞ Z ∞ Q = (σ ˙ σ¯ σσ¯˙ )d2x = 2πω σ 2rdr. (6.12) 2i −∞ − 0 | | Given that solutions are only assumed valid in the thin ring regime, the simple geometric argument captured by Figure 6.4 implies that the following integral approximation becomes applicable: Z ∞ Z ∞ 1 + 4χ x√1 4χ 2πω σ 2rdr 2πωR sech2 − dx. (6.13) 0 | | ' −∞ 2 2 which utilises the condensate profile appearing in the infinite planar wall scenario. Performing the integration, Z ∞ 2(1 + 4χ) σ 2rdr = , (6.14) | | √1 4χ 0 − leads to an exact expression for the charge, (1 + 4χ) Q = 4πωR (6.15) √1 4χ − The above formula may be manipulated into a far more beneficial form by taking its square and performing the substitution ω2 = χ + N 2/R2 to eliminate ω. This generates a cubic polynomial for χ, 2N 2 8N 2 Q2 Q2 1 16χ3 + 8 1 + χ2 + 1 + + χ + N 2 = 0. (6.16) R2 R2 4π2R2 − 16π2 R2

35 Figure 6.4: If the width of an annulus ∆W is suﬃciently smaller than the radius R, its area approximates to that of a rectangle of sides ∆W and 2πR.

It is worth noting that although this equation possesses 3 roots, there is always a unique real root, χ. Equipped with this, the physical characteristics and indeed behaviour of a vorton, may be distinguished entirely by the values of Q, N and R. Hence, with knowledge of this cubic and the associated ﬁeld equations, it is possible to enter any desired vorton as initial conditions in a numerical simulation. The cubic also allows one to formulate alternative descriptions of the various vorton regimes:

Chiral limit χ = 0 : • The condition necessary to produce a chiral vorton may be extracted from both (6.15) and (6.16) as Q = 4πN. It is also observed from the ﬁnal component of (6.16) that in taking the limiting case R the vorton asymptotically approaches the condition to → ∞ be chiral. An equivalent statement to make is that a vorton can be constructed to be arbitrarily close to chiral by specifying an appropriately large radius.

Electric Regime χ > 0 : • Operating under the assumption that χ > 0, validity of the cubic equation forces Q > 4πN as the only polynomial co-eﬃcient that may become negative in the cubic is the ﬁnal term. Hence, the onset of the electric regime may be characterized when Q > 4πN.

Magnetic Regime χ < 0 : • Similarly the converse scenario of χ < 0 necessitates the condition Q < 4πN. Collectively, the regime in which a vorton resides may be controlled exclusively by the parameters Q and N.

6.3 Energy minimising radius

Having established the functional form of kinky vortons, the next problem to address is which types are likely to emerge in a dynamical system, with the purpose of cataloguing the expected behaviours of the stable objects which where hinted upon in the numerical results. With such information, it then becomes possible to conclude whether or not these structures are expected to persist over time-scales greater than the scope of current numerical simulations. This issue may be approached by ﬁrst identifying parameter combinations Q, N and R that minimise the vorton energy, thus candidates to represent vacuum solutions. It is important to emphasise that for as long as the vorton structure remains, the winding number N is a conserved topological quantity1.

1As the ﬁeld must vary smoothly, changes in N are forbidden due to it being restricted to an integer value

36 Therefore, for given values of the conserved quantities Q and N the energy is controlled entirely by the vorton radius, R. The energy profiles as a function of vorton radius were obtained by placing the vorton configuration into the energy functional, and integrating numerically over a lattice. In the case of radially symmetric fields of the form (6.9), the energy functional reads,

Z ∞ 2 2 2 2 1 2 2 2 2 N 1 2 3 2 Q E = (∂iφ) +(∂i σ ) + (φ 1) + σ φ + + σ d x + , (6.17) | | 2 − | | r2 2| | − 4 R ∞ σ 2d2x −∞ −∞ | | where the previously declared values for the model parameters ησ, ηφ, λ, β have been accounted for. The ﬁnal term has used (6.12) to eliminate ω in favour of the conserved quantity, Q. Furthermore assuming validity of the thin ring approximation, the radial dependence appearing within the integrand may be suppressed by approximating with the vorton radius R. The energy minimising radius for a particular Q and N was exposed by performing numerical integration of the energy functional covering the range 10 R 300 and then de- ≤ ≤ termining the radial entry corresponding to the lowest recorded value. This scheme was repeated encompassing integers N spanning 0 N 250 for set values of the conserved charge ≤ ≤ Q = 0, 500, 750, 1000, 1500, 2500. The map of energy minimising solutions obtained is presented in Figure 6.5, those belonging to electric or magnetic regimes have been distinguished by adopting the formality discussed in the previous section. Knowing that chiral solutions lie on the interface between the two regimes, conditions for their construction may be directly extracted from the presented data. The plot for Q = 0 has been omitted as it remains at R = 0 irrespective of the value N. Reﬂecting on the implications of the data, it is observed that the vorton radius increases monotonically with both Q and N, agreeing with the interpretation that vortons are stabilised by both charge and current. In addition, it is apparent that if either Q or N are zero, the energy minimising radius is also zero, dictating that kinky vortons require both current and charge to exist.

300 Q = 2000 250 Q = 1500 R Q = 1000 200 Q = 750 150 Q = 500

100 Vorton Radius, 50

0 0 50 100 150 200 250 Winding Number, N

Figure 6.5: Energy minimising solutions for initial values of Q = 500, 750, 1000, 1500, 2000. Solutions residing within the electric regime are represented blue and those of the magnetic regime, green.

37 Chapter 7

Vorton Stability

7.1 Stability analysis

Prior analysis resolved the issue of what parameter values vortons would assume, should they form. It is possible however, that the object’s evolution departs from the vorton configuration entirely. This issue was addressed by exposing the projected vorton simulations to numerical evolution. The route into this follows primarily the account presented in chapter 5, the only difference residing in the initial conditions of the simulation. The resolution variables ∆t and ∆x and the model specific η’s, λ’s and β remain identical. The previously arbitrary initial field configurations are replaced by axially symmetric kinky vorton solutions of the forms described collectively by (6.9) and (6.11). The examination of the vorton constructions derived in the previous chapter was conducted in two stages. The objective was to first eliminate any vortons which decay. This aided in mapping the the region of solution space representing vorton configurations able to form in a dynamical system. The stability of the surviving vortons in response to radial perturbations was then studied, further confining the candidate vorton solutions to those capable of persisting in a physical environment. Attempts to map valid vorton solutions were limited to the cases of Q = 500 and Q = 1500 as the qualitative deductions presented to follow extend to arbitrary values of Q. Exploring the outcome for the example Q = 500, it was established that no persisting vorton solution exists within the confines of the Q N parameter space. This indicates that sufficiently large Q is − required to inhibit vorton collapse. The results for Q = 1500 allowed for, to reasonable precision, the identification of the parameter values corresponding to an onset of electric and magnetic instabilities.

Electric instability χ > 0.2122: • Deep into the electric regime, solutions of N < 10 or equivalently R < 50 are observed to suﬀer from ensuing collapse. The source of this instability can be attributed to the condensate becoming substantially delocalised, to such an extent that the loop cannot stabilise [43]. This reﬂects the limit at which the thin-ring approximation breaks down.

Magnetic instability χ < 0.08 : • − Evolving part-way into the magnetic regime, vortons corresponding to R > 244, N > 213 are found to exhibit a “pinching instability”. This signifies the region within which ’pinching’ effects manifest locally along the condensate, driving the field to zero and forcing the vorton to partially unravel. This leads to a reduction in the winding number, with the structure ultimately decaying toward a less magnetic solution. This process is visualised in Figure 7.1 which has captured the essence of the instability through several snapshots

38 of the vorton’s evolution. The evolution of the radius of the same vorton shown in Figure 7.2(a).

The range of surviving solutions, free from these these instabilities are presented in Figure 7.3.

Figure 7.1: Time-lapse (left to right) showing the evolution of a magnetic vorton admitting a pinching instability. The consequences of the pinching mechanism become evident in the second snapshot where the vorton begins to unwind. Following a period of rapid collapse, the ﬁeld conﬁguration oscillates between those shown in snapshots 3 and 4. The amplitude steadily diminishing as the condensate radiates energy. The process serves as to generate a less magnetic vorton which can be inferred from the transition to a more pronounced condensate structure, accredited to a larger value of χ.

250 80

65 200 60

55 Radius Radius 50 150 45

100 30 0 2000 4000 6000 8000 10000 0 2 4 6 8 10 12 14 16 18 4 Time Time x 10

Figure 7.2: Left (a): Time evolution of the radius of the same vorton as in 7.1. Following the pinching mechanism inducing a signiﬁcant decrease in the radius, the conﬁguration subsequently oscillates around an attractor state of R 145. Right (b): Radial evolution of an electric ' vorton of stable radius R = 50 perturbed along the x-axis by 10%. The radius displayed is the circumference (calculated using the length calculation algorithm) divided by 2π.

39 300

250 R 200 Q = 1500 150

100 Vorton Radius, 50

0 0 50 100 150 200 250 Winding Number, N

Figure 7.3: Vorton solutions for the case of Q = 1500, with the vortons exhibiting electric or magnetic instabilities removed.

To further probe stability, the remaining vortons were subjected to deformations. To isolate generic features, test solutions were distributed over the allowed range including both electric and magnetic regimes. For example, a perturbed electric vorton in Figure 7.2(b). In addition to demonstrating incredible resilience to axial and non-axial pertubations, there is a natural tendency for the structure to converge on the optimal radius following a long-lived phase of oscillation. This behaviour was realised for the entire set of test solutions, all exhibiting the capacity to stabilise.

7.2 Vorton candidates via energy minimisation

Towards the end of the study it was discovered that the stability and existence of vortons can be found in a more concise manner using a gradient descent method. This consists of an iterative technique which acts to evolve the system towards the nearest local minima of some desired function, which in the case of stable vortons is the total energy. The iterative process is implemented as follows, ∂ φn+1 = φn E − ∂φ (7.1) ∂ σ n+1 = σ n E . | | | | − ∂ σ | | with the parameter (> 0) some small quantity and the energy density. This formula can be E likened to a first-order finite difference scheme with ∆t: ≡ ∂φ ∂ = E = 2φ 2 φ(φ2 1) φ σ 2 , (7.2) ∂t −∂φ ∇ − − − | |

∂ σ ∂ N 2 Q2 | | = E = 2 σ 2 σ φ2 σ + σ 3 σ . (7.3) ∂t −∂ σ ∇ | | − | | − | | R2 | | − | |(R ∞ σ 2d2x)2 | | −∞ | | This approach is inherently less computationally taxing than evolving perturbed vortons, as it only demands expansion to ﬁrst order to perform an iteration 1. Supplementing this natural

1 ∂E This refers to the iterative mapping itself, here we have φn+1 = φn − ∂φ which contrasts the previous ﬁnite ˙ 1 ¨ 2 diﬀerence scheme which expands to second order φn+1 = φn + φ∆t + 2 φ∆t

40 advantage are the dynamic step sizes, being more substantial at larger distances from the neighbouring minima. Theoretically, this would enable the system to converge on stable attractors within a much more desirable time-scale, bypassing the large period of oscillation which appeared during the evolution via the ﬁeld equations. Unfortunately, due to the time limitations we were unable to employ this approach to identify the parameter region admitting vorton solutions. Hence, results presented were those acquired by evolving the ﬁeld equations. However, future work would likely favour this implementation.

7.3 Vorton collisions

Persisting vorton solutions become particularly relevant at late times, when they would represent the only surviving forms of charged domain walls. It is inevitable then that these objects will traverse space essentially unhindered and thus it becomes natural to question the outcomes of interactions between vortons. A requirement to simulate their collisions, is of course being able to induce motion in the stationary solutions describing kinky vortons (6.11). Exploiting the Lorentz invariance of the theory, it is possible to obtain propagating solutions through the act of performing a Lorentz boost. For the consideration of scalar fields, only co-ordinate variables need to be transformed. For example, to implant motion parallel to the x-axis, the relevant co-ordinate transformation is given by x vt x X = − → √1 v2 − (7.4) p p r = x2 + y2 X2 + y2, → for some configuration velocity v. It is also necessary to control the impact parameters and specify an initial separation between the vorton solutions, this is achieved by performing the spatial transformations, X X a, y y b, (7.5) → − → − which shifts the centre of a particular solution to co-ordinates (a, b). Equipped with these transformations, and in conjunction with the stationary solutions of (6.11), it is a simple task to generalise the initial conditions of a numerical simulation to accommodate for collisions. For reference, the analytic forms for a kinky vorton propagating at speed v centred on co-ordinates (a, b) are, r √1 4χ 1 + 4χ √1 4χ φ = tanh − ρ R , σ = sech − ρ R ei(Nθ+ωt), (7.6) 2 − 2 2 − p where ρ = (X a)2 + (y b)2. − − The various parameters distinguishing a collision can be decomposed into two categories, those specifying the structures of the colliding vortons and those relating to their trajectories. To isolate their respective influences on the outcome, the testing procedure was broken down accordingly. For computational ease two sample vortons were used, one in the magnetic regime (Q = 1500,R = 245,N = 150), and one in the electric regime (Q = 1500,R = 50,N = 10). Due to the winding number N appearing as a quadratic term in the dynamical equations, it is also perfectly valid obtain an additional solution by performing the substitution N N, → − which serves as to reverse the rotation direction of a given vorton (vortons with negative winding numbers are referred to as anti-vortons). Subsidising this with the previously declared sample solutions, four unique test scenarios were proposed (two of which are shown in Figure 7.4),

41 Electric - Electric collision • Electric - Anti-electric collision • Electric - Magnetic collision • Electric - Anti-magnetic collision •

Figure 7.4: Time-lapse (left to right) of an Electric-Electric collision and an Electric-Anti-Electric collision.

Aiming to draw parallels with classical collisions, these scenarios were studied for a range of velocities and impact parameters, cataloguing the various outcomes. The following graphics, Figures 7.5 and 7.6, illustrate how the collective size of the vorton structure contained within the simulation box evolves in time, for two of the test scenarios. The results presented here undoubtedly reﬂect the non-linearity of the theory, revealing only a very rough decipherable structure.

v1 = 0.5, v2 = 0 v1 = 0.9, v2 = 0.9 −100 500

−80 450 −60 −60 400 −40 −40 350 −20 −20 300

0 250 0

20 200 20 40 150 Impact parameter Impact parameter 40 60 100

80 50 60

100 0 500 1000 1500 2000 2500 3000 3500 Summed length 1000 2000 3000 4000 5000 Time Time

Figure 7.5: Collision spectrum for the ﬁnal system length in the outcomes of the collisions between two electric vortons for a range of velocities. One can track the moment of collision by the boundary of the dark red region of the plot.

42 Aligned rotation directions Opposing rotation directions 0.4 0.4 1000

0.5 0.5 800

0.6 0.6 600

0.7 400 0.7

0.8 200 0.8 Electric vorton velocity Electric vorton velocity

0.9 0 0.9 500 1000 1500 2000 2500 3000 Summed length 500 1000 1500 2000 2500 3000 Time Time

Figure 7.6: Collision spectrum for the outcomes of collisions between an electric and a magnetic vorton, for a range of impact parameters (b) between the two vortons.

Figure 7.5 depicting the influence of the impact parameter on the resultant structure conveys some peculiar behaviour. Given the initial symmetry of the vorton geometries, the remarkable non-linearity in the data suggests that the resulting configuration is incredibly sensitive to the alignment of the respective wave components of the two condensate fields. This hints towards an understanding that the interference phenomena between two interacting condensate fields constitutes a crucial ingredient in influencing the end product of a collision. Figure 7.6 illustrates the consequences of velocity variation on the surviving vorton structure. The ambition of this procedure was to liken kinky vortons to mechanical objects possessing a surface tension by identifying an associated critical velocity, above which the surface is pene- trated. One would anticipate that successful surface penetration would manifest as an abrupt change in the structure profiles of (7.6). For the case of aligned current circulation, it can be seen that this critical velocity is suggested at vc 0.42. On the contrary, for opposing polarisations ' the results portray no such sharp variations. This could possibly be attributed to contributions due to opposing circulation serving to shift the critical velocity outside the studied range. The outcomes of collisions that have been demonstrated here may be split into three clas- sifications: annihilation, combination and reflection. Annihilation occurs when the two vortons combine and immediately collapse, this can be seen happening frequently in Figure 7.6 in the case of oppositely aligned vortons. Combination occurs when a stable vorton solution is reached and persists after the collision takes place, this can be seen happening in Figure 7.5 in the case of aligned rotation. Finally, reflection occurs when the collision velocity is below the critical value for a given pair of vortons, this is seen in the collision spectra when the total length of the system appears to remain constant. This classification scheme appears to be the only sense of structure present in these scenarios. The difficulty faced here is similar to results obtained in 1 dimensional kink collisions performed previously in the literature [47]. The simulations that were carried in this study were severely limited due to the time and computational resources available. The large box sizes and extended simulation times needed were the limiting factor. In future it would of great interest to be able to perform a full study of a wide range of vorton types and collision parameters. However the evident non-linearity of the results that has been discovered here hints that there may not exist any underlying structure to the outcomes of collisions.

43 Summary and outlook

Our initial findings in the Z2 domain wall model confirmed scale invariant evolution as the asymptotic scaling law, as well as an insensitivity in this evolution to cosmological era and spatial dimension. We have also suggested that deviations from this solution found in the past literature are caused by the limited dynamical range of small-scale simulations. In future work with the use of enhanced computing power, considerations given to the wall producing phase transition and coupling to matter and radiation would help to further the study of domain walls. Following from this we have found that introduction of a conserved charge to the model adds a great deal more depth to the theory of domain walls. We have found that in agreement with past research, charged domain walls in a non-expanding frame have their scale invariance suppressed by an amount related to their charge. However we have discovered here that walls inside an expanding Universe exhibit the expected scale invariant evolution. We suggest that the model be reworked in an FRW background if departures from scale invariance are still desired, for example in dark energy models. It may also be interesting to study the scaling of this model in 3+1 dimensions, which has not been investigated here due to time and computing restrictions. We have also derived stable analytic solutions in this model which are 2+1 dimensional analogues to cosmic vortons. We have outlined the steps we have taken in finding which of these solutions are stable and have demonstrated their stability under perturbations and collisions. This is the first investigation of collisions between vortons and although this study was restricted, we have made conclusions regarding the non-linear nature of the outcomes and have classified the possible scenarios in the hope that it may lead to future research.

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