Primordial Galactic Magnetic Fields from the QCD Phase Transition by Michael McNeil Forbes B.Sc, The University of British Columbia, 1999 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Physics and Astronomy) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April 2001 © Michael McNeil Forbes, 2001 In presenting this thesis in partial fulfilment of the requirements for an ad-vanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics and Astronomy The University Of British Columbia Vancouver, Canada Abstract ii Abstract In this thesis I describe in detail a mechanism which we proposed to gener-ate large-scale primordial magnetic fields with correlation lengths of 100 kpc today. Domain walls with QCD scale internal structure form, coalesce and attain Hubble scale correlations. These domain walls subsequently align nucleon spins. Due to strong CP violation, nucleons in these walls have anomalous electric and magnetic dipole moments and the walls are ferro-magnetic. This induces electromagnetic fields with Hubble size correlations. The same CP violation also induces a maximal helicity (Chern-Simons) cor-related through the Hubble volume which may support an "inverse cas-cade" allowing the initial correlations to grow to 100 kpc today. Details of the physics and estimation methods are presented as well as necessary background and a discussion of the numerical methods used to obtain the classical domain wall solutions. In particular, a nice method for estimating properties of flat domain walls is presented. In addition, possible flaws with the argument are examined and other applications of QCD domain walls to astrophysical problems are discussed. Contents iii Contents Abstract i i List of Figures vi Forward v i i i Acknowledgements ix Part I Thesis 1 1. Introduction 2 2. Overview of the Mechanism 4 3. Evolution of Magnetic Fields 8 3.1 Helicity Conservation 8 3.2 Scaling Laws 9 3.3 Phase Transitions 10 3.4 Applicability of Scaling Laws 11 4. Generation of Magnetic Fields by Domain Walls 12 4.1 Hubble Size Correlations 12 4.1.1 Q C D Phase Transition -. . . 13 4.2 Essential Domain Wall Properties 15 4.3 Strong C P Violations in Domain Walls 17 5. QCD Domain Walls 20 5.1 Domain Wall Solutions 22 5.2 Domain Wall Decay 25 6. Alignment of Spins in the Domain Wall 28 6.1 Fermion Degeneracy in the Domain Wall 33 Contents iv 7. Generation of Electromagnetic Field 34 7.1 Helicity 35 8. Conclusion 37 8.1 Summary 37 8.2 Speculations and Future Directions 39 Bibliography. 42 Part II Appendices 48 A. Observations 49 A . l Techniques 49 A.1.1 Zeeman Splitting 49 A.1.2 Synchrotron Emissions 50 A.1.3 Faraday Rotation Measures (RMs) 50 A.2 Results 51 A.2.1 Local Fields 51 A.2.2 Clusters 52 A. 2.3 Intergalactic Media 52 A. 3 Summary 53 B. QCD Domain Wall Structure 54 B. l Introduction 54 B.2 Effective Lagrangian and 9 dependence in QCD 55 B. 2.1 The (7(1) Problem 56 B.2.2 Chiral Effective Lagrangian 57 B.3 Topological Stability and Instabilities 64 B.3.1 Higher Degrees of Freedom 66 B.4 Domain Walls 69 B.4.1 Domain Wall Equations 69 B.4.2 QCD Domain Walls 71 B.4.3 Axion Dominated Domain Walls 74 B.4.4 Axion-Pion Domain Wall 77 B.4.5 Axion-Eta' Domain Wall 77 B. 5 Decay of the QCD domain walls 79 C. Numerical Solution of Domain Wall 90 C. l Introduction 90 C.2 COLNEW 91 C.3 Purpose 91 Contents v C.4 Usage 92 C.5 Change of Variables 99 C.6 QCD Domain Walls 100 C.6.1 Example System 101 C.7 Original Documentation 102 C.8 Code Listings 102 C.8.1 Example Problem 102 C.8.2 QCD Domain Wall Problem 103 C.8.3 System Files 103 C.8.4 Automatic Differentiation 104 C.8.5 COLNEW 105 List of Figures vi List of Figures 5.1 Basic form of the QCD domain wall (soliton). The analytic approximation 5.10 is plotted as a dotted line to show the good agreement. We have taken Nc = 3 here. Notice that the wall thickness is set by the parameter \i 24 B . l Three examples of transitions that are topologically different. Paths A and B wind around the hold once (n = 1) and twice (n = 2) respectively whereas path C does not wind (n = 0). Path C can be shrunk to a point whereas the others cannot. Each path is said to belong to a different homotopy class. Only paths A and B are affected by the topology of the space. Path C might imagine that it is living in a space with no hole. 65 B.2 Here we show the same picture as in Figure B . l except that we show the third dimension. Here we can see that all the paths are now homotopically equivalent. We can deform the paths by "lifting" them over the obstacle so that we can un-wind them. If the paths were strings with some weight, then it would require some energy to "lift" the strings over the obsta-cle. If this energy was not available, then we would say that, classically, the configurations that wind around the "peg" are stable. Quantum mechanically, however, the strings could still tunnel through the "peg", and so the configurations are unstable quantum mechanically. The probability that one string could tunnel into another configuration would depend on the height of the "peg" 67 B.3 Basic form of the QCD domain walls. Notice that the scale for the pion transition is larger than for the eta' transition. Notice that the width of the rf wall is set by the scale m^. . 75 B.4 Same form as in Figure B.3 except in terms of the variables u and g = n to the left and (f>s = 0 to the right. The trough of the potential lowers from the cusp at p = 1, = TT where V = —ECOS(TT/NC) down to the vacuum state VM-M — —E. The hump where V = 0 is at the origin is where h = pexp(i(ps/NC) = 0 and hence the singlet field 4>s can have any value at this point. It is by passing across this point that a QCD domain wall can tunnel and a hole can form 86 Forward viii Forward The focus of this thesis is the discussion of a mechanism for forming large scale galactic magnetic fields from domain walls at the QCD phase transition in the early universe. As such, the purpose of the main body is to present this mechanism in as clear a manner as possible. The mechanism, however, includes concepts from several branches of physics. In order to elucidate the main presentation, several of the details, especially concerning the domain wall structure and QCD background, have been placed in appendices. The result is that the main body of this thesis serves as a presentation and tour of the mechanism for generating magnetic fields, referring to the (rather lengthy) appendices for technical details. By organising the thesis in this manner I hope to make it accessible to audiences who do not have the specific expertise, while providing the details for those readers who do. In particular, this thesis bridges both astrophysics and particle physics and it is my hope that the main body is accessible to members of both communities. The main body was written almost entirely by me and parallels the paper [1], though the ideas were developed jointly with my supervisor Ariel Zhitnitsky: the use of "we" generally refers to the two of us. Appendix B parallels the paper [2] which was written by both of us: I have removed or modified most of the content that was written by my supervisor, however, the arguments and estimates of the domain wall lifetime must be credited to him as must the detailed justification of the chiral effective Lagrangian. The rest of the thesis is exclusively written by me. The main ideas presented in this thesis were first communicated in the letter [3], and later in the papers [1, 4]. Details about QCD domain walls were first presented in.[2]. This thesis should be regarded as a collection and elucidation of these papers in a cohesive unit, with added explanations and background so as to be accessible to advanced undergraduate and graduate students working in the field. A cknowledgem ents ix Acknowledgements I am greatly indebted to my supervisor Dr. Ariel Zhitnitsky for the idea and much of the drive behind the ideas presented in this thesis. Without his support and contributions none of this would have been possible. I would like to thank him for exposing me to high energy physics and for showing me the possibilities in bridging high energy physics together with astrophysics and cosmology. I would also like to thank Robert H . Brandenberger and M . B . Voloshin for many useful discussions. My supervisor would like to thank: M . Sha-poshnikov and I. Tkachev for discussions which motivated this study; Larry McLerran and D. Son for discussions on Silk damping; and A . Vainshtein for discussions on the magnetic properties of domain walls. I also wish to thank F. Wilczek, K . Rajagopal, for useful discussions about high density matter and astrophysics, and A . Vilenkin and T. Kibble for discussions about phase transitions in the early universe. This work was supported by the N S E R C of Canada. Part I Thesis 1. Introduction 2 1. Introduction M a n y different observations suggest that there exist substantial (microgauss) magnetic fields i n the universe today [5, 6], however, there has yet to emerge a theory which adequately explains the origins of these fields. Most of the data on large scale astrophysical magnetic fields comes from the observa-t ion of synchrotron radiat ion emitted i n galaxies. T h i s radiat ion is plane polarized, and as it passes through magnetic fields, the plane of polar izat ion rotates due to the Faraday effect: an effect which depends on the frequency of the radiat ion, electron density, and the strength and orientation of the magnetic fields. B y comparing several sources or radia t ion w i t h different fre-quencies, one can extrapolate to determine the or iginal plane of polar izat ion and then estimate the magnetic field strengths. 1 W h a t is s t r ik ing, is not just the existence of magnetic fields, but that there appear to be microgauss fields which have correlations and field re-versals as large as 500 kpc in clusters. To put this into perspective, the luminous cores of galaxies have typical scales of up to 10 kpc while it is estimated that the galactic dark matter halos extend to 50 kpc. The current models for producing these fields involve two ma in processes: 1) dynamica l amplif icat ion and/or generation of magnetic fields by galactic processes (galactic mechanisms) and 2) p r imord ia l mechanisms which take place prior to gravitat ional structure formation. The galactic mechanisms are pr imar i ly based on gravitat ional dynamos, al though there are sugges-tions that supernovae or other stellar phenomena may play a role. W h i l e it is l ikely that galactic dynamos amplify fields, it seems difficult to account for the large scale correlations of the magnetic fields when only galactic mechanisms are considered. It is also not certain that galactic mechanisms can generate magnetic fields: instead they may serve only as an amplifier, requiring seed fields to be present for the dynamo to work. The inadequacies of the galactic mechanisms have lead to many proposals that the magnetic fields may have a pr imordia l origin. In this case, some process i n the early universe (typically at a cosmic phase transi t ion or dur ing 1 See Appendix A for details about the observational methods and results. 1. Introduction 3 inflation) is thought to generate magnetic turbulence. This turbulence then sustains itself as the universe expands and the fields that we observe are the remnants of this turbulence. Most primordial sources, however, also produce fields which end up with very small correlations today or which are very weak. Most likely, a complete picture of the history of astrophysical magnetic fields requires some primordial inputs as well as an amplification mechanism provided by gravitational dynamics. In this thesis, however, we discuss an-other primordial mechanism which may be able to produce fields of 100 kpc correlations today. In combination with dynamic amplification mechanisms, this mechanism might provide a solid foundation for the theory of large-scale astrophysical magnetic fields. At this point we would like to refer the reader to the several reviews and sample papers in this field. The primary discussions of observations, which contain reviews of the theory, are presented in [5, 6]. Good current reviews are given in [7-9]. Many different types of primordial mechanisms are dis-cussed, for example: Inflationary mechanisms [10], cosmic strings [11, 12], charge asymmetries [13, 14], and phase transitions [15, 16]. The evolution of primordial magnetic fields is discussed in [9, 17, 18]. In particular, the inverse cascades discussed will be important for our mechanism. 2. Overview of the Mechanism 4 2. Overview of the Mechanism The mechanism that we propose has the following core components: 1. Sometime near the Q C D phase transition, T Q C D ~ 1 GeV, domain walls form which can interact with Q C D scale physics. 2. These domain walls rapidly coalesce until there remains, on average, one domain wall per Hubble volume with Hubble scale correlations. 3. Baryons interact with the domain walls and align their spins along the walls. The Hubble scale correlation of the domain walls thus induces a Hubble scale correlation in the spin density. 4. The anomalous magnetic and induced electric dipole moments of the baryons generate helical electromagnetic fields also correlated on the Hubble scale. 5. The domain walls move rapidly and vibrate, effectively filling the Hub-ble volume with helical magnetic turbulence with a Hubble scale cor-relation. 6. The domain walls decay and the electric fields are screened leaving magnetic turbulence with Hubble scale correlations. 7. As the universe expands, an "inverse cascade" mechanism transfers energy from small to large scale modes, effectively increasing the re-sulting correlation lengths but diluting the field strengths. 8. Galactic dynamos amplify the fields in galaxies, but the fields should also persist in the extra-galactic media. The idea that domain walls might generate magnetic fields is not original. It was suggested that standard axion domain walls could be ferromagnetic in [19] , however mechanism discussed their seems to be flawed: The scale of the standard axion walls is of the order m~l which is at least some twelve orders of magnitude larger than the Q C D scale A Q £ D (ma is the axion 2. Overview of the Mechanism 5 mass.) It is hard to see how these walls can efficiently affect QCD physics at the temperatures that were present in the early universe where thermal fluctuations will destroy all coherence.1 Another problem with proposals including standard axion domain walls is that these walls must decay to prevent cosmological problems [22]. There are still questions about how the standard so-called N ^ 1 axion domain walls can decay. The N — 1 axion model, which has a known decay mode [23], may be relevant if it an be endowed with a QCD scale substructure. This thesis outlines the properties that domain walls must have to gen-erate sufficient magnetic seed fields via the proposed mechanism. The exact source could be one of several types of walls, including modified axion do-main walls, or something entirely different. To be concrete, we present our model in terms of a recently conjectured quasi-stable QCD domain wall [2] which may exist independently of axion physics or which can add additional QCD scale structure to the standard axion domain walls. These domain walls are characterized by a transition in the singlet rj field which has a size and energy scale set by A Q C D - Hence, QCD domain walls can directly couple to QCD physics. In addition, the singlet field transition at the center of the wall induces an effective non-zero CP violating 6 background which in turn will induce an electric dipole moment and alter the magnetic dipole moment in the fermions [24]. Thus, both the electric and magnetic dipole moments of all the particles are on the same order. In the presence of these anomalous dipole moments, the potential cancellations discussed in [25, 26] are no longer a problem and the walls are ferromagnetic. Another crucial aspect of our mechanism is some sort of "inverse cas-cade" mechanism which governs the evolution of the magnetic fields after they are formed. This mechanism was suggested by Cornwall [27], discussed by Son [28] and confirmed by Field and Carroll [29]. It is based on the idea that magnetic helicity (Abelian Chern-Simons number) H = J A • Bd 3 £ is approximately conserved in the universe where temperatures are higher than To ~ 100 eV. This conservation of helicity causes energy to cascade up the turbulent modes, increasing the energy in large scale modes, and in-creasing the effective correlation length of the turbulence. The importance of helicity was originally demonstrated by Pouquet and collaborators [30]. Without this helical inverse cascade, there is no known way to generate large correlations fields today from sub-Hubble scale fields formed in early phase 1 For some other discussions about the magnetic properties of the domain walls, see [20, 21] and reference therein. We should note, however, that in all these discussions, the most difficult problem of generating large scale correlations has not been addressed. 2. Overview of the Mechanism 6 transitions, and one must consider super-Hubble scale fields generated by inflationary mechanisms. It turns out, however, that if the inverse cascade functions, then Hubble scale correlations at the Q C D phase transition (the last major phase transition) result in ~ 1 0 0 kpc correlations today. Thus it is natural to consider Q C D physics as the source of primordial fields (earlier physics can only produce even smaller correlations). We must point out, however, that the mechanism of this inverse cascade is not well understood. Indeed, most of the results are based on simple scaling arguments with somewhat restrictive assumptions. We shall discuss this point later. In this thesis (and in our previous papers [1, 3 , 4 ] ) we have assumed what appears to be rather maximal efficiency for the inverse cas-cade. This mechanism relies on such a cascade to produce the correlations. More wil l be said about this later. Another little understood aspect of this mechanism concerns the dynam-ics of the domain walls and the interactions of the domain walls, nucleons and electromagnetic fields. As we shall show, all of these components inter-act on the same scale A Q C D and hence there are complicated back-reactions and nonlinear dynamics. Presently, we do not have the tools to fully analyze these features, but we show here in detail, quantitative calculations which we believe are good estimates of the scale of the effects. If there is an inverse cascade mechanism which functions at the maximal efficiency estimated in [ 2 8 , 2 9 ] , then the mechanism described here may produces magnetic fields today with correlations up to I ~ 1 0 0 kpc and with strengths of i Q-9Q Brms ~ 77 , 1 ~ 1 0 0 kpc ( 2 . 1 ) s A Q C D where the parameter 1 < A Q C D ^ 1 0 1 9 depends on the dynamics of of the domain walls as discussed in Chapter ( 4 ) . If the correlation £ turns out to be small, then this mechanism might generate detectable extra-galactic fields, otherwise we still require a galactic dynamo to amplify the fields. In any case, however, if the inverse cascade is sufficient to maintain the large scale correlations, then it seems that, even if £ is large, the resulting fields may be strong enough to seed the galactic dynamos [ 3 1 ] . We shall begin by discussing the inverse cascade mechanism in Chap-ter ( 3 ) and then give estimates of the field strengths in an idealized case of static, flat walls. Finally, We shall discuss the dynamics of the domain walls and describe the whole process, justifying the mechanism. The impor-tant details about the domain walls will be summarised in this body: A n 2. Overview of the Mechanism 7 extended discussion about the relevant physics and numerical methods is presented later in the appendices. 3. Evolution of Magnetic Fields 8 3. Evolut ion of Magnetic Fields Given a stable magnetic field configuration in the universe, one might naively expect the size of the correlations of the field to expand with space as gov-erned by the universe's scale parameter I oc R(T) and the field strength to be correspondingly diluted B oc R(T)~2. It was discovered by Pouquet and collaborators [30], however, that, if the magnetic fields have a non-zero he-licity (Abelian Chern-Simons number) H = f A • Bd 3 a; , then the fields will scale differently. Cornwall [27] suggested that helical fields might undergo an inverse cascade. The magnetohydrodynamic (MHD) equations were studied by Son [28] who derived the scaling relations (3.1) presented below. These have subsequently been confirmed by Field and Carroll [29]. We assume throughout this section, that helicity can be generated which supports the inverse cascade. 3.1 Helicity Conservation The basic idea behind the inverse cascade is that the magnetic helicity H is an approximately conserved quantity in the early universe. It is also known that the small scale turbulent modes decay more rapidly than the higher scale modes. In order to conserve the helicity, as the small scale modes decay, the helicity must be transfered to the larger modes. Wi th this transfer of helicity is a corresponding transfer of energy. This is the source of the inverse cascade. To understand the origins of the conservation of helicity, note that it is a topological quantity that describes the Gaussian linking number of the vector potential lines of flux (see for example [32]). In a perfectly conducting medium, these lines of flux cannot cross, and there is no way to unlink the flux lines: helicity is thus perfectly conserved. Even when the conductivity is finite, the helicity is also well conserved. There is a direct analogy between magnetohydrodynamics and fluid me-chanics. In fluid mechanics, the equivalent of helicity is vorticity ( = f v • (V x v)d 3a; which is the Gaussian linking number of the fluid flow lines. If the fluid has no viscosity, then ( is perfectly conserved because 3. Evolution of Magnetic Fields 9 the fluid flow lines are not allowed to cross.1 Even viscous fluids, however, approximately conserve vorticity. This is why, for example, smoke rings and tornados are quite stable. These relate the initial field strength Bvms(Ti) with initial correlation l(Ti) to the present fields today (T n o w % 2 x 10 - 4 eV) BTma(Tnow) with correla-tion Z(T n o w). During the period when the universe supports turbulence (as indicated by a large Reynolds number Re), the inverse cascade mechanism functions and we have the scalings B oc T 7 / 3 and I oc T ~ 5 / 3 as indicated by the second factors in (3.1). In the early universe, Re is very large and the turbulence is well supported. As the universe cools, eventually, for tempera-tures below some To, the turbulence is no longer well supported. Exactly at what effective temperature To the turbulence ceases is not clear: Son points out that, at To w 100 eV, the Reynolds number drops to unity and thus turbulence is not well supported because of the viscosity of the plasma [28]. We take this as a conservative estimate. Field and Carroll argue that the turbulence is force-free and hence unaffected by the viscosity. Thus, they take To ~ 1 eV, the epoch when the matter and radiation energy densities are in equilibrium, and argue that the cascade may even continue into the matter dominated phase of the universe. If this is true, then it might be possible to increase the correlation lengths of the fields by one or two orders of magnitude beyond the estimate (2.1). In any case, for temperatures lower than To, the turbulence and inverse cascade are not supported and so we assume that the fields are "frozen in" and experience only the naive scaling I oc T - 1 and B oc T 2 indicated by the first factors in (3.1). 1 If the flow lines could cross, consider the behaviour of a test particle sitting at the crossing. The trajectory would depend on the direction from which the particle came, but as the particle has no inertia (m = 0 ) , its behaviour is indeterminate. This is represented by a singularity in the fluid flow equations: if no singularity is present to start with, then one cannot develop. Once there is viscosity, one cannot discuss such test particles because the fluid now experiences shear forces and the argument breaks down. 3.2 Scaling Laws The main results of [28, 29] are: (3.1b) (3.1a) 3. Evolution of Magnetic Fields 10 3.3 Phase Transitions As pointed out by Son [28], the only way to generate turbulence is either through a phase transition T2 or through gravitational instabilities. Thus, until gravitational dynamos are active, the scalings (3.1) should be valid. In any case, galactic dynamos will amplify the fields, but will not affect the maximum correlation length. In particular, (3.1b) should be a good estimate, regardless of the role of galactic dynamics (the uncertainty coming from the transition period T ~ To when the scaling laws change). Now we consider, as the source of the magnetic turbulence, a phase transition in the early universe. As we shall show, our mechanism generates Hubble size correlations l{ at a phase transition Ti. In the radiation domi-nated epoch, the Hubble size scales as T^2. Combining this with (3.1b), we — 1/3 see that lnov/ oc T{ , thus, the earlier the phase transition, the smaller the resultant correlations. The last phase transition is the QCD transition, Ti = T Q C D ~ 0.2 GeV with Hubble size / ( T Q C D ) ~ 30 km. With our estimates (7.3) of the initial magnetic field strength BTms(Ti) « e A ^ C j D / ( £ A Q C D ) ~ ( 1 0 1 7 G)/(£A Q C D ) we use Equations (3.1) to arrive at the estimate (2.1). The meaning of the correlation length £ will be discussed in detail later in Chapter 4. The most important result here is that, as long as one has a mechanism to generate Hubble scale correlations and a maximally helical magnetic field at the QCD phase transition such that the inverse cascade proceeds via (3.1), magnetic turbulence of 100 kpc correlations is naturally produced. The questions: 'How can helical magnetic fields with Hubble-scale correlations be produced at the QCD phase transition' and, 'Are these fields strong enough to account for the observed microgauss fields?' will be addressed in the rest of this thesis. The estimate (2.1) suggests, however, that even in the worst case of almost maximal suppression £ A Q C D ~ 10 1 9, an efficient galaxy dynamo may be able to amplify the fields to the microgauss level. In the best case, the mechanism would produce measurable extra-galactic fields. In either case, the important result is the generation of the 100 kpc cor-relations: if observations show that the fields have much larger correlations, then the proposed mechanism can only be salvaged if a more efficient "in-verse cascade" mechanism is shown to work between T Q C D and now. Having said this, one might consider the electroweak or earlier phase transitions. As we mentioned, the earlier the phase transition, the smaller the resulting cor-— 1/3 relations / n o w oc T{ For the electroweak transition, the scaling (3.1b) suggests that Hubble scale helical fields could generate 100 pc correlations 3. Evolution of Magnetic Fields 11 today. Thus it might be possible that electroweak phenomena could act as the primordial source, but this presupposes a mechanism for generating fields with Hubble scale correlations. Such a mechanism does not appear to be possible in the Standard Model. Instead, the fields produced are corre-lated at the scale T~l which can produce only ~ 1 km correlations today which are of little interest. Thus, the previous analysis seems to suggest that, in order to obtain magnetic fields with 100 kpc correlation lengths, helical fields must be gen-erated with Hubble scale correlations near or slightly after the QCD phase transition T Q C D - The same conclusion regarding the relevance of the QCD scale for this problem was also reached by Son, Field and Carroll [28, 29]. 3.4 Applicability of Scaling Laws The applicability of the scaling laws (3.1) rests on several assumptions. In particular, that there be maximal helicity on all scales at which the inverse cascade functions. In our previous papers [3, 4] we assumed that Hubble scale correlations in the helicity sufficed to power the inverse cascade, but, as pointed out by A. Vilenkin [private communication] and confirmed by D. T. Son [private communication], it seems that this may not be sufficient and that the helicity must be correlated on all scales where the cascade is to occur. The mechanism as we present in this thesis naturally produces maximal helicity only on the Hubble scale near the QCD transition. Thus, as it stands, it may not produce the desired correlations. We still feel that the mechanism may have merit because: 1) The inverse cascade mechanism is far from well understood and thus, perhaps a similar mechanism will function with helicity generated on the same Hubble scale (this might be unlikely) and 2) Other CP violating physics may arrange for a net generation of helicity on larger scales which could power the cascade. This second point will be discussed in more detail in Section 7.1. Without further ado, we now present our picture of the mechanism and justify the the estimate (2.1) of the magnetic field strength. 4. Generation of Magnetic Fields by Domain Walls 12 4. Generation of Magnetic Fields by Domain Walls The key players in this mechanism are domain walls, which form shortly after the Q C D phase transition. Details of the walls were first presented in [2], wil l be described in Appendix B , and summarized in Chapter (5). In Chapters (6) and (7) we shall show that these walls tend to align nuclear magnetic and electric dipole moments along the plane of the wall. A n im-portant feature of the walls is that across the wall there is maximal strong C P violation due to an induced nonzero 6. Because of this, the electric and magnetic dipole moments of the nucleons are of the same order. Thus, both neutrons and protons will have non-zero electric and magnetic dipole moments and play a role in generating the electromagnetic fields. Because of the correlation between the electric and magnetic fields along the domain wall, the generated fields have an induced helicity as we shall examine in Chapter (7). This helicity has the same sign along the entire domain wall and we expect that the domain wall will fill the entire Hubble volume, thus the helicity will be correlated on the Hubble scale. C P asym-metry in the universe prior to domain wall formation may preferentially form walls of a particular CP. This would be transferred to the magnetic fields, extending the helicity to scales beyond the Hubble scale. Finally, the domain walls wil l decay so that the universe is not domi-nated by domain walls today. By this point, however, the helical magnetic turbulence has been generated. 4.1 Hubble Size Correlations The reason that we feel that domain walls may hold the key to explaining primordial magnetic seeds is that, in a short time, they can generate Hubble scale correlations. The initial fields must have a Hubble scale correlation or else there is no known way—even with an inverse cascade—to generate the huge correlations today. Let us briefly summarize the behaviour we expect of domain walls at the Q C D phase transition. For a nice description of general domain wall dynamics see [22] from which most of these results were derived. 4. Generation of Magnetic Fields by Domain Walls 13 1. Prior to the phase transition Ti = A Q C D I the fields are in random fluctuations on the scale Ti and domain walls are not present. 2. After the phase transition, the fields settle into their vacuum states. Domains are formed where the fields are settling into different1 vacuum states. These domains are separated by domain walls and have a scale set by A Q C D -3. Numerical studies suggest that these small-scale domain walls rapidly merge increasing the correlation length of the walls. This coarsening occurs simultaneously throughout space and the correlation length of the domain walls can increase faster than the speed of light. 4. The coarsening stops once the domain walls attain a Hubble scale. On average, one ends up with one domain wall per Hubble volume, but which curls and moves through space, essentially filling the volume. It is these Hubble sized domain walls that can generate magnetic tur-bulence with Hubble size correlations. As we shall see below, there are two types of domain walls corresponding to opposite field transitions. One we call a "soliton", and the other we call an "anti-soliton". Together a soliton and an anti-soliton can annihilate, but the coarsening essentially separates regions of solitons from anti-soliton regions by a distance of the Hubble scale so that they do not annihilate. In Chapter (7.1) we shall show that the soli-tons and anti-solitons are associated with helicity of the opposite sign. Thus, the domain walls effectively separate the helicity generating a Hubble scale correlation length in the fields and in the helicity. At this point, it may be possible to bring in some unknown C P violating physics which preferentially forms solitons over anti-solitons. Such an effect would convert this C P vi-olation into a net helicity in the magnetic fields on a scale larger that the Hubble scale. This would power the inverse cascade discussed in Section 3.2. 4.1.1 QCD Phase Transition There is some question about what conditions must be like at the Q C D phase transition in order for domain walls to form. Q C D lattice simulations suggest that at low densities (such as those present in the early universe), the 1 In the case of QCD domain walls, the vacuum states are actually the same but the field configuration, going from one domain to the next undergoes a classically stable transition. This behaviour is qualitatively similar to the sine-Gordon model Csa = (dn)2 — cosc/> where c/> is interpreted as a phase so that the vacuum states = 2irn are actually identical. 4. Generation of Magnetic Fields by Domain Walls 14 transition between the quark-gluon plasma and the normal hadronic phase is a smooth crossover, and that the critical point sits at some finite density (see the recent review [33]). If the rate at which the universe cools is sufficiently slow, then it is possible that no domain walls form. In the preface to the paperback edition [22], the authors discuss this scenario as the Kibble-Zurek picture: to estimate the size of the correlations produced, one must consider the relaxation timescale r (T) : the time it takes to establish correlations on the scale £ ( r ) . The freezout temperature Tf is determined by the condition T(TJ) ~ tr) = \Tj — T C | / | T / | , i.e. when the relaxation time is on the same order as the dynamical timescale of temperature variations. Since the true critical point is at a somewhat higher densities than the universe, to my be bounded from below and, if the cooling is sufficiently slow, it is possible that r — a), where (p is the dynamical field related to the rj' meson and a is the axion field. Thus, even when 9 = 0, CP will be violated in strong interactions in a domain wall background where or — a is non-zero over a macroscopically large region. Hence, QCD and axion domain walls induce strong CP violations over their central regions. One of the consequences of this strong CP violation is that nucleons have an induced electric dipole moment as well as a magnetic dipole moment4 [24]. We summarize these results here. In the chiral limit mq —> 0, and for small 9, the electric dipole moment is where gVNN is the strong nNN coupling constant and g^NN is the CP odd 3 In the language of Section 3.3, prior to some high-energy scale (Peccie-Quinn), the axion field is free. After this transition, the flat Mexican hat potential is formed restricting the degrees of freedom to a phase angle 9 except at the center of strings (This defect energy is associated with the string tension.) Through this spontaneous symmetry breaking, the massless phase excitation (the axion) acts as a Goldstone boson. As the temperature cools, this potential tilts and 9 = 0 is favoured. The tilt explicitly breaks the original symmetry and gives the axion a mass (thus it is usually referred to as a pseudo-Goldstone boson). This is when domain walls form. 4 Normally the electric dipole moment is suppressed to the same order as 9 in that C P is conserved. (4.2) 4. Generation of Magnetic Fields by Domain Walls 19 TTNN coupling constant which was estimated to be g^NN ~ O.O4|0|. In these formulae the 9 parameter should be treated as the singlet (f> domain wall solution (j){z) with nontrivial z dependence. From these formulae one can compute the following relation Thus, for all nucleons, including the neutron, both the electric and magnetic dipole moments are non-zero and of the same order in the domain wall background when 9(z) = (j>(z) ~ 1. 5. QCD Domain Walls 20 5. QCD Domain Walls We saw in Chapter (4.2) that several types of domain walls might act as sources for seed fields. To be concrete, we shall now restrict our attention to QCD domain walls to show how domain walls might produce magnetic seed fields. In this chapter, we shall present a short review of the results presented in Appendix B, simplifying the model for presentation. To describe these walls, we consider the low-energy effective theory of QCD, including the pions, and the rf singlet field. The rf field is not as light, but it represents the essential physics associated with the anomaly that is responsible for the QCD domain walls. The pions and rf enter the Lagrangian through the matrix representation U = exp r 7r aA a 2 rf tV2— Vi- (5.1) where 7ra are the Nj — 1 pseudo-Goldstone fields, A a are the Gell-Mann matrices for SU(Nf), and rj' is the singlet field. From now on, we limit ourselves to the simplest case of one flavour, Nf = 1, which contains only the rf field but which captures all of the relevant physics. The Nf = 2 case is described in [2] and presented in Appendix B. Although the models are quantitatively different, the phenomena described by both is the same. In this model, we see that (5.1) reduces to a single complex phase U = e^, =% (5.2) frf The effective Lagrangian density then reduces to _ frf with the effective potential C=J-^(d^)2-V{(j>) (5.3) V{(j)) = -min = 4>Q + 5$ we find the mass of the field1 The most important point to realize is that all of the ground states (5.5) in fact represent the same physical state U = 1. Thus, it is possible for the 0 field to make a transition 2irn -> 2irm for different integers n and m. Within this model (5.3), where all heavy degrees of freedom have been integrated 1 In the more general case of Nf quarks with equal masses, the right-hand side of Equation (5.6) should be multiplied by a factor Nf (Compare with (B.18).) 5. QCD Domain Walls 22 out, these transitions are absolutely stable and represent the domain walls. When one includes the effects of the heavier degrees of freedom, however, we find that the walls are unstable on the quantum level. This is described in detail in Appendix B and briefly reviewed in Chapter 5.2. To study the structure of the domain walls, we look at a simplified model where one half of the universe is in one ground state and the other half is in another. The fields wil l orient themselves in such a way as to minimize the energy density in space, forming a domain wall between the two regions. In this model, the domain walls are planar and we shall neglect the x and y dimensions: A complete description of this wall is given by specifying the boundary conditions and by specifying how the fields vary along z. We present here the two basic domain wall solutions. These are charac-terized by interpolations from the state (j> = 0 to: Soliton: — 2TT, Anti-soliton: (j) = —2ir. It is possible to consider transitions between further states (i.e. 0 —> 2rm) but these can be thought of as multiple domain walls. They also have higher energies, are less stable, and are thus less important for our discussion. To gain an understanding of the structure of the domain walls we look for the solution which minimizes the energy density. The energy density (wall tension) per unit area is given by the following expression where the first term is the kinetic contribution to the energy and the last term is the potential. Here, a dot signifies differentiation with respect to z: a = — To minimize the wall tension (5.7), we can use the standard variational principle to arrive at the following equation of motion for the domain wall solutions: 5.1 Domain Wall Solutions (5.7) hi = sine/) + E (5.8) AM MNC 5. QCD Domain Walls 23 Again, the last term of Equation (5.8) should be understood as the lowest branch of a multi-valued function as described by Equation (5.3). The general analytical solution of Equations (5.8) is not enlightening and we present the numerical solution in Fig. 5.1. In order to gain an intuitive understanding of this wall, we examine the solution in the chiral limit M 1, this is the limit field is governed by the differential equation: 72""'" 1^ * = *3I.si"w- (5'9) Now, there is the issue of the cusp singularity when (j> = n which results from switching branches of the potential (see Equation (5.3).) By defini-tion, we keep the lowest energy branch, such that the right-hand side of Equation (5.9) is understood to be the function sin(c/>/Arc) for 0 < < TT and sin([t/> — 2TT]/NC) for n < < 2n. However, since the equations of mo-tion are symmetric with respect to the center of the wall (which we take as z = 0), 4> = TT only at the center of the wall and not before, so we can simply look at half of the domain, z £ (—oo,0], with boundary conditions 4>{—oo) = 0 at z = —oo and 0(0) = TT at z = 0. The rest of the solution will be symmetric with {z)={ - L r C J - (5.10) ^ ' - 1 e-" 2 tan ^ - | , z > 0. which is a good approximation of the solution to Equation (5.8) when M 0 (see Equation (5.6)). Thus, we see that, indeed, the Q C D domain walls have a Q C D scale. Solution (5.10) describes the soliton. The anti-soliton can be found by taking z —> —z: thus, we have the transition soliton —> anti-soliton under the discrete C P symmetry. The numerical solution for the cj) field is shown in Fig. 5.1. It turns out that the approximation is reasonable even in the physical case where N^M/E ~ 1 0 _ 1 . 5. QCD Domain Walls 24 The wall surface tension defined by Equation (5.7) and can be easily calculated analytically in the chiral limit when the analytical solution is known and is given by Equation (5.10). Simple calculations leads to the following result:2 In the case when mq ^ 0, a is numerically close to the estimate (5.12). r>r-— , , -5 0 5 Z/J, Fig. 5.1: Basic form of the QCD domain wall (soliton). The analytic approximation 5.10 is plotted as a dotted line to show the good agreement. We have taken Nc = 3 here. Notice that the wall thickness is set by the parameter fi. 2 In the general case of Nf quarks of equal mass, the right-hand side of Equation (5.12) should be multiplied by the factor \/yJWf. In this case, we can compare (5.12) with (B.41) where Nf = 2. 5. QCD Domain Walls 25 5.2 Domain Wall Decay-Finally, we note that these domain walls are not stable: as mentioned ear-lier, the vacuum states (5.5) represent the same physical state. When one includes the heavier gluonic degrees of freedom, it becomes possible for the fields to "unwind" through this extra degree of freedom. Classically this is not allowed because the heavy degrees of freedom are constrained by a large potential barrier. It is still possible, however, for the field to tunnel through this barrier forming a hole in the domain wall. Once a large enough hole is formed, it will expand and consume the domain wall. This process is called "nucleation" and is similar to the mechanism consuming N = 1 axion domain walls [22, 23, 34, 50]. In Appendix B, we estimate the lifetime of these domain walls borrowing the same methods used to estimate the lifetime of axion domain walls in the N = 1 axion models [22, 23, 34, 50]. We should point out one major difference between the N = 1 axion model and our model. In the axion model, strings form first, and then the domain walls form, primarily bounded by strings [23] as discussed in Section 4.1.1. Q C D domain walls, on the other hand, would form simultaneously with strings at the Q C D transition. Some walls my form which are bounded by strings: these would probably disappear quite rapidly. Closed domain walls, however, could only decay through the nucleation process and would this would greatly enhance their lifetime. We do not fully understand the dynamics at the phase transition, but the picture we have is that many small closed domain walls may form and them merge to reduce the wall tension. This is the coarsening phenomenon discussed in Section 4.1 (see also [22] for a nice discussion). The tunneling probability can be estimated by computing the action So of an instanton solution of the Euclidean (imaginary time, t = ir) field equations, which approaches the unperturbed wall solution at T —>• ±oo. In this case, the probability P of creating a hole is proportional to the factor where So is the classical instanton action. If the radius Rc of the nucleating hole is much greater than the wall thickness, we can use the thin-string and thin-wall approximation. In this case, the action for the string and for the wall are proportional to the cor-responding world-sheet areas [50], P ~ e -So (5.13) S0 = 4irR2a (5.14) 5. QCD Domain Walls 26 Here o is the wall tension (5.12), and a ~ y2E is the string tension which we estimate based on dimensional arguments. The string tension a tries to close the hole while the wall tension a tries to widen the hole. Minimizing (5.14) with respect to R we find that p 2 a c 1 6 7 r c * 3 / M ^ Rc = —, o0= » o • (5.15) If a hole forms with radius R > Rc then the hole will expands with time as x2 + y2 = R2 + t2, rapidly approaching the speed of light and consuming the domain wall. Inserting numerical values for the phenomenological relevant case Nf = 2 we find that a ~ (0.28 GeV) 2 , a = (200 MeV) 3 , S0 ~ 120. (5.16) What is important is that So is numerically large, and hence the lifetime is much larger than the Q C D scale because of the huge tunneling suppression e~s° ~ I O - 5 2 . A more complete analysis is presented in Appendix B where we estimate the lifetime (Equation (B.56)) of the walls to be on the order •T ~ I O - 5 s (5.17) even though the walls are governed by the microscopic Q C D scale. This result should be interpreted with some caution: in the low energy regime, we do not have very good control over the quantitative physics.3 Arguments presented in Appendix B show, however, that it is at least possible for domain walls of purely Q C D origin to live for macroscopically large lifetimes. To summarize, we have a Q C D domain wall with all of the properties required to generate magnetic fields: 1. The walls form shortly after the Q C D phase transition and attain Hubble-scale correlations through the Kibble mechanism. 2. The Q C D domain walls have a structure on the scale of m~} ~ A Q C D and thus they can efficiently interact with nucleons and other Q C D matter. 3. The transition in the singlet n' field produces an environment near the wall where the effective 6 parameter is non-zero. Thus, across the wall, 3 It is possible to regain control of the calculation in the high density limit. See the discussion in Chapter (8.1). 5. QCD Domain Walls 27 there is maximal C P violation. In such an environment, it is known that the electric and magnetic dipole moments of the nucleons are of the same order [24]. Because of the induced electric dipole moments and anomalous magnetic moments, the cancellations discussed in [25, 26] do not qualitatively affect the physics and the domain walls are truly ferromagnetic. 4. The strong C P violation also provides a mechanism for generating helicity on a Hubble scale4 by aligning both the electric and magnetic dipole moments along the domain wall. 5. The decay mechanism renders the Q C D domain walls unstable such that the walls themselves do not pose a cosmological problem. How-ever, the suppression in the decay mechanism due to quantum tun-neling might extend the lifetime of the walls to a macroscopic scale (5.17) which is long enough to generate the required electromagnetic turbulence as we shall show. 4 To be precise, the domain walls separate the helicity into Hubble size regions to that globally the total helicity is zero, but within Hubble scale regions, the helicity is maximal and correlated with the same sign. 6. Alignment of Spins in the Domain Wall 28 6. Alignment of Spins in the Domain W a l l Now, by present a simplified method for estimating the magnitudes of bulk properties on the domain wall, we shall show that, indeed, the domain walls acquire a magnetization. This method makes the approximation that the domain wall is flat, and that translational and rotational symmetries are preserved in the plane of the wall (which we take to be the x-y plane). These approximations are valid in the case of domain walls whose curvature is large in comparison to the length scale of the pertinent physics. Once this approximation is made, we can reformulate the problem in 1 + 1 dimensions (z and t) and calculate the density of the desired bulk properties along the domain wall. To regain the full four-dimensional bulk properties, we must estimate the density of the particles in the x-y plane to obtain the appropriate density and degeneracy factors for the bulk density. Thus, the final results are not independent of physics in the x-y plane, but rather, these effects are only accounted for through degeneracy factors. We proceed to demonstrate this technique by calculating the alignment of fermionic spins along the wall. We take the standard form for the inter-action between the pseudo-scalar n' field and the nucleons which respects all relevant symmetries: Here 0 = cj>(z) characterizes our domain wall solution as expressed in Equa-tion (5.10) and is the nucleon mass. For our approximations, we assume that fluctuations in the nucleon field ^ do not affect the domain walls and treat the domain walls as a background field. 1 The strategy is to break (6.1) into two 1 + 1 dimensional components by setting dx = dy — 0 (this is the 1 A full account would take into account the effects of this back-reaction. We expect that such back-reactions would affect the potential (5.4) by altering the form of the last term and possibly adding higher order corrections. This may affect the magnitudes of some of the estimates, but would certainly not alter the topology of the fields and thus the domain walls would still form with a similar structure. Quantitatively this would alter the results, but not the order of magnitude. What may be substantially affected is the lifetime of the domain walls. In particular, the result [51] that domain walls are very (6.1) 6. Alignment of Spins in the Domain Wall 29 approximation that the physics in the z direction decouples from the physics in the x-y plane) and then by manipulating the system of equations that result. First, we introduce the following chiral components of the Dirac spinors:2 • • - ( £ ) • *-=(!)• ( 6 - 2 ) * = V2 V%2 - 6/ (6.3) Now we re-express (6.1) by noting that 7 2 = / : d0 -o-jdj The associated Dirac equation is do mN -do mN cos(0) 2sin(c/>) is\n(4>) cos(c/>) cos(c/>) isin(c/>) * = 0. s^in((/>) cos(c/>) This is equivalent to the coupled system: 2i(d0 + -(&)• *<2> = (S)- (6-7) the equations have the following structure: ( i f f y - mNe+i^)y{1) = 0 (6.8a) ( i f f y - m i v e - ^ 5 ) ^ ) = o (6.8b) where the index \i 6 {t,z}, the Lorentz signature is (1,-1) and we define the following 2-dimensional version of the gamma matrices: It = cri , 7z = -io~2, 75 = 0-3. These satisfy the proper 2-dimensional relationships 75 = jtlz and 7^ 7,/ = 9[LV + C/ii/75- We can reproduce equation (6.8) from the following effective 2-dimensional Lagrangian density, £ 2 = * { 1 ) ( i f f y - m N e ^ ) + + * ( 2 ) ( i f f y - mne-^) * ( 2 ) , (6.9) where two different species of fermion with opposite chiral charge interact with the domain wall background 4>{z). We have thus successfully reduced our problem to a two-dimensional fermionic system. It is known that for several systems in 1 + 1 dimensions, the fermionic representation is is equivalent to a 1 + 1 dimensional bosonic -mjve«* i(dt + dz)\ [xi i{dt-dz) -mtfe-^J V£i -mNe^ i(dt - dz)\ (X2 i(dt + dz) -mNe-^) U 2 6. Alignment of Spins in the Domain Wall 31 system through the following equivalences3 [54, 55]: * U ) W * < j ) \(W2, (6.10a) - 7 = ^ 5 " ^ , (6.10b) -> -ncos{2^0j), (6.10c) * ( j ) i 7 5 * ( j ) -> -/isin(2v/7r6'j). (6.10d) After making these replacements, we are left with the following 2-dimensional bosonic effective Lagrangian density describing the two fields 9\ and 02 in the domain wall background (z) £ = \(9A)2 + - u(9ue2) (6.ii) where the effective potential is U(61,62) = -mNri [cos(2v^6>i - ) + COS(2VTF6>2 + )] . (6.12) The next approximation that we make is to neglect the dynamics of the 6% fields: we assume that they relax slowly in the domain wall background such that their dynamics do not contribute appreciably to the final state4 which minimizes the potential (6.12). The classical minimizing solution is thus w = 57? < 6 ' 1 3 ) We are now ready to show that the domain walls align the spins of the fermions. The relevant spin operator is 5 = § 7 7 5 * = ^+a^>+ + tfLtr#_ (6.14) 3 The constant /i in the last two equations is a scale parameter of order TUN • The exact coefficient of this term depends on the model an is only known for exactly solvable systems but in all cases, is of order unity. This technique is well-known to the condensed matter and particles physics communities. See for example [52, 53]. 4 This is the same adiabatic approximation used by Goldstone and Wilczek [52]. 5 Here we use the convention that atj = § [ 7 i , 7 j ] , thus the spin operator = ^eijkaij-In matrix form with the standard representation, this becomes: In terms of the gamma matrices, this is Sj, — 707 /175. 6. Alignment of Spins in the Domain Wall 32 Let us consider the z component of the spin. We then have = x i x i - xlx2+eki - & 2 = * ( i)7t* (l) - *(2)7t*(2) (6-15) and so we see that the four-dimensional spin operator S z = 707*75 is ex-pressed in terms of a pair of two-dimensional fermion charge operators. We can calculate the expectation value of the spin operator in the domain wall background using this two-dimensional correspondence (6.15) and the bosonic representation of the fermions M/tS.vj, = * ( 1 ) 7 t * ( 1 ) - * ( 2 ) 7 t * ( 2 ) = -±dz(02 - 6X). (6.16) Finally, we use our minimizing bosonic solution (6.13) to obtain the following 4-dimensional average spin aligned along the domain wall: (tfts,*) = - I W (6.17) 7T OZ We will also need the following matrix elements later on: = < x k i - x k 2 + e l x i - d x 2 > = <*(1)*(1)> - <*(2)*(2)> = 0 (6.18) and = * ( - £ i X i + £2X2 + x k i - x k s ) = -<*(1)*75*(1)> + (*(2)*75*(2)> = 2^sin(0) (6.19) Remember that we have restricted ourselves to a 1 + 1 dimensional theory. We must now estimate the density and degeneracy of the nucleons along the wall so we can obtain a true 1 + 3 dimensional estimate of the spin density. 6. Alignment of Spins in the Domain Wall 33 6.1 Fermion Degeneracy in the Domain Wall We have assumed that, locally, the domain walls have only a spatial z depen-dence. There is still a 2-dimensional translational and rotational symmetry in the x-y plane. These translational degrees of freedom imply that momen-tum in the plane is conserved. Hence, we can treat the neglected degrees of freedom for the fermions as free. The degeneracy in a region of area S will simply be a sum over these degrees of freedom with a discrete factor g — 4 = 2 x 2 for spin and isospin degeneracy \\P\\• oo and thus no net field remains, as pointed out in [26]. The Q C D domain walls, however, are far from flat: the walls have many wiggles and high frequency dynamic excitations. Thus, the fields generated by the dipoles wil l not can-cel on the domain wall, but will be suppressed by a factor1 of ( £ A Q C D ) _ 1 where £ is an effective correlation length that depends on the dynamics of the domain walls. As an upper bound, the extent of the domain walls is limited by the Hubble scale. Typically, domain walls remain space filling, thus we expect A Q C D < £ <§; Hubble scale. Unfortunately, we presently cannot make a tighter bound on iota. We shall see, however, that even in the worst case, this mechanism can at least generate feasible seed fields for galactic dynamos to amplify. The result of Chapter (6) is a method for estimating the strengths of various sources in the domain walls. We now need to couple these to the generation of electromagnetic turbulence. To do this properly requires the solution to Maxwell's equations as coupled to the sources in (7.1). This is difficult, though no doubt important for accurate numerical estimates, and so for an order of magnitude estimate we make a dimensional estimate 1 The density of the dipoles is governed by the Q C D scale A Q Q D . 7. Generation of Electromagnetic Field 35 considering the sources as a set of dipoles sitting in the domain walls. The spacing between the dipoles is set by the Q C D scale A Q C D , and the strengths of the field can be estimated from (7.1) using dimensional arguments: (Fp,) ~ - r ^ — (