IT’S ALL ABOUT ELECTROMAGNETISM — FROM MAGNETIC MONOPOLES TO COSMOLOGICAL MAGNETIC FIELDS

by

YIFUNG NG

Submitted in partial fulfillment of the requirements

For the degree of Doctor of Philosophy

Department of Physics

CASE WESTERN RESERVE UNIVERSITY

January 2011 CASE WESTERN RESERVE UNIVERSITY

SCHOOL OF GRADUATE STUDIES

We hereby approve the thesis/dissertation of

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*We also certify that written approval has been obtained for any proprietary material contained therein. Contents

Contents ...... ii

List of tables ...... iv

List of figures ...... v

Acknowledgements ...... vii

Dedication ...... viii

1 Introduction ...... 1 1.1 Magnetic monopoles of Electromagnetism ...... 1 1.1.1 Duality of electromagnetism and the U(1) group ...... 1 1.1.2 Dirac’s monopoles and charge quantization ...... 3 1.1.3 ’t Hooft and Polyakov monopoles in non-Abelian theories ...... 4 1.1.4 Group manifold and topological defects ...... 5 1.1.5 Cosmological monopoles ...... 6 1.1.6 Current Investigations ...... 6 1.2 Baryogenesis and the sphaleron in the electroweak model ...... 8 1.2.1 Electroweak phase transition ...... 8 1.2.2 Baryogenesis ...... 8 1.2.3 Non-perturbative sphaleron ...... 9 1.2.4 Dynamo for astrophysical magnetic field ...... 11 1.2.5 Cosmological magnetic seed field ...... 13 1.2.6 Baryogenesis<=>Magnetogenesis ...... 14 1.2.7 Probes of cosmological magnetic seed field ...... 15 1.3 Gravity and electromagnetism studies ...... 16 1.3.1 EffectiveFieldTheory ...... 16 1.3.2 QEDincurvedspace-time ...... 18 1.3.3 Cosmological and Astrophysical probes ...... 18 1.4 Conclusion ...... 19

2 Magnetism from the particle physics side ...... 22 2.1 Introduction and context ...... 22 2.2 Model...... 24 2.3 Numerical implementation ...... 28 2.4 Results...... 34 2.5 Discussion...... 35

ii 3 Primordial magnetic field ...... 38 3.1 Introduction...... 38 3.2 Model...... 42 3.3 MonteCarloSimulation ...... 47 3.4 Results...... 51 3.5 Conclusions and Future Directions ...... 53

4 Coupling of Gravity with Electromagnetism ...... 56 4.1 Introduction...... 56 4.2 BasicTools ...... 59 4.3 Enumerationofterms...... 60 4.3.1 Dimension:2and4...... 61 4.3.1.1 Without Levi-Civita Contractions ...... 61 4.3.1.2 With Levi-Civita Contractions ...... 62 4.3.2 Dimension:6...... 62 4.3.2.1 Without Levi-Civita Contractions ...... 62 4.3.2.2 With Levi-Civita Contractions ...... 63 4.3.3 Absorption of Terms via Metric Re-definition ...... 64 4.4 ObservablesandConstraints ...... 65 4.4.1 Cosmological Constraints ...... 66 4.4.2 SolarSystemConstraints...... 69 4.4.2.1 Effective metric solution ...... 72 4.4.2.2 DeflectionAngle...... 73 4.4.2.3 Modified Shapiro Time Delay ...... 74 4.5 SummaryandDiscussion...... 76

5 Conclusion ...... 78

A Appendix to Chapter 2 ...... 82 A.1 Appendix: Topological charge ...... 82 A.2 SU(3) geodesicmatrix ...... 83 A.2.1 Zi = Z0 case: ...... 83 A.2.2 General Zi case:...... 85 A.3 Construction of the matrix S...... 86 A.4 Consistency of monopole and string numbers ...... 88 A.5 SU(2) monopolesandstrings...... 90

B Appendix to Chapter 4 ...... 92 B.1 Shapirodelaycalculations ...... 92

Bibliography ...... 94

iii List of tables

4.1 List of tensors for different dimensions ...... 60

iv List of figures

1.1 WMAP-CMB temperature anisotropies plot ...... 7 1.2 Electroweak bubble nucleation ...... 9 1.3 Plot of sphaleron energy versus Chern-Simons number change...... 9 1.4 Sphaleron’s magnetic dipole moment ...... 10

2.1 Confined monopoles in SU(3) model...... 23 2.2 Schematic picture of the cubic lattice ...... 28 2.3 Parallel transport of vector in manifold space to locate the string ...... 34 2.4 Log Nstring(annihilating) versus string length ...... 35 − 2.5 Log Nstring(clustering) versus string length ...... 35 2.6 Clustersofmonopoles− ...... 36 2.7 Monopole-string network from GUT SU(5) ...... 36

3.1 Schematic picture of sphaleron explosion on electroweakbubbles...... 44 3.2 Planaplotsofmagneticfield ...... 45 3.3 Plot of bubble nucleation number versus time ...... 49 3.4 Plot of sphaleron nucleation versus time ...... 49 3.5 ML(r) plots versus correlation length ...... 52 3.6 MN(r) plots versus correlation length ...... 52 3.7 MH(r) plots versus correlation length ...... 53 3.8 S(k) plotsversusk ...... 54 3.9 A(k) plotsversusk ...... 54

5.1 Feynman diagram with virtual monopole loop ...... 79

A.1 Parallel transport of the vector for string determination ...... 91

v Your children are not your children. They are the sons and daughters of Life’s longing for itself. They come through you but not from you, And though they are with you, yet they belong not to you. You may give them your love but not your thoughts. For they have their own thoughts. You may house their bodies but not their souls, For their souls dwell in the house of tomorrow, which you cannot visit, not even in your dreams. You may strive to be like them, but seek not to make them like you. For life goes not backward nor tarries with yesterday. You are the bows from which your children as living arrows are sent forth. The archer sees the mark upon the path of the infinite, and He bends you with His might that His arrows may go swift and far. Let your bending in the archer’s hand be for gladness; For even as he loves the arrow that flies, so He loves also the bow that is stable.

— Kahlil Gibran

You gain strength, courage, and confidence by each experience in which you really stop to look fear in the face. You are able to say to yourself, ‘I have lived through this horror. I can take the next thing that comes along.’ You must do the thing you think you cannot do.

— Eleanor Roosevelt

Alternatively, the task of estimating the length of human life is beyond our capacity, for directly we say that it is ages long, we know that it is briefer than the falling of rose onto the ground.

— Virginia Woolf

vi Acknowledgements

Foremost, I must thank my advisor, Professor Vachaspati, for bearing with and guiding me for the past 5 years. His dedication and enthusiasm in research has imparted shares on me, or else I would not have made it thus far. Many faculties, colleagues and departmental personnel at Case, as well as other international research institutions, have also lent their helping hands (i.e. technical, administrational, etc.) to me at various points in making my graduate life a bit smoother; I remain grateful for your generosity and kindness. I would also like to hereby acknowledge the financial support provided by the physics department of CASE, the grants from the DOE as well as the invaluable financial planning lessons provided by my short stint at Princeton. Friends who have come and gone but brightened my life through various stages of my graduate career/life: you know you all have a special place in my heart, even though I do not have the space here to list you all. I am delighted to have our life-paths crossed, and wish that we are all marching towards the destinations our hearts so desire. In particular, I would like to thank my girl-pal of 20 years, Fanny, for offering her free weekly counseling services throughout all my years abroad. We are marching towards our quarter-life friendship, and I look greatly forward to that, and many more years of sharing of life’s ups and downs. Mom, dad, sis and bro, you are my world. I thank you all for your presences gracing my life. Our times on earth are fleeting, and life events can be utterly unpredictable, but I am forever thankful to live with the solid certainty that your love would and will always stay with me, no matter what. It guides me to keep on marching no matter what. I love you all, more than words suffice.

vii Dad and Mom: whatever cultural revolution/history has denied you (i.e. freedom and access to higher education), you strive with all means in your lives to provide for me, often at pains and great cost. To that, your abundant unconditional love, the blessing of life and loving siblings, I am, forever, forever deeply grateful. This thesis is dedicated to my family.

viii IT’S ALL ABOUT ELECTROMAGNETISM— FROM MAGNETIC MONOPOLES TO COSMOLOGICAL MAGNETIC FIELDS

Abstract by YIFUNG NG This thesis is concerned with the studies of electromagnetic phenomena in high-energy particle physics and the early universe. We will first report the works of a numerical study on magnetic monopole network formed from a symmetry breaking model. The specific construction of the mapping between the fields in the physical group space and the topological space will be given in full and the results on the statistical properties of the final monopole-string network will be presented. Second, we will present in detail the motiva- tion, model construction as well as the numerical details of a Monte-Carlo study done to obtain the 2-point correlation function of the magnetic field arising from the cosmological electroweak symmetry breaking. Lastly, we will report on a study for a phenomenological model of gravity coupled with electromagnetism via the effective field theory construct. We will illustrate some examples that show the limitations electromagnetic observations in cosmology and astrophysics face in constraining such class of models. We will end with some thoughts on future directions to be taken regarding the three projects reported in the final section.

ix Chapter 1

Introduction

This is a thesis concerned with the study of electromagnetic phenomena: spanning from elementary magnetic monopoles arising from symmetry breaking, the magnetogenesis scenario in cosmological electroweak epoch, to the studies of non-minimal coupling of gravity and electromagnetism via effec- tive field theory. Since the range of topics covered is so wide, we will only focus here to give ample motivations for the studies carried out as well as summaries on some of the insights gained from the results obtained. We will comment more on future research directions in the conclusion chapter.

1.1 Magnetic monopoles of Electromagnetism

1.1.1 Duality of electromagnetism and the U(1) group

The studies of electromagnetism have a long winding history. Ever since Faraday performed his fa- mous conduction experiments during the Royal Academy’s evening lectures in the mid-1800s, the inseparable entities of electricity and magnetism took on a firm ground. Soon, the analogous picture of Newtonian gravitational theory was borrowed over: the electromagnetic field is interpreted as an invisible field that permeates space from a point electric charge.

The mathematical formulation as dictated by the Maxwell equations exhibits the intricate dual roles played by the electric and magnetic fields:

∇ E = ρe ∇ B = 0 (1.1) ∂ ∂E ∇ E = B ∇ B = J+ (1.2) × − ∂t × ∂t

1 For vacuum, in which there is no charge (i.e. ρe = Je=0), the above equations are invariant upon dual transformation of E and B

E B B E (1.3) → →−

An exact duality would mean invariance upon double-dual transformations, (E,B) ( E, B). → − − The presence of electric charge but no single magnetic charge (i.e. magnetic charge always comes in pairs, like the two poles of a compass) in real physical systems, though, breaks such a symmetry. It also helped introduce a vector potential term, Aµ , which for a covariant form of field strength F

F0i = Ei Fi j = ε Bk (1.4) − i jk is defined as F µν = ∂ µ Aν ∂ ν Aµ where Aµ =(φ,A). − Such a vector potential term though is not uniquely defined, and is said to have a U(1) gauge symmetry. To see this, consider the coupling of electromagnetism to quantum mechanics. When the momentum operator gets replaced by its covariant counterpart under the minimal-coupling scheme,

p = i∇ i(∇ ieA) (1.5) − →− − where e here refers to the electric charge; the resulting Schrodinger equation under such a change is invariant under both a gauge transformation of the vector potential Aµ as well as up to a phase multiplication of the wave function:

ieΦ ψ ψe− (1.6) → A A ∇Φ (1.7) → −

∇Φ ∇ ieΦ ∇Φ i ieΦ∇ ieΦ Writing as products of and e , i.e. = e e e− , we say that the fundamental quantity in such a gauge transformation of A is the element of U(1), eiΦ.

2 1.1.2 Dirac’s monopoles and charge quantization

In 1931, Dirac [43] offered a nice resolution of keeping both the principle of duality transform in electromagnetism and the consistency of the quantum theory. His result was that in principle single magnetic monopole with charge g can exist in the universe if its charge is quantized:

ge = 2πnh¯ (1.8) where again e is the electric charge and g is the magnetic charge.

A more explicit way of seeing this is to tie back the definition of magnetic charge with the definition of the vector potential A and a magnetized version of Gauss’s law, which is what Wu and Yang ([132],

[133]) have accomplished for the vector potential A in Yang-Mills field equations. For our purpose here, the main message is this: on top of preserving duality and consistency in quantum theory, the presence of even one single magnetic monopole is sufficient to explain the observed quantization of electric charge in the universe; an established observation of nature which remained an enigma that lacked any plausible theoretical explanation or justification.

On the other hand, another thread for the studies of electromagnetism at its quantum level was opened up during the 1960s as particle physicists probed the studies of elementary particles’ funda- mental interactions. The standard model so established ([55] and [127]) says that the three funda- mental forces of nature, strong, weak and electromagnetic are all described by gauge theory. They obey distinctive symmetries which are described by mathematical structures called groups (i.e. for detailed pedagogical introduction see [108]). Each individual force type as we know of today essen- tially surfaces after “symmetry breaking”. In short, the standard model has a SU(3) SU(2) U(1) × × symmetry-group structure corresponding to the three forces listed above. The U(1) here though is not the same U(1)EM mentioned above in our discussion of the vector potential A. It suffices to say that

U(1)EM is made out of a combination of the groups SU(2) and U(1) together, after symmetry breaking from what is termed the electroweak theory of the standard model. More will be said on this special

3 symmetry breaking and its intricate link with electromagnetism in the next section.

We have introduced the principles of duality and gauge invariance with compact U(1)EM group in electromagnetism and the concept of Dirac’s magnetic monopole from their historical perspectives. We have also alluded to the fact that electromagnetic force gets unified with other forces of nature at higher energies, and is dictated by gauge theory which gives rise to the U(1)EM via a mechanism called symmetry breaking. We will see shortly how all these loose threads are woven together.

1.1.3 ’t Hooft and Polyakov monopoles in non-Abelian theories

Magnetic monopole can be viewed as a type of solution to the gauge theory, subject to some constraints on its group structures. Even though the electroweak sector of the standard model has no such solution ([97]), ’t Hooft [111] and Polyakov [95] demonstrated in 1974 that such magnetic monopole solutions do exist if the standard model is embedded in some non-Abelian group of a Grand Unified Theory

(GUT) SU(5). Their proofs rely on deep mathematical insights on the topology of the non-Abelian Higgs field (i.e. the non-zero vacuum field when the electroweak symmetry is broken). Even though the derivation seems to be of a completely different origin from Dirac’s original proposal, there is in fact a deep connection between the two views ([32],[56]) 1, but the beauty is that the same quantization scheme holds true in both cases: eg = 2πnh¯.Kibble noted two years later that the vacuum manifold corresponding to the groups can lend itself to various other topological defects [66], based on how the order parameter aligns itself throughout group space during the symmetry breaking, and that the monopole is just one type of such defects. A brief discussion on the mathematical characterization of topological defects will be given now, as a prelude to the specific model of non-Abelian monopole- string network considered in Ch.2.

1We will not divulge on the deep connection between quantum excitations of fields and the solitons as solutions of their classical field theory here, see [114] for brief review of the vast literature.

4 1.1.4 Group manifold and topological defects

In essence, topological defects are solutions characterizing specific mappings of the topological man- ifold to the group structures of symmetry breaking models ([118]). The vacuum manifold mentioned earlier characterizes the plethora of lowest-energy states (represented by fields φ ) that is invariant un- der some symmetry group, G. If there exists subgroup H in G that keep φ invariant, then the vacuum manifold is isomorphic to the coset space G/H.

Mathematically, the homotopy group, πn(M;x0) characterizes the topology of a manifold M, where n=0,1,...etc, by mapping the n-sphere to M. Depending on the number of physical dimensions, different topological defects such as domain walls (n=0), cosmic strings(n=1) or monopoles (n=2) can be formed (see [125]). Thus, the ’t Hooft-Polyakov monopoles are really just a sub-class of such defects arising out from the topological structures of the group manifold. If the vacuum manifold’s topology is rich enough, hybrid structures of a combination of defects can arise ([59]). These monopoles defects are not necessarily the same magnetic monopole as in Dirac’s case, since they can carry different topological charge, depending on the symmetry breaking process.

Say a 1-form A “gauge potential” is constructed from the points on the manifold M, the correspond- ing “field strength” F is just dA, and the topological charge Q is just defined as:

1 Q = F (1.9) 2π ∂ V where for a monopole defect, the topological volume V is just a sphere. One can also write down the charge Q defined via the group elements of G. We will see explicit construction of such mappings for the numerical study of the two-stage symmetry breaking SU(3) > U(2)=[SU(2)XU(1)]/Z2 in − the next chapter, and show that they are equivalent.

5 1.1.5 Cosmological monopoles

Kibble’s original paper [66] emphasized the link between cosmological phase transitions and the pos- sible occurrence of topological defect remnants; and ’t Hooft and Polyakov’s papers also alluded to the presence of magnetic monopoles in the early universe should grand unified theory hold true. However, up till today, the search for them is not successful [69]. Other than invoking inflation to help push the monopoles beyond the horizon after creation, there exists other novel mechanisms that can alleviate the so called cosmological monopole problem. One example of such hybrid structures is a network of monopole-antimonopole connected by strings can come together and be annihilated via the so-called

Langacker-Pi mechanism [76]. If topological defects like magnetic monopoles and cosmic strings did exist in the early universe, another trace they could have left is their effect on structure formation ([45]). In short, defects seed isocurvature fluctuations as opposed to adiabatic fluctuations by inflation, and would lead to a predom- inance of vector and tensor perturbations to the cosmic microwave background ([93]). Since current data strongly favors inflation-driven adiabatic perturbation (i.e. see [70]), this alternative route to probe cosmological magnetic monopoles or other defects also seems to be not as fruitful.

1.1.6 Current Investigations

Even though the magnetic monopoles as original descendants from gauge theory of high energy so far eluded experimental detection [69], they remain fascinating objects of study for their distinctive theoretical and practical flavors as stated above. Currently there are two main routes of active research efforts for probing the existence and properties of magnetic monopoles:

(i) Mathematical modeling of solitons or monopole networks Since properties of magnetic monopoles are uniquely dictated by the topological properties of the manifold in which they reside, a viable way to probe the dynamics and properties of their presence is through “creation” via numerical simulation. This usually requires a precise modeling of the map- ping between the topological manifold and the symmetry breaking process. As hinted by the studies

6 Figure 1.1: WMAP-CMB temperature anisotropies plot CMB shows strong evidence of primordial adiabatic, as opposed to isocurvature, perturbation, thus dis- favoring topological-defect driven structure formation (Figure courtesy of NASA/WMAP science team; http://en.wikipedia.org/wiki/VerySmallArray in Ch.2, considerable complications will arise once the dimension of the vacuum manifold gets too big. e.g. the vacuum manifold of the symmetry breaking process SU(5) [SU(3)XSU(2)XU(1)] → is 12-dimensional. This will be computationally quite demanding, not to mention the extra technical intricacies involved in the identification of topological charge, etc. (ii)Experimental observation of magnetic monopoles as quasi/dual particles in analogous con- densed matter systems

The mechanism of symmetry breaking is not specific to high energy physics only, and in fact has proved to be vastly useful in the studies of condensed matter system also. That is, topological defects are not generic to the cosmos only, but can be found much closer to home ([30]),[120]). In fact, vast progress has been made in recent years regarding observation of analogous monopoles or other defects for experiments in laboratory ([51], [27], [98]).

Almost three quarters of a century later, Dirac’s proposal of the single magnetic charge remains as mysterious as ever. Yet, the combined theoretical and experimental efforts carried out to study its problems have enriched our understandings of the physics ranging vast scales: from here on earth to

7 that of the early universe.

1.2 Baryogenesis and the sphaleron in the electroweak model

The early universe abounds with enigmatic phenomena, and one of such is the asymmetry of matter over anti-matter that leads to the presence of , or baryogenesis (see reviews of [4], [116]).

1.2.1 Electroweak phase transition

As alluded to earlier, the electroweak phase transition marks the decoupling of electromagnetic and the weak forces in the standard model, i.e. the symmetries of SU(2) U(1) get spontaneously broken. × Before this epoch, it is difficult to uniquely distinguish the electromagnetic force from its U(1)EM. From the particles physicist’s point of view though, the electroweak symmetry breaking also holds great promise for breaking another important symmetry: the symmetry between matter and anti-matter, or “baryogenesis”.

1.2.2 Baryogenesis

The conditions for baryogenesis are nicely summarized by the Sakharov conditions (1967): (i)Parity (P) violation; Charge and Parity (CP) violation

(ii) out of equilibrium (iii)baryon violating

Electroweak forces are generically C and CP violating, and this prompted the question of whether the epoch of the cosmological electroweak phase transition epoch was responsible for baryogenesis or not [14]. Out-of-equilibrium physics can be easily achieved via a electroweak first-order phase transition, and this leaves the last condition to become the pivotal focus. This task was fulfilled soon when baryon-violating physics was discovered to be generically present in the electroweak model: a

8 Figure 1.2: Electroweak bubble nucleation Schematic picture of a bubble nucleation in first-order cosmological electroweak phase transition (Figure courtesy of “Baryogenesis” by J.Cline [31].)

Figure 1.3: Plot of sphaleron energy versus Chern-Simons number change Energy configuration of sphaleron gauge field versus its Chern-Simons number change (Figure courtesy of “Baryogenesis” by J.Cline [31].) non-perturbative object called sphaleron, which essentially represents a certain configuration of gauge fields.

1.2.3 Non-perturbative sphaleron

Non-perturbative physics share great similarities with defects discussed earlier: both are related to the topological structures of the vacuum manifold. In 1982, Manton and Klinkhammer [68] discovered the existence of sphalerons: the transitions between the topological minima of the vacuum manifold. Since these minima are characterized by different Chern-Simons numbers, and thus baryon numbers, sphalerons lead to baryon-number changing processes 2

2 The studies on electroweak baryogenesis got its real boost when Kuzmin, etc. ([74]) pointed out that finite-temperature effect was present in the real universe to achieve realistical physical effects from the sphaleron transitions. See reference [102]for recent discussions.

9 Figure 1.4: Sphaleron’s magnetic dipole moment Schematic model for sphaleron’s magnetic dipole moment (Figure courtesy of “The origin of the sphaleron dipole moment”, [60].)

Mathematically, the anomalous current and the topological transitions are encapsulated by the equa- tions below:

µ Nf 2 a aµν 2 µν ∂µ J = ( g WµνW + g′ Bµν B ) (1.10) 32π3 − g2 2ig N = d3xεi jkTr W ∂ W + W W W (1.11) CS 32π2 i j k 3 i j k

Here, J refers to the anomalous current that characterises the matter content (i.e. fermion number

Nf ), NCS is the Chern-Simons number, which characterizes the topological property of the electroweak a vacuum. Bµν and Wµν (a=1-3) refer to the B bosons and the three electroweak gauge fields, where g and g’ are the respective gauge couplings. εi jk is the Levi-Civita symbol that characterizes cyclic computation. How should one model a sphaleron? [68] has explicitly calculated the magnetic dipole moment of a sphaleron for a non-zero Weinberg angle, and it has been shown subsequently that the sphaleron can be modeled as a magnetic monopole connected to a magnetic antimonopole, with a twist of string between them [60].

10 This kind of twist can be quantified by a measure called the helicity. Since we are talking about twist of string coming from magnetic charge, we’d look at the magnetic helicity term:

H = A Bd3x (1.12)

It is worthwhile to note here that magnetic helicity density (A B) is gauge-dependent but the volume-integrated magnetic helicity is gauge-independent: say for A A + ∇Φ, where Φ is a scalar → function:

d3x∇Φ B = d3x∇ (ΦB) d3xΦ(∇ B) (1.13) − Both terms in the last expression are zero as there is no charge inside the enclosed volume and the magnetic field is a divergence-free quantity. The non-zero magnetic helicity of a sphaleron will be essential to the discussion that follows. Now let us revert to the line of discussion central to the theme of this thesis: the magnetic field.

1.2.4 Dynamo for astrophysical magnetic field

Although magnetic monopoles remain elusive, the universe we inhibit abounds with magnetic phe- nomena from compact neutron star with extreme field strengths of 1014 Gauss [58], to weaker fields

6 9 prevalent in the galaxy (10− Gauss), galaxy clusters (10− Gauss) ([124], [130]) and even very weak magnetic fields in the extra-galactic region ([72]). Historically, astrophysicists study the evolution of magnetic field in galaxy system via the dynamo studies.

The state of the early universe can be effectively modeled as a fluid and studied via hydrodynam- ics, and with the presence of magnetic fields, magnetohydronamics(MHD). Understandably, both these studies are quite involved, and we will not go into the details here (see for example [17] for an updated review). But for our purposes, there are two points worth pointing out:

(1) The galactic dynamo requires a seed field but the one generated generically either by the Bier-

11 mann battery (i.e. differential current arising from the relative motion between the light electrons and heavier ions from zero initial condition) or astrophysical charge flows can run into a saturation prob- lem and has difficulty to be amplified up to the observed value (see [73] for related discussion). Also, recent observations of both optical and radio telescopes ([10]) indicate that magnetic fields found in higher redshift galaxies are of much higher strength than conventionally expected. This probe called to question the conventional standard assumption about a seed field formed just prior to recombination and re-opens up the possibility of a much earlier cosmological magnetic seed field.[126]

(2)The evolution of the magnetic field is closely tied with its topological configurations. Recalling that the early universe exists as plasma with high conductivity, there are a set of mathematical equations that describe the plasma’s evolution:

∂ρm +∇ (ρmv) = 0 (1.14) ∂t ∂ρq +∇ j = 0 (1.15) ∂t ∂v ρm +v ∇ v = J B ∇P (1.16) ∂t × − J 1 E +v B = + (J B ∇Pe) (1.17) × σ enq × −

The first two equations are mathematical statements for mass (mass density ρm) and charge (charge density ρq) conservations. The third equation represents the transfer of momentum (mass m and veloc- ity v) via current J, magnetic field B and pressure term P. The last equation relates a generalized version of the Ohm’s Law for electric field E and plasma of conductivity σ with charge number density nq, and electric charge e. Note that for the early universe, where σ approaches infinity, and for length scale in which the Hall term and pressure gradient term is negligible, the right hand side of the last equation above can be effectively taken to be zero. i.e.

E = v B (1.18) − ×

12 It is thus clear that both the magnetic field and bulk velocity field lie orthogonal to the electric field. That is why magnetic flux is conserved, i.e. the magnetic fields are essentially transported as flux tubes.

Another conservation law is related to the knottiness of the flux configuration, or helicity, as mentioned earlier. Borrowing insights gleaned from plasma physics, helicity conservation is found to be a key property of the plasma evolution in the galactic dynamo, see [17] again for detailed descriptions.

1.2.5 Cosmological magnetic seed field

By our current knowledge, established particle physics models lead us to believe that multiple phase transitions corresponding to the breaking of symmetries at different energy levels happened [15], with implications and remnants that can last till today. In fact, back in the 1980s, Hogan proposed that the charge flow in cosmological phase transition could potentially act as the seed fields for cosmological magnetic fields [62], much like the Biermann battery. Being the most energetic physical mechanism, it comes as no surprise that people started more elaborate model-building of cosmological magnetic field generation in inflation (see e.g. [117],

[100],[39]). The main point is that normal electromagnetism is conformally invariant, and thus any acausal physical mechanism involving stretching of electromagnetic wave mode beyond the horizon

(i.e. adiabatic amplification) involves the breaking of such conformal invariance. However, such model-building is not risk-free: the back-reaction of the fields upon such violent action may in turn destabilize the model ([39], [40]). The next line of models focused on generic phase transition sce- narios for cosmological QCD ([99], [107] and [28]). However, lattice studies seem to suggest that the transition of QCD is of a smooth second-order kind. Also, the correlation lengths involved within the realm of causal physicsal mechanism at such high energy were usually too small for more detailed model building. Being the symmetry breaking that essentially bears out electromagnetism, the cosmo- logical electroweak transition seemed a promising candidate. However, the key again is to sustain any possible cosmological magnetic field against any dissipation in strength and correlation length upon subsequent primordial plasma evolution.

Central to this discussion is the concept of “cascade” in plasma evolution. In short, cascade refers

13 to the transport of a physical quantity, subject to the constraint of energy conservation, from larger to smaller length scales. A distinction is made when the transport goes instead from the smaller to larger lengths: “inverse cascade”. For magnetic field configuration with twists and knots, it has been found that the energetic barriers to untwisting do prevent the energy from saturating in the smaller length scales; instead the twists and knots, or helicity, favors inverse-cascade. That is, energy ends up being transferred from smaller to larger length scales. The insight earned is then that if the cosmological seed field gives rise to topologically non-trivial field configuration, such that inverse cascade is favored in its evolution, the chance of the field surviving subsequent dissipation as the plasma moves around is much boosted. If the fields were strong enough over a certain coherence length scale, they could indeed potentially act as the pre-recombination magnetic seed, which leads to the magnetic field seen in galaxies, galaxy clusters and the likes today.

1.2.6 Baryogenesis<=>Magnetogenesis

As before, all these loose threads come together in the discussion of magnetic field. Think of hav- ing all these conditions satisfied in the early universe: a first-order phase transition, the engulfing of sphalerons (and thus baryon number changing processes) by the real vacuum that eventually made up our present universe, and the decay of such sphalerons with time (since sphalerons are unstable con- figurations). However, the energy of the sphalerons, including their magnetic energies, must go into the primordial plasma. 3 And the twist of the magnetic dipole is important: combined with flux and helicity conservation of the magnetic field these dipoles carry, the distinctive magnetic power spectrum could get inverse-cascaded from the smaller length scales to larger length scales, and thus preserve the magnetic seed up until structure formation. Essentially, the sphalerons, if conditions prevail, are direct harbingers of both baryogenesis and magnetogenesis processes, but the key question is then how well-tuned is the helicity spectrum upon the phase transition dynamics? Does the ”twistiness” of the

3The literature on electroweak baryogenesis mainly focuses on the specific microphysics details of the phase transition to fulfill the combined constraints of laboratory electroweak precision measurement and sufficient baryon-asymmetry. Few have focused the attentions on the “magnetic” energy transfer of the gauge field configuration.

14 sphaleron field configuration survive over long-scale, and if indeed, how long is the resulting correla- tion length of the magnetic field?

1.2.7 Probes of cosmological magnetic seed field

The discussions covered thus far have touched on numerous subfields related with the studies of cosmo- logical and astrophysical magnetic fields, and the paragraphs alone cannot do justice to them all. Here, we will briefly point out two areas of observations and experiments which are decisive in resolving the astrophysical/cosmological magnetic field origin controversy:

(i) CMB polarization and non-gaussianity probes Up-coming experiments for CMB polarization measurements will undoubtedly tighten the limit on primordial magnetism right around the surface of last scattering(z 1000), through its effects on the ≈ vector source contribution([110]), Faraday rotation ([71]), as well as generic non-gaussianity signatures

([24],[105],[18]). The current constraints mostly border on a field strength of O(1)nG for a coherent length of around 1Mpc for power-law spectra magnetic fields.

(ii)Intergalactic magnetic field A key differentiator between primordial and astrophysical magnetic field origin will be a probe of the magnitude of magnetic field in the open void of the intergalactic medium [87]. It has been proposed that [94] high energy photons from say the TeV blazar source can initiate an electromagnetic cascade of charged positrons and electrons via quantum pair-production processes, which would then get “deflected” in the presence of a background of EGMF (extra-galactic magnetic

16 field). A lower bound of 10− G has been reported in the recent literature with the gamma-ray telescope 15 FERMI’s earlier data ([88]), but a recent claim of field strength of 3x10− G ([2]) is not conclusive yet because of systematic ambiguities.

15 1.3 Gravity and electromagnetism studies

We will now switch our attention from magnetic phenomena to the studies on electromagnetism at its quantum regime, which are experimentally well-established. In particular, we would look at the behav- ior of quantum electromagnetism in curved space-time, via a calculation tool called the effective field theory. Here, we could only afford to give a brief introduction on its use and relation to our specific project; the reader is referred to [20] and [101] for introduction to its vast applications.

1.3.1 Effective Field Theory

Quantum field theory has been extremely useful in exploration and explanation of interactions of fun- damental particles (i.e. QED). One key insight for the basis of its experimental success is that whenever there is a hierarchy of physical scales involved, it is always possible to study the low-energy Lagrangian as dictated by the symmetry principles, which is the effective description of the system where higher energy effect has been “integrated out”. This is called the Effective Field Theory (EFT) approach. In a more strict definition, EFT is defined as such:

“For a given set of asymptotic states, perturbation theory with the most general Lagrangian containing all terms allowed by the assumed symmetries will yield the most general S-matrix elements consistent with analyticity, perturbative unitarity, cluster decomposition and the assumed symmetries”. [129]

From the principles of gauge, Lorentz, charge conjugation and parity invariance, the most general

QED Lagrangian is:

6 1 µν a µν 2 b µν ρσ F L µν µν νρ σµ O e f f = F F + 4 (F F ) + 4 F F F F + ( 6 ) (1.19) −4 me me me

where Fµν refer to the electromagnetic strength tensor, me is the electron mass and a and b are the low-energy coupling constants that capture the dynamics of the theory. Note that this Lagrangian

16 captures only the information of the lighter fields (photons) when the energy for its interaction is much lower than me.This can be also calculated by essentially integrating out the electron field from the gen- erating functional of the full QED. Dimensional analysis can also tremendously simplify the problem: one can get an order of magnitude estimate on the coupling constant values simply by noting the cor- responding dimensionalities of the fields content, multiplied by any combinatorial and loop factors.

In short, EFT presents a general Lagrangian Lgeneric:

Lgeneric = ∑cioi (1.20) i

Here, ci is the coupling constant to operator oi, where the terms are usually grouped and summed up in series according to the dimensions of the operators oi. Essentially, while the operators oi consist of only the “light” fields in the theory, all the influences of the “heavier” fields are manifested in the values of the coupling constants ci. The dimensionality of the terms is important, because they spell out the weighting or contribution of the individual term at the specific energy where the effective Lagrangian holds. ∝ 1 4 For example, if oi has dimension di, then ci Λd 4 where i−

(i)di < 4 for relevant operator

(ii)di = 4 for marginal operator

(iii)di > 4 for irrelevant operator

di 4 Irrelevant operators refer to those interactions that are suppressed by E/Λ − , such that Λ is the high energy cut-off where the effective theory description is still valid. The word irrelevant does not preclude that these terms are interesting — they are in fact the ones that hint of interesting dynamics towards higher energy.

4Here we are assuming the space-time dimensionality is 4, which does not necessarily have to be so. I.e. Conformal field theory has 2+1 space-time dimension, and thus the number is replaced by 3, etc.

17 1.3.2 QED in curved space-time

Deser and van Nieuwenhuizen have done a study via EFT that involved coupling the Maxwell tensor to the Riemann and Ricci tensors [], but they skipped any parity-violating term. If one allows for such a term, as say from two-loop quantum processes in the electroweak at dimension-6, the most general form of electrodynamics in curved space-time would thus be

1 2 4 1 4 µν S = M d x g (R 2Λcc) d x g F Fµν −2 pl | | − − 4 | | 4 1 µναβ 1 µναβ + d x g Fµν Fαβ R + Fµν Fαβ R (1.21) | | Λ 2 Λ 2 1 2 Here, the Λ1 and Λ2 terms denote the unknown high-energy cut-off of the theory (electromagnetism coupled with gravity) to be determined “experimentally”, since we do not have a complete generating functional like the QED case. We will now turn to its discussion.

1.3.3 Cosmological and Astrophysical probes

There are two main reasons why one would want to turn to cosmological and astrophysical avenues for bounds on electromagnetism and gravity couplings:

(i)This approach has been proven to be very fruitful for studies of EFT with photons and unknown high-energy particles such as axions or chameleons [21]. Cosmological and astrophysical observations routinely collect photons from a wide continuum of electromagnetic wavelengths at different redshifts, and this in turn gives powerful handles on deciphering electromagnetic properties of unknown parti- cles, should they have any 5. An additional advantange of cosmological electromagnetic probes is its accumulative effect over huge spatial scales. Any minuscule effect has an increased hope of detection through such amplification. (ii)Higher energy EFT inevitably involves pair production from the vacuum, which is a routine

5Dark matter remains a huge cosmological enigma, partly because they are inert to electromagnetic couplings.

18 calculation for QED. Insights gleaned from the unique signatures of electromagnetism, say parity- breaking, may also provide unique complementary probes of the existing vast quantum phenomena available for studies.

1.4 Conclusion

The rest of the thesis is organized as follows.

Chapter 2 is on the numerical studies of the formation of non-Abelian magnetic monopoles from a two-stage symmetry-breaking process of SU(3). There we will introduce the explicit detailed mecha- nism for the numerical modeling of monopole-string network from SU(3) to SU(2)/U(1). The explicit winding numbers for the monopoles and anti-monopoles will be found. These are then linked by strings, and the statistical properties of their networks from annihilation or clustering will be presented in the end to give some tentative hints of the networks’ characterization and subsequent evolution. The bulk of this chapter is centered on the paper published, “Formation of non-Abelian monopoles connected by strings” by Y.Ng, T.W.B. Kibble and T.Vachaspati [89]. Chapter 3 is focused on the electroweak magnetogenesis model, as based on the electroweak baryo- genesis scenario: a first-order phase transition in which the baryon number changing magnetic-charge carrying agents, sphalerons, diffuse across the bubble wall from the false vacuum to the true vacuum. We have explicitly constructed a magnetically helical sphaleron model. The motivation is to obtain the ratio of the helical component to the non-helical (i.e. transverse and normal) parts of the correlation functions, and how the position or magnitude of the peak of the helical spectrum depends on the pa- rameters (i.e. bubble wall speed, number of bubbles, etc.) of the model. The question we are trying to answer is — whether the twisted magnetic dipole configuration of the sphalerons survives the phase transition dynamics to give a reasonable helical magnetic spectrum?

What is the magnetic field spectrum at injection, before it undergoes the subsequent decay and evolu-

19 tion in the plasma? [65]

For a statistically homogeneous and isotropic Gaussian-random field, the properties of the field are encoded in its two-point correlation functions:

In real space:

< Bi(x)B j(x +r) >=(δi j rˆirˆj)MN(r)+ rˆirˆjML(r)+ ε rˆ MH(r) (1.22) − i jl l

where ML(r) denotes the component of the spectrum withr ˆ along the direction of the vector joining the two points, perpendicular for MN(r), and MH(r) carries the helical component. In Fourier space:

1 3 i jl < Bi(k)B j(k ) >= δ (k k ) (δi j kˆikˆ j)S(k)+ iε kˆ A(k) (1.23) ′ (2π)3 − ′ − l where the symmetric piece is named S(k),and the asymmetric piece denotes the helical part of the field component. Here, the δ 3(k k ) arises due to the random phase approximation. We will obtain the − ′ generic plots of both the real and Fourier space spectra. This chapter is focused on works published,

“Spectra of magnetic fields injected during baryogenesis” by Y. Ng and T. Vachaspati [90]. Observationally, A(k) remains hard to be probed, but proposals exist [92] to measure the helicity content of astrophysical systems via a combination of synchrotron radiation and Faraday rotation mea- sures; there remain the formidable task of resolving the ambiguities on the intervening electron density in the IGM though.

In Chapter 4, we present a study on the potentials and limitations cosmological and astrophysical electromagnetic observations have for shedding light on any new physics, in particular, gravity, oc- curring above the established standard model energy level. First, general arguments on mathematical symmetry properties would be presented to account for all the terms that can generically arise from gravito-electromagnetism up to first perturbative order (or dimension-6) of EFT approach. Then, cal-

20 culations on possible observable effects on cosmological CMB and GR test in the solar system (in the form of birefringence angle and extra time-delay for different polarizations of gravity) will be pre- sented. Comparing with the sensitivities of current observational probes, we will conclude that the great disparity in the theoretical and current observational electromagnetic sensitivities (and the foreseeable future) involved and render such observations of little power to offer much constraining powers. This work is done in collaboration with Y.Chu, D. Jacobs and G.Starkman, and has been published “It is hard to learn how gravity and electromagnetism couple” [29].

21 Chapter 2

Symmetry Breaking

2.1 Introduction and context

Topological defects are formed in a vast array of laboratory systems and may also have formed during a cosmological phase transition [66]. The statistical properties at formation of the simplest of defects have been studied quite extensively in the context of cosmology [119] and more recently in a variety of different condensed-matter systems. Experiments have been performed to observe the spontaneous formation of defects in nematic liquid crystals [30, 13, 42], in superfluid 3He [103, 104] and in super- conductors [81, 85]. In most particle physics applications, the vacuum manifold can be quite complex, and hybrid topological defects may be formed. These may consist of monopoles connected by strings or walls that are bounded by strings (see for example [67]). In this paper we study the formation of non-Abelian monopoles that subsequently get connected by strings due to a second non-Abelian symmetry breaking. More specifically, we study monopoles formed in the symmetry breaking

SU(3) U(2) [SU(2) U(1)]/Z2. (2.1) → ≡ ×

The fundamental monopoles carry both SU(2) and U(1) charge and may be labeled by a pair of charges, (1, 1), where the first entry (with no sign) is the SU(2) charge, and the second entry is the U(1) charge. ± After the monopoles are formed, we consider the further symmetry breaking

SU(2) Z2. (2.2) →

Now all the monopoles will get connected by strings. However, the SU(2) charge is a Z2 charge, and so there are two types of monopole states connected by strings (Fig. 5.1). The first of these is a

22 Figure 2.1: Confined monopoles in SU(3) model Two types of confined monopoles in the SU(3) model. The picture on the left rep- resents a monopole and an antimonopole connected by a string. The picture on the right shows two monopoles with the same U(1) charge connected by a string. monopole-antimonopole bound state i.e. a bound state of (1,+1) and (1, 1). The confining strings − will then eventually bring the monopole and antimonopole together and lead to their annihilation. The second possibility is that the string confines a monopole to a monopole i.e. two (1,+1) or two (1, 1) − objects. In this case, the confining string will bring together the two monopoles to form a charge 2 object, (0, 2), that carries no net SU(2) charge but carries twice the basic U(1) charge. One of our ± aims is to determine the relative number densities of the two types of objects subsequent to the second symmetry breaking stage.

In the context of Grand Unification Theories (GUTs), fundamental magnetic monopoles also carry non-Abelian charges. For example, in the minimal GUT model with SU(5) symmetry, the fundamental monopoles carry SU(3) color, SU(2) weak, and U(1) hypercharge quantum numbers. The formation of magnetic monopoles in the grand unified context occurs due to the non-trivial topology of a very large vacuum manifold and our toy SU(3) model may be expected to capture some of the complications. One motivation for considering the formation of strings that connect non-Abelian monopoles is that the physics of confinement is not fully understood, and it is possible that non-Abelian magnetic

fields also get confined due to quantum or plasma effects [37, 78]. A second related motivation comes from the Langacker-Pi proposal to solve the cosmic monopole over-abundance problem [76]. The scenario assumes that electromagnetic gauge symmetry is spontaneously broken for a period in the early universe. As a result, magnetic monopoles carrying electromagnetic flux will get confined by strings and annihilate effectively. Later the electromagnetic symmetry is restored to be consistent with present observations. The breaking of SU(2) in our toy model performs a similar function for this non-

23 Abelian model as does the Langacker-Pi mechanism for the Abelian case, although it does not involve symmetry restoration at low energy. Monopoles again get connected by strings but here they can either annihilate or form charge 2 states. The corresponding scenario in GUTs is more complicated since the monopoles get connected by several different kinds of strings [37, 78], as we discuss in Sec. 2.5. We start in Sec. 3.2 by describing the field theoretic model under consideration, focussing on the topological aspects. In Sec. 2.3 we describe our numerical implementation to study defect formation in the model and the results in Sec. 3.4. We conclude in Sec. 2.5 by discussing defect formation in an

SU(5) GUT model.

2.2 Model

Our model contains an SU(3) adjoint field, Φ, whose vacuum expectation value (VEV) implements the symmetry breaking in Eq. (2.1). Two more SU(3) adjoint fields, Ψ1 and Ψ2, acquire VEVs to break the SU(2) subgroup of U(2) to Z2 as in Eq. (2.2). The Lagrangian for the model is

2 1 2 1 2 L = tr[(Dµ Φ) ]+ ∑ tr[(Dµ Ψi) ] 4 4 i=1 1 µν tr(Xµν X ) V(Φ,Ψ1,Ψ2), (2.3) −8 − where Dµ Φ = ∂µ Φ ig[Xµ ,Φ], Xµν is the field strength for the SU(3) gauge field Xµ , and the potential, − V, is assumed to have a form that is suitable to give the fields the desired VEVs. The first stage of symmetry breaking is achieved by the VEV

1 0 0 η 0 8   Φ = Φ( ) ηT 0 1 0 , (2.4) ≡ ≡ √ 3     0 0 2  −    where η is the energy scale at which the first symmetry breaking occurs and will be set to unity since its value has no effect on the topological structures we are considering. (We could also take Φ = gΦ(0)g†

24 for any global g SU(3).) The vacuum manifold at this stage is ∈

C 2 SU(3)/U(2) ∼= P . (2.5)

2 Points on CP are labeled by three complex numbers (z1,z2,z3), identified under a (complex) rescaling

T Z (z1,z2,z3) = κ(z1,z2,z3) , κ C, κ = 0. (2.6) ≡ ∼ ∈

It will be convenient for us to label the points, following [8], by a point on an octant of a two-sphere given by θ¯ and φ¯, and two phases, α and β:

α β ZT =(sinθ¯ cosφ¯ ei ,sinθ¯ sinφ¯ ei ,cosθ¯), (2.7) with 0 θ¯,φ¯ π/2 and 0 α,β 2π. ≤ ≤ ≤ ≤ The relation between the field Φ and a point on CP2 is

1 ZZ† Φ = 1 3 . (2.8) √3 − Z†Z

The second homotopy group of CP2 is known to be the set of integers Z. A topologically non-trivial configuration can be constructed explicitly by taking φ¯ = 0. The points on the φ¯ = 0 sub-manifold are

α ZT =(sinθ¯ ei ,0,cosθ¯) (2.9) and these describe a CP1 subspace of CP2. The points on a two-sphere in physical space, labeled by

(θ,φ), can be mapped onto this CP1 using

θ¯ = θ/2, φ¯ = 0, α = φ, β = 0. (2.10)

25 Equivalently, 3cosθ 1 0 3sinθ eiφ − − 1   Φ = 0 2 0 . (2.11) √ 2 3    θ iφ θ   3sin e− 0 3cos 1 − − −    This map represents a simple example of a monopole. An expression for the topological charge of a monopole can be derived by first constructing the 1-form “gauge potential” 1 Z†dZ dZ†Z A = − . (2.12) 2i Z†Z

Note that under the “gauge transformation” Z Zeiλ , which is a special case of (2.6), A transforms as → A A + dλ. The corresponding field strength 2-form is →

1 dZ† dZ dZ†Z Z†dZ F = dA = ∧ ∧ . (2.13) i Z†Z − (Z†Z)2

Since this 2-form is exact, its integral over a closed two-surface is a topological invariant — and moreover is zero unless the surface contains in its interior a point or points where Z = 0 (so that A is undefined). So the expression for the topological charge in a volume V with closed boundary ∂V is

1 1 2 i i jk Q = F = d S ε Fjk. (2.14) 2π ∂ 4π ∂ V V

There is another way to obtain the expression for the topological charge. We start with the expres- sion known for the ’t-Hooft-Polyakov monopole in SU(2) and extend it to SU(3):

1 2 i i jk a b c Q = d S fabcε n ∂ jn ∂kn , (2.15) 8π ∂ V where Z†T aZ na = , (2.16) Z†Z with a,b,c = 1,...,8. Here the T a are the generators of SU(3), normalized by tr(T aT b)= 2δ ab, the

26 a b c fabc are structure constants defined by [T ,T ]= 2i fabcT , and the integration is over the two sphere at infinity. Also note that the vector na satisfies nana = 4/3. In Appendix A.1 we show that the two forms for the topolgical charge are equivalent.

It is simple to check that Q = 1 for the monopole configuration in Eq. (2.10) and Eq. (2.11). The formula in Eq. (2.14) will be useful to locate monopoles in our numerical work described in Sec. 2.3.

The second stage of symmetry breaking is more involved. The fields Ψ j now also acquire VEVs, which are required to lie in the unbroken SU(2) subgroup, and hence commute with Φ. Their mag- Ψ2 nitudes tr( j) are fixed by the potential, and they are also required to be mutually orthogonal in the sense that tr(Ψ1Ψ2)= 0. Given a value of Φ at some spatial point P, we need to identify this unbroken subgroup. The standard procedure is to work out commutators of Φ with SU(3) generators and to find linear combinations of the generators that commute. In practice, it is easier to first rotate Φ, say by an

SU(3) rotation R, to the reference direction, Φ(0). We discuss how to choose R below. Then the gen- erators of the unbroken SU(2) sit in the 2 2 upper left corner while the generator T 8 of the unbroken × (0) (0) U(1) is in the direction of Φ itself. With respect to Φ , the VEVs of Ψ1 and Ψ2 can be written in terms of two orthonormal 3-vectors, a and b, as Ψ(0) = a T and Ψ(0) = b T where 1 2

σi 0   T i = , i = 1,2,3, (2.17)      0 0      σ Ψ(0) Ψ(0) and i are the Pauli spin matrices. Once 1 and 2 are constructed, we can rotate all the fields back to the original point using R†.

The VEVs of Ψ1 and Ψ2 break SU(2) down to Z2, which is the center of SU(2), 1, 12 , i.e. the { − } identity element of SU(3) and 12 diag( 1, 1,1). A string passes through a spatial contour if Ψ1 − ≡ − − and Ψ2 are such that, on going around the contour, these fields are transformed by the element 12 − and not by the identity element. The strings are of the Z2 variety and there is no distinction between a string and an anti-string. Also, there is no known integral formula that can be used to evaluate the winding around the contour.

27 Figure 2.2: Schematic picture of the cubic lattice Each cell of the cubic lattice is sub-divided into 24 tetrahedra. Only one cubic cell and representative tetrahedron are shown.

2.3 Numerical implementation

To simulate the formation of the monopole-string network, a 3-dimensional cubic lattice is chosen. Each cubic cell is further divided into 24 tetrahedral sub-cells, obtained by connecting the center of the cube to the 8 corners and the centers of the 6 faces (see Fig. 2.2).

The next step is to assign random points of CP2 at each point on the lattice, including the centers of the cubic cells and their faces. Now, the unique SU(3)-invariant metric on CP2 is the Fubini-Study metric dZ†dZ dZ†ZZ†dZ ds2 = , (2.18) Z†Z − (Z†Z)2 or, in terms of the parameter choice of (2.7),

ds2 = dθ¯ 2 + sin2 θ¯ dφ¯2

+sin2 θ¯ cos2 φ¯(1 sin2 θ¯ cos2 φ¯)dα2 − 2sin4 θ¯ cos2 φ¯ sin2 φ¯ dα dβ − +sin2 θ¯ sin2 φ¯(1 sin2 θ¯ sin2 φ¯)dβ 2. (2.19) −

28 Hence the SU(3)-invariant measure on CP2 is

√gdθ¯ dφ¯ dα dβ = sin3 θ¯ cosθ¯ sinφ¯ cosφ¯ dθ¯ dφ¯ dα dβ. (2.20)

Thus the assignment is done by drawing 0 sin4 θ¯ 1, 0 sin2 φ¯ 1, 0 α 2π and 0 β 2π ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ from uniform distributions, and then constructing Z as in Eq. (2.7). The four vertices of a spatial

2 tetrahedron then get mapped on to a tetrahedron in CP which we will denote by (Z1,Z2,Z3,Z4). To find out if this tetrahedron in CP2 is topologically non-trivial (i.e. incontractable) we use a discrete version of the charge formula in Eq. (2.14)

1 α Q = π ∑ i jk , (2.21) 2 i jk { } { } where the sum is over the four triangular faces of the tetrahedron (with positive orientation), and for each face, α † † † i jk = arg(Zi ZkZk Z jZ j Zi), (2.22) { } α π π where we require i jk to lie within the range [ , ]. We can explicitly check that small changes in { } − the Zi do not affect Q, thus showing that even the discrete formula is topological. One can also check that Eq. (2.21) agrees with Eq. (2.14). The charge Q is the integral of F/2π over a large sphere, which can be broken up into the sum of the four separate contributions from the individual faces of the tetrahedron. Each of these can be expressed as the integral of the 1-form A/2π † around the perimeter. In discretized form, the integral of A along the 1-2 link becomes arg(Z2Z1) (see Eq. (2.12)) and so the magnetic flux through the triangular plaquette 123 , is found by summing the { } contributions from the three edges,

† † † dx A = arg(Z Z1)+ arg(Z Z2)+ arg(Z Z3) 2 3 1 +2πn, (2.23)

29 where n is an integer and the extra term, 2πn, in Eq. (2.23) is included because each of the phases is ambiguous up to 2π. This can also be seen as a gauge ambiguity: a gauge transformation may ± change the value of n. It has a geometric interpretation as well. For the special case of triangles on a CP1 subspace of CP2 (isometric to a sphere of radius 1/2), we have shown that the flux through a triangle, found using Eq. (2.14), is equal to twice the area of the triangle. Thus the ambiguity in the

flux in Eq. (2.23) is equivalent to the ambiguity in choosing between the two complementary spherical triangles with this boundary. We choose the one with the smaller area, so that

α dx A = 123 . (2.24) { }

Thus Eq. (2.21) is the discretized version of Eq. (2.14). We conjecture that for a general triangle in CP2, not lying on a CP1 subspace, the flux through it may still be equal to twice the area of the minimal surface with that boundary. Choosing the minimal area may be seen as a generalization to areas of the “geodesic rule” for lengths [119]. The rule in general is to choose the minimal value of the integral in Eq. (2.23). Next we turn to the formation of strings that connect the monopoles. For this we need to consider a triangular face of a tetrahedron and determine if a string passes through it.

Each vertex of a triangular plaquette has already been assigned a point on CP2, equivalently a VEV of Φ. It is convenient to label the subgroup that leaves Φi invariant as SU(2)i U(1)i/Z2. Now we × also assign VEVs of Ψ1 and Ψ2, making sure that these lie in the unbroken SU(2) sector of SU(3) at

Zi, namely SU(2)i, and that they are orthogonal: tr(Ψ1Ψ2)= 0. The precise scheme is as follows.

2 The scheme is based on the construction, for each pair of points on CP , say Zi and Z j, of an • SU(3) transformation, R ji, that transforms Zi to some representative of the point Z j and moreover C 2 does so along a geodesic in P , i.e. R jiZi ∼= Z j. In fact the left-hand side is equal Z j times the † phase factor that makes the scalar product with Zi real (see Appendix A.2). In other words, we find † Z j Zi R jiZi = Z j † . (2.25) Z Zi | j | 30 The geodesic condition will be achieved if R ji can be written as

R ji = exp(iMs), (2.26)

where M is a suitably chosen normalized combination of the generators T a and s is the geodesic

distance between Zi and Z j, given by

Z†Z Z†Z 1 ( i j)( j i) s = cos− . (2.27)  † †  (Zi Zi)(Z j Z j)  

A more explicit construction of R ji is described in Appendix A.2.

Similarly, for each Zi, we define an SU(3) transformation Ri0 such that Zi = Ri0Z0, where Z0 is the reference point (0,0,1). (With our choice of representative in (2.7), no phase factor is needed † here.) The matrix R described in the previous section, above Eq. (2.17), will be one of the Ri0.

2 To each vertex of the triangular face is associated a point on CP (say Zi) and two uniformly • distributed orthonormal 3-vectors, ai and bi where i labels the vertex of the triangle (see Fig. 2.3).

If we wish, we can construct Φi from Zi using Eq. (2.8). The two remaining fields Ψ1,2 may be found from a and b. We first define

Ai0 = a T, Bi0 = b T, (2.28)

which are SU(3) matrices lying in the SU(2)0 subgroup, with generators T given by Eq. (2.17).

Then the fields are given by Ψ1 = η1A and Ψ2 = η2B, where η1,2 are the magnitudes of these fields, and the normalized SU(3) matrices A and B may be found by using the transformation

Ri0: † † Ai = Ri0Ai0Ri0 , Bi = Ri0Bi0Ri0. (2.29)

Note that by construction Ai and Bi belong to SU(2)i and hence commute with Φi.

31 Now we want to compare the symmetry-breaking fields at neighboring vertices. To do this we • transport them using the geodesic transformations R ji. Transforming Ai and Bi by parallel trans-

port along a geodesic from Zi to Z j, we obtain

† † A ji = R jiAiR ji , B ji = R jiBiR ji. (2.30)

Next we compare these transported matrices with the corresponding matrices A j,B j defined at

the vertex Z j. We seek a transformation S ji SU(2) j such that ∈

† † A j = S jiA jiS ji , B j = S jiB jiS ji. (2.31)

In Appendix A.3 we describe our construction of S ji in detail.

The net rotation of the pair Ai,Bi as we circumnavigate the triangular face from Zi to Z j to Z and • k back to Zi is

S i jk SikRikSk jRk jS jiR ji. (2.32) { } ≡

Note that since this combined transformation leaves invariant all the fields Φi,Ai,Bi, it must

belong to the unbroken U(1)i.

To determine whether or not a string passes through the ijk face, we have to compare S i jk • { } { } with the transformation RikRk jR ji without the intervening S factors. Since this transformation

leaves Φi invariant, it belongs to SU(2)i U(1)i/Z2. Moreover, in view of Eq. (2.22), we know × that α i i jk RikRk jR jiZi = Zie { } . (2.33)

Consequently, we know that the U(1)i factor in this product must be

1 α √ 8 exp( 2 i i jk 3Ti ). (2.34) − { }

Now let us return to S i jk . Since for example the transformation S ji SU(2) j leaves Z j unal- { } ∈ 32 tered, it is clear that, regardless of the choice of the S factors, the effect of S i jk on Zi must be { } exactly the same as that of the product in Eq. (2.33). Consequently, the combination

1 α √ 8 W i jk = S i jk exp( 2 i i jk 3Ti ) (2.35) { } { } { }

must leave Zi invariant, and also not contribute a phase to Zi, and hence it belongs to SU(2)i.

But we know that W i jk also belongs to U(1)i, since it consists of two factors each of which is { } an element of U(1)i. So W i jk must in fact be one of the two central elements that are common { } to both SU(2)i and U(1)i. If W i jk = 1, the winding is trivial and there is no string through the { } triangular face. If, however, W i jk = 12, then there is a string through the triangular plaquette. { } − It can be shown (see Appendix A.4) that if the monopole charge (2.21) within the tetrahedron is

non-zero, then there must be an odd number of faces with strings passing through, while if it is

zero there must be an even number. This follows from the fact that each edge, say (i j) appears,

with opposite orientation in two faces, and the relevant factors in say S i jk and S jil are inverses { } { } † of each other: (S jiR ji) = Si jRi j.

To get a better physical sense for this algorithm, it is useful to consider monopole and string forma- tion in the simpler symmetry breaking pattern

SU(2) U(1) 1. (2.36) → →

This example is discussed in Appendix A.5. We should also add that the natural language for our

3 2 discussion is in terms of fiber bundles since what we have in our model is an S /Z2 fiber over a CP base manifold. The topology of the base manifold, CP2, gives rise to monopoles while the topology of

3 the fiber, S /Z2, gives rise to strings that may end on monopoles.

33 (Zi, ai, bi) i

Rji Rik

k (Zj, aji, bji) j Rkj (Zk, ak, bk) (Zj, aj, bj)

Sji

Figure 2.3: Parallel transport of vector in manifold space to locate the string The algorithm to find strings requires parallel transport of the variables at vertex i along a geodesic on CP2to the vertex j. Then the transported variables are ro- tated to the assigned variables at j, by using an SU(2) geodesic transformation.

2.4 Results

The simulations were done on a cubic lattice of side 12 i.e. in 24 123 tetrahedral cells and was × repeated 10 times to gain statistics. The probability of having a monopole or antimonopole in a cell is 0.17. If N is the total number of string segments, then the relative numbers of segments in closed loops, string segments connecting like charge monopoles, and string segments connecting oppositely charged monopoles, are given by

Nloops = 0.4%. N N ±± = 4.2%. (2.37) N N+ − = 95.4%. N

This shows that roughly 4% of SU(3) monopoles will end up in the doubly charged state and survive annihilation due to strings.

The length distribution of + strings is shown in Fig. ??. Denoting the number density of these −

34 0 -4

-1 -5 -2

-3 -6

Ln(N/V) -4 Ln(N/V)

-5 -7 -6

-7 -8 0 5 10 15 20 0 5 10 15 20 l l

Figure 2.4: Log Nstring(annihilating) versus Figure 2.5: Log Nstring(clustering) versus string length − string length −

Logarithm of average number density Logarithm of average number density of strings connecting monopoles and of monopole-monopole(++) and antimonopoles versus string length. antimonopole-antimonopole( ) connections versus string length.−−

3 strings, i.e. number of segments divided by the volume (12 ), by n+ , the least-squares linear fit is −

(0.31 0.03)l n+ (l)=(0.46 0.08)e− ± (2.38) − ±

The corresponding distribution of ++ and strings is shown in Fig. ?? and the fit is −−

(0.23 0.07)l n (l)=(0.02 0.01)e− ± (2.39) ±± ±

2.5 Discussion

We have studied the formation of monopoles connected by strings in an SU(3) model and the results for the distribution of monopoles and strings are summarized in Sec. 3.4. Here we discuss qualitatively how a similar analysis in realistic grand unified models would proceed. Our experience with SU(3) helps us understand and appreciate the difficulties that are likely to be encountered. As an example, consider the minimal grand unified model based on a SU(5) symmetry group. The symmetry breaking

35 Figure 2.6: Clusters of monopoles Figure 2.7: Monopole-string network from GUT SU(5) A cluster of 6 monopoles can form a sin- glet of SU(3) and SU(2), as in ordinary Drawing of an infinite monopole-string net- baryons. A bound state of a monopole work that could result from SU(5) grand and antimonopole is also possible, as in or- unified symmetry breaking. The three dif- dinary mesons. The SU(3) charge on a ferent shades of circles represent the SU(3) monopole is shown in shades of grey (or color charge and the plus-minus symbols in color) and the SU(2) charge as a . within the circles the SU(2) charge. The ± We have not shown the U(1) charge. Z3 U(1) (hypercharge) charge has not been strings are shown as solid lines; Z2 strings shown. The isolated clusters of monopoles as dashed lines. have to occur in SU(3) and SU(2) singlets. pattern is

SU(5) [SU(3) SU(2) U(1)]/Z3 Z2. (2.40) → × × × and, if the non-Abelian magnetic charges are confined, the relevant symmetry breakings are

SU(3) Z3 , SU(2) Z2. (2.41) → →

The fundamental magnetic monopoles carry SU(3) and SU(2) charges in addition to the topological

U(1) charge. Therefore each monopole will get connected to a Z3 string and another Z2 string. Then isolated clusters of monopoles come in two varieties, similar to known baryons and mesons, as shown in Fig. ??. However, a likely outcome at formation seems to be that, in addition to some isolated baryonic and mesonic clusters, the monopole-string network percolates and we essentially obtain one giant structure, such as depicted in Fig. ??.

It seems hard to explicitly confirm if the network percolates, say by numerical simulation. For

36 example, the vacuum manifold at the first stage of symmetry breaking is 12 dimensional and it also does not fall into a straightforward category like CPn. Determining the distribution of strings is also more complicated since the SU(3) breaking leads to Z3 strings. These problems do not seem insurmountable but are hard enough that we have not attempted to solve them at the present time. If very few baryonic clusters form and instead an infinite monopole-string network forms, our experience with string networks [5, 123, 34, 61] suggests that the network energy density scales with time and never comes to dominate the universe. Processes such as monopole-antimonopole annihilation and meson formation could dissipate the energy of the network at a rate that is determined by the Hubble expansion. However, this scenario ignores the process of baryon formation from the network. Depending on the rate of this process, we could still have a monopole over-abundance problem coming from the production of baryonic clusters. graphicx

37 Chapter 3

Magnetic field from electroweak generation

3.1 Introduction

Our study of the universe relies on relics left-over from early cosmological epochs. The cosmic mi- crowave background brings information from the epoch when atoms formed, the light elemental abun- dances from the epoch when nuclei formed. Similarly, the electroweak phase transition may mark the epoch when a net baryon number was generated, or when net lepton number was converted into net baryon number, and this coincides with the generation of magnetic fields. The present paper is based on the hypothesis that primordial magnetic fields will inform us of the epoch when a net amount of baryons first formed, the so-called epoch of “baryogenesis” [36, 121].

The question of whether a primordial magnetic field exists is often raised in connection with the magnetic fields observed in galaxies and clusters of galaxies with strength µG and kpc-Mpc coher- ∼ ence scale. There is considerable debate whether primordial fields are essential to the generation of galactic fields, and what properties of the primordial field are necessary to turn them into observed magnetic structures. The arguments involve the coherence and amplitude of observed magnetic fields, the efficiency of galactic dynamos, the turnover time scales associated with galactic dynamics, espe- cially with the earliest known galaxies containing magnetic structures, astrophysical sources e.g. active galactic nuclei that may spew out magnetic fields, and the generation of large scale seed fields by the Biermann battery. We shall by-pass these issues since, in our view, a primordial magnetic field is of interest in itself, whether or not it is responsible for the observed magnetic fields in galaxies. If there are strongly motivated early universe scenarios, based on reasonably well-established particle physics, that lead to the generation of magnetic fields, they provide good reason to study and to look for these structures in cosmological data.

38 The connection between baryon number production and primordial magnetic fields can be under- stood intuitively in the following way. Baryon number violation in the standard model of the elec- troweak interactions is made possible due to a quantum anomaly. As a physical process, baryon num- ber is generated when many different particles come together to form an object called a “sphaleron” [82, 68], which then decays into a final state with baryon number that is different from that of the initial state. The sphaleron, and its deformations, are made up of scalar and electroweak gauge fields and may also be viewed as being an unstable bound state of an electroweak magnetic monopole and an anti- monopole, with an electroweak Z string confining them [122, 60]. The monopole and antimonopole − attract each other by the Coulomb force and the confining Z string also pulls them together, and a − static solution is in general not possible. In the sphaleron though, the antimonopole has a twist in field space relative to the monopole which prevents the monopole and antimonopole from annihilating, and allows for the existence of a static solution [113, 82]. The presence of magnetic charges also explains the large magnetic moment of the sphaleron calculated in Ref. [68]. Twisted solutions similar to the sphaleron also occur in the context of kinks in one spatial dimension and can lead to novel static phases containing a lattice of kinks and antikinks [118]. Once we appreciate that baryon number violation processes have intermediate states that consist of monopole-antimonopole pairs, it is not hard to see that magnetic fields must be produced when a sphaleron decays. But there is a further “twist” to this connection. The twist in the fields that prevents the monopole and antimonopole from annihilating is not stabilized, and the monopole can untwist and annihilate the antimonopole. The instability of the system causes the sphaleron to decay and radiate away its energy, releasing magnetic fields in the process [35]. The decay has been studied numerically and a very interesting feature emerges at late times. The released magnetic fields inherit the twist of the sphalerons and is measured by the magnetic helicity integral

H = d3xA B (3.1)

At late times, the magnetic field evolves such that the magnetic helicity is conserved. The conservation

39 of magnetic helicity is familiar in magneto hydro-dynamics (MHD) in plasmas with high electrical conductivity. Yet in [35] there is no external plasma; the only charges in the system are those result- ing from the decay of the sphaleron itself. Remarkably, magnetic helicity is still conserved during sphaleron decay. To summarize, baryon number violating processes in the electroweak model occur via the produc- tion and subsequent decay of sphalerons. Each sphaleron produces helical magnetic fields when it decays and the helicity of the magnetic field is conserved. Now, since the production of each baryon gives a certain amount of magnetic helicity, the cosmological baryon number density, nb, can be related to the magnetic helicity density (h) h n (3.2) ≈− b where the minus sign requires more detailed considerations [121] and indicates that the primordial magnetic field is left-handed. (In writing Eq. (3.2) we are assuming that the cross-helicity between the magnetic field produced by different sphalerons averages out to zero.) In principle, there could be a numerical pre-factor on the right-hand side and, in fact, early estimates suggested it to be α 1 ∼ − where α = 1/137 is the fine structure constant. However, explicit numerical evolution of the sphaleron suggests that (3.2) may be a better estimate [35]. Numerically, the cosmic baryon number density is

1037/cm3 at the electroweak epoch. ∼ The rough equality of magnetic helicity and baryon number gives us an estimate for the integral in Eq. (3.1) but does not provide much information about the characteristics of the magnetic field itself. Of interest are the two point correlation functions of the magnetic field. Assuming statistically homogeneous and isotropic magnetic fields, the spatial correlator at a fixed time can be written as

Ci j(r) Bi(x)B j(x + r) ≡ 1 = d3xB (x)B (x + r) V i j = MN(r)Pi j + ML(r)rˆirˆj + εi jkrˆkMH(r) (3.3)

where, V is the integration volume, i, j,k = 1,2,3, r = r ,r ˆ = r/r, ε is the Levi-Civita tensor, Pi j is | | i jk

40 the traceless projection tensor,

Pi j = δi j rˆirˆj (3.4) − and MN, ML and MH denote the “normal”, “longitudinal” and “helical” correlation functions respec- tively. Also note that the helical term in Eq. (3.3) has a factor ofr ˆk and not rk in it, as is sometimes used. The correlation functions will also depend on time but here we are only interested in the injected field at the end of the electroweak phase transition. So it is to be understood that the correlation functions are all evaluated at t = tew where tew denotes the time at which the phase transition is complete. The normal and longitudinal correlation functions are not independent since the magnetic field is divergenceless, ∇ B = 0, and the relation between MN and ML is,

dML 2 = (MN ML) (3.5) dr r −

It is more conventional to work in Fourier space, where the correlator can be written as

b∗(k)b j(k′) = i 3 (3) [S(k)pi j + iε kˆ A(k)] (2π) δ (k k′) (3.6) i jl l −

with no longitudinal term (proportional to kˆikˆ j) since the magnetic field is divergenceless. We have denoted the Fourier components of B(x) by b(k) with the convention

3 +ik x bi(k)= d xBi(x)e (3.7) and the momentum space projection tensor is

pi j = δi j kˆikˆ j (3.8) −

41 By explicitly Fourier transforming Ci j(r) it is possible to derive

∞ π 2 sin(kr) cos(kr) S(k) = 4 dr r ML 3 2 0 (kr) − (kr) sin(kr) sin(kr) cos(kr) + MN + (3.9) kr − (kr)3 (kr)2

Further simplification occurs on using the divergenceless condition in Eq. (3.5),

∞ 2 sin(kr) S(k)= 2π dr r cos(kr) ML (3.10) 0 kr −

2 where, to omit the boundary term, we have assumed r ML(r) 0 as r ∞. While Eq. (3.10) is → → 2 simpler, we still use Eq. (3.9) in our numerical work because r ML(r) is not negligible at the scale of our simulation box. Similarly we have the relation

4π ∞ sin(kr) A(k)= dr r cos(kr) MH (3.11) k 0 kr −

The goal of this paper is to develop a model for the generation of magnetic fields during baryo- genesis (Sec. 3.2), to Monte Carlo the model (Sec. 3.3) and to obtain the correlation functions of the magnetic field (Sec. 3.4). Our final results will be the functions MN(r), ML(r), MH(r), S(k) and A(k). The spectral properties of the injected field should be useful to study their evolution. We discuss limi- tations of our model and future prospects in Sec. 3.5.

3.2 Model

The relation between baryogenesis and magnetic fields exists as long as there is anomalous baryon number violation. Even in leptogenesis scenarios, sphaleron processes are required to convert lepton number into baryon number [38], and these conversions will produce magnetic fields. The power spectra of fields produced during leptogenesis, however, may be different from those produced during baryogenesis as considered here because, unlike baryogenesis, leptogenesis does not rely on a first

42 order phase transition for departures from thermal equilibrium. We assume that baryogenesis occurred at the electroweak scale and the necessary departures from thermal equilibrium are provided by a phase transition that was strong enough. The electroweak sym- metry is broken within bubbles which then grow, merge and eventually fill all of space, thus completing the phase transition. During this process, baryon number changes can occur relatively freely outside the bubbles since there is little energy cost associated with the sphaleron in the symmetric phase, but baryon number changes are highly suppressed within the bubbles since the sphaleron has mass

mW /α in the symmetry broken phase. (Here mW 100 GeV is the mass of the electroweak W gauge ∼ ∼ boson.) Also, in the symmetric phase that is outside the bubbles, the electromagnetic magnetic field is just a linear combination of the electroweak magnetic fields, and has no special significance. For example, all the electoweak gauge fields are massless, and we expect rapid interactions to maintain equilibrium between the different degrees of freedom. Inside the bubbles, the electromagnetic field is the only massless gauge field and its evolution is described by Maxwell’s equations. However, there are no sphaleron transitions inside the bubble and so no magnetic fields are generated there. It is only the sphaleron transitions occurring in a thin region right around the bubble wall which produce the magnetic fields that are then captured by the growing bubble. It is this magnetic field, generated by sphaleron transitions at the surface of bubbles, that is of interest to us.

A schematic picture of our model is shown in Fig. 3.1. Essentially, there are bubbles of the broken symmetry phase that nucleate and grow, and sphalerons are explosions that occur on the surfaces of these bubbles and blow out magnetic fields into the environment. We would like to find the correlation functions of the magnetic fields left-over after the phase transition has completed. Our picture of the phase transition is similar to that of cosmological large-scale structure formation in which cosmic voids nucleate and grow. Supernovae or other astrophysical activity occurs on the surfaces and intersections of the voids and expels magnetic fields and other elements into the cosmic environment.

The only element that is missing from the picture so far is a model for the “explosion” that expels the magnetic field. In Ref. [35], the decay of the sphaleron was studied numerically, with the result that magnetic helicity stayed constant at late times, and the energy density spread out as time progressed.

43













































































































































Figure 3.1: Schematic picture of sphaleron explosion on electroweak bubbles Schematic representation of the electroweak first order phase transition via growing bubbles (circles), and of the sphaleron explosion-like events that occur on the surfaces of the bubbles. The discs represent the magnetic fields from the explosion event, with earlier explosions having had more time to grow out further.

We shall model the magnetic field in a sphaleron explosion as

acosθ Br = (a2 + r2)3/2 asinθ 2a2 r2 Bθ = − (3.12) − 2 (a2 + r2)5/2 r r/a Bφ = e sinθ a3 − where (r,θ,φ) are spherical coordinates centered at the location of the sphaleron and the z-axis is chosen so that there is azimuthal symmetry. (Plots of two sections of the magnetic field are shown in

Fig. 3.2.) The r,θ components of the magnetic field are chosen to be approximately those of a current- carrying circular loop of wire of radius a, and with current proportional to 1/a. (The approximation in Eqs. (3.12) for the field of a circular loop of wire is taken from Sec. 5.5 of Ref. [63].) The azimuthal component of the magnetic field is chosen so that it is localized in a region of size a. The size of ∼ the whole system is chosen to grow with time to model the exploding sphaleron. Assuming that the magnetic fields generated by the sphaleron expand out at the speed of light, we can take a(t)= t t0 − where t0 is the epoch at which the sphaleron transition occurred. The scalings with a are important since the magnetic helicity in the aftermath of a sphaleron decay stays constant, as we now show explicitly. We first construct the gauge potential for the azimuthal

44 0.6 2

0.4 1 0.2 y z 0.0 0

-0.2 -1 -0.4

-0.6 -2 -2 -1 0 1 2 -2 -1 0 1 2 x x

Figure 3.2: Plana plots of magnetic field Plots of the magnetic field vectors in the xz-plane (top) and xy-plane (bottom) with a = 1 in Eq. (3.12). component of the magnetic field

r/a 2 (azim) e− r r Aθ = 2 + 2 + sinθ (3.13) − r a a

In other words, (azim) ∇ (Aθ θˆ)= Bφ φˆ (3.14) ×

Now we define

B = B(1) + B(2) (3.15) where,

(1) B = Brrˆ+ Bθ θˆ (3.16)

(2) B = Bφ φˆ (3.17) and, correspondingly,

A = A(1) + A(2) (3.18) so that B(i) = ∇ A(i). Then since the helicities in B(1) and B(2) are individually zero, the total helicity ×

45 is given by the cross terms

H = d3xA(1) B(2) + d3xA(2) B(1) (3.19)

An integration by parts is now used to obtain

H = 2 d3xA(2) B(1) (3.20)

The boundary term vanishes because the fields are localized. Next we insert the expressions for the gauge potential and magnetic field as given in Eqs. (3.12),

(3.13). A simple change of variables, u = r/a, shows that the magnetic helicity of one source is

8π H = 0.57 = 4.8 (3.21) 3 for any value of a. Therefore the magnetic helicity is conserved for our choice of magnetic fields and matches the conservation seen in sphaleron decay [35]. The choice of factors of a in Eq. (3.12) also ensures that the relative energy in each of the three components of the magnetic field stays fixed, while the net energy in the magnetic field decays as 1/a. The decay of the magnetic energy is a necessary consequence of the conservation of helicity because the energy density is B2 while the helicity density is A B and hence, simply by counting dimensions, the total energy is the total helicity divided by a length scale. Since the only length scale in the problem is a and the total helicity remains constant, the total energy must decay as 1/a. The magnetic field in Eq. (3.12) is axially symmetric about the chosen z-axis. However, different sphalerons will produce magnetic fields that are azimuthally symmetric with respect to different axes.

So a sphaleron is described by its location as well as its orientation. We will assume that the orientations of the sphalerons are isotropically distributed and, for a given sphaleron, choose the orientation from a uniform distribution on the two-sphere i.e. in spherical coordinates, cosθ and φ are chosen from a uniform distribution over the intervals ( 1,1) and (0,2π) respectively. In principle, the interaction of −

46 electroweak fields with the bubble wall could result in a preferential orientation of the sphaleron (e.g. normal to the bubble wall) but we shall disregard this possibility in the present paper.

We do not claim that an electroweak sphaleron produces the magnetic field in Eq. (3.12) when it decays. Instead, (3.12) is a convenient choice for the magnetic field and has the following desirable properties: (i) The field is smooth and divergenceless. (ii) The magnetic helicity is independent of a.

(iii) The relative energy in all three components of the magnetic field is independent of a. Furthermore, we are only interested in the correlation functions for the magnetic field at large separations and hope that these are not sensitive to the exact form of the model we choose for the sphaleron’s magnetic field. To put this work on a firmer footing, it will be necessary to study the process of sphaleron decay more carefully and to devise a more accurate model for the magnetic fields produced.

3.3 Monte Carlo Simulation

The procedure we follow is to throw bubble sites randomly with uniform distribution within our sim- ulation volume. Successful bubble sites are those that lie outside of all existing bubbles. Then the bubbles grow at speed vb. The bubble growth velocity depends on the ambient plasma and can be much smaller than the speed of light (e.g. [31], [12], [48]). However, to keep the number of parameters to a minimum in our simulations, we took vb = c = 1. As the bubbles grow, we randomly nucleate sphalerons on the bubble surfaces with angular density 1/As where As denotes the average area per sphaleron. Here also we take care to eliminate sphaleron sites that lie within pre-existing bubbles. Each sphaleron is described by its time of nucleation, location, as well as its orientation. The sphaleron nucleation time enters the factor a in Eq. (3.12), while the spatial location sets the origin for the axes, and the orientation of the sphaleron fixes the direction of the local z-axis. With time, the radius of the magnetic field grows with velocity vm which, for convenience, is also taken to be the speed of light, vm = c = 1. After the phase transition is complete – no more bubbles nucleate since almost all the simulation volume is occupied by pre-existing bubbles – we find the magnetic field on a lattice within our simulation volume due to all sphalerons in our simulation box. We then compute the correlation

47 function, Ci j(rrˆ) (Eq. (3.3), which has 3 3 3 components because of the 2 free indices i, j and the × × 3 choices for the direction ofr ˆ. The simulation is run many times with different seeds for the random number generator and ensemble averages are calculated. We have explicitly checked that the correla- tors are of the form in Eq. (3.3). Then the spectral functions are found as linear combinations of the

Ci j(rrˆ),

1 ML(r) = ∑rˆirˆjCi j(rrˆ) (3.22) 3 rˆ 1 MN(r) = ∑Pi jCi j(rrˆ) (3.23) 6 rˆ 1 MH(r) = ∑εi jkrˆkCi j(rrˆ) (3.24) 6 rˆ where the sum is over the 3 directions:r ˆ (xˆ,yˆ,zˆ). ∈ The simulation takes care that bubbles can only nucleate in the false vacuum region, that is, outside every other bubble. In practice, a certain number, nb, of bubbles are thrown down at every time step, and any bubbles that lie within pre-existing bubbles are rejected. Next we want to locate sphaleron events on the surfaces of existing bubbles. We let As denote the mean area occupied by a sphaleron on 2 the surface of the bubble. On the surface of a bubble of radius R, we throw 4πR /As sphaleron sites 5 where we take As = 32 so that there are a large number of sphalerons (order 10 ) in the simulation but still within computational limits. We reject those sites that lie within any other bubble. For large bubbles, the mean distance between neighboring sphalerons on the same bubble is 2 As/(4π) 3. ≈ ≈ The bubble nucleation rate is chosen over a range that gives 102 bubbles in the simulation. The ∼ sphaleron nucleation rate is kept fixed, while the bubble nucleation rate is taken to be 30, 40 and 50 bubbles per time step. The number of bubbles nucleated per time step for the three different nucleation rates are shown in Fig. 3.3. The error bars denote 1σ fluctuations about the mean taken over 20 runs. In Fig. 3.4 we plot the number of sphalerons nucleated per time step, also including 1σ error bars.

This plot is equivalent to plotting the surface area separating the true vacuum and the false vacuum as a function of time.

A subtle point about the simulation is that we have nucleated bubbles within a box but then the

48 50

40

30

20

10 Number of bubbles nucleated

0 0 5 10 15 20 25 t

Figure 3.3: Plot of bubble nucleation number versus time The number of bubbles nucleated at time t for 3 different nucleation rates. The nucleation rates correspond to throwing down 30 (black solid curve), 40 (dashed red) and 50 (dotted blue) bubble sites per time step. The bubbles rapidly fill the simulation box and even with the lowest nucleation rate the phase transition is essentially complete by t = 25 i.e. 25 time steps.

8000

6000

4000

2000 Number of sphaleron explosions 0 0 5 10 15 20 25 t

Figure 3.4: Plot of sphaleron nucleation versus time The number of sphalerons nucleated at a time t for the three different bubble nucleation rates as in Fig. 3.3.

49 bubbles subsequently grow beyond the box. The sphalerons are, however, nucleated only within the box at a fixed rate, rejecting those sphalerons that lie on the parts of the bubbles that are outside the box. So the magnetic field close to the boundaries of the box suffer from boundary artifacts due to the lack of sphalerons outside the box. Hence it is important that the magnetic field only be calculated in a sub-box that is smaller than the original box, at least by a margin that is larger than the size of the typical magnetic structure, given by a(t). In our simulations, the box size was 144 lattice spacings and the sub-box size was 108 spacings, i.e. we excluded a boundary layer of 18 lattice spacings all around the box. As can be seen from Fig. 3.4, most sphalerons nucleated at t 15 and had a 10 when ∼ ∼ we stopped the simulation (t = 25). So a for most bubbles is less than the thickness of the excluded boundary layer. The other relevant simulation parameters are the lattice spacing, dx = 1, and the time step, dt = 1. After the phase transition is complete, we know where all the sphalerons are located, their orienta- tions, and also their sizes because we know the times at which the sphalerons exploded. We take the expansion speed of the magnetic fields produced by a sphaleron to be the speed of light. This deter- mines the size, a(t), occurring in Eq. (3.12) for every sphaleron. Then at every point on a sub-lattice we sum over the magnetic field due to every sphaleron. (This is the computationally expensive part of the code since it involves roughly 105 (108)3 1011 computations.) Once we know the magnetic × ∼ field at each lattice site, we calculate spatial correlations by doing the volume integral in Eq. (3.3) and averaging over 20 ensembles. Projections of the correlation functions as in Eq. (3.24) immediately give the normal, longitudinal and helical power spectra. The integrals in Eq. (3.9), (3.11) finally lead to the

Fourier space power spectra, S(k) and A(k). Before proceeding to the numerical details and results, we summarize the electroweak and cosmo- logical parameters. The Hubble distance at the electroweak epoch is H 1 10 cm, while the ther- ew− ≈ mal length scale is T 1 10 16 cm. The sphalerons are exploding on the inverse electroweak mass ew− ≈ − length scale which is comparable to the thermal length scale. The ejected magnetic fields produced can spread out freely until the MHD frozen-in length scale at the electroweak epoch, lfrozen tew/4πσc ≈ ≈ 8 2 10 cm, where σc Tew/e is the electrical conductivity of the plasma. The present baryon number − ∼

50 7 3 37 3 density is n 10 cm and at the electroweak epoch this corresponds to nb ew 10 cm . b ∼ − − , ∼ −

3.4 Results

In Fig. 3.4 we have already shown the nucleation of sphalerons as a function of time for each of the different bubble nucleation rates. (We do not vary the sphaleron nucleation rate.) The number of sphalerons nucleated at any time is proportional to the net surface area of the bubbles. As expected, the plot shows that the surface area, and hence the sphaleron nucleation rate, grows to a maximum and then decreases. If we increase the bubble nucleation rate, the rate of growth of the surface area is larger initially, but then the turning point is at earlier times because it is determined by the merging of bubbles. The plot of the sphaleron rate versus time in Fig. 3.4 also tells us the size distribution of the expand- ing magnetic field distribution. In particular, lower bubble nucleation rates, as are relevant in strongly first order phase transitions, lead to a sphaleron rate that is larger at later times. (The peak in Fig. 3.4 is shifted to the right.) So at some fixed late time, the sphaleron explosions have had less time to grow and there are a larger number of smaller sphaleron explosion remnants. The correlation functions should therefore be larger at small distances when the bubble nucleation rate is smaller. This feature can be seen in Figs. 3.5 and 3.6 where we show the spatial correlation functions, ML and MN, versus r. In

Fig. 3.7 we show the helical correlation function, MH(r). The peak shifts to the left for smaller bubble nucleation rates in agreement with our observation above that more sphaleron explosions occur later if the bubble nucleation rate is small and have less time to grow.

The fluctuations in the correlation functions, denoted by the error bars, are quite large. To further reduce them would require increasing the sphaleron rate and would increase the computational time.

At present, each Monte Carlo with 20 runs takes about 10 days to run. The Fourier space correlation functions can be found using Eqs. (3.9), (3.11) and are shown in

Figs. 3.8 and 3.9. The spectra are dominated by peaks at k 0.05. This corresponds to a length scale ∼ l k 1 20. From Fig. 3.4 we see that most sphalerons were nucleated at t 15 and these would ∼ − ∼ ∼

51 0.007

0.006

0.005

0.004

0.003 M_L(r)

0.002

0.001

0 0 50 100 r

Figure 3.5: ML(r) plots versus correlation length

ML(r) for different rate of bubble nucleation. Solid (black), dashed (red), and dotted (blue) curves corre- spond to 30, 40 and 50 bubbles nucleated at every time step. Fluctuations are only shown for the run with 30 bubbles nucleated per time step.

0.006

0.004 M_N(r)

0.002

0

0 50 100 r

Figure 3.6: MN(r) plots versus correlation length

MN(r). Plots are made following the scheme of Fig. 3.5.

52 0.002

0.0015

0.001 M_H(r)

0.0005

0 0 50 100 r

Figure 3.7: MH (r) plots versus correlation length

MH (r). Plots are made following the scheme of Fig. 3.5. primarily be on bubbles that are also of size 15, since most bubbles are nucleated at the earliest ∼ times. Hence the peak of the correlation is given by the sizes of the bubbles when bubbles started to percolate. The width of the peaks in Figs. 3.8 and 3.9 are ∆k 0.05 and also given by the bubble sizes ∼ at percolation. Note that it would not be suitable to characterize the injected spectra by power law fits.

3.5 Conclusions and Future Directions

Following the general scenarios discussed in Refs. [36, 121] we have proposed a concrete model for the generation of helical magnetic fields during baryogenesis at a phase transition. The model takes into account magnetic field generation due to baryon number violating processes occurring on bubble walls. By Monte Carlo simulations, we have evaluated correlation functions of the injected magnetic field on completion of the phase transition. The Fourier space power spectra shown in Figs. 3.8 and 3.9 tell us the characteristics of the magnetic fields injected into the plasma in this model. Our results should be viewed as providing initial conditions for subsequent evolution which will also entail MHD and cosmological effects. If the cosmological fluid is turbulent, say due to the motion of bubble walls, that too will play a role. These effects did not enter our study because the magnetic

fields that are produced due to sphaleron events are on scales comparable to the inverse W-boson

53 1000

800

600 S(k) 400

200

0 0 0.1 0.2 0.3 k

Figure 3.8: S(k) plots versus k S(k) vs. k. Plots are made following the scheme of Fig. 3.5.

500

400

300 A(k) 200

100

0 0 0.1 0.2 0.3 k

Figure 3.9: A(k) plots versus k Plot of A(k), analogous to Fig. 3.8.

54 mass and far smaller than the scale at which the medium can be treated like a fluid. The microscopic production, however, occurs at a high rate, since it is also the rate at which baryons are produced, and the magnetic field due to different sphalerons will subsequently spread, merge and permeate space. MHD effects will come into play on length scales that are large compared to the thermal scale. As the magnetic field expands to larger scales, but still less than the frozen-in scale, MHD effects will change the linear expansion to diffusive expansion. On yet larger scales, the magnetic field expansion enters the frozen-in regime where it can only scale with the expansion of the universe. In addition to these considerations, the evolution needs to include the helicity of the magnetic field and any turbulence that may accompany the phase transition. It is known that helicity can be responsible for an “inverse cascade” that transfers power to larger scales. A discussion of some of these issues in the present context may be found in Ref. [121]. A long term goal of our model for generation of magnetic fields, is to connect the particle physics processes during baryogenesis (such as the phase transition) to characteristics of the magnetic field. The hope is that eventually the observation of a primordial magnetic field may say something about particle physics at the baryogenesis scale, the nano-second universe, and perhaps also the observed astrophysical magnetic structures.

55 Chapter 4

Probing higher energy effects via electromagnetism

4.1 Introduction

The coupled Maxwell-Einstein system has been extensively studied. Beyond the classical level, non- minimal coupling of the electromagnetic and gravitational interactions may be described by an effective Lagrangian built perturbatively from electromagnetic and geometric tensors. This is not merely an academic pursuit as such terms are known to be generated within quantum electrodynamics (QED) by the exchange of virtual charged fermions in a curved space-time.

Specifically, Deser and van Nieuwenhuizen [41] have attempted to enumerate all possible non- minimal mass dimension 6 actions coupling the Maxwell tensor to the Riemann and Ricci tensors, but excluded parity violating ones because QED is a parity conserving theory. Following that, Berends and Gastmans [9] computed to one loop order the photon-photon-graviton 3-point correlation function,

T Aµ Aν hαβ , for QED in a generic weakly curved spacetime; and later, Drummond and Hathrell { } [44] used their results to do a low energy limit “matching calculation” to determine exactly the coeffi- cients of the terms obtained by Deser and van Nieuwenhuizen.

With the inclusion of the rest of the electroweak model it is conceivable that the weak interactions would induce, starting at two loops, parity violating mass dimension 6 non-minimal terms; although we are not aware of any explicit calculation to determine their exact coefficients. It cannot be a one loop process because the relevant Feynman diagram would need to contain at least one parity violating current involving the W or Z boson.

1 Because the action is dimensionless , any such additional terms will be suppressed by some inverse power of a mass scale, often times associated with the mass of the virtual particle(s) exchanged. The

1In this paper, the units h¯ = c = 1 are employed.

56 parity preserving terms at mass dimension 6 would receive contributions from the standard model O 2 O 2 beginning at (Me− ) and the parity violating ones possibly starting at (MW− ); where Me and MW are the masses of the electron and W boson, respectively. These dimension 6 terms could also receive contributions from particles that have yet to be observed experimentally, if they are massive enough or if their interactions are sufficiently weak. Therefore, one may hope that constraining the coefficients of such non-minimal terms in the action may in effect probe the existence of new physics. Since these are gravitational interactions, two natural probes are the propagation over cosmological distances of the cosmic microwave background (CMB) photons and solar system tests of General Relativity (GR). Through explicit calculations, however, we will show that one would have to look beyond these avenues to obtain a useful bound.

In this paper, we shall relax the parity preserving assumption of Deser and van Nieuwenhuizen and simply ask what the entire range of possible couplings between the Maxwell tensor and its geometric counterparts is, up to mass dimension 6. As we will see, upon field re-definition, there are only two such terms, so that the most general form of electrodynamics in curved space-time is now given by the action

1 2 4 1 4 µν S = M d x g (R 2Λcc) d x g F Fµν −2 pl | | − − 4 | | 4 1 µναβ 1 µναβ + d x g Fµν Fαβ R + Fµν Fαβ R (4.1) | | Λ 2 2Λ 2 1 2 2 1 µ where M (8πG) , Fµν = ∇ µ Aν = ∇µ Aν ∇ν Aµ is the Maxwell field tensor, [∇α ,∇β ]V = pl ≡ − [ ] − µ λ 1 αβ R V defines the Riemann tensor, and the dual Maxwell tensor is defined as Fµν εµναβ F . λαβ ≡ 2 The 1/Λ 2 term has a well-known contribution from QED. While these effects are not new, it is in- 1 teresting to point out that there is really only one non-trivial term. We can compare this to the work of Drummond and Hathrell [44] wherein their coefficient c/M 2 = α /(360πM 2), where α is the e − EM e EM Λ 2 Λ 2 fine structure constant, is identified with our 1/ 1 . As already mentioned, the 1/ 2 term may be generated by the weak interactions within the standard model.

57 It is important to note that the mass dimension 6 actions in (4.1) should be viewed as the first terms in an infinite series expansion involving the ratios of the microscopic lengths 1/Λ1 and 1/Λ2 to either the wavelength of the photons or that of the characteristic length scale of the gravitation field. That is, the second line of (4.1) is written down with an implicit assumption that the typical energy scale of the photons described by such a theory has to be significantly lower than Λ1 and Λ2. (One may see this more explicitly by referring to, for instance, the matrix element in equation (2.8) of Drummond and Hathrell [44]. The mass dimension 6 contributions to the energy-momentum tensor θ µν – i.e. the terms

2 2 2 2 containing g1,g2 and g3 – are O(p /Λ ) or O(q /Λ ) relative to the lowest order mass dimension 4 µναβ Maxwell contribution V0 . Here p and q are the momenta of the gravitational field and photon respectively and Λ2 M2/α .) Just as real electron-positron pairs could be produced if the energy of ∼ e EM the photons were of O[few MeV], one would have direct access to the (hypothetical) new physics if the energy scale of the photons could reach Λ1 or Λ2 and the effective theory in (4.1) would then no longer be adequate. At the very least, one would have to include actions up to much higher mass dimensions. As we will see, cosmological and solar system observations are just not sensitive enough to constrain a Λ of the same order of magnitude as the photon energies involved ( 3000K for CMB photons and ∼ 10 GHz for solar system tests of GR). ∼ In section 4.2 we list the basic tools needed to construct all the possible non-minimal terms. Section

4.3 includes their enumeration from mass dimension two to six, as well as an explanation of why, via a re-definition of the gauge potential Aµ and the metric tensor gµν , many of these terms are in fact redundant as far as the Maxwell-Einstein system is concerned. In section 4.4 we calculate, for cosmological propagation of CMB light and solar system tests of GR, how accurate observations need to be in order to place useful bounds on Λ1 and Λ2; they appear impossible to be achieved in the foreseeable future. We conclude and discuss directions for further work in section 4.5.

58 4.2 Basic Tools

The most general Lagrangian of gravity and electromagnetism can be written as a sum of the Einstein- L L Hilbert with a cosmological constant ( EH,Λcc ) and the electromagnetic action ( EM) as defined above, and a perturbative expansion in the mass dimension of the Lagrangian density

4 1 1 d x g LEH,Λ + LEM + L5 + L6 + ... | | cc M M2 ⋆ ⋆ with M⋆ representing the lowest of the (possibly many) mass scales that are physically relevant. In order to systematically enumerate all the possible non-minimal terms coupling gravity to electro- magnetism, we start by listing the most rudimentary tensors available, before forming the general set of scalars out of them. The requirements of U(1) gauge invariance and general coordinate covariance lead us to the dimensionful tensors, the covariant derivative and field strengths

∇µ , Fµν , Rµναβ (4.2)

2 with mass dimension 1, 2, and 2 respectively. The primitive dimensionless geometric objects are

µν µναβ 1 µναβ g , ε ε (4.3) ≡ g | | where ε µναβ is the fully anti-symmetric Levi-Civita symbol, and we define ε0123 1. Scalars built ≡− from the covariant Levi-Civita tensor will violate parity since it transforms as a pseudo-tensor (even under parity). As far as the complete enumeration of the primitive tensors is concerned, the placement of their indices (upper or lower) is immaterial because we will be forming scalars out of them anyway.

The basic strategy for constructing the most general set of scalars is, then, to consider all possible contractions between appropriate products of (4.2) and (4.3). We need not consider derivatives on the

2Strictly speaking, the physical dimensions of different components of a given tensor are the same when computed in an orthonormal frame, where the metric, in particular, is then gµν ηµν diag[1, 1, 1, 1]. However, since we are forming scalars out of these tensors – the result is independent of whether→ the≡ coordinate− or− orthonormal− frame was chosen – we are delineating this construction within a coordinate frame, where calculations are easier.

59 Mass Possible Combinations of (a,b,c) in (4.5) Dimension 2 (0,1,0) 4 (0,1,1), (0,2,0), (2,1,0) (0,1,2), (0,2,1), (0,3,0), (2,1,1), (2,2,0), 6 (4,1,0)

Table 4.1: List of tensors for different dimensions The possible combination of tensors at various dimensions.

µναβ ε-tensor as ∇τ ε = 0. Furthermore, because

ε µναβ ε δ µ δ ν δ α δ β ρτσλ = [ρ τ σ λ] (4.4)

we see that it suffices to consider terms that contain zero or one ε-tensor only.

4.3 Enumeration of terms

Schematically, we seek to form combinations of the type

∇aFbRc (4.5)

2 As Fµν and Rµναβ are both of dimension [M] , we look for terms satisfying a + 2b + 2c 6. As ≤ all tensors except ∇µ have an even number of indices, in order to form a scalar we see that a must be an even number, therefore no odd mass dimension terms exist. Since we are considering couplings between both the electromagnetic and gravitational field, we require b > 0. We have summarized these possibilities in Table 4.1. As we enumerate the possible terms, we will not consider trivial numerical factors as these would be absorbed into coefficients of the Lagrangian anyway. The following Bianchi identities are quite useful

Rµ[ναβ] = 0 (4.6) ∇ [ν Rλσ]ρτ = 0 (4.7)

60 Using (4.6) it is not hard to show that

µνρτ µνρτ ε Rµνλσ = 2ε Rµλνσ (4.8)

The anti-symmetric nature of F µν implies

µν λν µν ∇µ ∇ν F = Rλν F = g Fµν = 0 (4.9)

In 4 dimensional spacetimes, the antisymmetrization of more than 4 indices always yields zero. One useful corollary [54] is α β µνρτ g [ ε ] = 0 (4.10)

4.3.1 Dimension: 2 and 4

µν At dimension 2, the only scalar that can be formed here is Fµν g , which is zero. Next we consider the dimension 4 terms Fµν Rαβγδ , Fαβ Fµν , and ∇µ ∇ν Fαβ , which we call type I, II and III, respectively. In addition to contractions with the metric, we must separately consider both contractions without and with the Levi-Civita tensor.

4.3.1.1 Without Levi-Civita Contractions

µν Type I: The only scalar is F Rµν = 0. Type II: The scalar this forms is the one found in the canonical Maxwell Lagrangian

µν Fµν F

Type III: This is not only a total derivative, which does not contribute to the dynamics, it is also identically zero from (4.9).

61 4.3.1.2 With Levi-Civita Contractions

Type I: Because of (4.6), at most 2 indices on the Riemann tensor may contract with the Levi-Civita, µναβ leaving only the possibility of Fµν Rαβ ε , which is zero due to the symmetry of the Ricci tensor. µναβ Type II: The only scalar that can be formed here is Fµν Fαβ ε . Using identity (??), we see this is a total derivative: ∇ ε µναβ αβ µ Aν Faβ = F Fαβ (4.11) Type III: Zero, by the Bianchi identity (??).

4.3.2 Dimension: 6

Here we consider the terms (A) Fµν Rαβγδ Rρτσλ , (B) Fµν Fρτ Rαβγδ , (C) Fµν Fαβ Fρτ , individual terms from (D) ∇ρ ∇τ Fµν Rαβγδ and (E) ∇µ ∇ν Fαβ Fρτ , and (F) ∇µ ∇ν ∇ρ ∇τ Fαβ . We again separately con- sider contractions without and with the Levi-Civita tensor. As illustrated in the previous section, there are many tricks that simplify such constructions greatly. While we will not discuss all of them, we have tried to give the most salient examples. We have therefore only listed the non-zero, non-redundant terms below.

4.3.2.1 Without Levi-Civita Contractions

Type A: All of these are zero because while Fµν is antisymmetric, all two-index tensors built from contracting two Riemann tensors are symmetric in the indices. As an example, using (4.6) it is possible to show that ρβτ 1 ρβτ Rµ Rνβρτ = Rµ Rνρβτ (4.12) 2 ρβτ where Rµ Rνρβτ is (µ ν) symmetric. ↔ Type B: The non-redundant terms are Type C: All of these are zero due to symmetry considerations.

62 µ ν Type D: These terms are either equivalent to type A (which are zero) or are of the type ∇ FµνV (for some appropriate V ν ), which can all be made to vanish at this order in mass dimensions by a suitable field re-definition, Aν Aν +δAν , where δAν ∝V ν . While it is true that this field re-definition → will induce a variation of all the mass dimension 6 terms, the new terms appearing are of even higher order, and so are irrelevant for this analysis.

Type E: As in [41], one can show using (??) that, up to a total derivative, there is an equivalence between the term α βγ ∇α Fβγ ∇ F and α βγ α γλ αβ λγ 2 ∇α F γ ∇β F + 2F γ F Rλα + F F Rαβγλ Therefore, the non-zero terms here are equivalent to type B and D. Type F: This is a total derivative, and does not contribute to the dynamics.

4.3.2.2 With Levi-Civita Contractions

Type A: Zero, for the same reasons as above. To show this rigorously, (4.10) may be useful.

Type B: As an example, using (4.10) it is possible to show In all, the non-redundant terms here are

µν Fµν F R

σ µν Fµ F Rνσ αβ ρσ F F Rαβρσ (4.13)

Type C: Zero or redundant, as above.

Type D: Zero, as above. µναβ σ Type E: These are either redundant or zero, as above. Consider the term ε ∇µ Fνσ ∇α F β . After

63 integration by parts, we find it proportional to

µναβ σ ∇α ε ∇µ Fνσ F β

1 µναβ σ λ λ = ε F (Rν αµ Fλσ + Rσ αµ Fνλ ) (4.14) 2 β Type F: This is a total derivative, and does not contribute to the dynamics.

At this stage, there are six possible non-minimal coupling terms; three (??) of which are consistent with those listed by Drummond and Hathrell [44], while the other three (4.13) are new, and are their parity-violating counterparts.

4.3.3 Absorption of Terms via Metric Re-definition

Up to this point, the general action is of the form

4 µν S = SEH Λ + SEM + d x g A Rµν + BR + ... (4.15) , cc | | where (...) indicates terms that contain neither the Ricci tensor nor scalar (but may contain the Rie- mann tensor). We will now demonstrate that Aµν and B can be eliminated through a suitable change of variables, i.e. a re-definition of the metric. We make the transformation, gαβ gαβ + δgαβ and → choose αβ 2 αβ αβ δg = λg + f (4.16) −M2 pl where λ and f αβ obey the relations

f fαβ = Aαβ , λ + = B (4.17) − 2

64 where f is the trace of fαβ . The overall modification to the action (4.15) is then

δ S = SEH,Λcc + SEM + S + ... (4.18)

4 µν δS d x g Λccg Aµν + Bgµν (4.19) ≡ | | where the (...) represent the same omitted terms. While it is true that the change of variables will affect all terms, it may be checked that the variation to the non-minimal terms occurs at least at mass dimension 8, and so will not have an impact on this analysis. µν Discarding the total derivative Fµν F , we see that

βΛ 1 4 cc µν δS d x g Fµν F (4.20) →−4 | | M2 ⋆ for an appropriate mass scale, M⋆ and constant, β. This re-definition could pose an issue if it were to flip 2 the overall sign of the Maxwell action, but will not be a problemaslongas M Λcc. This modification ⋆ ≫ to the canonical Maxwell Lagrangian can be absorbed by simply rescaling the electromagnetic potential as 1 βΛcc − 2 Aµ Aµ 1 + → M2 ⋆ Upon these field re-definitions, we obtain the advertised result (4.1).

4.4 Observables and Constraints

Varying the action (4.1) with respect to Aµ , the modified Maxwell equations are

µ 1 µ αβ ∇ Fµν + ∇ α Rβ ν + Rµναβ ∇ F Λ2 [ ] 1 1 ρσµ αβ ρσ + Rρσαβ ε ν ∇µ F + 2F ∇ ρ Rσ ν Λ2 [ ] 4 2 µ ρσ +2R νρσ ∇µ F = 0 (4.21)

65 Within the cosmological context, we will derive the general solutions to (4.21) in a spatially flat Friedmann-Robertson-Walker (FRW) metric and examine their implications for the propagation of the cosmic microwave background. We will then see, as also discussed by [96], although one might have hoped that the large distances involved would help accumulate effects from these non-minimal terms and render them discernible, the Hubble parameter of our universe is simply too small for cosmology to be a sensitive probe. We shall also show below that these non-minimal terms do modify the geodesics followed by pho- tons in a curved background, defining an effective metric, so that the travel time and deflection angles of light about massive objects will be altered from their standard values. Observations of the Shapiro delay of radio signals from the satellite Cassini currently provide the tightest bound on the PPN param- γ Λ eter PPN. Even though these Cassini observations yield tighter restrictions on 1,2 than the cosmological ones, they still lie significantly below the threshold necessary to provide physically useful constraints.

4.4.1 Cosmological Constraints

We start first with cosmological probes and work with a spatially flat FRW universe, where gµν =

2 a(η) ηµν . We then proceed to solve, in the Coulomb gauge (A0 = 0), the general solutions of the vector potential Aµ to the vacuum modified wave equations, using the JWKB approximation. To this end, if k is the spatial momentum vector of the photon, it helps to expand the spatial portion of Aµ , A, in terms of basis vectors where one of them is parallel to k and the other two correspond to left- and right-circular polarizations. That is, if we first consider an orthonormal basis defined by unit vectors,

k eˆI,eˆII, k , then define { | | }

ik x A(η,x) (A+ (η)eˆ+ + A (η)eˆ )e− (4.22) ≡ − − where

1 eˆ (eˆI ieˆII) (4.23) ± ≡ √2 ±

66 the resultant equation of motion is (from (4.21))

2 A′′ (1 + ψ)+ A′ ψ′ + k A (1 + χ) 2φ ′kA = 0 ± ± ± ± ± (4.24)

Here k k , the prime denotes derivatives with respect to conformal time, η, and we have used the ≡| | following definitions

2 a a 2 ψ(η) ′′ ′ (4.25) ≡−Λ2 a3 − a2 1 2 a 2 χ(η) ′ (4.26) ≡−Λ2 a2 1 1 1 a φ(η) ′′ (4.27) Λ2 3 ≡ 2 2 a

Following [25], we now attempt a JWKB solution by first requiring the solutions take the form

η A (η)= A exp i dη′ f (η′) (4.28) ± ± η ± 0

We next assume that time derivatives of A are negligible, f f 2, and proceed to insert (4.28) into ′ ≪ (4.24). As they are small for the cosmological eras of interest, we expand f to linear order in φ, χ and ψ to find (choosing a positive root)

ψ′ 1 f k + i + k(χ ψ) φ ′ (4.29) ≈ 2 2 − ± so that

1 η A (η) A exp ik ∆η + dη′ (χ ψ) (4.30) ± ≈ 2 η − 0 ∆ψ i∆φ − 2 ±

67 ∆η Λ 2 The ik is just the usual plane wave term. The parity-conserving 1/ 1 term contributes to the phase a real part, the integral of χ ψ, and a dissipative imaginary part, i∆ψ/2. The birefringent ∆φ − ± Λ 2 arises from the parity-violating 1/ 2 term. We may now write 1 A ∝ exp iθ ∆ψ , (4.31) ± ± − 2 with the same proportionality holding for the two circular polarizations of the electric field, E . Hence, ± the energy density of electromagnetic waves propagating through the universe will be suppressed by ∆ψ a factor e− . The QED contribution can be obtained by borrowing Drummond and Hathrell’s result [44], that tells us that 1/Λ 2 10 3α /M 2, which leads us to find an extremely small damping of 1 ≈− − EM e roughly exp[ 10 73], if QED is the most dominant contribution. − − For light coming from a linearly polarized source, over the course of its propagation the plane

1 of polarization rotates by an angle, ∆α = (θ+ θ )= ∆φ . Since (in observer time) we have | | 2 − − | | 2 2 3 φ(z) (H(z)/Λ2) and during the matter-dominated era, H ∝ (z+1) , bounding the observed rotation ∼ angle restricts the mass scale as 1 3 2 Λ (z + 1) 2 & H0 ∆α , (4.32) | | 33 where H0 2 10 eV. ≈ × − While there are astrophysical (0 < z < 4) sources of polarized radiation, such as radio galaxies (see

[1] for example) which provide O(1◦) limits on polarization rotation, the CMB turns out to give the best constraint because of its large redshift (z 1100). By rotating the plane of linear polarization, ≈ birefringence mixes the E and B polarization modes of CMB. Specifically, this induces a non-zero TB TE ∆α cross correlation between the temperature anisotropy and B modes, given by Cℓ = Cℓ sin(2 ) [22].

WMAP [70] has put O(1◦) limits on the (isotropic) rotation angle of linear polarization of the CMB. By 33 applying (4.32), we obtain a naive constraint of Λ2 & 10− MeV. However, one must remember that the energy of the CMB photons themselves are of O(3 3000)K (or, O(3 10 10 3 10 7) MeV), − × − − × − at least 23 orders of magnitude greater than this lower bound. To obtain a physically meaningful bound, one ought to ask instead, how accurate does ∆α need to be determined for a Λ2 of at least the same

68 energy scale as that of the photons? Setting Λ2 & 3000K, and inverting the inequality (4.32), the answer 55 is ∆α . O(10− ). It should be safe to assume this is out of observational reach. Although this effect is miniscule, it is interesting to find a possible standard model source of cosmo- logical birefringence that does not invoke any new degrees of freedom or extra dynamics (e.g. [91, 26]).

4.4.2 Solar System Constraints

Next, we would like to examine how our mass dimension 6 terms modify standard GR predictions, so that we may use observations to constrain their coefficients. In the following, we will employ the JWKB approximation to work out the photon’s modified dispersion relations, due to the addition of the non-minimal terms to Maxwell’s equations. We then extract the effective metric experienced by these photons, and compute the induced corrections to both deflection angle and the Shapiro delay of light propagating past a massive body.

Let us first consider the modified Maxwell’s equations with an example background geometry given by the Schwarzschild metric:

dτ2 = B(r)dt2 A(r)dr2 r2dθ 2 r2 sinθ 2dφ 2 (4.33) − − −

1 rs where B(r)= U(r), A(r)= U (r), with U(r)= 1 , rs = 2GM, and M is the mass of the object. − − r Note that we have neglected the cosmological constant, Λcc. Since the wavelength of the light considered here is much smaller than the background metric’s ψ radius of curvature, we will use the JWKB ansatz, Aµ = Re(aµ ei ), in which the amplitude aµ is slowly varying while the phase eiψ varies rapidly. Under these consideration, the modified Maxwell

69 equations (4.21) become

µ ν 2 ν µ α 0 = kµ k δβ R µαβ k k − Λ2 1 1 ρσ µν α ν ρσα µ β + Rρσαβ ε kµ k R µρσ ε k kα a Λ2 β 2 2 − ν β N β a (4.34) ≡

ν For the system of equations to have non-trivial solutions, we require detN β = 0. The eigenvalues of N give us the photon’s dispersion relations, and the corresponding null eigenvectors are the polarization vectors. To simplify the algebra, it is helpful to rewrite our equations in an orthonormal basis using the

b vierbeins e β , defined as a b gµν = e µ e ν ηab (4.35) so that

ν kb = kν eb (4.36)

c c ν β N b = e ν N β eb (4.37)

Our conventions here reserve the greek indices for the coordinate frame (t,r,θ,φ) and latin for the orthonormal frame (t,r,θ,φ). In matrix form, the vierbeins are

√U 0 0 0   0 1 √U 0 0 b / e µ =   (4.38)    0 0 r 0       0 0 0 r sinθ      µ b µ b δ µ a µ δ a and eb is just the inverse of e µ , i.e. eb e ν = ν , and e µ eb = b. Due to the spherical symmetry of the Schwarzschild metric, we can consider, without loss of generality, the light propagation to lie

70 θ π in the = 2 plane, i.e. kθ = 0. From here on we also choose, for simplicity, to assume that the two hypothetical energy scales are the same, namely, Λ1 = Λ2 Λ. Under these considerations, we find ≡

Σ + ∆11 ∆12 ∆13 ∆14 − − −   ∆12 Σ ∆22 ∆23 ∆24 b − N c =   (4.39)  ∆ ∆ Σ ∆   13 23 + 33 0     ∆ ∆ Σ ∆   14 24 0 + 44      where

2 2 2 2 (2kr kφ )rs (2kt + kφ )rs ∆ = − ∆ = 11 r3Λ2 22 r3Λ2 2 2 2 2 2 (kt kr + 2kφ )rs (k k )rs ∆ = − ∆ = t − r 33 r3Λ2 44 r3Λ2 2ktkrrs 3krkφ rs ∆12 = ∆13 = − r3Λ2 r3Λ2 ktkφ rs 3ktkφ rs ∆ = ∆ = 14 r3Λ2 23 r3Λ2 krkφ rs ∆ = (4.40) 24 r3Λ2

2 2 2 and Σ = k k kφ . t − r − b 2 2 2 Two of the eigenvalues of N are k k kφ = 0; this is the canonical light-cone dispersion rela- c t − r − tion. However, their corresponding polarization vectors are pure-gauge modes, and thus non-physical.

2 The other two, to O(Λ− ), are

3√2r k2 k2 k2 1 s 0 (4.41) t r φ Λ2 3 = − − ± r

At this order, if Λ1 and Λ2 had been kept distinct, the non-minimal modifications to the dispersion relations add in quadrature and are thus symmetric under the interchange of Λ1 and Λ2. Therefore, any physical effects derived from these relations will not distinguish between the two; however, this symmetry does not hold for their corresponding polarization vectors.

71 4.4.2.1 Effective metric solution

We wish to analyze the implications of this modified dispersion relation in terms of a modified metric,

ab following Myers and Lefrance [75]. The two dispersion relations (4.41) could be viewed as kakbg = 0 µν or, in a coordinate frame, kµ kν g = 0. This defines the effective metric, gµν , for each of the two dispersion relations, however, only up to an overall conformal factor. We choose to write it as

dτ2 = B(r)dt2 A (r)dr2 r2dφ 2 (4.42) − − where

r B(r) (1 δ) 1 s (4.43) ≡ ± − r r A (r) (1 δ) 1 + s (4.44) ≡ ± r 3√2r δ s (4.45) ≡ Λ2r3

To Ørs, the modification to the dispersion relations amount to a re-scaling of B(r) and A(r) found in (4.33) by a factor of 1 δ, as predicted by the standard result in GR. Notice that we continue to ± θ π work in the = 2 plane without loss of generality. Since there is still no time dependence, the metric also remains static. µ µν In this modified geometry the contravariant wave vector, k = g kν , is the tangent vector to the path normal to the surfaces of constant phase ψ, i.e. kµ = dxµ /ds, where s is an appropriate affine parameter. In order to be convinced that is the right interpretation, all that is needed is to show that the modified dispersion relation implies that xµ (s) is a geodesic of this modified spacetime. We show this, following [84], by first taking a covariant derivative of the dispersion relation, now with respect to the effective metric

µν µ ∇α kµ kν g = 2k ∇α kµ = 0 (4.46)

72 Since kµ = ∇µ ψ = ∂µ ψ, it is straightforward to show that

∇α kµ = ∇µ kα (4.47)

Thus µ µ α k ∇µ kα = 0 = k ∇µ k (4.48) which is none other than the geodesic equation.

Though this effective metric is defined only up to an overall conformal factor, if one were to mul- tiply it by any function of r, the trajectory of the null geodesics remains unaltered. Therefore, the predictions we will quote below for the deflection angle and modified Shapiro delay which are calcu- lated based on (4.42) are unambiguous.

4.4.2.2 Deflection Angle

For a general metric of the form (4.42), the total deflection angle of light passing by a massive object is (see, e.g. [128])

∞ dr A(r) ∆Φ = 2 π (4.49) B(r )r2 r0 r 0 1 − B(r)r2 0 − where r0 is taken to be the point of closest approach of the light from the object. Expanding the integrand in powers of rs/r, the integral above gives us

r 2√2r ∆Φ = 2 s s (4.50) Λ2 3 r0 ± r0

The first factor is just the standard contribution from the Schwarzschild metric, and the second term is the new contribution of the non-minimal terms. An unpolarized light ray traversing close to the massive body would incur a splitting due to the opposite signs arising in the form of (4.50). While there are observational limits on the deflection angle of light passing close to our sun, these limits are

73 not as stringent as those derived in the following section, and we will therefore not purse a constraint on our non-minimal terms here. The gravitational deflection of light computation here suggests that our non-minimal terms would also modify the weak lensing signals currently sought by large scale structure observations.

4.4.2.3 Modified Shapiro Time Delay

We now move on to consider the time-of-flight of a null light ray propagating between two points in space such that it passes close to a massive object. Such a light ray is known to experience a delay in its time of flight, relative to the same flight in Euclidean space. This is commonly referred to as Shapiro delay [106].

For ease of comparison to and discussion in reference to the literature, we will now switch to the isotropic gauge in calculating the modification to this delay. In this coordinate system, the Schwarzschild metric (4.33) is written as

dτ2 = B(r)dt2 A(r) dr2 + r2dθ 2 + r2 sin2 θdφ 2 (4.51) − where

1 rs 2 B(r)= − 4r (4.52) 1 + rs 4r r 4 A(r)= 1 + s (4.53) 4r

b To first order in rs, both the matrix N c and the form of the modified dispersion relations remains unchanged. The expressions that do change are the associated vierbeins and the effective metric. In particular, the latter becomes

dτ2 =(1 δ)B(r)dt2 (1 δ)A(r)dr2 ± − ± A(r) r2dθ 2 + r2 sin2 θdφ 2 (4.54) −

74 where δ was defined in (4.43). By approximating the null path to be a straight line in space, the delay in the round-trip time-of-flight between the two points (P1,2) is

4X2X1 4√2rs ∆t = 2rs log (4.55) b2 ± b2Λ2

If Q is the point on the straight line joining P1 and P2 closest to the massive object, then b is the distance between Q and the object and X1,2 are the distances from Q to P1,2, respectively. We note that b is not equal to the actual distance of closest approach, r0, used in (4.49). A discussion of this and the different ways of calculating ∆t that appear in the literature may be found in appendix (B.1). Within the Parametrized Post-Newtonian (PPN) formulation, the Schwarzschild metric is altered γ to quantify deviations from GR [131]. To date, the most precise measurement of the parameter PPN, which is equal to 1 in GR, comes from the observation of the Shapiro delay from the Cassini spacecraft

[11]. Under this parametrization, and in the idealized limit where the Earth and Cassini are stationary, the Shapiro delay is 4X X ∆t =(1 + γ )r log 2 1 (4.56) PPN s b2 The Cassini experiment measured the fractional Doppler-frequency shift of the radio carrier waves, which in turn is the time derivative of the Shapiro time delay y(t)= d∆t/dt. Since the most rapidly changing length scale in (4.56) is the straight-line closest approach distance, y(t) ∝ db/dt. In order to put a constraint on the energy scale Λ, we set y(t) as determined by (4.55) equal to y(t) as determined by (4.56), thus determining the lower bound on Λ. The actual interpretation and calculation involved for the timing measurement is quite involved and the reader is referred to [86] for further information on the details on the actual treatment. Using the experimental parameter, b 6R (see ≈ ⊙ both [11] and [6]) and the measured value γ = 1+(2.1 2.3) 10 5, we obtain a naive constraint of PPN ± × − 19 Λ & O(10− )MeV, which is 14 orders of magnitude better than the above naive cosmological bound on Λ2 alone. However, just like in the cosmological case, we need to recognize that the energy of the radio waves used in the Cassini observations is roughly 10 GHz 4 10 11 MeV, at least 8 orders of ≈ × − magnitude greater than the lower bound. Once again, we need to ask instead how accurate the timing

75 measurement needs to be to probe Λ1,2 & 10 GHz. Using the second term on the right hand side of 27 (4.55), the answer is ∆t . O(10− )s. This is at least 18 orders of magnitude more precise than the Casinni observation, if one estimates the fractional error for the latter observation to be given by the γ 3 current bound on PPN.

4.5 Summary and Discussion

We have constructed the most general effective Lagrangian coupling electromagnetism and gravity up to mass dimension 6, built from all possible contractions between tensors that obey the underlying gauge symmetries of both theories. There are many such non-minimal terms. However, after allowing for field re-definitions of the electromagnetic vector potential and the metric, it is seen that the number of non-redundant terms reduces to two. One represents the type of coupling already explored from one-loop quantum effects in QED. The other is parity-violating; if it is induced by the standard model, O 2 we expect it to come from the electroweak sector and to be suppressed by (MW− ). We have also discussed some of the phenomenology of these non-minimal terms, including birefrin- gence of the CMB, modified dispersion relations for the photon, as well as corrections to the Shapiro time delay, and deflection angle. Via detailed calculations, we came to see that cosmological and solar system probes do not seem likely, within the foreseeable future, to give any physically useful con- straints on 1/Λ1 and 1/Λ2. This is because, as already alluded to in the introduction, observations are unlikely to ever reach a level of precision to even probe Λ1,2-scales of the same magnitude of the photon energies involved.

We end with some suggestions on possible future work. Other than the weak lensing surveys already mentioned in the body of the paper, Drummond and Hathrell [44] have initiated the investigation of these modified photon dynamics on a gravitational wave background; it would be natural to extend

3One may consider using the timing measurements of pulsars, with masses of order M and radii on the order of several km, as was exploited by the authors of [96]. However, there the measured Shapiro delay (really⊙ r, the range of Shapiro delay) is governed not by the radius of the pulsar itself, as they have claimed, but rather the separation distance between pulsar and companion (see [7] and [109]), which is typically of order R . Considering the relative errors on such observations, solar system observations remain a superior test. ⊙

76 Λ 2 their analysis to include the effects of the parity violating 1/ 2 term. In this paper, we have only examined the dynamics of the photon itself; looking at how Einstein’s field equations are altered and their corresponding implications may provide alternate channels to constrain Λ1 and Λ2. One may 2 also want to seek perturbative solutions for Aµ or Fµν (with 1/Λ = 0) about exact solutions of the 1,2 Λ 2 Einstein-Maxwell system (with 1/ 1,2 = 0) containing pure magnetic fields, as a toy model of more realistic astrophysical systems. Finally, a stability analysis of the full system in (4.1) may also be Λ 2 performed to perhaps help constrain the range of physically reasonable values of 1/ 1,2 .

77 Chapter 5

Conclusion

As noted by its title, this thesis revolved multiple threads around the topic of electromagnetism in the early universe. Here, we will end with several separate comments touching on the further directions such threads may lead in the future.

The toy model of monopole-antimonopole network in Chapter 2 serves to provide a background study for the possible complex networks of monopole-string arising from the Langacker-Pi mecha- nism. It is worthy to point out that other novel alternatives do appear in recent literature (see i.e. [46],[47]). Even though we have stressed the wide resurgence of magnetic monopole studies, either as theoretical constructs in supersymmetry [115], quark confinement problems [80] or useful analogues in experiments of condensed matter systems in the thesis’ introduction, we will end with some numerical

figures reported for the most updated search of the heavy physical magnetic monopole arising gener- ically in Grand Unified Theory (with energy on the scale of 1016GeV) and other symmetry breaking particle physic models. Note that the GUT monopole is hopeless to be directly probed for in terrestrial experiment of any sort, and thus its bounds must be indirect, whereas the lighter monopoles could be bounded by direct accelerator experiments, assuming virtual quantum monopole loops (Reference [83] gives a detailed review on the different physical mechanisms involved for both searches). Combin- ing the figures reported ([49], [50]): collider experiments give an upper monopole cross-section limit

2 3 of O(10− ) to O(1)pb for center of mass energy ranging O(10)-O(10 )GeV in direct search and mass limits of 510-1580GeV for indirect search (with certain model dependence). For cosmology, the rate of flux of magnetic monopoles energy loss in our galaxy can be bounded via the Parker bound, in which the presence of the galactic magnetic field constrains the possible energy loss rate of monopole

78 Figure 5.1: Feynman diagram with virtual monopole loop Feynman diagram for rare virtual monopole(M) loop from virtual photon pairs in p-p ¯ collision . acceleration:

15 2 1 1 17 F < 10− cm− sr− sec− for Mm 10 GeV (5.1) ≤

Another bound can be obtained for GUT monopoles that effect baryon-number-changing processes: by evaluating the expected increase in luminosity of compact astrophysical objects (i.e. neutron stars) due to extra energy released from say proton decay of such process, contrasted with observations,

18 29 2 1 1 27 2 F < (10− 10− )cm− sr− sec− for σ∆ β 10− cm (5.2) − B ∼ β σ where is the usual relativistic factor, and ∆B denotes the cross-section for such catalysis. As seen, direct laboratory or indirect cosmological search today faces an obstacle of enormous high energy for successful monopole detection. Before its unambiguous detection, it should come as a solace that we can still study its statistical properties in formation via similar topological manifold method characterization here on Earth. Regarding the cosmological versus/and astrophysical magnetogenesis scenario of Chapter 3, there are multiple current avenues of experimental efforts that can shed light on such matters. In addition to the polarization of CMB and deflection of electromagnetic cascade of gamma ray in the open void of IGM mentioned in the introduction, huge radio surveys to probe the reionization, or the “first light”

[52], via the 21cm spin-temperature of the hydrogen atoms, could potentially give a 3-dimensional pic- ture of the magnetized state of the universe from redshift z=3 to the present [53]. As in the temperature anisotropies of the CMB, the presence of cosmological magnetic field would act as an extra energy

79 source and thus affect the 21cm emission/absorption spectrum ([112]). They can also potentially affect the star formation process (see [16], [92] for details). The two key issues to be resolved via all such observations is one, whether any cosmological magnetic seed is ruled out by observation, and two, the structure/configuration of the magnetic field lines over various length-scales throughout different “snapshots” of the universe, in particular, if they are “knotted” at all (i.e. a non-zero helical spectrum

A(k) for its two-point correlation function). A predominance of helical structures on smaller-length scales seem to be suggestive of physical processes like phase transition as studied, but fuller details must be worked out better. Since the magnetogenesis scenario worked out is intricately tied with the electroweak baryogenesis scenario via phase transition, it may be useful to mention in passing that the latter’s exact details are currently under stringent test with data from the new LHC accelerator at CERN. In particular, the mass of the higgs and the spectrum of supersymmetric particles (i.e. stop) discovered would give strong con- straints on say the self-coupling Higgs term in the effective potential, and thus, the order of the phase transition. Another avenue to look for possible phase transition remnants is to look for its imprints on the stochastic gravitational wave (GW) spectrum. [79] offers a nice introductory summary of the effect of general early universe phase transition on GW emission while [23] and [64] contain analytical calculations on realistic phase transition physics like turbulence on the peak and shape of the spectrum.

In particular, [3] has looked at some viable supersymmetric models for successful electroweak baryo- genesis and pointed out the respective parameter regions of interest to the LISA mission. Once the electroweak dynamics and TeV physics are mapped out with data from the LHC, more works are defi- nitely required to corroborate further model building of its cosmological counter-part. The knowledge on possible cosmological magnetic seed field as recounted in previous paragraph will offer comple- mentary information in such a quest.

Chapter 4 was centered on an investigation based on effective field theory approach on the gravito- electromagnetism coupling, and showed how cosmological and astrophysical observations are of little hope in dropping concrete hints along the vast unchartered energy land lying between established QED

80 physics and the supposedly Planck energy scale for quantum gravity. Yet, this seemingly discouraging conclusion should not be taken as a sign of hopelessness for further theoretical works on the use of effective field theory to probe quantum gravity effects at large. Rather, we would quote on similar views as expressed in Burgess’s review [19] on this subject matter; (1) Non-renormalizability of gravitational theory does not prohibit detailed calculation on its quan- tum effects, and effective field theory is a powerful calculation tool, provided one stays vigilant on the validity range where the perturbative order holds.

(2) The clear demonstration on the little constraining power of post-relativistic general relativity tests in our solar system or long-distance cosmological observations on quantum effects consolidate these tests as classical GR tests. Any claim on such deviations (i.e. cosmological birefringence) as be- ing due to quantum anomalies related to gravity should hence be subjected to careful scrutiny. Mean- while, this study also points to hinting at the look-out of quantum gravity effects arising from more exotic avenues with extreme gravity conditions such as cosmological singularity or primordial black holes.

(3) An explicit calculation on the dependence of quantum effects on variables such as “mass, phys- ical separation”, etc. is of enormous importance in charting new avenues say “larger-than-generic quantum systems” to probing generic quantum gravity. Continuing efforts in the lattice gravity com- munity may offer complementary handle, combined with the quantum gravity tests listed previously.

It is hereby fitting to conclude this thesis with the famed passage of Faraday on electromagnetism, as re-quoted in Schwinger’s 1969 Science article on “A Magnetic Model of Matter”— which half a century later, still rings true, “Nothing is too wonderful to be true, if it be consistent with the laws of nature, and in such things as these, experiment is the best test of such consistency”.

81 Appendix A

Appendix to Chapter 2

A.1 Appendix: Topological charge

We wish to show that the two expressions for the topological charge, Eqs. (2.14) and (2.15), are equiv- alent.

The demonstration follows by using the SU(3) identity

a b c f T T T = 2i(δinδ δ δ δ δm j). (A.1) abc i j kl mn k j ml − il kn

a a b ab where T are SU(3) generators normalized such that tr(T T )= 2δ , and fabc are the structure con- a b c stants defined by [T ,T ]= 2i fabcT . The above identity is a generalization of the better known identity for the SU(2) generators σ a:

a b c ε σ σ σ = 2i(δinδ δ δ δ δm j). (A.2) abc i j kl mn k j ml − il kn

Now, if we choose Z†Z = 1, Eq. (2.15) can be written

1 Q = f T aT bT c d2Sp 8π abc i j kl mn ε pqr ∂ ∂ (zi∗z j) q(zk∗zl) r(zm∗ zn).

Using (A.1) this becomes

i 2 p pqr Q = (δinδ δ δ δ δm j) d S ε 4π k j ml − il kn ∂ ∂ ∂ ∂ (zi∗z j)( qzk∗zl + zk∗ qzl)( rzk∗zm + zm∗ rzn). (A.3)

82 † † The contractions lead to factors such as Z Z = 1 or else similar factors with derivatives, such as Z ∂qZ, † † ∂qZ Z, or ∂qZ ∂rZ. Of the eight terms in (A.3), four cancel in pairs, and the other four are equal in pairs, yielding finally

1 2 p pqr † † † Q = d S ε (∂qZ ∂rZ ∂qZ ZZ ∂rZ), (A.4) 2πi − which, using Eq. (2.13), is precisely Eq. (2.14).

A.2 SU(3) geodesic matrix

Here we will construct the SU(3) matrix R ji such that

R jiZi ∼= Z j. (A.5)

There can be many such rotation matrices but we will be interested only in the geodesic rotation such that

R ji = exp(iMs), (A.6) where M is a linear combination of SU(3) generators and s is the geodesic distance between Zi and Z j as given in Eq. (2.27).

T The procedure we will adopt is to first consider the special case when Zi = Z0 =(0,0,1) . In this case, we can find R j0 and the corresponding M. Then we extend the result to include the case when Zi is arbitrary.

A.2.1 Zi = Z0 case:

Now

T Z0 =(0,0,1). (A.7)

Let us denote

T Z j =(z1,z2,z3), (A.8)

83 † where z1,z2,z3 are complex numbers and we assume Z j Z j = 1. We wish a matrix M such that

Z j = exp(iMs)Z0. (A.9)

The matrix M is a linear combination of SU(3) generators. However, the generators of the unbroken

SU(2) U(1) sub-group need not be included since they have no effect on Z0. So we need only × consider M of the form 0 0 iv −   M = 0 0 iw , (A.10)  −    iv∗ iw∗ 0      where v,w are complex numbers. M is normalized using tr(M2)= 2 and so v 2 + w 2 = 1. | | | | We want to find v,w in terms of z1,z2,z3. By the standard procedure of diagonalizing M or by using the formula M3 = M, one finds

iMs R j0 = e v 2 coss + w 2 vw (1 coss) vsins | | | | − ∗ −  2 2  = v∗w(1 coss) v + w coss wsins . (A.11) − − | | | |     v∗ sins w∗ sins coss   − −   

Now we can relate v,w to z1,z2,z3. We have

z1 vsins     Z j = z2 = R j0Z0 = wsins . (A.12)         z3  coss          and so, in terms of the parameterization (2.7),

α β s = θ¯, v = cosφ¯ei , w = sinφ¯ei . (A.13)

84 Note that, from Eq. (2.27), the distance between Z0 and Z j is s. This shows that the matrix exp(iMs) is indeed the SU(3) transformation (labeled by s) that traces a geodesic from Z0 to Z j. Note also that because in our convention (2.7) the third component of Z j is real, there is no need for an extra phase factor here.

It can also be verified by explicit substitution that one may write R j0 in terms of Z0 and Z j as

† † (Z0 + Z j)(Z0 + Z j ) † R j0 = 1 † + 2Z jZ0. (A.14) − 1 + Z j Z0

Next, we relax the condition Zi = Z0.

A.2.2 General Zi case:

We would like to find R ji such that

R jiZi ∼= Z j, (A.15)

2 where R ji = exp(iMs) and s is the geodesic distance between arbitrary points Zi and Z j in CP .

We already know how to construct the matrix Ri0 as in Eq. (A.11) that rotates from Z0 to Zi. Next find the point † Z j¯ = Ri0Z j (A.16) where the bar on the subscript j in Z j¯ denotes that the point is obtained by rotating Z j. It is important to note that the third component of Z j¯ may not be real. In fact, since scalar products are unchanged by † † SU(3) transformations, the third components is Z0Z j¯ = Zi Z j.

Next we find R j¯0 such that

R j¯0Z0 ∼= Z j¯. (A.17) where to use the result in Eq. (A.11) or (A.14) requires removing the phase factor, i.e.,

† Z j Zi R j¯0Z0 = Z j¯ † . (A.18) Z Zi | j |

85 Then it is straightforward to check that

† Z Zi R Z Z j Z (A.19) ji i = j † ∼= j, Z Zi | j | where † R ji = Ri0R j¯0Ri0. (A.20)

Note that the rotation R ji is the shortest such rotation since R j¯0 is the shortest rotation from Z0 to Z j¯.

The Ri0 transformations in Eq. (A.20) translate the geodesic path from Z0 to Z j¯ such that it now goes from Zi to Z j.

It is also possible to write an explicit formula analogous to (A.14) for R ji. In fact, we have simply † † to replace Z0 in that formula by Zi and Z j by Z j(Z Zi/ Z Zi ). j | j |

A.3 Construction of the matrix S.

2 The matrix S ji is an SU(2) geodesic rotation that transforms (A ji,B ji) to (A j,B j) at the point Z j on CP (see Fig. 2.3 and Eq. (2.31)). These are the well-known Euler rotations e.g. see Section 4.5 in [57]. † First we apply the rotations R j0 to parallel transport all quantities from Z j to Z0 where we know that the unbroken SU(2) lies in the 1-2 block of the generators. Quantities at Z0 will carry a (0) superscript (0) (0) (0) (0) (0) (0) (0) e.g. (A ji ,B ji ) and (A j ,B j ). Then we perform an SU(2) rotation S ji that rotates (A ji ,B ji ) to (0) (0) (A j ,B j ). There are two such rotations, each of which can be written as

in σΦ/2 (0) e 0 S ji =   (A.21) 0 1     where σ denotes the three Pauli spin matrices, and ψ, θ and φ are the Euler angles. The angle of rotation, Φ, is given up to a two-fold ambiguity,

Φ φ + ψ θ cos cos cos , (A.22) 2 ≡± 2 2

86 and e n = , (A.23) sin(Φ/2) with

φ ψ θ e = cos − sin , 1 2 2 φ ψ θ e = sin − sin , (A.24) 2 2 2 φ + ψ θ e = sin cos . 3 2 2

φ ψ θ (0) (0) (0) The Euler angles , and can be written in terms of the vector triads at Z0, (a ji ,b ji ,c ji ) and (a(0),b(0),c(0)) where c = a b: j j j ×

cosθ = c(0) c(0), i j j cosψ = a(0) ζ, j sinψ = (a(0) ζ) c(0), (A.25) j × j cosφ = a(0) ζ, ji sinφ = (a(0) ζ) c(0), ji × ji where ζ is a unit vector along the “line of nodes”

(0) (0) c ji c j ζ × . (A.26) ≡ c(0) c(0) | ji × j |

(0) Finally, the matrix S ji can be parallel transported back to Z j to obtain

(0) † S ji = R j0S ji R j0. (A.27)

The two-fold ambiguity in the rotation corresponds to two possible angles of rotation, by Φ or by

Φ 2π. We choose the rotation that is smaller i.e. Φ π. − | |≤

87 A.4 Consistency of monopole and string numbers

The topology of the symmetry breaking scheme described by Eqs. (2.1) followed by (2.2) requires that a cell with a nonzero monopole number has an odd number of strings through its faces, while one with zero charge has an even number. Here we demonstrate that the formalism described above respects this condition.

For this purpose it is convenient to rotate all the relevant quantities to the base point Z0. In particular, we consider, in place of (2.35) the quantity

(0) † W i jk = Ri0W i jk Ri0 { } { } (0) 1 α √ 8 = S i jk exp( 2 i i jk 3T0 ), (A.28) { } { } where (0) † S i jk = Ri0S i jk Ri0. (A.29) { } { } (0) (0) Clearly, W i jk must be one of the two central elements of SU(2)0, and consequently S i jk U(1)0 { } { } ∈ since the other two factors in (A.28) are in that subgroup.

Now consider the product of the W (0)s from all four faces, say

(0) (0) (0) (0) (0) W = W 123 W 142 W 134 W 243 . (A.30) { } { } { } { }

The order of the four factors is arbitrary but has been chosen for later convenience. This product is evidently again one of the two central elements of SU(2)0; which one determines whether the number of strings entering the cell is even or odd.

8 (0) Since T0 commutes with all the S i jk , when we substitute from (A.28) into (A.30), we can move { } all the exponential factors to the right, and so write W (0) as a product

(0) (0) π 8 W = S exp(i Q√3T0 ), (A.31)

88 where we have used Eq. (2.21), and

(0) (0) (0) (0) (0) S = S 123 S 142 S 134 S 243 . (A.32) { } { } { } { }

Moreover, using Eq. (2.32), we see that each factor here may be written as a product of three factors coming from the edges of the triangle, each transported to Z0:

(0) (0) (0) (0) S i jk = Uik Uk j Uji , (A.33) { } where, for example, (0) † Uji = R j0S jiR jiRi0. (A.34)

(0) (0) The key now is to compare the transformations Uji and Ui j . By construction, S jiR ji transforms

Φi, Ψ1i, Ψ2i into Φ j, Ψ1 j, Ψ2 j, whereas Si jRi j performs the inverse transformation. Moreover, the prescription for choosing between the two possible transformations is the same in each case. These two products are therefore inverses. Thus we learn that

(0)† (0) Uji = Ui j . (A.35)

Now when we substitute (A.33) into (A.32) we find

(0) (0) (0) (0) (0) (0) (0) S = U13 U32 U21 .U12 U24 U41

U(0)U(0)U(0) .U(0)U(0)U(0) . (A.36) × 14 43 31 23 34 42

These factors are six pairs of mutual inverses, although since they do not necessarily commute, it is not immediately obvious that they cancel. It is clear, however, that two pairs cancel at once, leaving us with

(0) (0) (0) (0) (0) (0) (0) (0) (0) S = U13 U32 .U24 .U43 U31 .U23 U34 U42 . (A.37)

89 (0) But now recall that the product of the last three factors is S 243 U(1)0. Consequently, this product { } ∈ commutes with all the U(0)s, so we may move these three factors together to any desired position in the product. Placing them after the first two we find

(0) (0) (0) (0) (0) (0) (0) (0) (0) S = U13 U32 .U23 U34 U42 .U24 .U43 U31 . (A.38)

But now it is clear that we can cancel these pairs successively, so that finally we obtain

S(0) = 1. (A.39)

So this factor may be cancelled from the right side of Eq. (A.31), which then becomes

(0) π 8 W = exp(i Q√3T0 ), (A.40)

This shows, as required, that the number of strings is odd or even according as Q = 1 or 0.

A.5 SU(2) monopoles and strings

Here we discuss monopoles connected by strings in the model

SU(2) U(1) 1. (A.41) → →

The first symmetry breaking is achieved by giving a VEV to an SU(2) adjoint, equivalent to choosing

2 a unit 3-vector (call it v). The vacuum manifold is SU(2)/U(1) ∼= S . The second symmetry breaking is achieved by giving a VEV to a second SU(2) adjoint, call it a, which is orthogonal to v. At this stage the vacuum manifold is S1. Therefore monopoles are formed in the first symmetry breaking and these get connected by strings in the second symmetry breaking. To simulate monopole formation, we assign unit vectors v, equivalently points on S2, to the points on our spatial lattice [33, 77]. A tetrahedral cell gets mapped to a tetrahedron in S2 and some of these

90 i

R k

j

S

Figure A.1: Parallel transport of the vector for string determination To determine if a string passes through a spatial triangular plaquette, we first take the corresponding triangle on S2, labelled i jk , and then determine if the vector in the tangent plane rotates by 2π in circumnavigating the spherical{ } triangle. To do this, we first parallel transport the vector from i to j along a geodesic, described here as a rotation, R. Then we find the rotation S within the tangent plane that takes the transported vector into the vector at the vertex j. In each case we choose the minimal- angle rotation. Then we do the same thing for the remaining sides. Since we end up with the same vector at i that we started with, the combined transformation is either the identity or a 2π rotation. mappings will be incontractable, implying the existence of a monopole within the tetrahedral cell. The formation of strings that connect the monopoles is more involved but easy to picture, as in Fig. A.1. Since a is orthogonal to v, we can view it as picking a direction on the tangent plane of the

S2. To determine if there is a string passing through a triangular plaquette of the spatial lattice, we have to parallel transport a between the vertices of the triangle using rotations R and then rotate the transported vectors at the vertices using S. This is explained in Fig. A.1. The scheme for SU(3) is just a generalization of the scheme for the SU(2) model. The complications are technical in that, instead of

3 the tangent plane, our “vectors” at every vertex lie on an S /Z2 fiber and the geodesics and rotations are harder to determine in practice.

91 Appendix B

Appendix to Chapter 4

B.1 Shapiro delay calculations

For the idealized case where the Earth, satellite and Sun are all motionless, the proper Earth-satellite time of flight for a light signal (as measured on the Earth) has been first computed by Shapiro [106] and elaborated in detail in the standard textbook by Weinberg [128] in the standard Schwarzschild gauge.

Subsequent calculations in isotropic (spatially-conformally-flat) coordinates have also been done (see

[84],[131],[11]). At order rs there appear to be differences amongst these calculations, but these can be attributed to either the choice of gauge or whether or not the straight line approximation is used. Reference [6] offers a nice discussion and interpolation between some of the methods.

Most of the calculations found in the literature mentioned above, except in [128], use the straight line approximation to compute the time delay. This method is, in fact, exact up to order rs, at least in the idealized case of motionless bodies, because of Fermat’s principle in a static spacetime (∂tgµν = g0i = 0). Namely, the coordinate time of flight

i j gi j dx dx ∆t = ds (B.1) −g ds ds 00

µ is extremized if x (s) is a null geodesic of the spacetime described by gµν . In a weakly curved space- time where rs is much smaller than all other length scales, both the null geodesics and ∆t can be developed as a power series in rs. The O(rs) accurate ∆t can be obtained by employing the lowest order solution to the null geodesic equation, which is simply a straight line. The contribution to ∆t due O 2 to the deviation of the null path from a straight line begins at (rs ), due to Fermat’s principle. One could show the equivalence between Weinberg’s [128] and Shapiro’s [106] formulas by an explicit calculation, in which the true distance of closest approach, r0, and the “straight line” distance

92 of closest approach, b, are related through the light deflection angle integral (4.49).

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