Dark Energy Thesis

Hamilton CollegePhysics Department

Exploring Symmetron Dark Energywith a Massive Electrostatic Analogy

Written by:Lillie Ogden

Under Supervision of:Katherine Brown, Ph.D.

Assistant Professor of Physics

December 14, 2015

Abstract

This paper explores the Chameleon and Symmetron scalar fields, which are two separate

models of dark energy. The equations of motion that govern these two fields are generally

complicated and intricate, however, under conditions of interest on terrestrial scales, the

equations of motion reduce to Laplace’s equation. As a result, the scalar fields can be

studied through the employment of electrostatic analogies because physical principles that

obey the same mathematical form of equation will have the same mathematical form of

solution. Consequently, solutions to classic electrostatic problems can be used to gather

simple and eloquent solutions to the more complex scalar fields. Understanding these fields

on terrestrial scales may lead to an increase in experimental sensitivity when attempting

to detect the predominantly veiled and undetectable Chameleon and Symmetron scalar

fields.

Acknowledgements

First and foremost, this research would not have been possible without the remarkable guidance

and unwavering encouragement from my thesis advisor Kate Brown. Over the past two years

she has inspired me with her extraordinary accomplishments and has shown me how to be a

great physicist and an even better person. She remains one of the most influential professors

in my life and it was a great pleasure and honor to work with her. I would also like to extend

my thanks to the rest of the Hamilton College physics department for their dedication to foster

growth and learning in their students as well as for their continuous compassion and valuable

advice. Next, I would not have earned my physics degree had it not been for the laughter and

tears shared with my fellow classmates in our journey as physicists. The support and love from

the students and professors in this department have made my four years at Hamilton the best

they can be and I am forever grateful. Finally, my enjoyment and success in this department

would not have been possible without the overwhelming enthusiasm and love from my family

and friends and the lessons and advice they have given me along the way. Although they might

not understand the contents of this document, their encouragement and support have made

this thesis possible.

Lillie Ogden Symmetron Dark Energy 12/14/2015

Contents1 Introduction 5

2 History and Discovery of Dark Energy 62.1 Hubble’s Discovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 The Friedmann-Walker-Robertson Metric . . . . . . . . . . . . . . . . . . . . . . 72.3 Discovery of the Cosmic Microwave Background . . . . . . . . . . . . . . . . . . 92.4 Type 1a Supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Models of Dark Energy 113.1 Quintessence Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.1.1 Chameleon Scalar Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.2 Symmetron Scalar Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Electrostatics and Electrostatic Analogies 174.1 Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.1.1 Conducting Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.2 Electrostatic Analogies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5 Chameleon Electrostatic Analogy 225.1 Field Produced by a Spherical Body . . . . . . . . . . . . . . . . . . . . . . . . . 235.2 Torque on an Ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.2.1 Lightning Rod Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.2.2 Shape Enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

6 Massive Electrostatics 286.1 Potential of a Point Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.2 Spherical Cavity: Determining the Potential . . . . . . . . . . . . . . . . . . . . 296.3 Spherical Cavity: Approximating µ . . . . . . . . . . . . . . . . . . . . . . . . . 30

7 Symmetron Massive Electrostatic Analogy 317.1 Field Produced in the Absence of Matter . . . . . . . . . . . . . . . . . . . . . . 317.2 Field Produced in Uniformly Homogeneous Fluid . . . . . . . . . . . . . . . . . 337.3 Field Produced by a Sphere of Matter . . . . . . . . . . . . . . . . . . . . . . . . 347.4 Interior Symmetron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

7.4.1 Thick Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367.4.2 Thin Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

7.5 Exterior Symmetron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.5.1 Thick Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387.5.2 Thin Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

7.6 Symmetron Force in the Exterior Profile . . . . . . . . . . . . . . . . . . . . . . 397.6.1 Force Analysis: Thick Shell . . . . . . . . . . . . . . . . . . . . . . . . . 397.6.2 Force Analysis: Thin Shell . . . . . . . . . . . . . . . . . . . . . . . . . . 39

8 Conclusion and Future Work 40

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1 IntroductionCosmology is as old as humankind. Not long after language and communication developed

in primitive peoples, man sought to understand the world around him. Over hundreds of years,the questions pondered by early civilizations gazing up at the night skies evolved into questionsof “how does the universe work?” People began to answer these questions with philosophicalconsiderations, astronomical conjectures and intuitive reasoning, but these explanations lackedconcrete data to support the claims. By 1915, the current paradigm for understanding thedynamics of the universe was put forth by Albert Einstein. His theory was called the GeneralTheory of Relativity (GR). It described the fundamental interaction of gravity as governedby curved spacetime and it revolutionized humanity’s thinking of the universe. The theoryadvocated for a dynamic universe, raising even more questions about the mysteries of theuniverse: If the universe is not static, what did it look like in the past? What is the origin ofour universe? As questions were raised about our beginnings, many people began to wonderabout the future: What will the universe look like in the future? Will gravity eventually causethe collapse of the universe? Will the universe expand indefinitely?

Modern cosmology aims at answering these types of questions and it involves the study of theuniverse and its complex components, how it was formed, how it has evolved, and what its futureholds. It seeks to develop a complete understanding of the universe as outlined by GR that isalso consistent with experimental observations. The field has a history replete with setbacks andreconditionings of the fundamental perceptions of the surrounding world. The tools to findingsanswers have rapidly improved in the past twenty years with technological advances in spaceobservatories and telescope stations. An influx of theoretical and experimental astronomersinto the field of cosmology has resulted in an abundance of recent discoveries. These newdiscoveries have provided radically new information that has far-reaching implications for thestructure, origin, and evolution of the universe. Important discoveries such as the cosmicmicrowave background (CMB) have been able to confirm the nature of the universe outlinedby general relativity and support the notion that the universe is flat. However, as more resultsrush in, there is a gap missing in the fundamental understanding of the universe. It appearsthat about 70% of the total energy density in the universe necessary to enable a stable universeis unaccounted for. In other words, there is not enough luminous, “baryonic” energy densitythat scientists can detect that would allow for a ‘flat’, stable universe. An attempt to explainthis puzzling lack of energy density is the theory of dark energy. “Dark energy” provides anexplanation for the deficient energy density in the universe, but additionally, it expounds thatthis energy is a force working to counteract gravity and cause the accelerated expansion of theuniverse. Several theories have been put forth in an attempt to explain dark energy but it isstill a field of study that lacks a compelling model and requires further extensive research byexperimentalists and theorists alike. This paper explores the symmetron model of dark energy.

In this paper, we will begin by exploring the history that led to the discovery of dark energyin the universe beginning with the Einstein Field Equations. We will introduce two separateclasses of models of dark energy before beginning an in depth analysis of the third type of model,the quintessence model. The quintessence model proposes that dark energy is a slowly evolvingscalar field that lives in a potential. We will introduce two of these scalar fields, the Chameleonand the Symmetron, and explain their basic characteristics and the equations that govern them.This will bring us to the bulk of the paper, which will elaborate on the scalar field model asit relates to electromagnetism via electrostatic analogies. We will review relevant scenarios in

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electrostatics in order to identofy that the Chameleon scalar field obeys an electrostatic analogy,allowing us to find solutions to this complex field under certain, simplifying conditions. Next,will will demonstrate how working with ellipsoidal objects as opposed to spherical objects mayincrease experimental sensitivity due to the “lightning rod effect”. We then will introduce atheoretical branch of electrostatics called massive electrostatics to show that, similarly, theSymmetron obeys a massive electrostatic analogy and explore the behavior of the field outsideof a spherical object. Finally, we calculate Symmetron forces to explore the field’s affect in aterrestrial environment.

2 History and Discovery of Dark EnergyEinstein’s theory of general relativity yielded a concise, mathematical tool for describing the

arrangement of matter in space and was immediately recognized by the scientific communityas having profound ramifications for the field of physics and cosmology. These implicationswere encapsulated in a packet of field equations and these set foundation for future research inthe field of cosmology. Similar to how Maxwell’s Equations describe the electromagnetic fieldsby evaluating the presence of charges and currents, the Einstein Field equations describe thespacetime geometry resulting from the presence of mass and energy:

Gµν = 8πGTµν . (1)

These equations determine a metric tensor of spacetime for a given configuration of energyand stress in the universe. The left hand side of the equation, Gµν , describes the geometryand structure of the universe. The right hand side of Equation (1), 8πGTµν , describes thecomposition of the universe including mass, energy, stress, and density. The equations indicatethat the composition of the universe determines how spacetime curves and in turn, curvedspacetime determines the behavior of the composition.

Further, the equations implied that the universe was dynamic because they consisted ofdifferential equations, changing in time and space. However, the common worldview at thetime believed that the universe was fixed and unchanging [1]. Thus, Einstein first attempted toa fabricate a solution with a static universe. What he found was that if the universe were staticat the beginning of time, the gravitational attraction in his equations would have resulted inthe collapse of the universe, suggesting that the universe was indeed dynamic. However, giventhe apparent stationary and stable nature of the universe, Einstein proposed that there mustbe some device that he had missed working to hinder and cancel out the gravitational force andcreate the static universe. He stabilized his theory to account for this anti-gravity by addinga simple, non-zero cosmological constant in his equation. The “cosmological constant” termrepresented only a hypothetical entity that could counteract gravity and therefore stabilize theuniverse against gravitational collapse. In fact in a paper written by Einstein in 1917, he stated“The term is necessary only for the purpose of making possible a quasi-static distribution ofmatter, as required by the fact of the small velocities of the star.”[2]

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2.1 Hubble’s Discovery

The overarching belief that the universe was static was invalidated by tangible data and ob-servations. In 1929, Hubble was studying light coming from galaxies at various distances fromearth and was able to determine that the further from earth the galaxy was located, the greaterthe receding velocity of a galaxy. Hubble’s observations that light showed a red shift that in-creased with distance ruled out the possibility of the Einstein static state model. In fact, withthe data, Hubble conjectured that the universe was not only dynamic but was expanding. This“cosmic expansion”, as it became known as, meant that the light from distant galaxies was red-shifted because the galaxies were in fact moving away from us and from all the other galaxiesin the universe. Space itself was expanding between the matter in the universe. Therefore,the farther any two galaxies were from each other, the faster they continued to move apartand separate. Hubble published this linear proportionality between distance and velocity andthe Hubble Constant is now the unit of measurement that is used to describe the expansion ofthe universe. Cosmologists quickly recognized that an expanding universe meant that in thefuture, the galaxies would lie farther apart. However, they also extrapolated that in the past,the galaxies must have been much closer together and the universe must have been far moredense. In fact, at some point in time, the universe would have been contained in the size of theatom. Thus this data led to the theory of the Big Bang, which was essentially confirmed withthe discovery of the CMB in the late 20th century.

2.2 The Friedmann-Walker-Robertson Metric

After Hubble’s eminent discovery, Einstein’s field equations had to be solved for new modelsthat allowed for a dynamic and complex universe. In the 1920’s, mathematician AlexanderFriedmann was credited with designing a set of possible mathematical solutions that gave a non-static universe [1]. Einstein’s original static state solution used the simplifying assumption thatthe universe was spatially homogeneous and isotropic, meaning that it appeared the same nomatter where in the universe a person stood and what direction they looked. This homogeneityand isotropy of the universe became known as the Cosmological Principle. Friedmann’s metricmaintained a universe that was homogeneous and isotropic, but that was no longer static. Thismetric, called the Friedmann-Walker-Robertson, metric is given as

−c2dT 2 = −c2dt2 + a2(t)

[dr2

1− kr2+ r2dθ2 + r2sin2θdφ2

], (2)

where k is an important parameter that describes the curvature of spacetime. In the 1930s,Robertson and Walker showed that there were only three possible spacetime metrics for auniverse that were consistent with the Cosmological Principle: k = ±1, 0 [3]. If k = 1, thenthe universe is said to be positively curved or closed. If k = −1, then the universe is said to benegatively curved or open. If k = 0 then the universe is said to be “flat”. If the geometry is flatthen, the universe will stop expanding after infinite time and spacetime geometry is euclideanon cosmic scales. Scientists began to explore these three cases in order to gain intuition aboutthe structure of the universe. Considering the case where the universe is flat (k = 0), one cansolve for the density of the universe that enables it to be flat. This is called the critical density

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and is given by

ρcrit =3H2

8πG(3)

where H = a/a is the Hubble constant and describes the evolution of the metric in terms ofexpansion. In the constant, a is called the scale factor and multiplies the spatial components inthe FRW metric and a is the time derivative of a. Cosmologists frequently describe the energydensity of the universe in terms of the density parameter Ω. It is defined as the ratio of thedensity of some configuration of spacetime relative to the critical density:

Ω =ρ

ρcrit(4)

We can now define flat, open and closed in terms of the density parameter and portray thescenarios graphically in Figure (1). A flat universe is when k = 0, ρ = ρcrit and Ω = 1.This means that the universe is flat on a large scale and contains the critical density. Anopen universe has values k = −1 , ρ < ρcrit and Ω < 1. This means that the universe isnegatively curved and the density of the universe, related to the amount of mass and energyin the universe, is less than the critical density required to have a flat universe. Thus, in anopen universe, there is insufficient energy density to counteract or reverse the expansion dueto gravity and the universe expands forever as galaxies fly apart from each other, named the“Big Freeze” because the universe will slowly cool as it expands. A closed universe is whenk = 1, ρ > ρcrit and Ω > 1. This means that the universe is positively curved and excess energydensity in the universe counteracts the tendency of the universe to expand, causing the universeto eventually collapse back on itself due to gravitational attraction, called the “Big Crunch”.

Figure 1: The fate of a universe as it evolves according to Einstein’s field equations under situations with different amounts ofdensity, form NASA and WMAP. If the density of the universe is less than the critical density, then the universe will expand forever,like the red curve in the graph above. This is also known as the“Big Freeze”. If the density of the universe is greater than thecritical density, then gravity will eventually win and the universe will collapse back on itself, the so called “Big Crunch”, like thegraph’s orange curve. [4]

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Cosmologists contend that this theoretical line of thinking points to a universe that mustbe flat. First, the fact that the universe is thirteen billion years old and it looks the way it doestoday, with stars and galaxies populating the night sky, points to a flat universe. Consider auniverse that started with an initial density slightly less than the critical density, Ω = .999999ie. an open universe. Then, the universe would have evolved according to the Einstein fieldequations and by the time it was 13.7 billion years old, it would not have formed structuresthat are bound by gravity. In other words, it would have expanded indefinitely and wouldappear nothing like the universe we encounter today. On the other hand, consider a universethat started with an initial density slightly larger than the critical density, Ω = 1.0000001 ie.a closed universe. Then, the universe would have already collapsed after 13.7 billion years.Therefore, the only way to have a universe that is stable and occupied by structures, galaxies,and stars is if it started flat and has always been flat. The Friedmann-Walker-Robertson metric,a solution to the Einstein field equations, enabled a theoretical line of thinking that demandeda stable universe to be flat. Today, Friedmann is applauded for his ingenuity but during the1920s, neither Einstein nor anyone else took any interest in Friedmann’s work, which they saw asmerely an abstract theoretical curiosity [1]. However, the metric has proofed a crucial solutionthat permits concrete models of the mathematical composition of matter in the universe.

2.3 Discovery of the Cosmic Microwave Background

In the 1960s, new technologies enabled the discovery of the cosmic microwave background(CMB), which helped to promote the notion of a flat universe and finally wipe out steadystate models. The detection of the CMB radiation was the most impressive piece of evidenceconfirming the Big Bang theory. The CMB is the ancient, constant light source that permeatesthrough and saturates the universe. The start of our universe was a Big Bang 13.7 billionyears ago; a small, hot, dense event that sent the universe into a rapid inflationary epoch. Thisinflationary epoch was immense enough to flatten the geometry of the universe. After the initialburst of expansion, the rapid inflation disappeared and the universe resumed a more constantexpansion rate. This allowed the universe to cool and particles to form atoms. This coolingleft an imprint that permeated through space-a constant background radiation that glows ata temperature just above absolute zero, about 2.7 K, and is uniformly distributed. However,strictly speaking the CMB it is not entirely uniform and improved technologies and instrumentshave detected tiny variations in the early temperature of the universe, which are produced byvariations in the early distribution of matter, shown in Figure (2).

With CMB data of the early temperature fluctuations, scientists can detect slightly denserspots in the early universe where galaxies were eventually born out of and these regions wereextremely sensitive to the initial conditions of the geometry of the universe. The tempera-ture fluctuations are consistent with an initial geometry corresponding to a primitive universethat was flat. With the discovery of the CMB, and previous theoretical reasoning from theFRW solution to Einstein’s equations, the flat universe became the dominant paradigm withincosmology.

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Figure 2: Temperature fluctuations of the CMB in the early universe, taken by the WMAP. [5]

A thorough comprehension was beginning to come together regarding the arrangement andstructure of the universe as data from new technology and cosmologists continued to confirm theflat, dynamic nature of the universe. However, one large gap in the understanding of the cosmoswas entirely perplexing. Since the universe appeared flat, the contents in the universe mustsum to the critical density, Ω = 1, according the the FWR metric. However, measurements ofilluminating energy density in the form of baryonic matter such as stars, planets, galaxies, etc.only make up 4.6% of the energy density required for a flat universe. Scientists were bewilderedby the apparent lack of energy density in the universe. Either the universe must not be flat andcosmologists had the daunting, nearly impossible task of explaining the temperature fluctuationsin the CMB or the current technology could not detect the missing energy density in theuniverse. Many scientists favor the latter option because current technology only enables us toexplore the luminous matter. The CMB made observations in the 1960’s of dark matter, whichscientist believe composes 23% of the density. Dark matter is believed to resemble ordinarymatter, differing only in that it has a reduced frequency of interacting with its surroundings.That leaves about 72% of critical density unaccounted for. This 72% of the composition of theuniverse is what scientists call dark energy. Many people were skeptical of the theory of darkenergy and worked at finding alternative explanations of the apparent lack of density.

2.4 Type 1a Supernovae

In the late 1990’s, Saul Perlmutter, Riess, and Schmidt were able to counteract the skepticismregarding dark energy with his research on light emission from supernovae [6]. A supernovaeis the event of a dying white dwarf that explodes after it has reached a critical density andthus, releases a large amount of radiation. In particular, he was examining type 1a super-novae, which are especially unique because they emit the same amount of radiation for everyexplosion. Therefore, the brightness observed on earth from a supernovae is proportional tothe distance that the star is away from earth since radiation goes as 1/r2. Perlmutter andothers found that all the nearby supernovae had the predicted luminosity based on our un-derstanding of nuclear processes, but the supernovae in a certain far range were dimmer than

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the science could explain. At the time of this discovery, many scientists suspected that theuniverse was slowing in its expansion due to the force of gravity. However, the results fromthe supernovae data, presented in Figure (3), showed that distant type 1a supernovas weredimmer than expected, which was interpreted as the accelerated expansion of the universe.This is called “cosmic acceleration” and was yet another marvel in cosmology. Some forcewas dominating the expansion of the universe only in a certain redshift distance range of theuniverse causing increased expansion that could not be explained with only matter and darkmatter. However, when dark energy was included into the equation, the data made sense.Thus, type 1a supernovae data provided enormous support for the theory of dark energy, andPerlmutter, Riess, and Schmidt were awarded the Nobel Prize in 2011 for their discovery [6].

Figure 3: A plot of the distance and luminos-ity of type 1a supernovae. For far distances, theluminosity is dimmer than predicted. [7]

Dark energy has become a model that not only wouldaccount for the extra density needed for a flat universe,but also provide the force that works to counteract grav-ity in the Einstein Field Equations and cause the cosmicexpansion observed by Perlmutter, Riess, and Schmidt.There have been several other credible arguments and ob-servations that support the theory of dark energy. Con-sequently, the current acceptance of the composition ofthe universe is one in which baryonic matter constitutesroughly 5%, dark matter constitutes roughly 25%, anddark energy constitutes the remaining 70% of the energydensity [3]. Thus, the fundamental interpretation of thecosmos once again changed and cosmologists were forcedto delve into the world of dark energy, an important fieldin modern cosmology. Such a force would explain newrealms of cosmology and fill in the gaps in the field as awhole, but it currently lacks a compelling model and sev-eral scientists still doubt its existence. If dark energy doesexist, it has three defining characteristics. First, it doesnot interact electromagnetically, thus why it is called dark.Second, it must possess energy, as it is a form of energydensity. And lastly, it has the opposite effect of gravityand can dominate in distant regions of space, causing cosmic acceleration. Apart from thesebroad features, the current knowledge of dark energy is fairly limited.

3 Models of Dark EnergyThere are three classes of models for dark energy but they all have significant drawbacks and

none is universally accepted. One requirement of any dark energy model is that it must possessthe correct equation of state. This is the ratio of pressure p to the density ρ, w = p/ρ. Scientistshave shown that in order to have the correct cosmological properties ie. cosmic acceleration,the dark energy component must have w ≈ −1 [3]. There are three classes of models of darkenergy that have w ≈ −1. The first model is called the cosmological constant model, and isbased on the “vacuum energy” in the universe. This is, in some ways, the most tenable model.

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Consider the vacuum state of the universe, which scientist understand to contain no particlesand no fields. However, the Heisenberg uncertainty principle for energy and time, ∆E∆T ≥ ~/2, says that the change in energy (E) multiplied by the change in time (T) must be greater thanor equal to a constant. Theoretically, at small enough time intervals, ∆T , the vacuum energy,∆E, would be sufficient enough to create matter particles and antiparticles out of nothing. Thismodel postulates that particles and anti-particles are created during small time intervals and astime moves forward, they rejoin and annihilate. This process would create a backward pressureto balance and counteract gravity. Thus, the model contends that the process of creation andannihilation is what is causing cosmic acceleration and acting as dark energy. The big problemwith this model is that, mathematically, it predicts and energy density that is 120 orders ofmagnitude greater than the observed amount inferred from astrophysical observations, thusruling it out as a viable model.

The second type of model counters that dark energy doesn’t truly exist and rather, is anartifact of something that scientists have misunderstand about the fundamental physics in ourworld. A subcategory of this model is a gravity modification model, which postulates thatscientists have misconstrued the equations governing gravity and that, for example, gravitygoes as 1/r2+α where α << 1, as opposed to 1/r2. If α were a small enough number, it wouldonly be noticeable on cosmological scales, accounting for the balancing of gravity. Anothersubcategory is a model that suggests that earth is located in a relatively under-dense regionof space, resulting in the calculation of an inaccurate metric for spacetime. Models that denythe existence of dark energy have shortcomings such as necessitating the revision of Einstein’sequations, which most leading scientists have come to accept as fact.

3.1 Quintessence Model

The final type of model for dark energy is called the quintessence model and it predicts thatdark energy is a slowly evolving, nearly massless scalar field φ living in a potential V (φ) thatgives rise to negative pressure. This is the class of model that we will examine in detail inthis thesis. The main limitation with this model, as with any scalar field model, is that itis difficult to experimentally prove its existence. Take the Higgs model, which suggests thatparticles in the Standard Model acquire their mass through coupling to the Higgs scalar field.The existence of the Higgs took fifty years and the most expensive particle accelerator on theplanet to detect. Detecting a dark energy scalar field could proof even more difficult as newtechnology that could span into space would be required to definitively detect the field.

The model involves a nearly massless scalar field because often scientists work in units wheremass is inversely proportional to length. Thus, the more massive a scalar field, the shorter itsrange and the lighter the scalar field, the longer its range. This makes sense when considering aphoton, which is fundamentally massless and has an infinite range. So, a nearly massless scalarfield allows for a long range making it a good candidate for dark energy, which needs to operateon a cosmological scale. It is also plausible, that a more massive scalar field would be harderto detect because it has a shorter range and thus a shorter corresponding lifetime. Recall theLagrangian of a massive scalar field:

L =1

2(∇φ)2 − 1

2m2φ2 (5)

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The mass of the scalar field is just the square root coefficient in front of the φ2 term. Nowsuppose the scalar field φ lives in a potential V (φ) that has a minimum Vmin at φ = φ0, shownin Figure (4).

Figure 4: A potential, V (φ) with a minimum.

Taylor expanding the field about this minimum yields

V (φ) = V (φ0) +dV

dφ

∣∣∣∣φ=φ0

(φ− φ0) +1

2

(d2V

dφ2

)∣∣∣∣φ=φ0

(φ− φ0)2 + ... (6)

Because it is a minimum, dV/dφ = 0 and we can choose Vmin = 0 so that

V (φ) ≈ 1

2

(d2V

dφ2

)∣∣∣∣φ=φ0

(φ− φ0)2 (7)

We can then define

m =

((d2V

dφ2

)∣∣∣∣φ=φ0

) 12

(8)

as the mass of the scalar field. This means that for a scalar field to be close to its minimum, themass is given by the second derivative of the potential with respect to the field. Rememberingcalculus, we can interpret the second derivative of the potential as the curvature of the potentialin the vicinity of its minimum. Thus, a scalar field which stays flat towards its minimum haslittle curvature and little mass, whereas a potential that rises very steeply in the vicinity ofits minimum has more curvature and more mass. This behavior is illustrated in Figure(5) andFigure(6).

Although a nearly massless scalar field is a good candidate for dark energy on a cosmologicalscale, a scalar field with any amount of mass would cause problems on a terrestrial scale thatwould enable scientists to detect their presence. In order for the model to be feasible, it mustcontend with experimental constraints observed on earth. A slightly massive scalar field wouldconflict with the gravitational force going as 1/r2 and the Equivalence Principle, which havebeen studied extensively on earth. Although a slightly massive scalar field is necessary tooperate over the long range of the universe, a massless scalar field is necessary to operate overterrestrial scale so as not to disrupt the fundamental physics observed near our earth. It seemsimpossible that a scalar field could have the mass necessary to travel the cosmos as well asconform to terrestrial constraints. Nonetheless, cosmologists have mathematically describedseveral scalar fields that do just this, including the “Chameleon” and the “Symmetron” scalarfields. It is important to note that scientist have recently shown that some scalar field modelsare equivalent to modifications of gravity models, discussed above. However, it is most relevant

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Figure 5: V (φ) versus φ for the Chameleon field in thevicinity of low density matter. The low density reducesthe curvature of the potential as the scalar field remainsflat towards its minimum resulting in a smaller mass. [8]

Figure 6: V (φ) versus φ for the Chameleon field in thevicinity of high density matter. The high density in-creases the curvature of the potential resulting in a largermass. [8]

in this thesis to understand the “Chameleon” and the “Symmetron” as scalar fields.In 2003, two physicists Khoury and Weltman proposed a scalar field called the Chameleon

that was able to avoid the problems associated with mass [8]. The scalar field was able to changeits mass depending on the surrounding matter density, resulting in both a field that could avoiddetection on a terrestrial scale, allowing for the correct gravitational force and equivalenceprinciple theory, as well as operate on large scales. A closely related scalar field is called theSymmetron model. It varies from the Chameleon scalar field only in the potential term, butmaintains the ability to change its mass. We will describe the details of these models belowbefore beginning the analysis of these fields as they relate to electrostatics and electrostaticanalogies.

3.1.1 Chameleon Scalar Field

The Chameleon model, so named for its ability to ‘blend in’ with it’s surroundings byconformally coupling to ordinary matter, is a model used to explain cosmic acceleration. Thisconformal coupling allows the Chameleon to preserve certain aspects of the field but distortthe mass. There are two different space time metrics operating in a chameleon model: the truemetric gµν of general relativity and the conformally equivalent metric, gµν . gµν is the metricof spacetime governing the universe and it determines the behavior of the scalar field. gµν isconformally equivalent to the ordinary metric and determines the behavior of ordinary matterfields. The two metrics are related by the conformal transformation given by

gµν = e2βφ/MPlgµν (9)

where β is a coupling constant and MPl is the Planck mass (∼ 1019GeV ). This conformalcoupling is important because it allows the scalar field to obtain a linear, density-dependenteffective potential. Consequently, the potential of the scalar field is dictated by the ambient

14


matter density:Veff = V (φ) + A(φ)ρm, (10)

where V (φ) = M4+nφ−n and Aφ = eβφ/MPl . This in turn means that the mass of the scalarfield, which is just the second derivative of the potential from Equation(8), is also determinedby the ambient matter density. So, through this coupling, the chameleon is sensitive to thedensity of matter which surrounds it, permitting the Chameleon to change its mass. This isplausible from knowing that gµν consists of the Tµν component in the Einstein field equationsand elements of Tµν are density and pressure.

Figure 7: V (φ) versus φ for the Chameleon field in thevicinity of low density matter. This would correspond toa cosmological scale, where there is relatively low den-sity and lots of empty space. The low density makesthe density-dependent potential term of the potential onlyslightly curved, indicating that the mass of the scalar fieldis small in regions of low density. [8]

Figure 8: V (φ) versus φ for the Chameleon field in thevicinity of high density matter. This would correspond toa terrestrial scale, where there are high densities in theform of planets and stars. The high density makes thedensity-dependent potential term of the potential excep-tionally curved, indicating that the mass of the scalar fieldis large in regions of high density. [8]

Exploring the effect of this new potential term on the scalar field provides insight as to howthe potential enables a scalar field to adapt its mass, as shown in Figure (7) and Figure (8).At low densities, when the bare potential, Vφ, and the density dependent term, A(φ)ρm, aresummed they yield an effective potential that has small curvature and therefore, small mass.On the cosmic scale (Figure(7)), where density is extremely low, the mass of the chameleon isextremely small, enabling the scalar field to travel long distances and generate the present dayacceleration of the universe. This means that where density is relatively low, the Chameleon canplay the role of dark energy. Meanwhile, here on earth where the density is roughly 30 ordersof magnitude greater (Figure(8)), the chameleon acquires mass and is hard to detect. In highdensity regions, the bare potential remains unchanged, but now the density dependent termhas gained curvature from the increased ambient matter density, causing a steeper effectivepotential. So, the scalar field obtains a large mass, meaning it can travel shorter distancesand has a much shorter lifetime. Consequently, these scalar fields are difficult to detect ona terrestrial scale. That is why is has the name chameleon because it can blend in with itssurroundings and conform to terrestrial observations.

15


The field dynamics of the Chameleon field are governed by a complicated non-linear partialdifferential equation:

∇2φ =∂V

∂φ. (11)

It is clear that the potential V dictates the evolution of the scalar field. However, since thechameleon field is defined with a conformal transformation, the potential gets replaced by theeffective potential. Consequently, the chameleon evolves as

∇2φ =∂Veff∂φ

. (12)

Generally, this non-linear differential equation is complicated to solve, however in regionswith certain simplifying approximations, we can obtain easy and insightful solutions for theChameleon profile.

3.1.2 Symmetron Scalar Field

The ability to change mass in varying ambient densities is not unique to the Chameleon scalarfield: an alternate scalar field model is called the symmetron scalar field. The symmetron isanother hypothetical scalar field that, similar to the Chameleon, interacts with ordinary matter.Under suitable circumstances, including the assumption of time independence, the symmetronfield φ satisfies similar equation of motion,

∇2φ =∂

∂φVeff (φ), (13)

where the effective potential, Veff = V (φ) + ρA(φ). ρ represents the density of matter, as inthe Chameleon scalar field model. The only difference from the Chameleon scalar field is thatthe Symmetron lives in a different potential, V (φ). The symmetron potential, V (φ) is given by

V (φ) =λ

4

(φ2 − µ2

λ

)2

− µ4

4λ2(14)

If we define φ0 = µ2

λthen we have V = λ

4(φ2 − φ2

0)2 − µ4

4λ2. The conformal factor A(φ) in the

effective potential is given by

A(φ) = 1 +1

2M2φ2, (15)

where M,µ, and λ are parameters of the symmetron model. The units are where ~ = c = 1,causing φ, M and µ to have dimensions of L−1. λ is a dimensionless coupling constant.

Similar to the Chameleon, the symmetron field is determined by the density of matter ρ.Given ρ, in principle, one can determine the symmetron profile φ by solving the field equation∇φ = d

dφVeff (φ).The symmetron field manifests itself by exerting forces on matter. Consider a

test massm0 immersed in a symmetron field. This test mass will experience a force proportionalto the gradient of the symmetron field. Assuming the test mast is moving non-relativistically,the force is given by

m0d

dtv = m0

(∂A

∂φ

)∇φ = m0

1

M2φ∇φ. (16)

16


4 Electrostatics and Electrostatic AnalogiesBefore we begin our exploration of these two scalar field models in greater depth, it is necessaryto explore the heavily understood branch of electrostatics. Electrostatics is a well-exploredarea of physics that deals with the phenomena and properties of stationary electric chargesand the resulting electric and magnetic fields. As was shown by Jones-Smith and Ferrer,electrostatics plays a major role when working with scalar field models of dark energy [9]. Infact, solutions that are derived from equations in the simple electrostatic regime are relevant todeftly solve equations that arise in the Chameleon and Symmetron model. Thus, we will beginby introducing the static regime Maxwell’s equations, which govern the behavior of the electricand magnetic field. We will then solve the electrostatic equations in the vicinity of a sphericalconductor to find straightforward solutions to Laplace’s equation. We will then introduce theidea of electrostatic analogies as outlined by Feynman, which are useful in determining solutionsto complex physical phenomena that obey the same mathematical form as Laplace’s equation.The discussion of electrostatics and electrostatic analogies will enable us to assert that theChameleon scalar field obeys an electrostatic analogy, as it obey’s Laplace’s equation. In latersections, we will introduce a branch of electrostatics called massive electrostatics in order todeclare that the Symmetron obeys a massive electrostatic analogy.

4.1 Electrostatics

Maxwell’s equations are a set of partial differential equations that form the foundation ofclassical electrodynamics and electrostatics. They describe how the electric and magnetic fieldsare generated and altered by charges and currents, as well as their effect on each other. In thestatic regime Maxwell’s equations are as follows

∇ · E =ρ

ε0∇× E = 0

∇ ·B = 0

∇×B = µ0J.

∇×E = 0 and∇·B = 0 are called the “source free" Maxwell Equations because the 0 on theright hand side of the equation implies that there is no source from which the field originates.The source free equations allows us to express the electric and magnetic fields in terms of thepotentials, φ and ξ, at every point in space. However, we are interested in working with theequations obeyed solely by the electric field. Hemholtz decomposition says that a vector fieldcan be decomposed into two orthogonal components and expressed as f = ∇φ+∇× ξ where φand ξ are the potentials of the field. Thus, ∇×E = 0, implies that E is completely composedof an expressive aspect and thus there is no curl of the field. In other words, it is irrotational.

17


∇× E = ∇× (∇φ+∇× k) = 0

= ∇×∇φ+∇×∇× k = 0

= ∇×∇× k = 0 (since∇×∇ϕ = 0)

⇒ k = 0.

This means that E = ∇φ +∇× k = −∇φ, where the - sign arises by convention. Thus, it ispossible to express the electric field as the gradient of a scalar function φ called the electrostaticpotential.

Now let us consider the sourced Maxwell Equations obeyed by the electric field, ∇ ·E = ρε0.

The source for the electric field is various electric charges, which can be expressed in terms ofthe charge density, ρ. We can rewrite this equation since, from above, we know that E = −∇φ:

∇ · E = ∇ · (−∇φ)

⇒ −∇2φ =ρ

ε0.

This is known as Poisson’s Equation. If the charge density is absent, meaning ρ = 0, then wecan reduce Poisson’s equation to

∇2φ = 0. (17)

This is called Laplace’s equation. We can express or approximate a more general form ofLaplace’s equation,

∇ · (k∇φ) = ρ (18)

which says that gradient of the potential, ∇φ, multiplied by a scalar k has a divergence that isequal to a different scalar function ρ. The solutions to Laplace’s equation have been extensivelystudied for electromagnetic fields in various situations. Below, we will derive the solutions toLaplace’s equation in important, relevant circumstances. We will calculate the solutions to thepotential φ and electric field E for the interior and exterior of a solid, conducting sphere andthen examine these quantities in a conducting sphere with several, various shaped cavities.

4.1.1 Conducting Sphere

Consider a solid, spherical conductor of radius R held at a potential V0 which is placed in aa uniform electric field of magnitude E that points along the z axis, shown in Figure(9). Thepotential obeys the boundary conditions that ψ → V0 as r → R and ψ → −Ercosθ as r →∞.We wish to find the potential φ and electric field E at a point inside and outside of a sphericalconductor. We can quickly determine the solutions inside the conducting sphere since electriccharges are free to move around causing the interior electric field to be zero. A zero electricfield implies that the potential in the interior of the conducting sphere is constant, or ψ = V0

for r<R.

18


Figure 9: A solid, spherical conductor of radius R held at a potential V0 placed in a a uniform electric field of magnitude E thatpoints along the z axis.

In order to determine the solution outside of the conducting sphere, we will begin withLaplace’s equation in spherical coordinates

∇2φ =1

r2

∂

∂r

(r2∂φ

∂r

)+

1

r2sinθ

∂

∂θ

(sinθ

∂φ

∂θ

)+

1

r2sin2θ

(∂2φ

∂ϕ2

)= 0 (19)

However, we can assume azimuthal symmetry, meaning that φ does not vary in the ϕ directionor in other words, ∂φ/∂ϕ = 0. Then, the Laplacian becomes

∇2φ =1

r2

∂

∂r

(r2∂φ

∂r

)+

1

r2sinθ

∂

∂θ

(sinθ

∂φ

∂θ

)= 0 (20)

By using separation of variables we find that a general solution to Laplace’s equation in sphericalcoordinates, outside of a conducting sphere is given by

ψ = a+b

r+

c

r2cosθ + drcosθ

We can determine the coefficients of this solution for the conducting sphere by imposing theboundary conditions stated above. Consider ψ as r →∞.

ψ(r →∞) = −Ercosθ

ψ(r →∞) = a+b

∞+

c

∞2cosθ + drcosθ

= a+ drcosθ

⇒ a = 0, d = −E

19


Now, consider ψ as r → R, with a = 0 and d = −E,

ψ(r → R) = V0

ψ(r → R) = 0 +b

R+

c

R2cosθ − ERcosθ

⇒ V0 =b

R+

c

R2cosθ − ERcosθ

At r = R, the potential must maintain continuity. Note that if b = V0R and c = ER3, we have

ψ(R) = V0 =V0R

R+

(ER3

R2− ER

)cosθ = V0 + (ER− ER)cosθ = V0.

Thus, we can conclude that b = V0R and c = ER3 and that for r > R the potential of the fieldgoverned by Laplace’s equation is

ψ =V0R

r+

(ER3

r2cos− Er

)cosθ =

V0R

r+ E

(R3

r2cos− r

)cosθ.

In order to complete our calculations we must determine the behavior of the electric field outsidethe spherical conductor. The electric field is given by

E = −∇ψ = −∂ψ∂rr − 1

r

∂ψ

∂θθ − 1

rsinθ

∂ψ

∂ϕϕ.

We can take this partial derivatives to find the three components of the electric field for r > R.

Er = −∂ψ∂r

=V0R

r2+ E

(1 +

2R3

r3

)cosθ

Eθ = −1

r

∂ψ

∂θ= −E

(1− R3

r3

)sinθ =

(R3

r2− 1

)Esinθ

Eϕ = 0

On the surface of the sphere when r = R, Eθ = 0, so there is no tangential component onthe surface. This means that the normal component of the electric field on the surface is justEn = Er(r = R) = V0/R+ 3E0cosθ. This ensures that the only component of the electric fieldin the exterior of the conducting sphere is perpendicular to the surface of the sphere. We cansummarize our results as follows

E(r) = 0 r < b

ψ(r) = V0 r < b

E(r) =V0R

r2+ E

(1 +

2R3

r3

)cosθ; r > b

ψ(r) =V0R

r+ E

(R3

r2cos− r

)cosθ r > b.

20


We also note that in the absence of the constant electric field the results simplify to

E(r) = 0 r < b

ψ(r) = V0 r < b

E(r) =V0b

r2r r > b

ψ(r) =V0b

rr > b,

which the reader can recognize as the classic results for a conducting sphere which is not placedin an electric field. So, in the exterior space of a spherical conductor in an electrostatic regime,the potential falls off as 1/r and the electric field falls off as 1/r2. Additionally, in the interiorof a spherical conductor in an electrostatic regime, the electric field is equal to zero and thepotential is constant. These solutions were found by solving Laplace’s equations in sphericalcoordinates and consequently we have procured solutions to any equation that has the samemathematical form of Laplace’s equation near a spherical conducting body.

Figure 10: A spherical conductor of radius R with a spherical cavity of radius a carved out of the center.

Let us quickly consider the case in which the spherical conductor has a spherical cavity ofradius a carved out of the center (Figure (10)). We noted earlier that in a conductor, chargesare free to move around creating the absence of the electric field in the interior of the body. Thismeans that all of the excess charge is on the surface of the conductor, and we can effectivelyimagine the solid conductor as a shell of charge, ρ. Thus, the conditions for the sphericalconductor with a carved out cavity are the same as the solid spherical conductor. The electricpotential can be represented as,

φ(r) =1

4πε0

∫ρ(r′)

Rdτ ′ (21)

where R = r − r′and ρ is the charge density. Inside of the cavity the charge density is zero,ρ = 0 so E = φ = 0. Outside of the conductor, the sphere is still held at the potential V, sothe Laplacian governs the behavior of the electric field and potential. Therefore, we obtain anidentical result to the spherical conductor without a cavity. Furthermore, it does not matterwhat the shape of the cavity is-the result will always be that of an ordinary solid, sphericalconductor.

21


4.2 Electrostatic Analogies

The vast amount of information that scientists have discovered about the physical world isfar too large for someone to have gathered even a sensible selection of it. However, peoplemanage to draw connections and intuitions about the universe that make it drastically simplerto comprehend the many principles of physics. There are important laws that apply to all typesof phenomena such as the conservation of energy and angular momentum. These quantities andlaws governing the universe limit the possibilities that one could encounter in physics. Moreimportantly though, the equations for many different physical phenomena have the exact samemathematical form or appearance. Principles with the same mathematical form of equationmust have the same mathematical form of solution. Thus, identical mathematical forms allowfor a direct translation of the solutions to solve problems in other fields. This is extremelyprevalent in the field of electrostatics, which is outlined in detail by Richard Feynman in TheFeynman Lectures on Physics [10]. Many physical phenomena appear to obey electrostaticequations in the sense that many physics problems have the form of a potential φ whose gradientmultiplied by a scalar function k has a divergence equal to another scalar function, ρ,∇·(k∇φ) =ρ. Recalling the discussion of electrostatics, this is simply the general form of Laplace’s equation.When a physical phenomena obeys Laplace’s equation, it is called an electrostatic analogy. Thisenables scientists to quickly derive solutions to more complex situations from the simple case ofelectrostatics. Therefore, while learning electrostatics, physicists have simultaneously learnedabout a large number of other subjects. Below, we will follow the work outlined by Jones-Smithand Ferrer to demonstrate how the Chameleon obeys an electrostatic analogy by performingseveral calculations and exploring the profile of the field in relevant problems. Next we willdiscuss a field of electrostatics called massive electrostatics before investigating the Symmetronmodel and arguing that it obeys a massive electrostatic analogy.

5 Chameleon Electrostatic AnalogyThe Chameleon Scalar model of dark energy obeys an electrostatic analogy, enabling us toeasily derive solutions in relevant configurations. Under conditions relevant to terrestrial ex-periments, the Chameleon field obeys the same equations as the electrostatic potential. Recallthat ordinary matter follows the geodesics of

gµν = e2βφ/MPlgµν .

where φ is the Chameleon scalar field. Due to this conformal coupling the effective potential ofthe field includes a term that depends on the density of matter, ρm: Veff = V (φ) +A(φ)ρm. Inthe Chameleon model, the bare potential V (φ) is non-increasing and A(φ) is non-decreasing.This results in a minimum for the potential, which is dependent on the presence of ambientmatter density. Recall, that for static configurations of the field, the chameleon obeys ∇2φ =∂Veff∂φ

. This is called the Klein-Gordon equation. As we mentioned early, this is generally a non-linear differential equation that is challenging to solve, however, in some regions approximationsallow for relatively simple solutions.

22


5.1 Field Produced by a Spherical Body

Consider the field produced by a solid sphere of of radius Rc and uniform density ρc immersedin a uniform background medium [8]. Let us explore the Chameleon profile in the vicinity ofthis massive spherical body such as a planet in space, presented in Figure (11).

Figure 11: A solid sphere of of radius Rc and uniform density ρc immersed in a uniform background chameleon gradient.

The conformal coupling establishes that the Chameleon field varies when the density changes.The sphere is uniformly dense, so the Chameleon field inside the sphere is also uniform, giv-ing us φ = φc for r < R. Far away, some other density prevails ρ∞ giving rise to a differentchameleon φ∞. As a result, the only place where the chameleon discerns the density contrast isa thin shell of material just underneath the surface of the sphere. This thin shell of mass is theonly mass that contributes to and sources the outside field. The thin shell suppression factoris given by

∆Rc

Rc

=(φ− φ∞)

6βMPlΦc

1,

where ∆Rc is the width of the thin shell and Rc is the radius and Φc is the Newtonian gravi-tational potential. Virtually all terrestrial objects fall into this thin shell regime, under whichthe density contrast is great enough between the inside of the sphere and the boundary of thesphere so as to make the field inside of the body impenetrable to the field outside of the body.That is, throughout the core of the spherical body, the Chameleon field rests at the minimum ofVeff , and only the thin shell of matter on the boundary provides a density contrast large enoughto source the Chameleon field outside the body. Thus, the Chameleon field only perceives thebody as r → Rc and only over the course of the thin shell, just underneath the surface, doesthe field begin to vary. Once outside the body, the Chameleon is governed by its equation ofmotion, ∇2φ =

∂Veff∂φ

. Solving this equation for the spherical body yields

φ(r) ≈(−β

4πMρ,l

)(3∆Rc

Rc

)Mce

−m∞(r−Rc)

r+ φ∞ (22)

This form of solution, (e−λr)/r, is called the Yukawa profile and under circumstances of interest

23


to terrestrial experiments, the exponential term is negligibly small causing the Chameleon fieldoutside of the body to goes as 1/r and reducing the outside field profile to

φ = φ∞ + (φc − φ∞)Rc

r∝ 1

r. (23)

In summary, in the thin shell regime φ is constant throughout the core of the body and a thinlayer of matter on the boundary of the body contributes to the exterior field, where the profileis adequately described by the Yukawa profile but approximately, goes as 1/r for terrestrialexperiments. Furthermore, these conclusions for the Chameleon field φ of a massive sphericalbody are precisely the solutions we obtained when exploring the behavior of the electric potentialV of a spherical conductor: inside of the sphere, the electric potential was constant and outsidethe sphere the potential was sourced by a thin film of charge and V fell off as 1/r.

To make the analogy between electrostatics more clear, we can obtain the behavior of φin an alternate way. The spherical body contains a large density, resulting in a highly curvedpotential of the Chameleon scalar field. Consequently, the massive body corresponds to aminimum of the effective potential of the field (Recall Figure (8)). We can Taylor expandaround this minimum value to reduce the equation of motion to ∇2φ = m2(φ − φ∞). Forterrestrial experiments, however, the mass at infinity is effectively zero because the sphericalbody is in empty space. This reduces the equation of motion to ∇2φ = 0, which is justLaplace’s equation for electrostatics. Hence, the behavior of the Chameleon directly outsidethe massive, spherical body is governed by Laplace’s equation and we recognize the solution tothis as φ ∝ 1/r. Extrapolating, we can infer that the chameleon field may be approximated byPoisson’s equation in the thin shell regime, ∇2φ = βρc/MPl.

Due to the electrostatic analogy, the behavior of the chameleon field for a thin-shelled sphereis the same as the electrostatic potential for a conducting sphere because they both obeyLaplace’s equation outside of the massive body. The physics involved in the two situationsdescribes different phenomena, but the equations are of the same mathematical forms. Inelectrostatics, a conducting sphere contains a thin layer of electric charge on the surface thatsources the external electric field and similarly, in the Chameleon model, a massive spherecontains a thin shell of mass that sources the external Chameleon field. Since the two physicalprinciples have the same mathematical form of equations, they have the same form of thesolution.

In the electrostatic regime, the electric charge, σ(θ) is defined as ∂φ/∂n = −σ(θ)/ε0, where∂φ/∂n is the external field gradient and n is the direction normal to the surface. Similarly, forthe Chameleon field, we can write ∂φ/∂n = %δ where % = βρc/MPl is the volume density of the‘Chameleon charge’ and δ is the thickness of the thin layer over which this chameleon chargeis distributed. For a sphere, we can plug in % to get ∂φ/∂n = (βρc/MPl)δ and solve for δ toobtain

δ =(φ∞ − φc)Rc

6βMp,lΦC

, (24)

which is identical to the thickness of the shell, ∆Rc. This suggests that % is the volume densityof the ‘Chameleon charge’ and all terrestrial objects possess the thin shell effect. Then, theChameleon “charge" is represented by the material within the body that interacts with theChameleon field outside and it is confined to the shell layer. The main difficulty with theChameleon field, is that the thin shell suppression factor ∆Rc

Rcis so small that it causes the

24


scalar field φ(r) itself to be very small. Furthermore, the force on a test particle due to thechameleon field is equally as small since F ∝ ∇φ. This introduces the problem of detection inthe vicinity of spherical bodies, because the force acting on a test mass due to the Chameleonis adequately small enabling the field to avoid discovery in experiments involving sphericalobjects.

5.2 Torque on an Ellipsoid

There is reason to believe that an elliptical test mass could experience an extremely smalltorque due to the Chameleon field that experiments would be able to observe. Consequently,using a solid ellipsoid test mass as a replacement for a solid spherical test mass may allow forthe detection of the Chameleon field in a terrestrial lab. This is permitted due to the concept ofthe “lightning rod effect”, that originates from electrostatics. Since the Chameleon field obeysan electrostatic analogy there is no reason to doubt the presence of the lightning rod effectin a Chameleon model, and therefore it must be true. In fact it was shown mathematicallyby Jones-Smith and Ferrer [9]. Below, we will discuss the lightning rod effect briefly beforeexamining the behavior of a solid ellipsoid in the presence of a Chameleon field.

5.2.1 Lightning Rod Effect

Figure 12: A depiction of a sphere which has been stretched into an ellipsoid. This object exploits the lightning rod effect as thereis a build-up of charge in regions of high curvature.

Imagine if we stretched the spherical body from Figure(11) into an ellipsoid as in Figure(12).The lightning rod effect states that the field at the polar region of an elongated object isenhanced relative to the polar region of a sphere, meaning there are more field lines at the polesof an ellipsoid than at the equator. This enhancement arises due to the fact that extendedobjects such as ellipses have a build-up of charge in regions of high curvature. The build-up ofcharge causes a preferred axis for the elongated object in an external field, which the spherelacks. This is a general characteristic for of all systems that obey Laplace’s equation and theboundary conditions that we have assumed here.

25


Figure 13: A conducting ellipsoid immersed in a uniform electric field experiences a torque to align itself with the electric field.

In electrostatics, the electric field at the polar region of a conducting ellipsoid is enhancedrelative to the equator due to build-up of electric charge. If we place a conducting ellipsoid ina uniform electric field as in Figure(13), a dipole moment forms along the axis of the ellipsoidand if it is misaligned with the ambient electric field, the ellipsoid will experience a torque in anattempt to align itself with the electric field. Since the Chameleon is an electrostatic analogy,we should be able to exploit this effect by calculating the torque on a massive ellipsoid.

5.2.2 Shape Enhancement

Although the thin shell effect of the Chameleon profile arises due to the density contrast andboundary conditions of a spherical body, it stands to reason that a less symmetric shape wouldstill possess a thin shell effect. Consider a massive, uniformly dense ellipsoid that is placed intoa uniform Chameleon field gradient [9].

Ellipsoids are three dimensional figures that can be described by prolate spheroidal coordi-nates (ξ, η, ϕ). ξ = 1/ε where ε is the eccentricity and further, the surface of an ellipsoid hasthe radial coordinate ξ = ξ0. η is a measurement of the latitude, where the poles are defined atη = ±1 and the equator at η = 0. Ellipsoids are convenient objects to work with due to the factthat they can be compared with spherical results when we impose the limit that the eccentricityε → 0. It is useful to introduce an equivalent radius Re for the ellipsoid such that the volumeof the ellipsoid is given by 4/3πR3

e. We can assume that the ellipsoid consists of an arbitrarymaterial and only possesses a thin shell. The interior field value is constant due to the uniformdensity of the ellipsoid, ρc, and the exterior field is a solution to Laplace’s equation, since theChameleon obeys an electrostatic analogy. We can also define a as the interfocal distance ofthe ellipsoid and r as the radial spherical coordinate. Assuming that r >> a, the Chameleonprofile can be written as

φ = φ∞ + f(ξ0)(φc − φ∞)Re

r∝ 1

r. (25)

wheref(ξ0) =

(2

[ξ(ξ20 − 1)]1/3

)(1

ln[(ξ0 + 1)/(ξ0 − 1)]

)(26)

is the shape enhancement factor and we have chosen a = 2Re/[ξ(ξ20 − 1)]1/3. We note that the

ellipsoid has a shape enhancement factor f(ξ0) > 1 that diverges as the ellipsoid flattens to a

26


line. More importantly, this shape enhancement changes the field profile of the Chameleon inthe exterior of the body. We can compare the new field profile in the exterior of an ellipsoidwith the field profile in the exterior of a sphere, where there was unmitigated suppression of theforce imposed on a test mass due to the thin shell effect. If we now use an ellipsoidal source asopposed to a spherical one, the addition of the shape enhancement factor dominates and is ableto overcome the thin shell suppression factor, which was responsible for causing the Chameleonto impose an undetectable force on test masses.

Figure 14: A depiction of the lightning rod effectfor a solid sphere which has been stretched into anellipsoid. This object exploits the lightning rod ef-fect as there is a build-up of mass in regions of highcurvature.

Figure 15: A massive, uniformly dense ellipsoid im-mersed in a uniform Chameleon field gradient ex-periences a torque to align itself with the ambientChameleon field.

Equating this to electrostatics, this is comparable to the lightning rod effect described above.When a massive, uniformly dense ellipsoid is placed into a uniform Chameleon field gradient asFigure (15), the shape enhancement factor causes the “charge” to gravitate towards the polarregions of the ellipsoid, where the curvature is greatest. Instead of a build up of electric chargein regions of high curvature, there will be a build up of mass at the poles. In other words, theshell thickness would be enhanced at the poles of the solid ellipsoid, inferring that ellipsoidalobjects would be able to source stronger Chameleon fields. The stronger fields may be ableto overcome the thin shell suppression present in the object, which could lead to a non-zeroforce on the solid ellipsoid and a corresponding non-zero torque on the ellipsoid in a Chameleonfield gradient. The fact that a material could cluster in the polar regions of an elongatedobject in a Chameleon field gradient is not intuitive but the Chameleon model obeys the sameequations as electrostatics so this must be true as well. So, when we place a uniformly denseellipsoid in a uniform chameleon field gradient, a matter dipole moment forms along the majoraxis of the ellipsoid and if this dipole moment is misaligned with the ambient field gradientthen it will experience a non-zero torque. Just like an electric dipole can cause a conductingellipsoid to torque in an electric field, a matter dipole can cause a solid ellipsoid to torque in achameleon field. Simple estimates reveal that the torque is on the order of 10−15 Newton meters.The central finding is that the chameleon field outside of elongated bodies such as ellipsoidsis enhanced relative to the spherical bodies typically considered in terrestrial experiments [9].This shape enhancement can be exploited by experimenters to probe new regions of chameleonparameter space, even in the experimentally unfavorable thin shell regime. This may allow

27


scientists to rule out models of scalar field and swaths of parameter space or even detect thescalar field itself, increasing the overall sensitivity of the experiment.

6 Massive ElectrostaticsIt is now relevant to introduce the concept of massive electrostatics. Massive electrostatics is ahypothetical version of electrostatics in which Maxwell’s equations are modified by the presenceof mass terms. More specifically, in ordinary electrostatics (and electrodynamics) the photon ismassless, and in a region of no free charge, the electrostatic potential obeys Laplace’s equation.In massive electrostatics, the electrostatic potential obeys a modified Laplace’s equation, inwhich the ordinary Laplacian operator is supplemented with a mass parameter called µ, asdiscussed below. Massive electrostatics declares that the mass of the boson is represented bythe mass parameter µ. It follows that when µ = 0, we will return to the equations laid outin classic electrostatics. The units used in massive electrostatics are called “natural units” inwhich ~ = c = 1. Therefore, the mass parameter µ will have units of inverse length. Wecan achieve new insights by performing the previous calculations of a spherical conductor inmassive electrostatics, which will enable us to draw the correlation to the massive electrostaticanalogy we explore for the symmetron model. Thus, we will begin this section by performingthe calculations of a spherical conducting body in massive electrostatics. Our discussion ofmassive electrostatics will become relevant as we assert that the Symmetron model obeys amassive electrostatic analogy. Thus, by procuring a solution to the conducting sphere in themassive electrostatic regime, we will be able to quickly derive solutions to the Symmetron field.

6.1 Potential of a Point Charge

In massive electrostatics the electric field is given by E = −∇φ and the scalar potential obeys

−∇φ+ µ2φ =ρ(r)

ε0(27)

where ρ(r) is the charge density and µ is the mass parameter. This equation is the massiveversion of Poisson’s equation. As stated above, we can observe that when µ = 0 we obtain−∇φ = ρ

ε0, which is the classic electrostatic form of Poisson’s equation. In the relevant source

free regime, the above equation reduces to the massive electrostatic version of Laplace’s equa-tion,

∇φ+ µ2φ = 0. (28)

Using Fourier transforms, we can determine that the massive electrostatic potential of a pointcharge q located at the origin, corresponding to a charge density ρ(r) = qδ(r) is given by thefollowing Yukawa profile:

φ(r) =q

4πε0

e−µr

r.

From this we can quickly conclude that the electric field of a point charge in the electrostaticregime is given by

E =ke−µr

r2.

28


6.2 Spherical Cavity: Determining the Potential

We now wish to return to our example of a conducting sphere of radius R with a sphericalcavity of radius a carved out of the center in order to determine the corresponding solutions inthe massive electrostatic regime.

Figure 16: A spherical conductor of radius R with a spherical cavity of radius a carved out of the center.

In ordinary electrostatics, the potential inside the cavity was simply equal to zero, whichfollowed from the fact that ρ inside of the cavity was zero, causing the potential and electricfield to also be zero. However, in the massive electrostatic regime, since µ 6= 0, we cannotuse the logic that ρ = 0 implies E = φ = 0. Thus, the potential inside the cavity in massiveelectrostatics is non-zero and we will calculate this value below. This will enable us to determinean approximation for the mass parameter µ. Doing so will help us to repeat the same processfor the symmetron field when performing similar calculations. We will begin by rewritingthe Laplacian term in spherical coordinates, noting that spherical symmetry still guaranteesφ = φ(r):

∇φ =−1

r

d2

dr2(rφ)

By letting u = rφ and using ∇φ+ µ2φ = 0 we acquire

−1

r

d2(u)

dr2+ µ2φ = 0

⇒ −1

r

d2(u)

dr2= −µ2φ

d2(u)

dr2= µ2rφ = µ2u,

for which we can deduce the solutions. The second order differential equation suggests thatthe solutions involve an exponential decay or growth. We can choose to use the hyperbolic sineand cosine functions:

sinh(µr) =eµr − e−µr

2cosh =

eµr + e−µr

2.

Therefore, we can express u as u = e±µr = Asinh(µr) + Bcosh(µr). This gives us a general

29


solution of the potential in the cavity

φ(0) =A

rsinh(µr) +

B

rcosh(µr). (29)

We now must look at boundary conditions to determine the particular solution. At r = 0,the potential cannot approach ∞, inferring that B = 0. Thus, u = Asinh(µr) implyingφ = A

rsinh(µr). Additionally, at r = a, φ = V , so

φ(r = a) = V ⇒ V =A

asinh(µa)

⇒ A =V a

sinh(µa).

We can plug in the specific value of A to rewrite the potential inside the cavity as

φ(0) =

(V a

sinh(µa)

)(sinh(µa)

r

)= V

(µa

sinh(µa)

)(sinh(µa)

µr

).

This is clearly non-zero, as was the case in ordinary electrostatics. But, note that if µ = 0,then the potential φ inside the cavity is also equal to zero, which is what we would expect.

6.3 Spherical Cavity: Approximating µ

We now wish to express µ in terms of V. Let us define the potential difference between thesurface of the cavity and the center of the cavity as ∆V = φ(a)−φ(0). We know that φ(a) = Vand φ(0) is the potential inside the cavity, which was determined in the preceding calculation.Additionally, note that as r → 0, sinh(µr)/r → 1. Therefore we can rewrite the potentialdifference as

∆V = V

(1− µa

sinh(µa)

).

Dividing both sides of this equation by V allows us to determine the ratio of the potentialdifference to the potential:

∆V

V=

(1− µa

sinh(µa)

)⇒ sinh(µa)

µa=

(1− ∆V

V

)−1

.

If the potential difference is much smaller than the potential itself, then the ratio is extremelysmall and we can say, ∆V/V << 1. This allows for the approximation(

1− ∆V

V

)−1

≈ 1 +∆V

V.

30


We can also use the Taylor series expansion of sinh(x) to simplify the sinh term. We knowthat sinh(x) ≈ x+ x3/6 + ..... Thus, we can express the sinh term as

sinh(µa)

µa≈ µa

µa+µa3/6

µa+ ... = 1 +

µ3a3

6.

Combining this all together, we obtain

1 +µ3a3

6= 1 +

∆V

V

Basic algebra allows us to express µ in terms of the cavity radius a, the potential differencebetween the surface of the cavity and the center of the cavity as ∆V , and the potential V :

µ =1

a

√6∆V

V. (30)

We can use a very similar methodology to obtain solutions to the equations of a Symmetronscalar field.

7 Symmetron Massive Electrostatic AnalogyIn this section, we will gain some intuition about the symmetron field by performing variouscalculations under simple circumstances using similar techniques acquired in massive electro-statics. This will allow us to show that the Symmetron obeys a massive electrostatic analogy,in that it obeys the massive form of the Laplacian outside of a spherical conductor.

7.1 Field Produced in the Absence of Matter

Recall the potential of the symmetron, Veff = V (φ) + ρA(φ). Suppose there is no matter; inother words the mass density ρ = 0, which creates a vacuum. In this case Veff = V . Theeffective potential becomes a double well with a minima at φ = ±φ0 where φ0 = µ/

√λ, shown

in Figure (18).

Figure 17: A double well Symmetron potential in the absence of matter.

Assuming the potential remains close to φ0, the potential inside of the cavity, we can Taylor

31


expand around the minima to yield

Veff = V(φ=φ0) +V ′(φ=φ0)

1!(φ− φ0) +

V ′′(φ=φ0)

2!(φ− φ0).

Looking at the first term, we have

V (φ) =λ

4

(φ2 − µ2

λ

)2

− µ4

4λ2

V (φ = φ0) =λ

4

((µ√λ

)2

− µ2

λ

)2

− µ4

4λ2

= − µ4

4λ2.

Looking at the second term, we have

V ′(φ) =λ

2

(φ2 − µ2

λ

)× 2φ

V ′(φ = φ0) =λ

2

((µ√λ

)2

− µ2

λ

)× 2φ = 0

Looking at the third term, we have

V ′′(φ) = 3λφ2 − µ2

V ′′(φ = φ0) = 3λ

(µ2

λ

)2

− µ2

= 3µ2 − µ2

= 2µ2

Putting this all together we obtain

Veff ≈ −µ4

4λ2+

2µ2

2!(φ− φ0)2 =

µ4

4λ2+ µ2(φ− φ0)2

We can plug this into the field equation, ∇φ = ddφVeff (φ) to procure

∇2φ =d

dφVeff

=d

dφ

(µ4

4λ2+ µ2(φ− φ0)2

)= 2µ2(φ− φ0)

⇒ ∇2φ− 2µ2(φ− φ0) = 0,

Since, φ0 is constant, we can define φ = φ0 + ξ and thus, acquire

∇2ξ = 2µ2ξ, (31)

32


which is just the massive version of the Laplacian. Thus, within our vacuum approximation, thedeviation ξ of the symmetron field from its vacuum value φ0 is governed by the same equationas massive electrostatics. We have just shown that the Symmetron obeys a massive electrostaticanalogy.

7.2 Field Produced in Uniformly Homogeneous Fluid

Another simple circumstance to imagine is that the space is filled with a homogeneous fluid ofuniform density ρ. Thus

Veff = V (φ) + ρA(φ)

= −1

2µ2φ2 +

1

4λφ4 + ρ

(1 +

1

2M2φ2

)= ρ+

1

2

(ρ

M2− µ2

)φ2 +

1

4λφ4

If the density is sufficiently high, ρ/M2 > µ2 and Veff has a single minimum at φ = 0.

Figure 18: A single well Symmetron potential in a uniformly homogeneous fluid.

Assuming that the field remains close to this minimum, we can approximate the effectivepotential as

Veff ≈ ρ+1

2

(ρ

M2− µ2

)φ2

This produces a new field equation,

→ ∇2φ =d

dφVeff =

d

dφ

(ρ+

1

2

(ρ

M2− µ2

)φ2

)=

ρφ

M2− µ2φ

=

(ρ

M2− µ2

)φ

Thus in a homogeneous fluid of high density the symmetron also obeys the same equation as

33


the massive electrostatics but with a different value of the mass parameter:

∇2ψ =

(ρ

M2− µ2

)φ. (32)

7.3 Field Produced by a Sphere of Matter

As a prelude to analyzing the symmetron field in a more interesting configuration let us recallthe spherically symmetric solution to the massive electrostatic equation:

∇2ψ = m2ψ

If ψ is assumed to depend only on the distance from the origin, r, then

d2

dr2(rψ) = m2(rψ)

This is a second order differential equation and must have two independent solutions. Evidentlyrψ is an exponential function. Thus we may write the general solution as

ψ =A

remr +

B

re−mr

Alternatively, we may take our independent solutions to be hyperbolic functions and write

ψ =C

rsinh(mr) +

D

rcosh(−mr).

Now let us consider the symmetron field produced by a sphere of matter of radius Rc surroundedby otherwise empty space shown in Figure (19).

Figure 19: A sphere of matter of radius Rc in empty space with density ρ.

We assume that the matter has a uniform density ρ. This could be considered a caricatureof the field around a planet, star, or dwarf spheroidal galaxy. Thus we must solve Equation(31)for r>R and Equation(32) for r<R. Assuming that the solution is spherically symmetric and

34


depends only on r and making use of the results shown directly above it follows that

φ = φ0 +A

re−√

2µr +B

re√

2µr for r > R

φ =C

rsinh

(√ρ

M2− µ2r

)+D

rcosh

(√ρ

M2− µ2r

)for r < R

(33)

Assume that the field relaxes to its vacuum value sufficiently far from the sphere. In otherwords, φ→ φ0 as r →∞. This leads to the conclusion that B = 0. There is no reason for thesymmetron field to be singular at that center of the dense body. This leads to the conclusionthat D = 0. In order to determine A and C, we impose continuity of the symmetron field atthe surface of the sphere. Specifically, at r = R, rφ and d/d(rφ) are continuous yielding twoequations that must be satisfied by A and C:

φ0R + Ae−√

2µr = Csinh

(√ρ

M2− µ2R

)rφ continuous (34)

φ0R−√

2µRAe−√

2µr = C

√ρ

M2− µ2cosh

(√ρ

M2− µ2R

)d/d(rφ) continuous (35)

If we assume the spherical mass is sufficiently dense, then ρ/µ2M2 >> 1. Also we can defineα = ρR2/M2, enabling us to write

ρ

µ2M2>> 1

⇒ ρ

M2>> µ2

⇒√

ρ

M2− µ2 ≈

√ρ

M2

⇒(√

ρ

M2− µ2R

)≈√

ρ

M2R

=

√ρR2

M2

=√α.

Thus, we can use α as a simplification to rewrite Equation (34) and Equation (35) as

φ0R + Ae−√

2µr = Csinh√α rφ continuous

φ0R−√

2µRAe−√

2µr = C√αcosh

√α d/d(rφ) continuous

(36)

Solving Equation (36) for A and C we obtain

A = φ0R

(sinh√α−√αcosh

√α√

2µRsinh√α +√αcosh

√α

)C = φ0R

(1 +√

2µR√2µRsinh

√α +√αcosh

√α

).

35


Plugging these into Equation(33), we find the potential for the sphere is given by

φ = φ0 + φ0


√α√

2µRsinh√α +√αcosh

√α

)R

re−√

2µ(r−R) for r > R

φ = φ0

(1 +√

2µR√2µRsinh

√α +√αcosh

√α

)R

rsinh

(√αr

R

)for r < R

(37)

By utilizing the derivation techniques that we used for massive electrostatics, we could accu-rately approximate the potential of the Symmetron scalar field both inside and outside a sphereof radius R and density ρ. Now we will look at several approximations for α that enable us toexplore the thin and thick shell regimes for both the interior and exterior symmetron profile.

7.4 Interior Symmetron

First let us analyze the Symmetron profile in the interior of the shell. We will assume thatµR is negligible for all cases of interest in Equation(37) for r<R. We will also assume thatα = ρR2/M2 is a key parameter that characterizes the dense body and it defines two differentregimes. For

√α << 1, we have the “thick shell regime” and for

√α >> 1 we have the “thin

shell regime”. We will explore both of these limits below.

7.4.1 Thick Shell

In the thick shell approximation, we expect that the dense sphere will have no affect at all on theSymmetron field and further, the field will retain its vacuum value φ0 throughout the interior ofthe body. This is because bodies that fall in the thick shell regime essentially possess the samedensity as the ambient matter density such as a ball of gas. We can show this mathematicallyby imposing the thick shell limit, which states that

√α << 1. First we can neglect µR in

Equation(37) for r<R to obtain

φ = φ0

(1√

αcosh√α

)R

rsinh

(√αr

R

). (38)

Using Taylor expansions of sinh and cosh, and neglecting higher orders of α we find

sinh(x) = x+x3

3!+x5

5!+ . . . .⇒ sinh

(√αr

R

)≈√αr

R+

(√αr

R

)31

6

cosh(x) = 1 +x2

2!+x4

4!+ . . . .⇒ cosh

√α ≈ 1 +

α

2.

Plugging this back into Equation (38) yields

φ = φ0

(R

r

)1√α

(1

1 + α2︸︷︷︸)(√

αr

R+

1

6

(√αr

R

)3

︸︷︷︸),

cosh term sinh term

36


where we can explicitly see the cosh and sinh terms. We can use the binomial expansion forcosh term, 1/(1 + x)n ≈ 1− nx, and perform basic algebra on this equation to find

φ = φ0

(1− α

2+

1

6αr2

R2

)for r < R in thick shell (39)

to first order in α. Since,√α << 1, we can conclude that in the thick shell regime for the

interior of the sphere, the Symmetron essentially retains its vacuum value φ0.

7.4.2 Thin Shell

For the thin shell profile we have√α >> 1. Thin shell bodies have densities that vary greatly

from the ambient density such as a planet. Still neglecting µR in Equation(37) for r<R givesus Equation(38).Now, using the definition of sinh and cosh in terms of exponentials, we canapproximate them when the argument is large:

coshx =ex + e−x

2≈ ex

2for x >> 1

⇒ cosh√α ≈ e

√α

2for√α >> 1

sinhx =ex − e−x

2s ≈ ex

2for x >> 1

⇒ sinh

(√αr

R

)≈ e(

√αr)/R

2for√α >> 1.


φ = φ0

(1√α

)(1e√α

2

)R

r

)e(√αr)/R

2

We can reduce this down to the following,

φ =φ0√α

(R

r

)e−√α(R−(r/R)). for r < R in thin shell (40)

Thus, in the shin shell regime, the dense body has a large effect on the Symmetron field asexpected. On the surface of the sphere, the field starts with a value φ0/

√α, which is significantly

less than the vacuum value of φ0 since√α >> 1. As we move towards the center of the sphere,

the field value decays exponentially. The length scale of the decay is R/√α. Deep within

the interior of the dense sphere, the Symmetron field is exponentially small. As a result, theSymmetron field only penetrates a thin shell of thickness R/

√α near the surface of the body.

7.5 Exterior Symmetron

Now let us analyze the thick and thin shell regime of the Symmetron field in exterior of thedense spherical body.

37


7.5.1 Thick Shell

Again, we expect that the Symmetron field will not substantially deviate from its vacuum valueφ0 in the thick shell approximation since bodies in this regime possess a similar density to theambient density. We can show this mathematically by once again imposing that

√α << 1.

Now, neglect µR in Equation(37) for r>R to obtain

φ = φ0 + φ0


√α√

αcosh√α

)R

re−√

2µ(r−R). (41)

Using Taylor Series expansion for the sinh and cosh term we can find the numerator to be

sinh√α−√αcosh

√α u

(√α +

√α

3

6

)−(√

α

(1 +

α

2

))=−√α

3

3

and the denominator to be

√αcosh

√α u√α

(1 +

α

2

)≈√α.


φ = φ0 + φ0

(−√α

3

3√α

)R

re−√

2µ(r−R), (42)

which can be simplified to

φ = φ0 + φ0

(α

3

)R

re−√

2µ(r−R). for r > R in thick shell (43)

As we expected, this result is not very far from the vacuum value φ0 since√α << 1. At the

surface of the dense body the field is lower than the vacuum value by a factor of 1− (α/3). Thesmall deviation from the vacuum value decays exponentially with distance as we move outwardsfrom the surface. The decay has a length scale of 1/

√2µ.

7.5.2 Thin Shell

In the thin shell regime we take√α >> 1 and we can approximate the numerator as

sinh√α−√αcosh

√α ≈ e

√α

2−√αe√α

2= −(

√α− 1)e

√α

2

and the denominator as√αcosh

√α ≈

√αe√α

2.

Plugging this back into Equation (41) for r>R yields

φ = φ0 − φ0

(1− 1√

α

)R

re−√

2µ(r−R). for r > R in thin shell (44)

38


The field begins on the surface of the dense body with value φ/√α, which is significantly smaller

than the vacuum value. This is exactly what we saw for the interior profile.

7.6 Symmetron Force in the Exterior Profile

A test particle will experience a force due to the Symmetron field in the exterior of the densebody. This force could provide experimental relevance when attempting to detect the field. Wewill look at the force exerted on the test mass in both then thick and thin shell regime.

7.6.1 Force Analysis: Thick Shell

Assume that we have a test particle of mass m0 placed at a distance r from a dense sphereof radius R. Also assume that R << r << 1/µ and that the dense body is in the thick shellregime meaning that

√α << 1. With these assumptions we find that we can reduce Equation

(43) to

φ ≈ φ0 − φ0αR

3r. (45)

Additionally, we can make use of Equation(16) to show that the test particle experiences a forceof

Fφ =m0φ

20αR

3M2r2r. (46)

We can compare this to gravity and show that the ratio of the Symmetron force to the gravi-tational force is given by

FφFgrav

= 2

(φ0MPl

M2

)2

, (47)

where MPl = 1/√

8πG is the Planck mass and we employed the fact that α = ρR2/M2. In apaper published in 2010, Khoury and Hinterbichler found that gravity and the Symmetron arecomparable in the thick shell regime [11]. Presumably, this is demanded by the Symmetronfield if it is to serve as dark energy. This requirement results in the condition that(

φ0MPl

M2

)∼ 1. (48)

7.6.2 Force Analysis: Thin Shell

Again, assume that we have a test particle of mass m0 placed at a distance r from a densesphere of radius R with R << r << 1/µ. Now assume that the dense body is in the thin shellregime meaning that

√α >> 1. This simplifies Equation (44) to

φ ≈ φ0 − φ0R

r. (49)

The test particle will experience a force of

Fφ =m0φ

20R

M2r2r (50)

39


due to the gradient in the Symmetron field. Again, we can compare this to gravity to find that

FφFgrav

= 6

(φ0Mp

M2

)21

α. (51)

We can take into account the condition in Equation(48) to assert that, in the thin shell regime,the Symmetron force is weaker than the gravitational force by a factor of 1/α. Further, since√α >> 1, the Symmetron force is much smaller than the gravitational force. Thus we can assert

that tese forces exerted by the Symmetron on a spherical test mass are effectively negligible.

8 Conclusion and Future WorkModern cosmology aims to study the universe and its complex components. An influx of the-oretical and experimental astronomers into the field of cosmology coupled with improvementin observational technology has resulted in an abundance of recent discoveries. The most per-plexing of these discoveries is the apparent lack of energy density in the universe that manycosmologists strive to explain by dark energy. This theory attempts to explain the deficientenergy density in the universe, and further, it proposes that this energy is a force working tocounteract gravity and cause the accelerated expansion of the universe. Several classes of modelshave been put forth in an effort to understand dark energy. The type of model looked at in thisthesis is a type of quintessence model, which presents dark energy as a slowly evolving scalarfield living in a potential. Several of these scalar fields, such as the Chameleon and Symmetron,are able to blend in with terrestrial settings, eliminating the potential problems of a genericscalar field model. Typically, the equations that govern the Chameleon and Symmetron areinvolved and cumbersome. However, under conditions of interest on the terrestrial scale, theseequations reduce down to Laplace’s equation, thus representing electrostatic analogies. Exploit-ing electrostatic analogies when working with dark energy scalar field model simplifies complexmathematical equations to well understood and manageable forms. For both the Chameleonand Symmetron models, we find that they both obey Laplace’s equations in the vicinity of amassive spherical body on terrestrial scales. By utilizing the approach that physical phenomenadescribed by the same mathematical form of equation possess the same mathematical form, itis easy to derive valid, insightful profiles for the behavior of these fields.

The Chameleon scalar field was found to obey a classic electrostatic analogy [9]. Thus,source free, static, and massless electromagnetism provides an analog under which to solve theChameleon field in systems of interest. However, for the Symmetron, an unfamiliar and marginaltype of electromagnetism called massive electrostatics seems to provide the analog. We showedthat, although the usefulness of massive electrostatics itself is contentious, the techniques usedto solve problems in the field are nevertheless crucial to solving similar problems with theSymmetron field. Thus, without the convenience of electrostatic analogies, the solutions to boththe Chameleon and Symmetron scalar fields would be laborious, if not impossible to obtain. Itis also worth noting that these electrostatic analogies are more than a attractive mathematicaltrick. These analogies have powerful ramifications with regards to increasing experimentalsensitivity. Scientists hope that we can employ the results obtained from exploring scalar fieldmodels dark energy via electrostatic analogies to either detect the field or rule out its existenceas an explanation for dark energy.

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The next step will be to calculate the Symmetron profile for an ellipsoidal object to observewhether it exploits the lightning rod effect. There is no reason to believe that it will not, as itobey’s Laplace’s equations and maintains similar boundary conditions. It will also be relevantto calculate the force exerted on a test particle by the Symmetron field in the vicinity of thepolar region of the ellipsoidal object, which we expect to be increased relative to that of aspherical object. These calculations will enable the exploration into measuring and testing theSymmetron field itself. The possibility for testing the Chameleon field were made possible afterthe calculations performed by Jones-Smith and Ferrer, who introduced the idea that shapeenhancement of elongated objects could probe the Chameleon field [9]. Thus, it is probablethat there exists a similar shape enhancement for the Symmetron field that can be used toincrease experimental sensitivity.

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References[1] Hetherington, Norriss S., and W. Patrick McCray.“Cosmic Journey: A History of Scientific

Cosmology.” Center for History of Physics: American Institute of Physics, (2015). Web.

[2] Einstein, Albert. “Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie (Cos-mological Considerations in the General Theory of Relativity).” Koniglich PreußischeAkademie der Wissenschaften, Sitzungsberichte (Berlin), (1917): 142-152. Print.

[3] Jones-Smith. “Cosmology Primer”, (2015). Print

[4] “The Expansion of the Universe”. Digital Image. NASA and WMAP.http://map.gsfc.nasa.gov/universe/uni_fate.html. Web

[5] “The Cosmic Microwave Background Radiation Map”. Digital Image. Physics-Database.com, (17 Nov, 2012). Web.

[6] Perlmutter, Saul, Brian P. Schmidt, and Adam G. Riess. “Written in the Stars.” The 2011Nobel Prize in Physics Press Release. Nobelprize.org, (4 Oct. 2011). Web.

[7] High-Z SN search team. Digital Image. http://supernova.lbl.gov/. Nov. 2009. Web.

[8] Khoury, Justin and Amanda Weltman. “Chameleon Cosmology.” Physical Review D. 69.4,(2004). Print.

[9] Jones-Smith, Katherine and Francesc, Ferrer. “Detecting Chameleon Dark Energy via anElectrostatic Analogy.” Physical Review Letters. 108.22, (2012). Print

[10] Feynman, Richard P., Robert B. Leighton, and Matthew L. Sands. “Electrostatic Analogs.”The Feynman Lectures on Physics. Vol. II. Reading, MA: Addison-Wesley Publishing Com-pany, (1964). Print

[11] K. Hinterbichler and J. Khoury. “Screening Long-Range Forces through Local SymmetryRestoration”. Phys. Rev. Lett.104, 231301, (2010). Print.

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