Foundations of Massive Gravity

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FOUNDATIONS OF MASSIVE GRAVITY

ANDREW MATAS

Submitted in partial fulﬁllment of the requirements for the degree of

Doctor of Philosophy

Dissertation Advisor: Prof. Claudia de Rham

Department of Physics

CASE WESTERN RESERVE UNIVERSITY

August 2016 CASE WESTERN RESERVE UNIVERSITY

SCHOOL OF GRADUATE STUDIES

We hereby approve the dissertation of

Andrew Matas

candidate for the degree of Doctor of Philosophy∗

Committee Chair

Claudia de Rham

Committee Member

Stacy McGaugh

Committee Member

Glenn Starkman

Committee Member

Andrew J. Tolley

Date of Defense

May 19th, 2016

∗We also certify that written approval has been obtained for any proprietary

material contained therein.

i Contents

List of Figures vii

1 Introduction 1

1.1 Notions of gravity ...... 1

1.1.1 Gravity is an attractive force ...... 2

1.1.2 Gravity is curved space-time ...... 4

1.1.3 Gravity is a fundamental interaction ...... 5

1.2 A massive graviton and its implications ...... 9

1.3 Building models of massive gravity ...... 16

1.3.1 Stability ...... 17

1.3.2 Strong-coupling and continuity ...... 19

1.3.3 Extra dimensions and massive gravity ...... 22

1.4 Overview of dissertation ...... 24

2 Basics of Massive Gravity 29

2.1 Setting the stage ...... 31

2.1.1 Consistent m → 0 limit of a massive spin-1 ...... 31

ii 2.1.2 Ghost modes ...... 33

2.1.3 Massless spin-2 ...... 35

2.2 Toward a theory of massive gravity ...... 37

2.2.1 Non-interacting massive spin-2 ﬁeld ...... 37

2.3 Non-linear formulations of massive gravity ...... 41

2.3.1 Ghost-free potential ...... 43

2.3.2 Vielbein formulation ...... 44

2.3.3 St¨uckelberg for gravity ...... 45

2.4 Absence of the Boulware-Deser mode ...... 47

2.4.1 ADM form ...... 48

2.4.2 Counting degrees of freedom ...... 49

2.4.3 Mini-superspace ...... 51

2.5 Minkowski decoupling limit ...... 52

2.5.1 Interaction scales ...... 54

2.5.2 Explicit form of decoupling limit ...... 56

2.5.3 Galileon interactions ...... 58

2.6 Realization of the Vainshtein mechanism ...... 59

2.6.1 Vainshtein mechanism ...... 59

2.6.2 Boulware-Deser ghost ...... 61

2.6.3 Vainshtein mechanism for stars ...... 62

2.7 Bi-gravity and multi-gravity ...... 65

2.8 Summary ...... 66

iii 3 Dimensional deconstruction 68

3.1 Deconstruction of a scalar ...... 71

3.2 Deconstruction for gravity ...... 74

3.3 Deconstruction of the metric ...... 76

3.4 Consistent deconstruction procedure ...... 79

3.4.1 Diﬀerent discretization of derivative ...... 84

3.4.2 Multi-gravity ...... 85

3.5 Strong-coupling from deconstruction ...... 86

3.5.1 Strong-coupling scale at large N ...... 87

3.5.2 The 5D origin of the strong-coupling scale ...... 91

3.5.3 Gauge ﬁxed continuum theory ...... 91

3.5.4 Generic gauge for continuum theory ...... 92

3.6 Outlook ...... 93

4 Kinetic interactions in massive gravity 95

4.1 Kinetic term for massive spin 1 ...... 98

4.2 Candidate kinetic interactions ...... 100

4.2.1 Leading order interactions ...... 101

4.2.2 Non-linear St¨uckelberg decomposition ...... 103

4.2.3 Non-linear completions ...... 104

4.3 Deconstruction-inspired ansatz ...... 106

4.4 Systematic argument ...... 109

4.4.1 St¨uckelberg criteria ...... 110

iv 4.4.2 Decoupling limits ...... 112

4.4.3 Algorithm ...... 114

4.4.4 Quadratic order ...... 116

4.4.5 Cubic order ...... 116

4.4.6 Quartic order ...... 119

4.4.7 Extension to higher order ...... 122

4.5 Relationship to the coupling to matter ...... 124

4.5.1 Eﬀective vielbein coupling ...... 124

4.5.2 Eﬀective metric ...... 126

4.6 Outlook ...... 128

5 Galileon radiation from binary systems 130

5.1 Galileons with diﬀerent scales ...... 133

5.2 Perturbations around a spherically symmetric background ...... 134

5.2.1 Keplerian source ...... 134

5.2.2 Background Galileon ﬁeld ...... 137

5.2.3 Perturbations for Galileon ...... 138

5.3 Formula for the power emission ...... 141

5.4 Power emission for the cubic Galileon ...... 144

5.5 Power emission for the general Galileon ...... 147

5.5.1 Mode functions ...... 148

5.5.2 General form for the power ...... 149

5.5.3 Power in quadrupole ...... 151

v 5.5.4 Breakdown of perturbation theory ...... 152

5.5.5 Hierarchy of masses ...... 157

5.6 Hierarchy between two strong coupling scales ...... 159

5.6.1 Power emission ...... 160

5.6.2 Quadrupole radiation ...... 162

5.7 Outlook ...... 162

6 Conclusions 165

6.1 Summary ...... 165

6.2 Outlook ...... 167

A Lovelock interactions 169

vi List of Tables

4.1 Coeﬃcients for ghost-free kinetic term at quartic order ...... 120

vii List of Figures

1.1 Newtonian T-shirt...... 3

1.2 Fundamental Forces Comic...... 5

1.3 Scattering processes...... 7

1.4 Gravitational wave polarizations...... 11

1.5 Hulse-Taylor pulsar...... 12

1.6 Ghost instabilities...... 17

1.7 Vainshtein radius...... 20

1.8 Kaluza-Klein compactiﬁcation...... 22

viii Foundations of Massive Gravity

ANDREW MATAS

General Relativity (GR) is a relativistic theory of gravity which has a large number

of theoretical and observational successes. From the perspective of quantum ﬁeld

theory, GR can be thought of as a theory of a massless spin-2 particle called the

graviton. It is a fundamental question to ask how the graviton behaves if it has a

small but non-zero mass. In this dissertation I shall study the recently constructed

theory of ghost-free massive gravity, which avoids the pernicious Boulware-Deser ghost that had thwarted previous attempts to study massive gravity.

In Chapter 1 I will give a broad overview of the context, motivation, and model- building issues underlying massive gravity. Then in Chapter 2 I will give a detailed pedagogical introduction to ghost-free massive gravity focusing on concepts that will be used throughout the remainder of the work.

In Chapter 3 I will derive the ghost-free structure of massive gravity from an extra dimensional perspective through a process known as Dimensional Deconstruction.

Before my work it had been an open question whether Deconstruction could be consistently applied to gravity. The key insight relies on using the elegant formulation of General Relativity in terms of the vielbein.

Inspired by Deconstruction, in Chapter 4 I will discuss the possibility of non- standard kinetic interactions in massive gravity. I will show that the only consistent derivative interactions for a massive spin-2 particle must be the same as in GR. This is remarkable because there is no known symmetry reason for this to be the case,

ix since massive gravity breaks diﬀeomorphism invariance.

Finally, in Chapter 5, in order to connect with observations I shall consider the radiation emitted from binary systems in Galileon theories, which are scalar theories that can mimic the behavior of a massive graviton. This work extends the understanding of the Vainshtein screening mechanism into a time-dependent situation.

x Chapter 1

Introduction

In this work I will be concerned with an extension of General Relativity (GR) known as massive gravity. Before diving into the details of this theory, it is useful to recall the basic ideas underlying our modern understanding of gravity.

1.1 Notions of gravity

On the one hand, gravity is a familiar phenomenon. Gravity makes apples fall from trees and is responsible for the tides.

On the other hand, gravity is mysterious. Gravity connects us to the universe.

Newton taught us that the same force that pulls us to the Earth also holds the Earth to the Sun, the Sun to the Galaxy, the Galaxy to the Local Group, the Local Group to the

Local Supercluster, and that holds galaxy clusters together in an enormous network of ﬁlaments mapped by large scale structure. Gravity collapses stars into black holes, regions so dense that light cannot escape. Under the inﬂuence of gravity, the universe

1 expands. The expansion rate has recently been observed to be accelerating, and if this acceleration continues forever then ultimately all matter and energy will be diluted away to almost nothing.

In this opening section I will give three different perspectives on the nature of gravity. These different pictures look quite different, but all of these pictures are useful and relevant in different regimes, and whenever two or more are valid they agree in their quantitative predictions.

1.1.1 Gravity is an attractive force

The theory of gravity published by Isaac Newton in 1687 states that gravity is an attractive force between massive objects. Between any two objects is an attractive force proportional to the masses of the objects, and inversely proportional to the square of the distance between them. The proportionality constant is Newton’s gravitational constant G, which sets the strength of the gravitational force. These statements can be summarized in an equation so simple and profound that it can be found on T-shirts

(see Figure 1.1).

From this one (apparently) simple equation, one can make an enormous array of precise predictions: planets orbit the sun in elliptical orbits and a bowling ball and a basketball dropped simultaneously from the top of the Tower of Pisa will hit the ground at the same time.

Despite its success and continued utility, Newton’s theory of gravity is unsatisfying in at least one respect. In Newton’s theory the gravitational force is instantaneous:

2 Figure 1.1: This T-shirt illustrates the Newtonian view of gravity as a force. Image from http://www.nerdytshirt.com/physics-tshirts.html. the force between two objects at a given time depends on the distance between the objects at that same time. This raises puzzles in extreme situations with fast moving objects. For example, if the sun were to suddenly disappear1, would this information be transmitted to earth instantaneously, or would the information take a finite time to propagate? Intuitively we might say that there should be a finite time. This problem is even more compelling from the perspective of special relativity. In special relativity there is no notion of absolute time, and the need to have a finite time of propagation goes from being intuitive to being mandatory.

Our modern understanding of gravitation is given by the theory of GR, developed by Einstein, to which we now turn.

1For readers rightly uncomfortable with the non-conservation of mass in this example, we can imagine that a mischievous alien race has very suddenly and quickly moved the sun to another place.

3 1.1.2 Gravity is curved space-time

Within the framework of GR, gravity is interpreted as the curvature of space-time.

Freely falling masses follow straight paths through curved space-time. When you

stand on a scale, the scale is not supporting you against the force of gravity, the scale

is preventing you from following the natural straight line path through space-time.

The curvature of space-time is aﬀected by matter and energy through Einstein’s

Equations

1 8πG R − Rg = T . (1.1) µν 2 µν c2 µν

On the left hand side are quantities describing the curvature, and on the right hand

side the quantity Tµν describes the mass (and energy) of matter living in space-time.

We can also see that this equation relates gravity (through the presence of Newton’s constant G) and relativity (through the speed of light c).

Whenever space-time is approximately ﬂat, GR approaches Newton’s theory. How-

ever, there is a key diﬀerence, even in this limit. The gravitational force is no longer

instantaneous. Einstein’s equations allow for propagating wave solutions. In the

case when space-time is approximately ﬂat, Einstein’s equations reduce to a linear

wave equation describing propagating disturbances in the metric. Solutions to this

linearized equation describe gravitational waves2, which travel at the speed of light

c. The speed can be seen directly from the linearized wave equation. The existence

2The notion of a gravitational wave is not limited to linearized GR, however linearized GR provides a very useful description of freely propagating gravitational waves.

4 Figure 1.2: ”Of these four forces, there’s one we don’t really understand.” ”Is it the weak force or the strong–” ”It’s gravity.” From https://xkcd.com/1489/. of gravitational waves propagating at c provides an answer to the question posed about the sun’s disappearance: the earth would orbit for eight minutes before careen- ing oﬀ into space. Shortly before the time of this writing, gravitational waves have been directly detected by the Laser Interferometer Gravitational-Wave Observatory

(LIGO) [1].

GR has been summarized by Wheeler, “Space-time tells matter how to move; matter tells space-time how to curve” [2]

1.1.3 Gravity is a fundamental interaction

A different picture of forces has developed through the development of particle physics and the Standard Model (see Figure 1.2). While gravity governs the motion in the heavens and keeps us fixed on earth, the forces responsible for keeping the earth solid and warm apparently behave quite differently. There are three (known) such forces, the electromagnetic force and the strong and weak nuclear forces. Our modern understanding of these forces is that they are described by the exchange of force-

5 carrying particles, associated with quantum mechanical ﬁelds.

It is possible, and fruitful, to formulate GR in similar terms, at least in special

situations such as when space-time is asymptotically ﬂat and the curvature is not too

large (for example see [3], for a more modern eﬀective ﬁeld theory approach see [4]).

This allows us to think of gravity on similar footing as other known forces, and allows

us to use techniques from other areas in physics to understand gravity. We emphasize

that while this gives us a diﬀerent picture of gravity, it agrees quantitatively in terms

of physical observables.

A familiar example of a fundamental interaction is electromagnetism. Like charges

repel and opposites attract because charges interact with the electromagnetic ﬁeld.

An electric charge generates an electric ﬁeld, and this ﬁeld spreads through space

until it pushes or pulls on another charge. Associated with the electromagnetic ﬁeld

spreading is a propagating wave, which we experience (at least for a range of wave-

lengths) as light. The color of light is related to the wavelength of this wave. Light

travels at the speed of light3 c.

Associated with the electromagnetic ﬁeld is a quantum mechanical particle (or

indivisible packet of energy) called the photon. Quantum mechanically, the photon

is the basic force-carrying entity of the electromagnetic force (see Figure 1.3). Two

negatively charged electrons repel as they exchange photons. The fact that electro-

magnetic waves travel at the speed of light translates into the fact that the photon is

3It may sound tautological to say light travels at the speed of light, but strictly speaking calling c the ‘speed of light’ in special relativity is misleading. In special relativity, the parameter c describes the space-time geometry and is not directly related to any particle. In practice this does not usually matter since the photon is massless (or at least has a very tiny mass) and so it travels at (or at least very near) c.

6 Figure 1.3: To illustrate the picture of a force as being due to an exchange of particles, in the left ﬁgure we see electrons e− scattering by exchanging a photon γ. In the right ﬁgure, the earth and moon interact by exchanging a graviton hµν. Time runs vertically upward in these diagrams.

a particle with zero mass.

We can make similar statements about the gravitational ﬁeld. At the classical

level, as we have discussed, masses interact with the gravitational ﬁeld. Gravitational

waves propagate through this ﬁeld. Quantum mechanically, there is expected to be

a particle of the gravitational ﬁeld that is called the graviton. One can think of

(and calculate) the force between the earth and the moon as being mediated by an

exchange of gravitons (see Figure 1.3).

This framework allows us to incorporate quantum eﬀects into gravity in a calcula-

ble way, when the energies involved are not too large. Scattering processes involving

exchanges of gravitons, with typical energy E, can be computed as a perturbative series in E/MPl, where MPl is an energy scale known as the reduced Planck scale

r c M ≡ ~ = 2.4 × 1018 GeV/c2. (1.2) Pl 8πG

This is a remarkable quantity which combines quantum mechanics (through Planck’s

7 constant ~), relativity (c), and gravity (G). The Planck scale is very large (the

−15 Large Hadron Collider (LHC) probes energies that are a ‘measly’ 10 MPl) and therefore the gravitational interaction is very weak (the electrical repulsion between two electrons is about 1042 times larger than the gravitational attraction).

When fundamental particles collide at energies E & MPl, the perturbative picture we are describing of graviton exchange breaks down and a more complete description is needed (indeed the very notion of a particle may cease to make sense in this regime).

We will not discuss this issue further in this work.

In macroscopic situations relevant for astrophysics, gravity is of course an important force even though quantum eﬀects are not relevant. This can occur because the systems involved (such as the sun) are macroscopic and contain many particles. As a result, even though the mass of the sun is enormously larger than MPl (by a factor

38 of around 10 ), E/MPl is negligibly small for each individual particle in the system.

We emphasize that this picture underlies much of modern cosmology. Future observations may detect primordial gravitational waves from the early universe. Within the framework of cosmic inflation, these originate as quantum fluctuations in the gravitational field in an inflating background [5]. Primordial gravitational waves may be observed by looking at B-mode polarization in the cosmic microwave background [6,7], or may contribute to a stochastic gravitational wave background [8].

8 1.2 A massive graviton and its implications

In this work we will be interested in extending GR. The graviton is normally taken to be a massless particle with spin-2. However, intuitively, since mass is a continuous parameter, one may expect that there should only be a small diﬀerence in observable quantities in a theory of a massless graviton, and a theory of a graviton with a small mass.

In this section I will discuss some of the phenomenological and theoretical implications of massive gravity (the theory of a massive graviton). Massive gravity introduces one new dimensionful parameter, the mass of the graviton m (which I will often quote in units of eV/c2). While I report some observational bounds, this is not meant to be a comprehensive review of different bounds on the graviton mass (see [9,10] for some discussions), but rather to give a sense of how different observations fit together.

Gravitational waves. As we have discussed, gravitational waves are propagating disturbances in the gravitational ﬁeld. The eﬀects of the mass can be seen very directly in terms of these waves.

The mass of the graviton aﬀects gravitational waves in two ways.

First, a mass modiﬁes the speed of propagation. The speed of the waves is determined by its dispersion relation, which relates the frequency and wavelength of a gravitational wave. The dispersion relation (on a ﬂat background geometry) is

9 determined by the mass of the particle associated with that wave by4

m2c4 ω2 = c2k2 + , (1.3) ~2

where ω is the angular frequency and k is the wavenumber5. A packet of gravitational

waves travels at the group velocity

dω m2c5 vg ≡ = c − + ··· < c, (1.4) dk 2~2ω2

where ··· refers to terms that are negligibly small when the mass is small mc2 ~ω.

When m 6= 0, waves of diﬀerent frequency travel at diﬀerent speeds. This can lead to an experimental test of the graviton mass, [9,12].

Recently gravitational waves have been directly detected by LIGO from the inspiral of a binary black hole system. When m is nonzero, low frequency gravitational waves travel more slowly than high frequency waves. This modiﬁes the predicted gravitational waveform by inducing a frequency dependent phase delay relative to the prediction for GR. Performing a search for this kind of modiﬁcation to the waveform, LIGO has placed a bound on the graviton mass m < 1.2 × 10−22 eV/c2 at 90%

conﬁdence [13].

Second, a mass introduces new polarization states (see Figure 1.4). While the

waves associated with a massless spin-2 particle have 2 polarizations, the waves asso-

4“This is the sophisticate’s version of the layperson’s E = mc2.”—Zee, Quantum ﬁeld theory in a nutshell [11]. 5Note that even though ~ appears, here we are talking about classical gravitational waves. Quan- tum mechanics comes in through interpreting the parameter m in the dispersion relationship as the mass of a particle.

10 Figure 1.4: This figure shows the effect of different gravitational wave polarizations acting on a ring of test particles. The direction of motion of the gravitational wave is shown in each figure. The top row displays the tensor modes (+, ×). The second row shows the two vector modes (note that the x and y axes are marked on the figure). The last row shows the two possible scalar modes, the left is the longitudinal mode and the right is the conformal mode. GR has the two tensor modes only. A healthy theory of massive gravity has two tensor, two vector, and one scalar polarization. The sixth scalar mode appears as a ghost if it is present in massive gravity. This figure is based on Figure 8 of [9].

11 Figure 1.5: The decay of the orbital period of the binary pulsar system PSR B1913+16, compared with the prediction from General Relativity. Figure from [14] ciated with a healthy theory of a massive spin-2 particle have ﬁve. As we will review in greater detail in Chapter 2, understanding the physics of these additional states

(in particular the scalar mode) is crucial to understanding massive gravity.

Gravitational waves have been indirectly detected by studying binary pulsars.

In 1974 Hulse and Taylor discovered the ﬁrst binary pulsar system PSR B1913+16

[14,15]. This system consists of two neutron stars in orbit around a common center.

GR predicts that as the neutron stars orbit, the system emits gravitational waves and the orbit slowly decays. Subsequent observations of PSR B1913+16 (and other systems discovered later, for a review see [16]) conﬁrm that the rate of the spin- down is consistent with the prediction from the emission of gravitational waves (see

Figure 1.5).

In massive gravity we expect the spin-down rate of pulsars to be (ever so slightly)

12 larger because the system can radiate into the additional polarization states. I will discuss this further in Chapter 5 where we discuss calculating the gravitational wave emission from binary pulsar systems.

Gravitational force. Newton’s law of gravitation can be modified by adding a mass. Within the Newtonian approximation we expect the mass will effectively cut off the gravitational force at distances larger than the Compton wavelength λg ≡ ~/mc.

The smaller the mass, the larger the range of the gravitational force. We can express this in terms of the gravitational potential sourced by a star of mass M

e−r/λg V (r) = GM . (1.5) r

For distances that are short compared with the Compton wavelength r λg, this looks like the potential ∼ 1/r of Newtonian gravity. However for large distances r λg, the force is exponentially suppressed. This eﬀect is known as the Yukawa suppression. Solar system tests of planetary motion bound the Compton wavelength

12 −22 2 to be λg < 2.8 × 10 km or m < 4.4 × 10 eV/c [9].

In addition, the new polarization states mediate additional forces, often called

ﬁfth forces since it is often said that there are four forces in fundamental physics

(recall Figure 1.2). Physically this occurs because there are more polarization states that can be exchanged contributing to an attractive force in a Feynman diagram such as Figure 1.3. These polarization states couple to matter diﬀerently than the tensor modes present in GR and their eﬀect cannot simply be absorbed into a rescaling of

13 Newton’s constant, as we will describe in more detail in Section 1.3.2. Since gravity

is well tested in the solar system, these forces need to be screened using a screening

mechanism. Screening mechanisms allow the gravitational forces to depend on the

local environment in a way that recovers GR in regions where it is successful [17].

Despite the existence of screening mechanisms, solar system tests are stringent

enough to put tight bounds on the graviton mass from this additional contribution

to the Newtonian force. In particular these forces predict an anomalous precession

of the moon’s orbit around the earth, which is tightly constrained by Lunar Laser

Ranging experiments [18]. This bound is model dependent, but can be as strong as

m < 10−32 eV/c2 in some models [19,20].

Cosmic acceleration. Over the past 15 years a wide array of observations have conﬁrmed that the universe’s rate of expansion is accelerating [21]. In GR, in the absence of spatial curvature the rate of expansion of the universe, called the Hubble parameter H, is related to the energy density in the universe ρ by the Friedman equation

8πG H2 = ρ. (1.6) c2

Within GR, an explanation for the acceleration of the universe can be given by the

existence of a cosmological constant. The cosmological constant contributes a con-

stant amount of energy density to the Friedman equation, which leads to a constant

rate of expansion and therefore an exponential growth in the scale factor of the uni-

14 verse. Physically, we can interpret this constant energy density ρvac as energy density associated with the vacuum itself. The data suggests that the vacuum energy has a

obs −3 4 value ρvac ∼ (10 eV) [21]. However, there is still room for alternative explanations, and ultimately experiment should decide between diﬀerent possibilities.

The possibility of alternative explanations of cosmic acceleration is especially tantalizing in light of the cosmological constant problem (for reviews see [22,23]). Quan- tum mechanically, every ﬁeld has a zero point motion. The ﬁeld will contribute some energy even in vacuum. The vacuum energy we observe is a combination of the ‘bare’

(intrinsic) vacuum energy, plus contributions from the vacuum energy of each ﬁeld6

obs bare 4 4 ρvac = ρvac + α1m1 + α2m2 + ··· , (1.7)

bare where ρvac is the bare value (which is not computable although may in principle be determined in a fundamental theory) α1,2 are constants, and m1,2 are the masses of particles associated with the ﬁelds present in nature, and the ··· refers to a sum over all ﬁelds. The heaviest fundamental particle that has been observed is the top quark,

2 4 mtop ∼ 170 GeV/c , leading to a contribution to the vacuum energy ∼ (170 GeV) ∼

57 obs 10 ρvac. In order to produce the small observed value, the bare value of the vacuum

bare energy ρvac must cancel the contribution to the top quark, as well as the corrections from other particles, to enormous precision. This is logically possible, but extremely uncomfortable, and such a high degree of ﬁne-tuning of the theory suggests there may

6More precisely there is an equation of this form when one performs a matching between a low energy effective field theory valid at energies below the mass of (say) m1 and a UV theory valid above the mass of m1. In this setup the coefficients α1, α2, ··· are calculable in terms of parameters of the UV theory. See [23] for more discussion.

15 be more to the story.

As a result there has been a push to ﬁnd alternative explanations for cosmic acceleration. Massive gravity can address cosmic acceleration in various ways. It may be possible for a massive graviton to degravitate a large cosmological constant [24,25].

Roughly speaking we expect gravity to be cutoﬀ on large distances because of the

Yukawa suppression. As a result massive gravity might not see the full strength of a cosmological constant. An alternative scenario is called self-acceleration where the graviton forms a condensate whose energy density drives the cosmic acceleration [26].

In order to be relevant for cosmic acceleration, the graviton should have a mass

m ∼ H ~ ∼ 10−33 eV/c2, (1.8) 0 c2

where H0 is the Hubble parameter today. We will often take this as our ﬁducial value for m.

1.3 Building models of massive gravity

Having described some of the potential applications of a massive graviton, I will now discuss issues in constructing realistic models of a massive graviton. Finding viable theories is not a trivial task. Indeed, a consistent interacting theory of a massive graviton was not discovered until recently [27, 28]. Here we describe some of the major themes that we will discuss throughout the rest of this thesis.

16 Figure 1.6: Illustrating different ways to think about a ghost instability. The left figure shows how a ghost manifests itself classically. The right figure illustrates an example quantum process where the vacuum decays into ghosts and ordinary matter, in this case a photon. As an example of a specific theory which would contain this 1 2 −2 2 2 vertex, we may consider the lagrangian L = + 2 (∂φ) + Λ φ Fµν, where Λ is a scale needed for dimensional reasons and where Fµν = ∂µAν − ∂νAµ is the electromagnetic field strength tensor. The real scalar field φ is a ghost field because it has the wrong sign kinetic term.

1.3.1 Stability

The ﬁrst challenge in constructing a healthy theory of massive gravity is related to the fact that there are six possible polarizations for a symmetric two index tensor hij

(see Figure 1.4), but a healthy theory of a massive spin-2 ﬁeld should only have ﬁve

polarizations, corresponding to the 2S + 1 = 2(2) + 1 = 5 spin states [29].

When the sixth mode appears, it appears as a ghost instability. This problem was

recognized already for non-interacting massive spin-2 particles in [30]. As recognized

in that paper, there is a speciﬁc choice of mass term in the Lagrangian which removes

the instability. However, a realistic model of massive gravity necessarily involves

an interacting massive spin-2 particle. Early attempts to build models of massive

gravity found that the ghost that was vanquished at the linear level returned when

interactions was included, this instability is known as the Boulware-Deser (BD) ghost

[31].

17 A ghost is a particle with negative kinetic energy. This is extremely problematic

as can be seen from multiple points of view (see for example [32]). At the classical

level, we can see how a ghost behaves by considering a particle released from height

on a potential well (see Figure 1.6). Since the total energy E = K + V (where K

is the kinetic energy and V is the potential energy) must be conserved, an ordinary

particle with K > 0 will roll in the potential without ever rising above the height from which it was dropped. Energy conservation binds the particle to a ﬁnite region of the potential. However for a ghost particle with K < 0, the particle can rise up

the potential, increasing V , while simultaneously moving faster and faster to make K

more and more negative, keeping the combination E = K + V constant. This strange behavior is clearly unphysical. Of course, analogous results hold in ﬁeld theory when ghost-like and ordinary ﬁelds interact.

The most serious version of the problem is that if the ghost interacts with ordinary matter, then a process is possible where the vacuum spontaneously decays into combinations of ghosts and ordinary particles. This can happen because the positive energy of the ordinary particle(s) is compensated by the negative energy of the ghost(s). Furthermore, if we trust the theory to arbitrarily high energies, then there is an infinite amount of available phase space for this process and so the decay rate is infinite. Even if we consider a more realistic scenario where we do not trust the theory to arbitrarily high energies and we regulate the divergence by cutting off the allowed energy, one still finds that a ghost is a very fast instability [33].

The BD ghost can be avoided by carefully choosing the potential interactions

[27, 28]. I will make use of these stability requirements in Chapter 4 when I will use

18 these requirements to constrain extensions to the kinetic structure of massive gravity.

While this discussion can become technical, there is also something beautiful about

how the internal consistency of a theory is so constraining.

1.3.2 Strong-coupling and continuity

Intuitively, as long as the mass of the graviton is suﬃciently small, all predictions for

physically observable quantities in massive gravity should be close to those in GR.

While this is widely thought to be the case, showing how this occurs in detail is not

straightforward.

In GR, a standard approach is to treat the gravitational ﬁeld as weak, or more

precisely to take the metric to be approximately ﬂat. Various observable quantities,

such as the orbit of the moon around the earth or the bending of light by the sun,

can be computed within this regime.

However, when one applies the same weak ﬁeld approximation to massive gravity,

one ﬁnds there is a discrete diﬀerence between the predictions of observable quantities

in massive gravity, and the predictions of observable quantities in GR, even when the

mass m is arbitrarily small. For example, fixing Newton’s constant G from non- relativistic terrestrial probes, one finds that the predictions for the bending of light by the sun are different in the two theories. This is known as the van Dam-Veltman-

Zakharov (vDVZ) discontinuity [34, 35]. Naively, it implies that the graviton mass must be identically zero, which is an extremely strange conclusion.

A possible resolution to this puzzle was proposed by Vainshtein [36] (see also

19 Figure 1.7: An illustration of the Vainshtein mechanism. For distances less than the Vainshtein radius r?, strong-coupling effects suppress the interactions of extra graviton modes and leads to the recovery of GR. Outside of this radius, the fifth force effects are unsuppressed and there are in principle large deviations from GR. The boundary is smeared to represent the fact that there is a continuous transition between these different regions.

20 [37,38] for a more modern take).

Vainshtein found that around a matter distribution with mass M, it is only consistent to treat the gravitational ﬁeld as weak at large distances. In ghost-free theories of massive gravity larger than a characteristic scale now known as the Vainshtein radius

GM 1/3 r ∼ . (1.9) ? m2

−33 2 7 For the sun, taking m ∼ 10 eV/c , the Vainshtein radius is r? ∼ 100 pc.

It may seem strange that the gravitational ﬁelds can be treated as weak in GR, but are very strong in massive gravity. However, while the calculation of an observable quantity may be diﬀerent in the two theories, the theories should agree on physically observable quantities. In fact, we will see that the strong-coupling is responsible for decoupling the helicity-0 polarization state and the usual helicity-2 modes present in

GR remain weakly coupled.

While the Vainshtein mechanism is well understood in static, spherically symmetric situations (for example see [39,40]), it is an ongoing research eﬀort to understand how the Vainshtein mechanism operators in more general settings. In Chapter 5 I will study the Vainshtein mechanism in a time dependent situation, binary pulsars.

21 Figure 1.8: On the left hand side we see how extra dimensions can be wrapped up so that they are hard to see at large distances. On the right hand side we see the associated Kaluza-Klein tower of gravitons. The mass scale is set by the size of the extra dimension ∼ a.

1.3.3 Extra dimensions and massive gravity

Massive gravity is intimately connected to GR when formulated with extra dimensions, where extra dimensions refer to spatial dimensions beyond the three we observe.

Since GR in more spatial dimensions is a consistent theory, we can use extra dimensions as a tool to understand consistent theories of massive gravity. Indeed ghost-free massive gravity theory was discovered by using extra dimensional models [41–43].

Historically, an important extra dimensional model giving rise to a theory of massive gravity is the Dvali-Gabadadze-Porrati (DGP) model [44]. The DGP model is a brane-world model consisting of a brane (or membrane) living in a bulk space with four spatial dimensions of infinite extent. Ordinary standard model fields are confined to the brane, but gravity can explore the bulk.

DGP exhibits many features of massive gravity. At suﬃciently short distances

7In fact this is likely an underestimate as it assumes that the sun is in a vacuum with no other matter present. As one considers spheres of larger size around the real sun, the spheres will enclose additional sources of matter which will in turn contribute to the Vainshtein screening, and so in reality the Vainshtein regime may be even larger.

22 (smaller than the Hubble scale), an observer on the brane sees a graviton that has a soft (resonance) mass [45, 46]. It avoids the Boulware-Deser ghost, and the theory exhibits a Vainshtein mechanism. Studying DGP led to many insights that have been relevant for massive gravity. For example, Galileon scalar ﬁeld theories [39] (see [47] for an introduction), which are a class of scalar theories which exhibit the Vainshtein mechanism and play a large role in massive gravity, were discovered by studying

DGP [48,49].

Another much older approach to extra dimensions, that does not use branes, is to have compact (ﬁnite size) extra dimensions (see Figure 1.8). This is known as a Kaluza-Klein (KK) compactiﬁcation [50, 51]. The KK model consists of GR in a

4 + 1 dimensional space-time, with one dimension compactiﬁed on a circle. From the perspective of a 3+1 dimensional observer, this theory looks like a theory of a massless graviton (as well as additional massless states8, and an inﬁnite tower of massive gravitons, with masses given by the inverse size of the compact dimension, in appropriate units.

It is tantalizing to use this infinite tower to find a consistent theory of a massive graviton. However, in the simplest KK compactification, the tower is evenly spaced in mass so there is never a regime where only one graviton is relevant. At energies much below the mass of the lightest graviton in the tower, none of the massive states are relevant. When we work at energies large enough to produce the first massive graviton, however, there are many other states with a similar mass that cannot be neglected.

8In particular these massless states are a massless spin-1 (which could be associated with electromagnetism) and a massless spin-0 (which is not associated with any observed particle).

23 A ‘holy grail’ of massive gravity would be to ﬁnd an explicit compactiﬁcation which produced a single massive graviton widely separated from the other states in the tower.

I will apply some of these to studying massive gravity using an approach known as Dimensional Deconstruction. Deconstruction consists of taking a theory deﬁned in extra dimensions, and discretizing the extra dimension to produce a theory of interacting ﬁelds in one dimension lower. Dimensional Deconstruction has been suc- cessfully applied to Yang-Mills theories [52, 53]. However an outstanding issue has been to apply Deconstruction to gravity in a way that removes the BD ghost [54–59] .

In Chapter 3 I will resolve this by providing a procedure that shows how a consistent theory of massive gravity emerges from Deconstruction, using the elegant vielbein formulation of GR.

1.4 Overview of dissertation

Publications during dissertation work

During my dissertation work I have been involved in multiple research projects which have led to several publications and developments. Some of these publications form the basis of chapters in this work. I have marked these with a dagger (†).

• Extra dimensions

I have shown how to derive ghost-free massive gravity using Dimensional De-

construction, as I describe in Chapter 3. Previously it had not been known how

24 to apply Deconstruction to gravity to produce a ghost-free theory. This work

formed the basis of follow-up work on extensions of massive gravity that I will

also describe.

– † “Deconstructing Dimensions and Massive Gravity.” C. de Rham, A. Matas and A. J. Tolley, Class. Quant. Grav. 31, 025004 (2014) arXiv:1308.4136 [hep-th].

In addition, I have studied the motion of classical point particles in extra dimensions. This work is completely separate from massive gravity. Compactifications can break symmetries globally, for example a standard Kaluza-Klein compactification breaks 5D Lorentz invariance, and if the extra dimensions have the topology of a Klein-Bottle, the compactification breaks translation invariance as well. This breaking of symmetries can lead to additional forces on point particles moving through a compact space which are not present in flat space, which we describe. These results may have implications for brane-world scenarios.

– “Point particle motion in topologically nontrivial space-times” A. Matas, D. Muller and G. Starkman, Phys. Rev. D 92, no. 2, 026005 (2015) arXiv:1505.01937 [hep-th].

• Extensions of massive gravity

Given the construction of ghost-free massive gravity, there has been a lot of interest in extending the theory and exploring the full space of consistent related theories. I have been particularly interested in extensions to the kinetic structure (interactions involving derivatives), as I will describe in Chapter 4. My work on this has implications for other models in the literature, such as to generalized matter couplings and to charged spin-2 ﬁelds.

– † “New Kinetic Interactions for Massive Gravity?” C. de Rham, A. Matas and A. J. Tolley, Class. Quant. Grav. 31, 165004 (2014) arXiv:1311.6485 [hep-th].

25 – “Interactions of Charged Spin-2 Fields” C. de Rham, A. Matas, N. Ondo, and A. J. Tolley, Class. Quant. Grav. 32, 175008 (2015) arXiv:1410.5422 [hep-th]. – “New Kinetic Terms for Massive Gravity and Multi-gravity: A No-Go in Vielbein Form” C. de Rham, A. Matas and A. J. Tolley, Class. Quant. Grav. 32, 215027 (2015) arXiv:1505.00831 [hep-th]. – † “Cutoﬀ for Extensions of Massive Gravity and Bi-Gravity,” A. Matas, Class. Quant. Grav. 33, no. 7, 075004 (2016) arXiv:1506.00666 [hep-th].

• Galileons and phenomenology

I have studied Galileon scalar ﬁeld theories, which emerge in the decoupling limit of massive gravity as I describe in Chapter 2. These theories are useful toy models which capture many non-trivial features of the full theory of massive gravity. As a result they are useful for studying phenomenology, such as radiation from binary pulsars which I describe in Chapter 5.

– † “Galileon Radiation from Binary Systems” C. de Rham, A. Matas and A. J. Tolley, Phys. Rev. D 87, no. 6, 064024 (2013) arXiv:1212.5212 [hep- th]. – “Superluminality in the Bi- and Multi- Galileon” P. de Fromont, C. de Rham, L. Heisenberg and A. Matas, JHEP 1307, 067 (2013) arXiv:1303.0274 [hep-th].

This work has led to an invited talk at a workshop which led to a published review

– “Testing General Relativity with Present and Future Astrophysical Obser- vations” E. Berti et al, Class. Quant. Grav. 32, 243001 (2015) doi:10.1088/0264- 9381/32/24/243001 arXiv:1501.07274 [gr-qc].

Organization of dissertation

The rest of this dissertation is organized as follows.

• In Chapter 2, I will give a more technical introduction to massive gravity, con-

centrating on the issues and techniques that will arise in later chapters.

26 • In Chapter 3, I will discuss Dimensional Deconstruction, which is a way to

derive the ghost-free structure of massive gravity from GR in extra dimensions.

This is based on [60].

• In Chapter 4, I will discuss kinetic interactions in massive gravity. I will show

that the kinetic structure must be the same as in GR in order to avoid a ghost

instability, and I will also discuss implications of these results for the coupling

to matter ﬁelds. This is largely based on [61] and parts of [62].

• In Chapter 5, I will discuss binary pulsar tests. This is based [63], as well as

the closely related work [64].

• I will summarize my results and provide an outlook in Chapter 6.

Conventions

I will use units where Planck’s constant ~ and the speed of light c are set to unity,

~ = c = 1 (which to paraphrase Sidney Coleman is ﬁne so long as you remember that

when LeBron James shoots a free throw the ball has a tiny velocity but an enormous

angular momentum).

I use Greek indices µ, ν, ··· to refer to space-time indices, and Latin letters a, b, ···

to refer to local Lorentz indices, so the metric has the form gµν and the vielbein

a has the form eµ. Unless otherwise speciﬁed I will assume that space-time has four

dimensions. I use the mostly plus metric convention so that ηµν = diag(−1, 1, 1, 1). I p use MPl to refer to the reduced Planck mass which is given by MPl = ~c/8πGN =

27 2.4 × 1018 GeV. I use coordinates xµ = (x0, x1, x2, x3) where x0 = t, and wavenumber

µ 0 1 2 3 0 µ ~ k = (k , k , k , k ) where k = ω is the angular frequency, so kµx = −ωt + k · ~x.

I will symmetrize and antisymmetrize tensor indices with unit weight, so for example T(µν) ≡ (Tµν + Tνµ)/2 and T[µν] ≡ (Tµν − Tνµ)/2. I use square brackets to refer

µ to the trace of a matrix, [M] ≡ Mµ . The Levi-Civita symbol is normalized so that

ε0123 = +1.

28 Chapter 2

Basics of Massive Gravity

In this section I will review the basic ideas that underlie the construction of ghost-free massive gravity. This serves as an introduction to the theory, as well as allowing me to introduce many diﬀerent kinds of arguments and issues that will be important in the remaining chapters.

The central idea of massive gravity as an extension of General Relativity (GR) is that the limit in which the graviton mass m vanishes,

m → 0, (2.1)

is a smooth limit, and the physical predictions approach those of GR. This is why we expect massive gravity to be a ‘good’ modification of GR: there are small corrections under control and we can put bounds on m. As we will see, understanding this in detail is actually very subtle and not fully understood. Indeed as written, the limit is not well defined because we have not specified what we are holding fixed. As we will

29 see, while observable quantities may approach GR, the theory itself does not approach

GR but rather GR plus a set of decoupled degrees of freedom. Nevertheless, the continuity of this limit is the basic physical idea in the back of our minds. It informs how we will formulate the theory since we will want to write the theory in a form where the m → 0 limit can be taken in a clear way.

The rest of this Chapter is organized as follows. First in Section 2.1 I will set the stage by providing a few useful examples to introduce terminology and to highlight issues that will be important later. In particular I will show how to carefully take the limit m → 0 for a spin-1 ﬁeld, I will give an explicit example discussing ghost instabilities, and I will brieﬂy recall some facts and terminology about massless spin-2

ﬁelds (General Relativity).

After setting the stage I will work toward building an interacting gravitational theory of a massive spin-2 field. In Section 2.2 I will discuss the free theory, and discuss the issues that arise in particular the vDVZ discontinuity and the possibility of ghost degrees of freedom. I will also describe why self-interactions are a necessary part of a gravitational theory. In Section 2.3 I will extend this discussion to the non- linear level, explaining the generic form of a massive gravity theory, as well as giving explicit forms for the mass term in different languages. In Section 2.4 I will briefly describe the Hamiltonian (ADM) argument for the absence of a ghost, focusing on the mini-superspace which I will make use of in later chapters.

After having built up the non-linear theory, I will then describe how to approximate the theory at low energies on a Minkowski background. In Section 2.6, I will give an explicit realization of the Vainshtein mechanism in a static spherically symmet-

30 ric setup. Lastly in Section 2.7 I will describe certain extensions to massive gravity

known as bi-gravity and multi-gravity, which will play a role in some later chapters.

I conclude this chapter in Section 2.8 with a summary describing the most important

points.

2.1 Setting the stage

This section is meant to be introductory. We review several well-known results in

order to introduce some working examples of a scalar and vector ﬁeld that we will

return to several times, for example when we discuss the kinetic terms in Chapter 4,

and to deﬁne terms and ideas that we will use.

2.1.1 Consistent m → 0 limit of a massive spin-1

The free theory of a massive spin-1 particle Aµ is given by the Proca Lagrangian

1 1 L = − F F µν − m2A Aµ + A J µ, (2.2) 4 µν 2 µ µ

where Fµν ≡ ∂µAν − ∂νAµ and indices are raised and lowered with respect to ηµν.

J µ is an external source. An external source allows us to probe the theory, without explicitly introducing new dynamical ﬁelds that obey their own equations of motion.

We are interested in the m → 0 limit of this theory. The most direct approach is to simple set m = 0 in the action given in Eq. 2.2. However, this is not a consistent limit.

The reason is that if we set m = 0, the number of degrees of freedom discontinuously

31 changes from 3 to 2. Formally this occurs because when we set m = 0 in Eq. 2.2, the action is invariant under a U(1) gauge symmetry Aµ → Aµ + ∂µλ (provided that the

µ source is conserved ∂µJ = 0). We can use the freedom to choose λ to remove one degree of freedom in Aµ.

Instead, we want to take a decoupling limit where the third degree of freedom

(which we can identify as the helicity-0 mode of the photon) decouples from the system. To implement the decoupling limit, we ﬁrst re-introduce the broken U(1) gauge invariance of the massless theory by introducing a new ﬁeld via the so-called

St¨uckelberg procedure

∂µπ Aµ −−−−→ Aµ + , (2.3) replace m where the factor of m is chosen for later convenience. We now have a gauge invariance under transformations

Aµ → Aµ + ∂µλ, π → π − mλ. (2.4)

The Proca Lagrangian given in Eq.2.2 becomes (up to total derivative terms)

1 1 π∂ J µ L = − F F µν − (∂π)2 − mπ∂ Aµ + m2A Aµ + A J µ − µ . (2.5) 4 µν 2 µ µ µ m

Note that π appears with a canonically normalized kinetic term, meaning that the

2 1 coeﬃcient of (∂π) is equal to − 2 . In that case as we send m → 0 the kinetic term

32 for π remains ﬁxed. We may now consistently take the limit m → 0, resulting in1

1 1 L = − F F µν − (∂π)2 + A J µ. (2.6) m→0 4 µν 2 µ

Here we nicely see that we have taken a limit where the number of degrees of freedom

has not changed, and where π has completely decoupled from the system.

A question that might arise is, why should the kinetic term be built out of the gauge invariant ﬁeld strength tensor Fµν even though gauge invariance is broken. I will return to this issue in Chapter 4. The real reason is that any other kinetic term would result in a kinetic instability. Since the notion of a ghost is at the core of this work, in the next section we will introduce this class of pathologies.

2.1.2 Ghost modes

A ghost is a ﬁeld with a wrong sign kinetic term. In order to understand the problem, it is useful to consider a simpler model with two scalar ﬁelds φ and χ where χ has

the wrong sign kinetic term

1 1 1 L = − (∂φ)2 + (∂χ)2 + m2 χ2 − V (φ, χ). (2.7) 2 2 2 ghost

The Hamiltonian (or energy) density of this system is

1 1 H = φ˙2 + (∂ φ)2 − (χ ˙ 2 + (∂ χ)2 + m2 χ2 + V (φ, χ), (2.8) 2 i 2 i ghost

1 −1 µ Assuming m ∂µJ → 0, which is needed for the limit to be consistent. In other words, matter should couple to Aµ in a U(1) covariant way at least up to O(m).

33 2 i where (∂iφ) ≡ ∂iφ∂ φ. The main thing to note is that even if V (φ, χ) is positive

deﬁnite, the energy is unbounded below because of the kinetic energy of χ. The fact

that φ and χ interact means that the system can explore regions with arbitrarily large

gradients consistent with energy conservation. This is the ﬁeld theory version of the

discussion in Figure 1.6.

Quantum mechanically, the interactions in V (φ, χ) allow for processes where the

vacuum spontaneously produces φ and χ particles. The only constraint from energy

conservation is that the energy of the particles be ≥ mghost. If we trust the theory to be a valid description at all energies, then there is an infinite phase space volume allowed for this process and the decay rate is infinite. Alternatively if we only trust the theory to be a good description up to some cutoff in energy Ec.o., then the rate will be finite and depend on Ec.o.. If Ec.o. < mghost (that is, we do not trust this theory to be a good description of physics for modes with energy E > mghost), then we may consistently treat the decay process as simply an artifact of using the Lagrangian beyond its regime of validity. The full theory will include some new physics at Ec.o.,

and it may be that the new physics can remove the ghost.

Lastly we give the example of an Ostrogradsky ghost, which is a special case

associated with higher derivatives. As we will see in more detail in Chapter 4, an

Ostrogradsky ghost on the helicity zero mode occurs where the kinetic term was taken

2 to be diﬀerent from Fµν. A simple case of an Ostrogradsky ghost is given by the two

ﬁeld model

1 1 2 L = − (∂φ)2 − φ φ − V (φ). (2.9) 2 2 Λ2

34 We can rewrite this action in a completely equivalent form by introducing a new ﬁeld

1 1 L = − (∂φ)2 + φ χ + Λ2χ2 − V (φ). (2.10) 2 2

Integrating out χ (that is, eliminating χ using its own equations of motion2) reproduces Eq. 2.9. However we can now write the Lagrangian in an equivalent form by doing a ﬁeld redeﬁnition φ = φ˜ − χ which yields

1 1 1 L = − (∂φ˜)2 + (∂χ)2 + Λ2χ2 − V (φ˜ − χ), (2.11) 2 2 2

Now the ghost mode is manifest since the action has the same form as Eq. 2.7, with mghost = Λ.

2.1.3 Massless spin-2

Finally before turning to a massive spin-2 it is useful to state several facts about a massless spin-2 to ﬁx conventions.

The Lagrangian for a massless spin-2 can be written as

µν 1 µν αβ hµνT L = − hµνE αβh + , (2.12) 4 2MPl

where Tµν is an external source (representing the energy and momentum tensor,

2The language comes from the path integral formulation of quantum field theory. For fields that appear in the Lagrangian quadratically and whose equations of motion are algebraic, eliminating a field using its equation of motion is equivalent to doing a path integral over that field.

35 analogous to the external current in Eq. 2.2), and where the Lichnerowicz operator is

deﬁned by its action on a symmetric tensor ﬁeld hµν as

1 E αβ h = − h − 2∂ ∂ hλ + ∂ ∂ h + η ∂ ∂ hαβ − h µν αβ 2 µν λ (µ ν) µ ν µν α β 1 = − ε αγσε ∂ ∂λhβ . (2.13) 2 µ νβλσ γ α

The Lichnerowicz operator is normalized to agree with the linearized Einstein tensor,

2 Gµν = (Eh)µν + O(h ) when gµν = ηµν + hµν. The Lichnerowicz operator has the following properties

µν νµ µν µν E ρσ = E ρσ = E σρ = Eρσ

µραβ µναβ ∂µE = E ∂αχβ = 0, (2.14)

where χµ is an arbitrary one-form. Note these properties are easier to see directly

from the εε structure.

The action is invariant under the linear diﬀeomorphisms (linear diﬀs), a gauge

symmetry

hµν → hµν + ∂µξν + ∂νξµ. (2.15)

At the non-linear level, gravity is described by the Einstein-Hilbert action, which

36 is the action for General Relativity

M 2 Z √ S = Pl d4x −gR, (2.16) EH 2

−1 where R is the Ricci curvature scalar and where gµν = ηµν + MPl hµν. The linear gauge transformations given in Eq. 2.15 become invariance under general coordinate transformations, or diﬀeomorphisms (diﬀs), under which the metric transforms as

∂xρ ∂xσ g (x) → g˜ (y) = g (x(y)), x → y = y(x). (2.17) µν µν ∂yµ ∂yν ρσ

Much like in the spin-1 example, we expect a mass term to break the invariance of the theory under these transformations.

2.2 Toward a theory of massive gravity

2.2.1 Non-interacting massive spin-2 ﬁeld

I will discuss the Fierz-Pauli theory in the St¨uckelberg language in some depth, because I will use the logic presented here when I discuss kinetic interactions in Chap- ter 4.

The starting point is the free theory (which is described by a Lagrangian quadratic in the ﬁelds). The simplest tensor capable of describing spin-2 degrees of freedom is

37 a rank-2 tensor hµν. The Fierz-Pauli theory of a massive spin-2 ﬁeld is given by

µν 1 µν αβ 1 2 µν 2 hµνT L = − hµνE αβh − m hµνh − h + , (2.18) 4 8 2MPl

µν where h ≡ η hµν. The normalization of the mass term is chosen so that the propagating degrees of freedom (mass eigenstates) have a mass m.

Following the logic in Section 2.1.1, we now introduce a Stückelberg field to re- introduce linear diffs

1 hµν −−−−→ hµν − (∂µBν + ∂νBµ) , (2.19) replace m where the factor of m−1 is chosen for later convenience.

Then the action to quadratic order becomes

1 1 L = − h (Eh)µν − F F µν 4 µν 8 µν µν m µν ν 1 2 µν 2 hµν Bν∂µT + Bν (∂µh − ∂ h) − m hµνh − h + − , (2.20) 4 8 2MPl mMPl

where Fµν ≡ ∂µBν − ∂νBµ. Note that the scalings in Eq. 2.19 were chosen precisely so that these ﬁelds are canonically normalized.

We still cannot take the m → 0 limit, because the helicity-0 mode is still contained inside the massive vector Bµ. We therefore introduce another St¨uckelberg ﬁeld

∂µπ Bµ −−−−→ Bµ + , (2.21) replace m

38 µ µν where ∂ π = η ∂νπ.

In this form we can take the limit m → 0 in a safe way, that is without losing any

degrees of freedom.

To summarize, starting from Eq. 2.19 we make the St¨uckelberg replacements

2∂(µBν) 2∂µ∂νπ hµν −−−−→ hµν − − , (2.22) replace m m2 and take the limit m → 0. In the future, when doing St¨uckelberg replacements we will now simply use an arrow instead of explicitly writing replace, but it is important to keep in mind the distinction between a St¨uckelberg replacement and a gauge transformation. We can write the entire decoupling limit m → 0 as3

µν 1 µν αβ 1 µν 1 µ ν µν hµνT L = − hµνE αβh − FµνF − hµν (∂ ∂ π − πη ) + . (2.23) 4 8 4 2MPl

Note that π only gets a kinetic term from mixing with hµν. We can make this explicit

by doing a ﬁeld redeﬁnition

˜ hµν = hµν + πηµν, (2.24)

leading to

˜ µν 1˜ µν ˜αβ 1 µν 3 2 hµνT πT L = − hµνE αβh − FµνF − (∂π) + + . (2.25) 4 8 4 2MPl 2MPl

3 −2 µν Provided that the matter source satisﬁes m ∂µT → 0 in this limit.

39 vDVZ Discontinuity Note the coupling

πT (2.26) 2MPl survives even in the limit m → 0, even for a conserved source. As we send m →

0, π remains coupled to T . This leads to an apparent ﬁnite diﬀerence in physical predictions no matter how small the value of the mass. This is the van Dam-Veltman-

Zakharov (vDVZ) discontinuity.

At ﬁrst sight, this seems like a disaster. No matter how small m, there is no way we can have continuity with GR. In fact, the resolution proposed by Vainshtein is that the free theory is a bad approximation in a regime that becomes larger and larger as we send m → 0.

2 µν Ghost mode In the Fierz-Pauli mass term, the terms h and hµνh appear with a relative coeﬃcient of −1. This is no accident, as any other combination would not give rise to a consistent theory of massive gravity. In order to see this more precisely, we can allow for a more general mass term

µν 1 µν αβ 1 2 2 2 hµνT L = − hµνE αβh − m αhµν − βh + . (2.27) 4 8 2MPl

Introducing St¨uckelberg ﬁelds as in Eq. 2.22 and focusing on the scalar-tensor sector

(which is the most relevant for this discussion), and focusing on the part of the action

40 that is nonvanishing as m → 0 we ﬁnd,

1 α − β L = − h E µν hαβ − h (∂µ∂νπ + (α − β) ηµν π) − ( π)2 .(2.28) scalar−tensor 4 µν αβ µν 2m2

From here we see the presence of a dangerous higher derivative operator. I will use

similar techniques to identify the presence of a ghost in Chapter 4.

Comparing with Eq. 2.9, we see that the mass of the ghost

3 m2 m2 = , (2.29) ghost 2 α − β where the 3/2 comes from accounting for the normalization of π in Eq. 2.25. The special Fierz-Pauli potential occurs for α = β, in which case mghost is inﬁnite.

So far we have described the linear theory of a massive graviton in some detail.

When the spin-2 ﬁeld couples to matter, the spin-2 ﬁeld necessarily develops additional interactions. We now turn to a more realistic theory of an interacting massive graviton.

2.3 Non-linear formulations of massive gravity

A generic interacting theory of Lorentz invariant massive gravity has the form

M 2 Z √ m2 S[g ] = Pl d4x −g R − U(gµν, η ) + S [g , ψ]. (2.30) µν 2 2 µν matt µν

41 The action depends on a dynamical metric gµν. The potential U contains contractions

of the dynamical metric with the Minkwoski metric ηµν.

When perturbed to leading order around Minkowski, the potential should have the form of the Fierz-Pauli choice. At higher order it is less clear what the mass term should be. A natural ﬁrst guess for the potential is to covariantize the Fierz-Pauli mass term

1 U = gµνgρσ (H H − H H ) , (2.31) 2 µρ νσ µν ρσ

where

Hµν ≡ gµν − ηµν, (2.32)

and we use a capital Hµν to recognize that we are not assuming that H is a small

quantity. However, it was shown by Boulware and Deser that this combination, and

any function of this combination, is plagued by a return of the ghost instability [31]. It

has recently been shown that there is a choice of potential which avoids this problem,

to which we now turn.

42 2.3.1 Ghost-free potential

The general form of the potential which avoids the Boulware-Deser ghost was given

in [28]

4 X µν Ug.f. = − αnLn[K(g , ηρσ)], (2.33) n=0

where

√ µ µ µα K(g, η) ν ≡ δν − g ηαν, (2.34)

and where the mass terms are given by

L0[K] = 4!

L1[K] = 3![K]

2 2 L2[K] = 2! [K] − [K ]

3 2 3 L3[K] = [K] − 3[K][K ] + 2[K ]

4 2 2 2 2 3 4 L4[K] = [K] − 6[K] [K ] + 3[K ] + 8[K][K ] − 6[K ], (2.35)

µ where [K] ≡ Kµ.

Before moving on to describe the physics of this theory, it is useful to state some

equivalent forms of the theory that we will use later.

43 2.3.2 Vielbein formulation

A particularly simple way to write the action is provided by the vielbein formulation of massive gravity [65,66]. The vielbein is related to the metric by

a b gµν = eµeνηab. (2.36)

There is also a ﬁxed Minkowski (ﬂat) vielbein 1a

a a µ 1 ≡ δµdx . (2.37)

a where δµ is the Kronecker delta.

In terms of the vielbein, the action has the form

M 2 Z S = Pl ε Rab ∧ ec ∧ ed + m2Aabcd(e, 1) , (2.38) mGR 4 abcd g.f.

abcd where the potential structure is deﬁned by the tensor Ag.f. (e, 1),

abcd a b c d a b c d Ag.f. (e, δ) = c0e ∧ e ∧ e ∧ e + c1e ∧ e ∧ e ∧ 1

a b c d a b c d +c2e ∧ e ∧ 1 ∧ 1 + c3e ∧ 1 ∧ 1 ∧ 1

a b c d +c41 ∧ 1 ∧ 1 ∧ 1 . (2.39)

This formulation is equivalent to the metric formulation so long as the vielbein obeys

44 the symmetric vielbein condition or Deser-van Nieuwenhuizen condition [67,68]

a b a b ηabδµeν = ηabδν eµ. (2.40)

2.3.3 St¨uckelberg for gravity

It is often useful to write the theory in a way where diﬀeomorphisms are restored.

We can recover diﬀ invariance by introducing 4 new scalar ﬁelds φa. We make the formal replacement

a b ηµν → ηab∂µφ ∂νφ . (2.41)

Note that in this formulation there is invariance under a global symmetry in the ﬁeld space

a a b φ → Λ bφ . (2.42)

Then the metric ﬂuctuation is

a b Hµν = gµν − ηab∂µφ ∂νφ . (2.43)

45 Then the action is invariant under coordinate transformations where gµν transforms as a tensor and φa transform as scalars

∂yα ∂yβ g → g (x(y)), φa → φa(x(y)), (2.44) µν ∂xµ ∂xν αβ

and therefore Hµν transforms like a tensor.

We mention in passing that there is another way to do the St¨uckelberg procedure which we will use later. A geometric picture for what we have done is that we have two manifolds Mg and Mf on which the two metrics live. In order to describe ﬁeld

a theory on these manifolds we introduce maps φ that map Mg to Mf , and then we can ‘pull back’ ηµν along this map. We say that we have introduced the St¨uckelberg

ﬁelds through the reference metric.

It is also possible to go the other way and introduce the St¨uckelberg ﬁelds through

µ gµν. (or in other words to work with maps that go from Mf to Mg). Let y be the coordinates for the f metric so x = ψ(y) (Note that ψ and φ are inverses in the sense that φ(ψ(y)) = y and ψ(φ(x)) = x). Then we have

∂ψα ∂ψβ g → g (ψ(y)), (2.45) µν ∂yµ ∂yν αβ and

α β Hµν = ∂µψ ∂νψ gαβ(ψ(y)) − ηµν, (2.46)

46 which tranxforms as a tensor under

µ0 ν0 ∂y ∂y α α η → η 0 0 , ψ → ψ (y(˜y)). (2.47) µν ∂y˜µ ∂y˜ν µ ν

The global symmetry in Eq. 2.42 becomes

α α β ψ → Λ βψ . (2.48)

This method is sometimes more complicated to deal with because of the dependence of gµν on the St¨uckelberg ﬁeld in Eq. 2.45, however it always leads to the same physics in the end.

Now that we have developed diﬀerent ways of describing the ghost-free interactions at the non-linear level, it is time to turn to the physics of the theory.

2.4 Absence of the Boulware-Deser mode

The crucial breakthrough in ghost-free massive gravity was to realize how to avoid the Boulware-Deser ghost. In this section we brieﬂy review how this occurs using a

Hamiltonian framework.

Much of my work in later chapters builds on the logic in this section, particularly with the use of the mini-superspace analysis I describe in 2.4.3 and the ADM split in

2.4.1.

47 2.4.1 ADM form

Following the Arnowitt-Deser-Misner (ADM) 3+1 approach to gravity [69], we ﬁrst

foliate the space-time with hypersurfaces that are space-like with respect to the dy-

namical metric gµν. The foliation produces a natural time direction. We write the

(dynamical) metric as

2 µ ν dsg = gµνdx dx

2 2 i i j j = −N dt + γij dx + N dt dx + N dt , (2.49)

where N is known as the lapse and N i are known as the shift vectors (or shifts).

For our purposes, this is simply a convenient parameterization of the metric on this foliation. In terms of these variables the action takes the form

M 2 Z √ m2 S = Pl d4xN γ R + [K]2 − [K2] − U(N,N i, γ ) , (2.50) 2 3 2 ij

ij where R3 is the Ricci scalar of the spatial metric γij and where [K] ≡ γ Kij and

where Kij is the extrinsic curvature

1 K ≡ γ˙ − 2D N , (2.51) ij 2N ij (i j)

where Di is the covariant derivative with respect to γij, and where the dot refers to a time derivativeγ ˙ ij = ∂tγij.

48 2.4.2 Counting degrees of freedom

We can always write the action in a form which is linear in time derivatives (ﬁrst order

form) by introducing additional variables. Since the potential does not introduce any

time derivatives that are not already present in GR, the action may be written as

Z ij 4 ij 0 i 2 S[γij, π ] = d x πijγ˙ − NC (γ, π) − NiC (γ, π) − m Hm(γ, π, N, Ni) ,(2.52)

0 i where C and C are functions of γij and πij and their spatial derivatives given by

1 1 C0 = −R + πijπ − π2 , 3 γ ij 2 πij Ci = 2D √ . (2.53) j γ

ij We think of γij and π as being independent variables here, but of course we can return to the original form by integrating out πij which yields

δL √ πij ≡ EH = γ Kij − Kγij . (2.54) δγ˙ ij

ij When we vary the action in Eq. 2.52 with respect to γij and π we will ﬁnd 12

ﬁrst order equations of motion for the 12 variables. As a result we naively need 12 initial conditions to specify a solution to this system of equations, corresponding to 6 propagating degrees of freedom. However this may be too fast as we must account for the possibility that there are constraint equations which remove degrees of freedom.

Indeed, in GR with m = 0, C0 and Ci appear as constraints (the Hamiltonian and

49 momentum constraint, respectively) which constrain the initial data.

Varying the action with respect to the lapse and shift produces four equations

δS δ E (N)(γ , πij,N,N i) ≡ = Cµ(γ , πij) − m2 H (N,N i, γ , πij) = 0, (2.55) µ ij δN µ ij δN µ m ij where we deﬁne N 0 ≡ N for convenience. For a generic choice of potential, we will be able to solve these four equations for N,N i. After doing this, there will be no constraints placed on the initial data. As a result, in massive gravity we generically expect 6 propagating degrees of freedom. This corresponds to the ﬁve degrees of freedom for a massive graviton, plus a BD ghost.

There is a loophole to this argument: for special choices of Hm it might not be possible to solve the equations 2.55 for the lapse and shift. To see this more clearly, imagine that we can solve three of the equations for the shift

i i ij N = N (γij, π ,N). (2.56)

After eliminating the shift this we can write the action in the form

Z 4 ij ˜ ij S = d x πijγ˙ − Hm(γij, π ,N) . (2.57)

˜ ˜ ij If Hm is a linear function of N, then it can be written in the form Hm = f(γij, π ) +

˜ ij NC(γij, π ), then N generates a constraint which removes a phase space degree of

freedom. This constraint leaves us with 5 propagating degrees of freedom4 corre-

4Technically one must also show the existence of a secondary constraint, which comes from

50 sponding to the absence of a BD ghost.

The ghost-free potential given in Eq. 2.33 has this property, as has been shown

(in diﬀerent but equivalent) ways in [27,28,70,72–79].

2.4.3 Mini-superspace

We can see this explicitly by working in the mini-superspace We can consistently restrict the metric to have the form5

2 2 2 2 i j dsg = −N (t)dt + a (t)δijdx dx . (2.58)

In this case the shift vector is zero and the action is automatically written in the form of Eq. 2.57. Thus if the constraint exists, it must be that the action is linear in the lapse in the mini-superspace. This provides a very quick and practical tool to diagnose the presence of the BD ghost. However, the existence of a constraint in the mini-superspace is a necessary but not suﬃcient condition, and the existence of a constraint in the mini-superspace does not guarantee the constraint will exist beyond the mini-superspace.

In the mini-superspace the ﬁrst-order form of the action for a generic potential demanding that the time derivative of the constraint we are discussing vanishes when the equations of motion and constraints are imposed. The secondary constraint has been shown to exist [70], and its explicit form is known in some special cases such as in three dimensions [71]. 5Meaning, if we only vary the action with respect to N and a, we get the same equations of motion we would get if we vary the action with respect to all components of the metric and then set the metric to the form of Eq. 2.58.

51 has the form

2 Z 2 MPl πa 2 2 S = dt aπ˙ a − N 2 − m MPlUm.s.(N, a, πa) , (2.59) 2 12MPla

2 −1 where πa ≡ 6MPlN aa˙. For the naive covariantization of Fierz-Pauli given in

Eq. 2.31,

3 (a2 − 1) (2a2 − 1) 3a (a2 − 1) U = N + , (2.60) m.s. 4a N

and we see that the second term is non-linear in N, signaling a BD mode.

Conversely, the ghost-free mass terms must be ghost-free in the mini-superspace.

To illustrate this, consider the mass term α2 = 1, α3,4 = 0 which is linear in N in the mini-superspace

2 Ug.f. = 3 (a − 1) a + N (a − 1) (2a − 1) a. (2.61)

This is (of course) consistent with the absence of a BD mode.

2.5 Minkowski decoupling limit

The fully non-linear theory given by Eqs. 2.30 and 2.33 is formidable. It is useful to

develop an approximation scheme in order to understand the physics around special

backgrounds of interest. In particular, in this work we will be concerned with physics

about the Lorentz invariant vacuum, which we expect to be relevant for solar system

52 (or binary pulsar) physics.

We now perturb

a hµν a a B gµν = ηµν + , φ = x − . (2.62) MPl mMPl

When the background of the dynamical metric is Lorentz invariant, the action is invariant under simultaneous Lorentz transformations of the St¨uckelberg ﬁelds and the coordinates,

a a b µ µ ν φ → Λ bφ , x → Λ νx simultaneously. (2.63)

As a result, on the Lorentz-invariant background, we can think of Ba as a space-time- vector, even though it came from a set of 4 scalars. We therefore write Bµ instead of

Ba.

The metric perturbation then becomes

a b Hµν = gµν − ηab∂µφ ∂νφ

hµν 2∂(µBν) 2∂µ∂νπ = + + 2 MPl mMPl m MPl ∂µBα ∂µ∂απ αβ ∂νBβ ∂ν∂βπ − + 2 η + 2 . (2.64) mMPl m MPl mMPl m MPl

Comparing with the St¨uckelberg replacement in the Fierz-Pauli theory given in Eq. 2.22, we see that there are additional non-linear terms.

As we have discussed, at energies large compared to m the St¨uckelberg ﬁelds π

53 and Bµ capture the dynamics of the helicity-0 and -1 modes.

2.5.1 Interaction scales

The non-linearities in the potential will manifest themselves as interactions for the helicity -2, -1, and -0 ﬁelds hµν,Bµ, and π. The full set of interactions is rather complicated. However, we expect that at low energies, the most important contributions should come from the interactions at the lowest energy scale.

Because the kinetic term of the ﬁelds are canonically normalized and decoupled in this limit, we can read oﬀ the scale at which a given interaction becomes important by using dimensional analysis. Of course dimensional analysis cannot see the possibility of cancellations, but it gives us a way to organize the possible interactions.

In terms of the canonically normalized ﬁelds, a generic interaction has the form

h nh ∂B nB ∂∂π nπ L(int) ∼ m2M 2 . (2.65) nh,nB ,nπ Pl 2 MPl mMPl m MPl

This interaction comes in at the scale

1−p 1/p Λp = m MPl , (2.66)

where

n + 2n + 3n − 4 p = h B π . (2.67) nh + nB + nπ − 2

54 −60 Since m MPl (for cosmology we have in mind m/MPl ∼ 10 ), larger values of p

correspond to interactions at a lower scale.

Interactions at Λ5. The lowest interaction has p = 5 and comes from nh = nB = 0

4 1/5 and nπ = 3 at the scale Λ5 = (m MPl) and is of the form

(int) 1 3 L0,0,3 ∼ 5 (∂∂π) . (2.68) Λ5

These interactions involve higher derivatives on π. The equations of motion are

generically higher order, signalling an Ostrogradsky instability.

The ghost-free potentials were constructed precisely so that the dangerous interac-

tions at Λ5 form have vanishing equations of motion, so that the equations of motion

for π remain second order. I will use similar criteria when searching for ghost-free

kinetic interactions in Chapter 4.

For example, the mass term L3 has the following interactions at Λ5

1 3 2 3 L3 = 5 [Π ] − 3[Π ][Π] + 2[Π] , (2.69) Λ5 where the ··· refer to interactions at a higher scale. Since these interactions are a total derivative and therefore have vanishing equations of motion, L3 does not have a BD ghost.

There are in principle other interactions between Λ5 and Λ3. However, for the

ghost-free potentials all interactions below Λ3 arise in total derivative combinations.

It is useful to get a sense of the sizes of these diﬀerent interaction scales. We can

55 −33 express them in length scales, assuming m ∼ H0 ≈ 70km/s/Mpc ≈ 8 × 10 eV

H 4/5 Λ−1 = 1011 km 0 . (2.70) 5 m

Interactions at Λ3. These interactions have the form

int 1 nπ 1 2 nπ L1,0,n ∼ h(∂∂π) , (∂B) (∂∂π) . (2.71) π 3(nπ−1) Λ Λ3 3

I will give the explicit form of these interactions. However, the crucial thing is that the equations of motion are second order.

Numerically, Λ3 is given by

H 2/3 Λ−1 = 103 km 0 . (2.72) 3 m

There are an inﬁnite number of interactions above Λ3, however these are less important than the interactions at Λ3 at low energies.

2.5.2 Explicit form of decoupling limit

As we have seen, the interactions arising at the scale Λ3 are the interactions that are most important at low energies. In order to focus on these interactions, it is useful to take a Λ3 decoupling limit

2 1/3 m → 0,MPl → ∞, Λ3 = (m MPl) ﬁxed. (2.73)

56 Any interaction that arises at a higher scale is sent to zero in this limit.

The Λ3 decoupling limit of massive gravity has the form

µν hµνT LD.L. = Lhπ[hµν, π] + LBπ[Bµ, π] + . (2.74) 2MPl

The explicit form of the vector-scalar interactions LBπ is given in [80, 81]. I will not be concerned with it in this work, as the vectors are not directly sourced by matter

ﬁelds in the decoupling limit.

The scalar-tensor sector Lhπ of the decoupling limit, on the other hand, is of great interest because it can be used to describe the Vainshtein mechanism. Explicitly, the scalar-tensor sector (including the coupling of hµν to matter) is given by

µν hµνT Ls−t = Lhπ[hµν, π] + 2MPl µν 1 µν 1 µν (1) 2α2 + 3α3 (2) α3 + 4α4 (3) hµνT = − hµνEh + h 2α2Xµν + 3 Xµν + 6 Xµν + , 4 8 Λ3 Λ3 2MPl (2.75)

µ µα where Π ν ≡ η ∂µ∂νπ, and where the X tensors are deﬁned as

(1)µ µ µ X µ0 ≡ δµ0 [Π] − Πµ0

(2)µ 2 2 µ µ 2 µ X µ0 ≡ [Π] − [Π ] δµ0 − 2 [Π]Πµ0 − (Π )µ0

(3)µ 3 2 3 µ X µ0 ≡ [Π] − 3[Π][Π ] + 2[Π ] δµ0

2 µ 2 µ 2 µ 3 µ −3 [Π] Πµ0 − 2[Π](Π )µ0 − [Π ]Πµ0 + 2(Π )µ0 . (2.76)

57 2.5.3 Galileon interactions

In this form, h and π are mixed. We can diagonalize the interactions to some extent

by performing a ﬁeld redeﬁnition

˜ 2α2 + 3α3 hµν = hµν + πηµν − 3 ∂µπ∂νπ. (2.77) 2Λ3

(compared with the analogous ﬁeld redeﬁnition in Fierz-Pauli, Eq. 2.24 there are

additional non-linear interactions). This leads to a scalar tensor Lagrangian

5 1 3 3 X cn (n) L = − h˜Eh˜ − (∂π)2 − L + (α + 4α ) h˜ X(3),µν s−t 4 4 4 (3−n) Gal 3 4 µν n=3 Λ3 ˜ µν µν hµνT πT ∂µπ∂νπT + + 2 2 − (2 + 3α3) 2 2 , (2.78) 2MPl 2m MPl 2m MPl

where the constants cn are given by

1 c = (2 + 3α ) 3 2 3 3 c = (2 + 3α )2 + 4 (α + 4α )2) 4 2 3 3 4 5 c = (2 + 3α )(α + 4α ) . (2.79) 5 24 3 3 4

The Lagrangians LGal are known as Galileon interactions [39] and are given ex-

58 plicitly by

(3) 2 LGal = (∂π) [Π]

(4) 2 2 2 LGal = (∂π) ([Π] − [Π ])

(5) 2 3 2 3 LGal = (∂π) ([Π] − 3[Π][Π ] + 2[Π ]). (2.80)

We refer to the Galileon interactions by the order in π in which they appear in the

(3) (4) Lagrangian. We refer to LGal as the cubic Galileon, LGal as the quartic Galileon, and

(5) LGal as the quintic Galileon.

2.6 Realization of the Vainshtein mechanism

Now that we have derived an explicit form of the decoupling limit, it is possible to establish several important physical results.

2.6.1 Vainshtein mechanism

The Vainshtein mechanism provides a way to resolve the vDVZ discontinuity.

The interactions for the helicity-0 mode in massive gravity allow for a Vainshtein screening mechanism around non-trivial backgrounds. While perturbation theory breaks down and it becomes inconsistent to work with a weak ﬁeld approximation, the non-linear interactions can resolve the vDVZ discontinuity.

Let us show see schematically why we might expect this to work in the simplest

59 case of the cubic Galileon coupled to matter

1 2 1 2 πT L = − (∂π) + 3 (∂π) π + . (2.81) 2 Λ3 2MPl

If we perturb this theory around a generic backgroundπ ¯

π =π ¯ + ϕ, (2.82)

then to quadratic order, the action for the ﬂuctuations φ has the schematic form

2 ∂ π¯ 2 ϕT L ∼ 1 + 3 (∂ϕ) + . (2.83) Λ3 MPl

Deﬁning

2 ¯ ∂ π¯ Z ∼ 1 + 3 , (2.84) Λ3

√ we can write this in terms of a canonically normalized ﬁeldϕ ˆ ≡ ϕ/ Z¯

ϕTˆ L ∼ (∂ϕˆ)2 + √ . (2.85) ¯ ZMPl

√ ¯ We see thatϕ ˆ couples to matter with a redressed coupling ZMPl.

2 3 ¯ When the background is large in the sense ∂ π¯ Λ3, then Z 1, and the coupling to matter is greatly suppressed.

60 This is the Vainshtein mechanism. The non-linear interactions cause the coupling between the helicity-0 mode and matter to become suppressed in the strong-coupling regime. Meanwhile, the helicity-2 modes behave as in GR, since its interactions come in at a much higher scale.

2.6.2 Boulware-Deser ghost

Given our understanding of the Vainshtein mechanism it is useful to return to the

Boulware-Deser ghost.

If the interactions at the scale Λ5 are present then they have the form

1 3 4 (∂∂π) . (2.86) m MPl

Indeed in this form we can see that the degree of freedom will appear as a ghost around generic backgrounds This is because we can give π a background

π =π ¯ + ϕ. (2.87)

On this background the Lagrangian for ϕ will have the form

2 (∂∂π¯) 2 L ∼ (∂ϕ) + 5 (∂∂ϕ) . (2.88) Λ5

Note that ϕ now appears with higher derivatives in the Lagrangian. This will lead to

61 a ghost with mass

Λ5 m2 ∼ 5 , (2.89) ghost (∂∂π¯)

3 provided that ∂∂π¯ . Λ5 so that we can trust the approximation we have made to ignore interactions at larger scales. In principle if the mass of the ghost is large then we can neglect the ghost so long as we are comfortable with saying that we do not trust the theory to give a good description of physics for energies larger than mghost.

We can assume (or hope) that there is some new physics that kicks in at energies below mghost that removes the problem.

However, the mass of the ghost is background dependent, and the mass becomes smaller as the background becomes larger. Indeed we have seen that the Vainshtein mechanism requires ∂∂π¯ to be large. This in turn will make mghost small, and we would not be able to trust the Vainshtein mechanism within the regime of validity of the theory.

This is why it is so important to construct a ghost-free theory of massive gravity.

The interactions which are forced on us by working with a gravitational theory, and which cause the theory to become strongly coupled at a low scale, will in turn make the theory break down at an absurdly small scale.

2.6.3 Vainshtein mechanism for stars

I can now give an explicit realization of the Vainshtein mechanism in a static, spherically symmetric example (for example, a star).

62 We focus on the Galileon interactions

5 3 3 X cn (n) πT L = − (∂π)2 − L + . (2.90) 4 4 Λ3(n−2) Gal 2M n=3 Pl

These interactions provide a useful model for understanding how the Vainshtein mechanism behaves.

Let us consider a matter source given by a point mass M

µ 0 (3) T = −Mδ0 δνδ (~r). (2.91)

On a spherically symmetric background, the equations of motion are a polynomial equation for E ≡ π0

2 3 E 2c3 E 2c4 E M + 3 + 6 = 3 . (2.92) r 3Λ r Λ r 12πMPlr

Note that the quintic Galileon does not contribute to the background equation, this is related to the fact that we have essentially reduced the dimensionality of the system by not allowing the source to be time-dependent. We can solve this cubic equation exactly. However it is more useful to study it in two diﬀerent regimes.

There is a characteristic scale associated with this problem

(M/M )1/3 r ≡ Pl , (2.93) ? Λ which is known as the Vainshtein radius. This is more manifest if we rewrite the

63 equation in the form

2 1 r 3 y + c y2 + 2c y3 = ? , (2.94) 3 3 4 12π r where the dimensionless ﬁeld y is y ≡ E/rΛ3.

The behavior of the Galileon depends very sensitively on the ratio r/r?.

• Weakly coupled regime. At large distances r r? this equation gives

M 1 E = 2 + O 5 . (2.95) MPlr r

• Strongly coupled regime. At small distances r r? on the other hand we

have

r p M E = 2 + O(r), (2.96) r? MPlr

where p = 3/2 if only the cubic Galileon is present (that is if c4 = 0) and p = 2

if the quartic Galileon is present.

~ ~ Since the Galileon exerts a force Fπ ∝ ∇π = E, we see that for distances smaller than the Vainshtein radius the Galileon force is suppressed compared to the usual

Newtonian force, |Fπ| |FN | and we recover the predictions of GR. At distances larger than the Vainshtein radius the Galileon exerts a ﬁfth force which can be comparable to the Newtonian force.

64 2.7 Bi-gravity and multi-gravity

Finally, before closing this chapter we brieﬂy discuss a natural generalization of mas-

sive gravity that I will use when describing Deconstruction. Massive gravity is for-

mulated with two metrics, the dynamical metric gµν and the ﬁxed Minkowski metric

ηµν. In bi-gravity, one allows the second metric to become a dynamical ﬁeld [76].

Bi-gravity is formed by simply adding an Einstein-Hilbert term for the reference

metric, which is traditionally called fµν

M 2 Z √ M 2 Z S [g , f ] = Pl d4x −gR[g] + f d4xp−fR[f] bi−gravity µν µν 2 2 m2M 2 Z √ − −g U [gµν, f ], (2.97) 4 g.f. µν

−2 −2 −2 where M ≡ MPl + Mf .

The degrees of freedom of bi-gravity consist of a massive graviton with 5 degrees of freedom and a massless graviton with 2 degrees of freedom.

Massive gravity itself is a decoupling limit of bi-gravity, where the massless graviton decouples. We can see this by perturbing

δfµν fµν = ηµν + . (2.98) Mf

Then the decoupling limit consists of

Mf → ∞. (2.99)

65 In this limit, the full bi-gravity action in Eq. 2.97 has the form

M 2 Z √ m2 1 Z S = Pl d4x −g R − U [gµν, η ] − d4xδf E αβ δf . (2.100) 2 2 g.f. µν 4 µν µν αβ

Here we see that ﬂuctuations of fµν decouple from the system. By including matter

ﬁelds on the f metric and scaling the matter ﬁelds in the decoupling limit, we can

get massive gravity with other reference metrics. This gives us another picture of

massive gravity, it is the decoupling limit of an interacting bi-gravity theory.

It is also possible to construct theories of multi-gravity [66,82], with Ng metrics

Ng 2 Z 2 Ng ! X MPl,(i) p m X S = d4x −g(i) R[g(i)] − U [K(g(i), g(j))] , (2.101) 2 2 g.f. i=1 j=1

which consists of 1 massless graviton and Ng −1 massive gravitons, so has 2+5(Ng −1) degrees of freedom.

2.8 Summary

To summarize, let us recall the main points.

• By studying the linearized Fierz-Pauli theory, we saw that a healthy theory of

massive gravity propagates ﬁve degrees of freedom. We discussed the vDVZ

discontinuity which apparently rules out the linear theory, but we also de-

scribed how a gravitational theory necessarily contains interactions.

These basic issues inform much of the discussion.

66 • I gave an explicit form for the ghost-free mass term in several diﬀerent languages.

The vielbein form will appear when I discuss dimensional deconstruction in

Chapter 3. I outlined an ADM argument for the absence of the BD ghost and

showed explicitly how this worked in the mini-superspace. The ADM split will

be used in Chapter 3 and the mini-superspace argument will be used when I

discuss kinetic interactions in Chapter 4.

• By looking at the scales of the non-linear interactions, I gave a qualitative

argument for how the interactions can be responsible for a Vainshtein

screening mechanism which resolves the vDVZ discontinuity. These ideas

underlie massive gravity, and in Chapter 3 on dimensional deconstruction I will

show how this strong-coupling emerges from GR in higher dimensions. I also

showed how the BD ghost spoils the Vainshtein mechanism when it is present,

showing that this is a major problem. I also discussed how the BD ghost

manifests itself as an Ostrogradsky instability in the decoupling limit,

which spoils the Vainshtein mechanism. I will use these methods as a tool to

identify ghost modes when discussing the kinetic interactions in Chapter 4.

• Finally, I gave an explicit realization of the Vainshtein mechanism in

Galileon theories for static spherically symmetric sources. This will be

the basis for calculating pulsar radiation in Chapter 5.

67 Chapter 3

Dimensional deconstruction

In the previous chapter I reviewed how the ghost-free potential of massive gravity

1 avoids the BD ghost by cancelling the interactions at Λ5 . The ghost-free potential interactions are clearly not arbitrary, and the fact that they take such a simple form when written in terms of the vielbein language (see Section 2.3.2) is highly suggestive.

It is therefore natural to ask whether there is a deeper structure underlying the ghost- free potential ineractions.

In this chapter, we will discuss one approach to understanding the origin of the ghost-free structure. The approach we will take is dimensional deconstruction, which is a procedure to generate a 4D ﬁeld theory for multiple interacting ﬁelds by starting from a 5D theory. We will take GR in a 4+1 dimensional space-time (4 spatial and

1 time dimension), with one spatial dimension compact. We will then discretize the extra dimension. After discretization, a single five-dimensional field becomes become a finite, indexed set of four-dimensional fields, and derivatives along the y direction

1This chapter is based on my work in [60] with Claudia de Rham and Andrew J. Tolley.

68 become differences of fields. To give an explicit example, a generic field φ(x, y) (where xµ are coordinates for the 3+1 infinite directions and y is a coordinate the compact direction) becomes a labeled set of 4D field φ1(x), φ2(x), ··· , and the derivative along

the y direction becomes

2 2 2 (∂yφ(x, y)) → m (φ2(x) − φ1(x)) , (3.1)

where m−1 is the lattice spacing of the discretization. This must be done with some

care. Since the action for gravity is non-linear we will need to ﬁnd the appropriate

variable to discretize.

Dimensional deconstruction has been applied to massive gravity previously in [54–

59]. While many insights have emerged from this perspective, previous deconstruction

procedures always resulted in a BD ghost.

Our basic insight is to discretize using the vielbein instead of the metric

a 2,a 1,a ∂yeµ → m eµ − eµ . (3.2)

The vielbein is a particularly elegant and natural choice of variable for a gravitational

theory. The Einstein-Cartan formulation of gravity, in which gravity is a gauge the-

ory of the Poincar´egroup, uses the vielbein formulation. Additionally, while it is

impossible to couple fermions to the metric, fermions can be coupled to the vielbein.

Deconstruction will provide a way of interpreting many of the features of massive

gravity. The process will naturally produce a theory of ghost-free bi-gravity (or multi-

69 gravity), with two (or more) interacting spin-2 ﬁelds. Massive gravity can be obtained

as a decoupling limit as discussed in Section 2.7.

The strong-coupling scale of massive gravity may also be seen directly from 5D GR.

Previous works which applied deconstruction to gravity considered the low strong-

coupling scale as an obstruction to obtaining GR as the continuum limit.

The rest of this chapter is organized as follows. First, to give the basic idea, in

Section 3.1 we will brieﬂy review Kaluza-Klein compactiﬁcation in the simplest case of

a scalar ﬁeld and discuss how deconstruction works in this case. Then we will discuss

new issues that arise in the case of gravity in Section 3.2. I will review how a naive

deconstruction prescription in the metric language leads to a BD ghost in Section 3.3.

Then I will present the main result, an explicit deconstruction procedure in the viel-

bein language which produces the ghost-free potential interactions in Section 3.4. I

will show how generalizing the procedure in various ways one can recover diﬀerent

the diﬀerent ghost-free mass terms and multi-gravity. Finally we will discuss the

continuum limit of our procedure, paying close attention to the strong-coupling scale

in Section 3.5. I will close with some comments about the outlook for deconstruction

in Section 3.6.

Throughout this chapter I will use capital letters M,N, ··· to refer to ﬁve-dimensional

tangent-space indices and lower-case Greek letters µ, ν to refer to four-dimensional

tangent-space indices (so the 5D metric is GMN and the 4D metric is gµν). I will use upper-case A, B, ··· to refer to 5D local Lorentz indices, and lower-case a, b, ··· to refer to 4D local Lorentz indices.

70 3.1 Deconstruction of a scalar

To illustrate the basic idea of dimensional deconstruction, I will apply the process to

a scalar ﬁeld, and show that it produces a ﬁnite tower of massive scalars. This will

also serve to explain all the factors of N and 2π when we discuss the continuum limit

in Section 3.5. We start with the action for a free massless scalar in 5 dimensional

ﬂat space

Z Z 2πL (5D) 4 1 M S = d x dy − ∂M φ∂ φ , (3.3) 0 2

where the metric is ηMN . First I will brieﬂy recall the Kaluza-Klein procedure to

compare it with deconstruction. We split the 5D coordinates XM into 4D coordinates

xµ = (x0, x1, x2, x3) and a coordinate along the compact spatial dimension, y. We take

the dimension to be compact with length 2πL, so the points (xµ, y) and (xµ, y + 2πpL) are identiﬁed for any integer p. Then introducing the Fourier series of φ along the y direction,

∞ 1 X φ(x, y) = √ φ˜(n)(x)einy/L, (3.4) 2πL n=−∞

and using the orthogonality of the mode functions einy/L, the action in Eq. 3.3 can

be written as

" ∞ # Z 1 X n2 S(5D) = d4x − (∂φ˜(0))2 + −|∂ φ˜(n)|2 − |φ˜(n)|2 , (3.5) 2 µ L2 n=1

71 where |φ˜|2 ≡ φ˜?φ˜ and we have used the fact that φ is real so that (φ˜(n))? = φ˜(−n).

We see that we have a massless (real) scalar φ˜(0) and an inﬁnite tower of (complex)

−1 massive scalars with masses set by the size of the extra dimension mn = nL .

To perform dimensional deconstruction we discretize the y coordinate into N sites, labeled by I, and with lattice spacing m−1. The eﬀective size of the extra dimension is then 2πL = Nm−1.

After discretizing, the 5D field φ(x, y) becomes a labeled set of fields of 4D fields

φ(x, yI ) → φI (x),I = 1, ··· ,N. (3.6)

Additionally, derivatives along the y direction become diﬀerences

∂yφ(x, y) → m(φI+1(x) − φI (x)) on site I. (3.7)

While there are other possible discretizations of the derivative, here I will focus on the simplest one. I will also suppose that the boundary conditions are periodic so that φI+N = φI .

The net result of deconstruction (decon.) applied to Eq. 3.3 is a 4D action with

N ﬁelds

Z N (4D) 1 X 1 1 2 S = d4x − ∂ φ (x)∂µφ (x) − m2 (φ (x) − φ (x)) . (3.8) decon. m 2 µ I I 2 I+1 I I=1

The overall factor of m−1 accounts for converting the integral over dy into a sum. It √ can be absorbed into the scalar ﬁeld by rescaling φI → φI / m and accounts for the

72 3 fact that a scalar ﬁeld in 5 dimensions has mass dimension 2 while in 4 dimensions

a scalar has mass dimension 1. I will assume that this rescaling of the ﬁeld has been

done in what follows.

The spectrum of the theory can be determined by diagonalizing the mass term.

The mass term can be done by using the discrete Fourier transform. For odd N, this

is given by

(N−1)/2 1 X ˜(n) 2πinI/N φI (x) = √ φ (x)e . (3.9) N n=−(N−1)/2

In terms of the transformed ﬁelds, the theory takes the form

  Z (N−1)/2 (4D) 1 X S = d4x − ∂ φ˜(0)∂µφ˜(0) − |∂φ˜(n)|2 + m2 |φ˜(n)|2 , (3.10) Decon.  2 µ n  n=1

where the mass spectrum is given by

πn N − 1 m2 = 4m2 sin2 , n = 1, ··· , . (3.11) n N 2

For nmL 1 (that is, for modes that are very light compared to the discretization

−1 scale), we get a spectrum which is evenly spaced mn ∼ nL as expected from the

continuum theory. There is one real scalar and N/2 complex scalars, leading to N degrees of freedom.

When N is even, the expression for the Fourier transform is not as symmetric as for the case of odd N as in Eq. 3.9. Nevertheless, it is still true that given N sites, there is a massless mode as well as a tower of N − 1 massless modes. We can give an

73 simple, explicit example of this when N = 2. The action from deconstruction is

Z 1 1 S(4D,N=2) = d4x − (∂φ )2 − (∂φ )2 − m2(φ − φ )2 , (3.12) Decon. 2 1 2 2 1 2 where we recall that the mass term comes from a sum over the two distinct sites. We can diagonalize this action using the Fourier transform. When N = 2, taking the

˜(0) −1/2 Fourier transform results in two real scalar ﬁelds, given by φ ≡ 2 (φ1 + φ2) as

˜(1) −1/2 well as φ ≡ 2 (φ1 − φ2). In terms of these variables, the action is

Z 1 1 S(4D,N=2) = d4x − (∂φ˜(0))2 − (∂φ˜(1))2 − 2m2(φ˜(1))2 . (3.13) Decon. 2 2

In this form we can see that there is a real massless scalar ﬁeld φ˜(0) as well as a real massive scalar ﬁeld φ˜(1) with mass 2m.

To summarize, deconstruction provides a way of producing a ﬁnite tower of massive modes from a 5D theory. We now turn to applying deconstruction to gravity, where there are several additional subtleties to account for.

3.2 Deconstruction for gravity

Before moving into speciﬁc discretization procedures for gravity we address some general issues that we will have to face.

Non-linearities. We want to describe a theory of gravity, coupled to matter, which forces us to consider self-interactions. Because gravity is inherently non-linear, the

74 deconstruction procedure becomes ambiguous. The Leibniz rule, which holds for derivatives acting on ﬁelds A, B

∂y(AB) = (∂yA)B + A∂yB, (3.14)

no longer holds for discretized derivatives. As a result, there are many possible inequivalent discretizations. Of course, if we are interested in questions that involve length scales much larger than the discretization scale, we don’t expect this ambiguity to be very important. However, we will not be interested in this limit. Indeed we will

ﬁrst discretize a model that has only two sites. Rather we are interested in ﬁnding a prescription which reproduces the ghost-free structure. In the end discretizing in the vielbein, which is inequivalent to discretizing in the metric, produces the correct ghost-free structure. It is remarkable that choosing the vielbein, a simple and natural choice of variable, produces the ghost-free structure.

Gauge invariance. GR is a gauge theory. When we discretize we will be forced to compare fields at different space-time points from the point of view of the continuum theory. There are two ways to deal with this that turn out to be equivalent (up to one subtle point). We can fix a gauge and then discretize, or we can write the theory in a non-local but gauge invariant way before discretizing. While both methods are possible, it is more convenient to gauge fix first, and then recover the lost gauge symmetries via the Stückelberg procedure. The one subtlety is that our discretization procedure does force us to fix one coordinate freedom. Indeed we will see that from

75 the perspective of deconstruction, the strong-coupling scale of massive gravity can be

understood as a consequence of this gauge choice in higher dimensional GR.

Constraints. 5D GR has 5 ﬁrst class constraints which remove unphysical degrees of freedom. When we discretize, these constraints will generically be lost, as the discretization will cause the action to become non-linear in the lapse and shift. Indeed

our primary interest is in ﬁnding a discretization prescription that removes

the BD ghost (while still allowing for a Poincar´einvariant vacuum). One

method we could use to guarantee that the constraints are present is to perform a

4+1 split into time and 4 spatial directions, and then discretize the resulting Hamil-

tonian. Because the constraints are manifest at the level of the Hamiltonian, this

procedure guarntees that the constraints will remain after deconstruction. However,

this procedure runs the risk of losing 4D Lorentz invariance, since Lorentz invariance

is not manifest in the Hamiltonian language.

3.3 Deconstruction of the metric

A natural ﬁrst attempt at deconstruction is to discretize the derivative acting on the

metric. We write the 5D metric as GMN and the 5D Ricci scalar as R.

The action for 5D GR is

3 Z Z 2πL √ (5D) M5 4 SGR = d x dy −GR. (3.15) 2 0

Splitting the coordinates into the 4D coordinates xµ and the extra coordinate y, we

76 can write the action in an ADM form along the y (not t) direction as

M N µ µ ν ν 2 2 GMN dX dX = gµν (dx + N dy) (dx + N dy) + Ny dy . (3.16)

The action takes the standard ADM form

M 3 Z √ S(5D) = 3 d4xdy −gN R + [K]2 − [K2] , (3.17) GR 2 y

where R is the 4D Ricci scalar, and where the extrinsic curvature Kµν is given by

1 Kµν = (∂ygµν − ∇µNν − ∇νNµ) . (3.18) 2Ny

Fixing a gauge Ny = 1,Nµ = 0 we have

1 K = ∂ g . (3.19) µν 2 y µν

We then perform a two-site discretization. We refer to the metric on the ﬁrst site as gµν and the metric on the second site as fµν. We discretize the derivative in the sense

∂ygµν → m (gµν − fµν) on site 1,

m (fµν − gµν) on site 2. (3.20)

77 Applying this procedure to the action Eq. 3.17 leads to the bi-gravity action

M 2 Z √ S(4D) = Pl d4x −g R[g] + m2A (g, f) + (f ↔ g), (3.21) Decon. 2 metric

2 3 where MPl ≡ M5 /m and where

1 A (g, f) = gµνgρσ [(g − f )(g − f ) − (g − f )(g − f )] . (3.22) metric 4 µρ µρ νσ νσ µν µν ρσ ρσ

It is useful to see how this looks in the massive gravity decoupling limit2 where the background of fµν is frozen to ηµν and the ﬂuctuations decouple. The action for gµν is then

M 2 Z √ m2 S(4D) = Pl d4x −g R − [H2] − [H]2 , (3.23) decon.+D.L. 2 4 g˜ g˜

where decon.+D.L. refers to having done deconstruction and taken a decoupling limit,

µν and where Hµν ≡ gµν − ηµν and where [H]g˜ ≡ g˜ Hµν is a trace with respect to

the metricg ˜µν = gµν + (−g)−1/2ηµν. This potential is closely related to the naive

covariantization of Fierz-Pauli (Eq. 2.31), and of course has a BD ghost which can

be seen, for example, using the same kind of mini-superspace argument as in Section

2.4.3. 2 ˜ 2 2 This can be acheived by writing fµν = λf fµν and deﬁning Mf ≡ λf MPl. Then we can write ˜ −2 fµν = ηµν + Mf δfµν and take the limit Mf → ∞.

78 3.4 Consistent deconstruction procedure

In the previous section, we saw that a discretization procedure applied directly to the

metric failed to produce a ghost-free theory. In this section we will show that the

correct prescription which does reproduce the ghost-free potential is given by the very

natural choice of the vielbein as the fundamental variable. The procedure outlined in

this section is the heart of this chapter and the main result.

The 5D metric GMN is related to the 5D vielbein by

A B GMN = ηABEM EN . (3.24)

In terms of the vielbein, the action (ignoring the Gibbons-Hawking-York boundary term) can be written as

M 3 Z S(5D) = 5 ε RAB ∧ EC ∧ ED ∧ EE , (3.25) GR 12 ABCDE

where the Riemann curvature two form is related to the spin connection ΩAB by

AB AB AC DB R = dΩ + ηCDΩ ∧ Ω . (3.26)

A AB C The spin connection is ﬁxed by demanding the torsion-free condition dE +ηBCΩ E =

0, and is given explicitly by

1 ΩAB = EC OAB − O AB − OBA , (3.27) M 2 M C C C

79 where EM is the inverse vielbein (in the sense that EA EN = δN and EA EM = δA) A M A M M B B and where

C M N C C OAB ≡ EA EB (∂M EN − ∂N EM ), (3.28)

and local Lorentz indices A, B are raised and lowered with ηAB.

Now just as in the previous section, we will gauge fix before we discretize. It is always possible to re-introduce the broken gauge invariance via the Stückelberg procedure. We parameterize the vielbein and spin connection into quantities that will have nicer behavior under 4D diff and local Lorentz transformations

  a a eµ N A   EM =   . (3.29)  5  Eµ Ny

As discussed in the previous section, it is useful to ﬁx the gauge in the continuum theory. The gauge freedom can be recovered after discretization by introduce St¨uckelberg

a ﬁelds. Using our 5 diﬀ symmetries we will set N = 0 and Ny = 1. We will also use

5 four of the LLTs to set Eµ = 0.

It is useful to make a brief side comment about this choice of gauge. In a Kaluza-

Klein compactification, we typically expect to find additional massless states beyond the four-dimensional graviton. With one compact direction, in addition to the graviton there is typically a massless scalar field (the radion) and a massless vector field

(sometimes called the gravi-photon). We can account for these massless states by allowing Ny and Nµ to depend on the four dimensional coordinates, taking Nµ = Nµ(x)

80 and Ny = Ny(x). However, since these massless modes are well understood and do not aﬀect the structure of the mass term, we will neglect these modes in what follows

a by ﬁxing the gauge N = 0 and Ny = 1.

After performing this gauge ﬁxing, the vielbein has the form

  ea 0  µ  EA =   . (3.30) M   0 1

The torsion-free condition, plus the above fixing of diffs, leaves us with the nonvanishing spin connection components which are fixed by Eq. 3.27

ab ab µ ab Ω = ωµ dx + Ωy dy,

5a a µ Ω = Kµdx . (3.31)

5a In particular, note that Ωy = 0 in the gauge we have ﬁxed. We still have 6 remaining

gauge transformations, which are associated with the 4D Lorentz transformations.

We ﬁx this freedom by imposing the 6 conditions

ab Ωy = 0. (3.32)

5a a Meanwhile, the spin connection components Ω ∼ ∂ye are related to the extrinsic

curvature by

1 Ω5a = (eν∂ ea + ea∂ eν) e eb dxµ = eνK dxµ. (3.33) 2 b y ν ν y b µb µ a µν

81 We see that the extrinsic curvature Kµν can be written in terms of this component of the spin connection

a 5b a b Kµν = ηabeνΩµ = ηabeνKµ. (3.34)

The curvature two-form RAB may be written in terms of previously deﬁned quantities as

ab ab a b ab R = R − K ∧ K − ∂yω ∧ dy

5a a a b a R = dK + ω bK − ∂yK ∧ dy. (3.35)

Using this, we can write the action in the simple form

Z ab a b a b c d S = εabcd R − K ∧ K + 2K ∧ ∂ye ∧ e ∧ e ∧ dy, (3.36)

where the 4D Levi-Civita symbol is related to the 5D one by εabcd ≡ εabcd5. Note that we have been able to write the action in a form where there are only ﬁrst order y derivatives. This is a crucial property, as we will see in Chapter 4.

1,a Now we discretize. We deﬁne two sites, 1 and 2. We deﬁne a vielbein eµ on site

82 2,a 1 and a vielbein eµ on site 2. Then we discretize the derivative in the sense

a 2,a 1,a ∂yeµ → m e − e on site 1,

→ m e1,a − e2,a on site 2,

2 Z 1 X Z f (x, y)dxµ ∧ dy → f (x)dxµ. (3.37) µ m I,µ I=1

After discretization, the extrinsic curvature becomes

Ka → m(e2,a − e1,a). (3.38)

This leads to the 4D action

M 2 Z S = Pl ε R1,ab ∧ e1,c ∧ e1,d + m2Aabcd(e1, e2) + (1 ↔ 2), (3.39) 4 abcd

where

Aabcd(e, f) = (f a − ea) ∧ (f b − eb) ∧ ec ∧ ed. (3.40)

As a result, by discretizing 5D GR using our prescription, we have arrived at a theory

of ghost-free bi-gravity with a speciﬁc choice of potential.

The condition in Eq. 3.32, after discretization, becomes the symmetric vielbein

condition

1,a 2,a 1,a 2,a ηabeµ eν = ηabeν eµ , (3.41)

83 which we saw was related to the equivalence between the metric and vielbein formulations in Section 2.3.2. Here the condition has followed as a natural consequence of our procedure, related to a higher dimensional gauge choice.

Given the replacement rule Eq. 3.42, we can rephrase this procedure directly in the metric language as

a b α Kµν = ηabeµKν → mgµαK ν, (3.42)

µ µ µ p −1 where Kν = δν − g f ν . Applying this discretization directly to the action in the

ADM form in Eq. 3.17 yields the ghost-free mass term. This is not easy to guess in the metric language, which explains why previous attempts did not uncover a ghost-free mass term.

We can generalize this procedure in various ways. First, we may discretize the derivative diﬀerently. Second, we may discretize with multiple sites leading to a theory of multi-gravity rather than bi-gravity. We now turn to these generalizations.

3.4.1 Diﬀerent discretization of derivative

As a ﬁrst generalization of our deconstruction procedure, we consider alternative discretizations of the derivative. It is possible to discretize products of ﬁelds and derivatives such as A∂yB in multiple ways, even given a prescription for discretizing

84 ∂yB in isolation. Applied to the vielbein, we may discretize

a b c d 2 1,a 2,a 1,b 2,b e ∧ e ∧ ∂ye ∧ ∂ye → m re + (1 − r)e ∧ se + (1 − s)e

∧ e1,c − e2,c ∧ e1,d − e2,d , (3.43)

where r and s are real valued weights. Applying this discretization method to 5D GR leads to a more general ghost-free mass term. The action after discretization has the form of Eq. 3.39, but now with

abcd a a b b c c d d Ar,s (e, f) = (re + (1 − r)e ) ∧ se + (1 − s)e ∧ (e − f ) ∧ e − f . (3.44)

In fact, this is the most general mass term consistent with having a Poincar´einvariant

a a a vacuum solution eµ = fµ = δµ. In terms of the parameterization in Section 2.3.2, we have c0 = rs, c1 = (r + s − 4rs), c2 = (1 − 3s − 3r + 6rs), c3 = (−2 + 3s + 4r − 4rs), and c4 = (1 − s)(1 − r).

3.4.2 Multi-gravity

We may also generalize the deconstruction procedure easily to produce a theory of multi-gravity. To do this, we discretize the extra dimensions into N sites, with N

(I),a vielbeins eµ , where I = 1, ··· ,N. Then we apply similar discretization rules as in

the previous section. We keep a label I when we discretize, using

a (I),a a (I+1),a (I),a eµ(x, yI ) → eµ (x), ∂yeµ(x, yI ) → m eµ − eµ , etc. (3.45)

85 Taking this prescription and applying it to Eq. 3.36, we are led to a multi-gravity theory with N vielbeins

N 2 Z (4D) X MPl h (I) ab (I),c (I),d 2 abcd (I) (I−1) i S = εabcd R(e ) ∧ e ∧ e + m A (e , e ) , (3.46) Decon. 4 rI ,sI I=1

where N is the number of sites and where the parameters rI and sI depend on the site index I.

We need to specify boundary conditions when we do this. For technical reasons when N > 2 the simplest case of periodic boundary conditions is not appropriate

[83], so we can use an analogue of Dirichlet boundary conditions where we impose

a ∂ye(x, y1) = ∂ye (x, yN ) = 0 before discretizing.

3.5 Strong-coupling from deconstruction

Since any sensible discretization of the derivative will reproduce the derivative at large N, we expect the large N limit of multi-gravity to reproduce 5D GR. On one level this is a trivial observation. However, there is a subtlety with this statement. In massive gravity, there is a strong-coupling scale Λ3 MPl, yet there is no such scale in 5D GR. Thus it is interesting to see what happens to the strong-coupling scale in multi-gravity in the large N limit. This question was posed in the literature before the form of the ghost-free mass terms was known [55,56]. In this section we will give a somewhat diﬀerent perspective on the nature of the strong-coupling scale than is given in those work, with the modern understanding of how the strong-coupling is

86 related to the Vainshtein mechanism.

3.5.1 Strong-coupling scale at large N

As a ﬁrst step to understanding the strong-coupling scale in the continuum limit, it

is useful to see how the strong-coupling scale scales with the number of sites N at

large N.

In order to take the decoupling limit, we should first introduce Stückelberg fields to

carry the helicity-1 and -0 modes of the massive gravitons. To perform this analysis,

it is convenient to work in the metric language. While we have derived the mass

terms in the vielbein language, we can of course return to the metric language at any

point since they are equivalent formulations of the theory. There is a lot of freedom

for how to introduce the St¨uckelberg ﬁelds, but of course the physics will not depend

on the speciﬁc way we carry out the St¨uckelberg procedure. A simple choice is to

introduce N − 1 St¨uckelberg ﬁelds to act as maps from site 1 to the sites I for I 6= 1.

µ µ More precisely we introduce the N − 1 ﬁelds Φ(I)(x) = xI for I > 1 through the

St¨uckelberg replacement

(1) (1) (I) α β (I) gµν (x1) → gµν (x) and gµν (xI ) → ∂µΦ(I)∂νΦ(I)gαβ (Φ(I)(x)) for I > 1, (3.47)

µ µ where we have deﬁned x ≡ x1 . Let us perturb around the Poincar´einvariant vacuum

(I) (I) hµν gµν = ηµν + 2MPl µ µ µ µ BI ∂ πI Φ(I) = x − − 2 . (3.48) mMPl m MPl

87 In order to determine how the interaction scales depend on N, the crucial step is to determine how to canonically normalize the ﬁelds. In the site language (where the mass terms are not diagonalized), the free part of the Lagrangian has the schematic form

N X h 2 2 2 2 2i Lfree = (∂hI ) + m (hI+1(x) − hI ) + (hI+1 − hI )∂ πI + (∂BI ) . (3.49) I=1

Much like the scalar example in Section 3.1, we can diagonalize this by introducing the discrete Fourier transform

N ˜ 1 X 2πinI/N Tn = √ TI e . (3.50) N I=1

√ The N is chosen so that the modes are canonically normalized.

N−1 X h mn i L = |∂h˜ |2 + m2 |h˜ |2 + h˜? ∂2π˜ + |∂B˜ |2 , (3.51) free n n n m n n n n=0

where mn = 2m sin(πn/N) are the mass eigenvalues. The correctly normalized scalar

ﬁeld for n N is

m n πˆ = n π˜ ∼ π˜ . (3.52) n m n N n

We are interested in determining the lowest scale of interactions. To do this we note that the interactions can be written (schematically) in as powers in nh, nB, nπ (these powers are not to be confused with the discrete Fourier transform index n), expressing

88 the interactions in terms of the canonically normalized ﬁelds

nh nB 2 nπ (nh,nB ,nπ) 2 2 X hJ ∂BJ ∂ πI Lint = m MPl 2 MPl mMPl m MPl J 1 X ˜ nh ˜ nB 2 nπ ∼ n +n +n −2 (hn) (∂Bn0 ) (∂ πˆn00 ) . N (nh+nB −nπ−2)/2M h B π mnB +2nπ−2 Pl n,n0,n00,··· (3.53)

The scale associated with this interaction is

n +n −n −2 (n ,n ,n ) B h π h B π 2(n +2n +3nπ−4) Λint = N h B Λp, (3.54)

where Λp is given in Eq. 2.66. By inspection we can see that at large N, the lowest

−1/2 interaction scale occurs for nh = 1, nπ = 2. In this case, Λsc = N Λ, that is, the

interactions are given by

1 ˜? 2 2 Lint,lowest ∼ h2 ∂ πˆ1 , (3.55) Λsc where the strong-coupling scale is given by

2 1/3 r (m MPl) M5 Λsc = √ = . (3.56) N 2πL

We should note that of course this argument is purely based on counting powers of

N. To truly determine the interaction scale one should compute physical quantities

such as S-matrix amplitudes. However, we do not expect to be able to cancel the

interactions at this scale.

89 As a result, as we take the continuum limit N → ∞ holding 2πL = mN ﬁxed, the strong-coupling scale remains below the 5D Planck scale M5. In previous work, this was considered evidence that deconstruction was an ill deﬁned procedure. From our modern perspective, the strong-coupling scale is responsible for the Vainshtein mechanism in multi-gravity.

Indeed there is a nice way to see what is going on. We can introduce the Planck scale associated with the lowest n = 0 mode, which we call M4, as

NM 3 M 2 = M 2 N = 5 = M 3L, (3.57) 4 Pl m 5

3 2 where we used N = mL and M5 = MPlm. This relation between M4 and M5 is a familiar result from Kaluza-Klein theory. Then

2 1/3 Λsc = (m1M4) , (3.58)

−1 where m1 ≡ mN is the mass of the lightest massive Kaluza-Klein mode. In other words, the strong-coupling scale occurs at the Λ3 scale associated with the lightest

Kaluza-Klein mode.

However, it is worth seeing if we can understand the origin of the strong-coupling scale directly in terms of the continuum theory.

90 3.5.2 The 5D origin of the strong-coupling scale

We can see the origin of the strong-coupling in the continuum theory. In this section we will work again with 5D GR.

3.5.3 Gauge ﬁxed continuum theory

Working in a gauge where Ny = 1 corresponds to the gauge we have chosen in the discretization process. We can write the extrinsic curvature as

1 K = (∂ g − ∇ N − ∇ N ) . (3.59) µν 2 y µν µ ν µ ν

Writing Nµ = −∇µφ to capture the dynamics of the scalar St¨uckelberg mode in the continuum theory, the continuum version of the mass term becomes

2 2 Lm = [K ] − [K]

µ ν µν = ∂ygµν (∇ ∇ φ − g φ) 1 + (∇ ∇ φ)2 − ( φ)2 − ∂ g ∂ gµν + (gµν∂ g )2 . (3.60) µ ν 4 y µν y y µν

Thus in the continuum theory we see that φ gets a kinetic term only by mixing with

∂ygµν in this gauge. Thus, schematically, in order to canonically normalize this mode we need to rescale by 1/∂y. The interactions

2 2 µν (∇µ∇νφ) − (φ) = −R ∇µφ∇νφ + total derivatives, (3.61)

91 correspond precisely to the dangerous h(∂2φ)2 interactions.

As a result, 5 dimensional GR has a low ‘strong-coupling’ scale, in a gauge where

Ny = 1. In GR this is a statement that perturbation theory will break down at a low scale. For backgrounds with very small y dependence, perturbation theory will break down earlier. In GR, this is not a fundamental statement about physics, it is simply a statement about this coordinate system. As we will see in the next section by changing coordinates we can remove this strong-coupling. However, after discretizing this becomes a physical scale. This is not necessarily a problem, but simply a statement that ghost-free massive gravity theories are not equivalent to higher dimensional GR (even though they derive much of their structure from GR).

3.5.4 Generic gauge for continuum theory

We can see explicitly that the choice of gauge in the continuum theory is related to the strong-coupling scale by choosing a diﬀerent gauge where the strong-coupling does not arise. To see this explicitly, in this section we will allow the lapse to be arbitrary. The action for GR, keeping the lapse, is given by

3 Z M5 5 √ 1 2 2 SGR,5D = d x −g NyR − [K ] − [K] . (3.62) 2 Ny

−1 After a conformal transformation gµν = Ny g˜µν this becomes

M 2 Z 3 S = 5 d5xp−g˜ R˜ − (∂ N )2 − N −3 [K[g]2] − [K[g]]2 . (3.63) GR,5D 2 2 µ y y

92 2 Here we see the lapse appears with its own kinetic term ∼ (∂Ny) . Thus the scalar mode which now lives inside of Ny is not strongly coupled. In principle we could then discretize the theory in Eq. 3.63, and avoid the strong-coupling scale in the discretized theory. However, we expect such a discretization procedure to introduce a ghost, precisely because this is not the same procedure as the one which produced ghost-free potential interactions. This points to a tension between discretizing in a way which produces ghost-free potential interactions, and a discretization scheme which avoids a low strong-coupling scale. However, from the point of view of massive gravity, the low strong-coupling scale is not a sickness of the theory, but rather an essential component which allows for an active Vainshtein mechanism.

3.6 Outlook

In this chapter I have given an explicit demonstration of how to derive ghost-free massive gravity from ﬁve-dimensional GR, using dimensional deconstruction. This resolves an open problem in the literature as it was previously unknown how to apply deconstruction to GR and ﬁnd a consistent theory. Through this procedure we can view the constraint which removes the BD ghost as ultimately arising from GR itself.

Deconstruction provides a convenient way to think about many of the features of massive gravity. For example, the fact that there are two metrics is naturally interpreted as the existence of two sites. Intriguingly we have also shown that the strong-coupling in massive gravity is also present in ﬁve-dimensional GR in a particular gauge. In the past, the strong-coupling scale was interpreted as a cut-oﬀ for an

93 effective field theory, and so the fact that the strong-coupling remained in the large N limit of deconstruction was interpreted as a sign that the continuum limit of deconstruction was ill-defined. However, from the modern perspective, the strong-coupling scale is interpreted as being related to the Vainshtein mechanism, and is not a sign that the theory is breaking down. In principle, it may be possible to see an analogue of Vainshtein screening in five-dimensional GR in a gauge where the ‘lapse’ Ny is set

to 1.

A natural next step is to apply the deconstruction procedure to other set-ups

in higher dimensions, to look for additional consistent interactions. I applied this

logic to look at electrically charged spin-2 ﬁelds in [84]. A very intriguing possibility

along these lines is to apply deconstruction directly to Lovelock interactions, which

are consistent higher-derivative (higher-curvature) interactions for a massless spin-2

ﬁeld (which we review in Appendix A). One would naturally expect deconstruction

applied to these terms to generate higher derivative interactions for a massive spin-2

ﬁeld. We now turn to studying this idea.

94 Chapter 4

Kinetic interactions in massive gravity

The unique consistent theory of an interacting massless spin-2 ﬁeld, up to second order in derivatives on the metric, is given by the Einstein-Hilbert action plus a cosmological constant.1

For a massive spin-2 field we have seen that the mass terms are fixed (up to two dimensionless parameters) by demanding that a Boulware-Deser mode does not propagate. However, is this the most general consistent interacting theory of a massive spin-2 field? So far we have been assuming that the kinetic interactions (meaning the interactions with two derivatives in unitary gauge) are the same as in GR.

When constructing a theory of a massless spin-2 field, diffeomorphism invariance is a guide, which restricts the possible interactions. For massive spin-2 fields, there

1This chapter is based on my work in [61,85] with Claudia de Rham and Andrew J. Tolley. Some parts, especially Section 4.5, come from my single author paper [62].

95 are many possible non-linear interactions consistent with the symmetries involving

two derivatives. For example, consider these derivative interactions involving gµν and

the Lorentz invariant reference metric ηµν

µν µν µ ρσ ηµνR , ηµνg R, ∂µgρσ∂ g ··· (4.1)

where the partial derivative ∂µ appears because it is the covariant derivative with respect to ηµν.

If ghost-free kinetic interactions of this form exist, they should be considered part of a theory of massive gravity on par with the mass terms. On the other hand, if ghost-free kinetic interactions do not exist, this tells us something special about the

Einstein-Hilbert structure that was not completely obvious a priori. The Einstein-

Hilbert structure is needed not just on symmetry requirements for a massless spin-2

ﬁeld, but is needed for consistency for a massive spin-2 ﬁeld with less symmetry.

Furthermore kinetic interactions can be useful from a model-building perspective.

For example massive gravity with a non-standard kinetic term could allow for exact

FLRW solutions without introducing new degrees of freedom, or without breaking any symmetries like Lorentz or translation invariance. Beyond cosmology, one is often led to consider kinetic interactions in natural extensions of massive gravity.

For example, a consistent theory of an electrically charged spin-2 ﬁeld coupled to gravity requires the existence of ghost-free kinetic interactions [84]. Finally, kinetic interactions are intimately related to generalized couplings to matter in which matter

ﬁelds simultaneously couple to both metrics (for examples, see [86–91]).

96 A candidate kinetic interaction was proposed by [92], which is cubic in the metric perturbation Hµν

(der) 1 µραγ ν σ β λ L3 = ε ενσβλH µH ρ∂ ∂αH γ, (4.2) Λder

where indices are raised and lowered with ηµν and where Λder is a scale needed for dimensional reasons. This interaction propagates 5 degrees of freedom when added to the Fierz-Pauli Lagrangian Eq. 2.18. This is not a free theory in the sense that it is cubic in H. Nevertheless it is not satisfactory as a gravitational theory in its present form. As we have seen when we include a coupling to matter we are forced to include an inﬁnite number of possible interactions including all powers of H. It is natural to ask whether this interaction can be extended to a gravitational theory without introducing new degrees of freedom.

As we will show in this section, in massive gravity there are no ghost-free extensions of the kinetic structure. Furthermore this result also applies to bi-gravity since massive gravity is a consistent decoupling limit of bi-gravity.

This result may be compared with recent work deriving the Einstein-Hilbert structure as the unique theory which is spatially covariant and propagates two degrees of freedom, appropriate for a massless graviton [93–95]. In that case there are many possible interactions allowed by symmetry, but the Einstein-Hilbert term is required in order to have two propagating degrees of freedom. As a result, this work into a larger story that the Einstein-Hilbert structure can be derived by consistency requirements.

While the results of this chapter are a null result in the sense that we do not

97 ﬁnd a new kinetic interaction, we emphasize that this is not obvious in advance and shows that there is more to the Einstein-Hilbert structure than diﬀeomorphism invariance. It also makes massive gravity in some sense very special. Once we ask for a consistent ghost-free theory of massive gravity that is Lorentz invariant and that does not introduce new degrees of freedom, there are only a handful of interactions that can be written down.

The rest of this chapter is organized as follows. In Section 4.1 we will illustrate the basic ideas for a spin-1 ﬁeld. Then we formulate the issues that will arise in the spin-2 case in Section 4.2. In Section 4.3 we look at candidate kinetic interactions arising from dimensional deconstruction. However, we will ﬁnd that these terms have a ghost by using a mini-superspace argument. We interpret the failure of deconstruction as evidence that new ghost-free kinetic interactions do not exist. In Section 4.4 we give a systematic argument based on the decoupling limit which shows that there are no ghost-free kinetic interactions for a massive graviton besides the Einstein-

Hilbert term, in four space-time dimensions. In Section 4.5 I apply this result to an extension of the non-linear matter couplings in massive gravity. Finally I discuss the implications of these results in Section 4.6.

4.1 Kinetic term for massive spin 1

As a warm up and to clarify some of the issues that will arise, let us ask the analogous question of a massive spin-1 field. For a Poincaréinvariant massive spin 1 field the

98 most general kinetic term has the form

1 1 − α (∂ A )2 + β(∂ Aµ)2 − m2A2 . (4.3) 2 µ ν µ 2 µ

The canonically normalized Maxwell Lagrangian has α = −β = 1 but it is not

necessarily obvious a priori that we need to accept this tuning for the massive theory.

One way to see that there is a problem is to construct the Hamiltonian. The conjugate momenta are given by

∂L π ≡ = 2 (α + β) A˙ 0 ˙ 0 ∂A0 ∂L π ≡ = −2 αA˙ + β∂ A . (4.4) i ˙ i i 0 ∂Ai

We are able to invert for all 4 velocities if α + β 6= 0, leading to a Hamiltonian

π2 1 H = − 0 + π2 + π X(A ,A ) + π Xi(A ,A ) + H0(A ,A ). (4.5) α + β α i 0 0 i i 0 i i 0

From the ﬁrst term, we see that for α + β 6= 0 the Hamiltonian is unbounded below,

and furthermore this happens through the kinetic energy signaling a ghost.

To connect with the methods we shall use in this section, we can see the ghost in

the St¨uckelberg language by introducing a helicity-0 mode π

1 A → A + ∂ π. (4.6) µ µ m µ

99 Then the focusing on the action for the helicity-0 mode we ﬁnd

1 1 L = α(∂ ∂ π)2 + β( π)2 − (∂π)2. (4.7) π m2 µ ν 2

The terms ∼ 1/m2 signal a ghost unless α + β = 0, in which case that term becomes a total derivative.

The net result is that for a massive spin-1 particle, we are forced to take the gauge invariant Maxwell kinetic term if we want to avoid a ghost, even though the mass term breaks gauge invariance, as we have seen in two diﬀerent languages.

We will argue that there is a similar conclusion for a massive spin-2.

4.2 Candidate kinetic interactions

The defining property of a spin-2 field is that it propagates five degrees of freedom around any background. This property must be preserved by any viable theory describing massive gravity, and in particular by a consistent kinetic interaction. The starting point for understanding kinetic interactions is an analogue of the Fierz-Pauli theory, namely the theory of spin-2 field Hµν living on a fixed, Minkowski background.

In this chapter, we will make a conceptual distinction between a fully non-linear in- √ teraction (for example, the Einstein-Hilbert term −gR) and the leading order piece

µνρσ of that interaction when perturbed around ﬂat space (for example, HµνE Hρσ). We

will assume that the non-linear interaction is analytic, when perturbed around ﬂat

space. This guarantees that the theory can be quantized in a consistent way around

100 the Minkowski vacuum state. A fully non-linear (analytic) interaction involves an in-

finite series in H of the schematic form ∂2Hn +∂2Hn+1 +··· . The leading order term in the expansion of a kinetic interaction. Physically this represents the interactions for a spin-2 field propagating on a fixed, Minkowski background.

4.2.1 Leading order interactions

In [92] one candidate kinetic interaction was proposed for a spin-2 ﬁeld Hµν on a

Minkowski background. Indeed, it was proposed to be the only possible kinetic interaction could be written in the form

(der) (der) Lg.f. = L2 + L3 , (4.8)

where

(der) µν αβ 1 µνρσ µ0ν0ρ0σ0 L = H E H = ε ε η 0 ∂ H 0 ∂ 0 H 0 (4.9) 2 µν αβ 2 σσ ρ µµ ρ νν

(der) µνρσ µ0ν0ρ0σ0 L3 = ε ε Hσσ0 ∂ρHµµ0 ∂ρ0 Hνν0 . (4.10)

(der) Of course, L2 is the leading order term in an expansion of the Einstein-Hilbert

(der) term. The interaction L3 , meanwhile, is a genuine two-derivative interaction for a spin-2 field. This interaction is perfectly valid for a massive spin-2 field propagating on Minkowski space-time, in the sense that five degrees of freedom propagate when this interaction is included.

This can be seen, for example, using a Hamiltonian analysis. At the linear level

101 we can identify H00 as the lapse and H0i as the shift. The εε structure guarantees

that the Lagrangian takes the form

00 0i ˙ 0i ˙ L = H C(Hij, ∂iHjk) + H Ci(Hij, ∂kHij) + H0iH K(Hij, ∂kHij)

˙ +L(Hij, Hij, ∂kHij). (4.11)

The constraint to remove the BD mode is present because when we integrate out the shift H0i, then the lapse H00 remains a Lagrange multiplier. Furthermore the

εε structure guarantees we can write the Lagrangian in a way where only ﬁrst order time derivatives appear, so we don’t generate higher derivatives in this process.

Stated more physically, the interactions in Eq. 4.8 are healthy interactions for a spin-2 field on a fixed Minkowski background because all equations of motion are second order when we perform the linear Stückelberg decomposition

hµν 1 2 Hµν = + (∂µBν + ∂νBµ) + 2 ∂µ∂νπ. (4.12) MPl mMPl m MPl

Physically, this is because hµν, Bµ, and π carry the helicity degrees of freedom and so

their equations of motion should be second order. Indeed, recall that we have already

described how higher derivatives on π are a manifestation of the BD ghost in Section

2.2. However, these interactions in Eq. 4.8 are still not suﬃcient for a gravitational

theory, as we will now describe.

102 4.2.2 Non-linear St¨uckelberg decomposition

While the interactions described in the previous section are satisfactory interactions for a massive spin-2 field on a fixed Minkowski background, they are not (by themselves) acceptable as interactions in a full theory of gravity. The reason is that once the spin-2 field couples to matter, one necessarily needs to include self-interactions for the spin-2 field. More precisely, matter fields must inevitably couple covariantly to a metric ηµν + Hµν.

For the case of a massless spin-2 field, this requirement ultimately forces the linear diff gauge symmetry of the linear theory to be promoted to a non-linear diff symmetry in the full theory. For the massive case, the analogous statement is that we should introduce non-linear Stückelberg fields [54,96]

hµν 1 2 Hµν = + ∂(µBν) + 2 ∂µ∂νπ MPl mMPl m MPl ρσ ∂µBρ ∂µ∂ρπ ∂νBσ ∂ν∂σπ +η + 2 + 2 , (4.13) mMPl m MPl mMPl m MPl which are patterned off of non-linear diffs. Under this decomposition, when we take the Λ3 decoupling limit as described in Section 2.5.2, the fields hµν,Bµ, and π decouple and live on a fixed Minkowski background. As a result, these fields represent the physical, propogating degrees of freedom (the helicity eigenstates of the massive graviton), and their equations of motion should be second order.

This leads us to a similar problem we saw in Chapter 2. Much as the naive covariantization of Fierz-Pauli had a BD ghost, the naive covariantization of Eq. 4.2

103 does as well. Consider

(der) 2 √ µνρσ µ0 ν0 (g)σ (g) ρ0 L3 = Λder −gε εµ0ν0ρ0σ0 Hµ Hν ∇ ∇σ0 Hρ , (4.14)

where now indices are raised and lowered with respect to gµν. Performing the non-

2 −4 linear St¨uckelberg we ﬁnd an interaction at the scale Λder(mMPl)

2 Λder µραγ ν σ λ β ω 4 ε ενσβλh µ∂ρB ∂α∂ ∂ωπ∂γ∂ ∂ π, (4.15) (mMPl) which leads to higher order equations of motion.

4.2.3 Non-linear completions

As we have seen, we need to add higher order interactions to make a consistent kinetic interaction. The most general non-linear completion for kinetic interactions in massive gravity will have the form

2 Z 2 2 MPl 4 √ m Λder S = d x −g R − U(g, η) + 2 Lder , (4.16) 2 2 MPl

where we can expand Lder perturbatively in Hµν = gµν − ηµν

(n) (n+1) Lder = Lder + Lder + ···

= ∂H∂HHn−2 + ∂H∂HHn−1 + ··· . (4.17)

104 Here we use a schematic notation, where each term in this sum represents every possible contraction of indices, with each contraction arising with an arbitrary coeﬃcient.

To determine whether Lder propagates ﬁve degrees of freedom or not, there are several diﬀerent methods we can use

1. We could do a fully non-linear Hamiltonian analysis of a general action of the

form Eq. 4.16 to determine the degrees of freedom. For example we could work

in unitary gauge, determine the naive phase space and determine all the ﬁrst

and second class constraints, and ask how many degrees of freedom there are

with the full Hamiltonian.

2. Alternatively in the St¨uckelberg language we could compute the Hessian (either

fully non-linearly or perturbatively around an arbitrary background) and deter-

mine the size of the phase space before imposing the ﬁrst class constraints. It

is both necessary and suﬃcient for a healthy kinetic term to have 5 degrees of

freedom when subjected to this analysis.

3. A method which provides a necessary but not suﬃcient condition, and is much

easier to check, is to study the interactions for the helicity degrees of freedom

around Minkowski space (following the same logic we used when considering

interactions for the mass term in Section 2.5.1). We introduce the St¨uckelberg

ﬁelds and take a decoupling limit around Minkowski space. Then the equations

of motion should be second order. As with the mass term, phrasing this re-

quirement in terms of the equations of motion allows us to bypass subtleties

about total derivatives in the Lagrangian formulation.

105 4. As a corollary to the above, the leading order (in H) part of the Lagrangian

must have second order equations of motion when we do the linear St¨uckelberg

decomposition.

We will apply these methods in this chapter.

4.3 Deconstruction-inspired ansatz

A particularly simple guess can be motivated by dimensional deconstruction. We will consider this guess and apply a non-linear ADM analysis to the result.

In 5 dimensions there is a four-derivative ghost-free kinetic structure for a massless graviton, the Gauss-Bonnet term (this is a special case of the Lovelock interactions, which are reviewed in Appendix A)

M 3 Z √ S = 5 d5x −G R2 − 4R2 + R2 GB m2 ABCD AB 3 Z √ M5 5 µναβ µ0ν0α0β0 h 1 = d x −GE E R 0 0 R 0 0 − K 0 K 0 K 0 K 0 4m2 µνµ ν αβα β 12 µµ νν αα ββ i +Kµµ0 Kνν0 Rαβα0β0 , (4.18)

where E µνρσ is the covariant form of the Levi-Civita symbol. Then we can apply our deconstruction rules given in the previous chapter. In the four-dimensional theory that occurs as a result of deconstruction, the ﬁrst term in the second line of Eq. 4.18 is a total derivative and can be neglected. If we follow the discretization precription

106 in Eq. 3.42

µ µ K ν → mK ν, (4.19)

then the second term ∼ K4 becomes a ghost-free mass term, and the third term

∼ K2R becomes a kinetic interaction for a massive graviton

Z 4d 2 4 √ ∗ µανβ SKK∗R = MPl d x −g KµνKαβ R , (4.20)

∗ αβγλ where Rµνρσ ≡ EµναβEρσγλR is the double dual of the Riemann tensor. We can

also generate another kinetic interaction by changing the discretization prescription,

as we did in Section 3.4.1. Or, more directly, by shifting Kµν → gµν +Kµν we generate

the interaction

M 2 Z √ S4d = − Pl d4x −g g K ∗Rµανβ (4.21) KG 4 µν αβ Z 2 4 √ µν = MPl d x −g Kµν G , (4.22)

where Gµν is the Einstein tensor.

4d In fact SKK?R is a candidate non-linear completion for the interaction in Eq. 4.2 since writing gµν = ηµν + Hµν and perturbing to cubic order,

Z Z 4 √ ∗ µανβ 4 µνρσ µ0 ν0 σ ρ0 4 d x −g KµνKαβ R = d xε εµ0ν0ρ0σ0 Hµ Hν ∂ ∂σ0 Hρ + O(H ). (4.23)

This is not the only non-linear interaction that completes Eq. 4.2, but it is striking

107 that deconstruction had identiﬁed the ‘most promising’ candidate.

4d Meanwhile, SKG linearizes to the kinetic term of the Fierz-Pauli Lagrangian given

in Eq. 2.18. It provides an alternative non-linear completion to Fierz-Pauli, beyond

the Einstein-Hilbert term.

4d 4d Unfortunately, all linear combinations of SKG and SKK?R lead to a BD ghost. This can be seen directly by working in the mini-superspace

2 2 2 2 i j ds = −N (t)dt + a (t)δijdx dx . (4.24)

If the theory has a constraint to remove the BD ghost, then the lapse N should appear

linearly the mini-superspace.

The action in the mini-superspace for the deconstruction-inspired interactions is

given by

Z aa˙ 2 a˙ 2 aa˙ 2 S = 3 dtd3x 2 − + KG N N N 2 Z 2 2 2 2 3 aa˙ a˙ aa˙ a˙ S ∗ = 24 dtd x − + − . (4.25) KK R N N N 2 N 2

Because of the terms ∼ N −2 in the action, the resulting Hamiltonian will not be

linear in the lapse as can be veriﬁed with a short calculation, signaling that there is

no constraint to remove the BD ghost.

It is interesting to ask why deconstruction succeeded in producing a ghost-free

theory when applied to the Einstein-Hilbert term in ﬁve dimensions, but failed for

Gauss-Bonnet. This is ultimately due to the fact that the Gauss-Bonnet term involves

108 2 higher powers of curvature ∼ Rµνρσ, and the action naively involves higher derivatives acting on h such as (∂2h)2. The Gauss-Bonnet term relies on total derivative combinations to cancel these interactions. However, since the Leibniz rule is broken by a discrete derivative, these total derivative combinations become physical after discretizing. A diﬀerent way to express this idea is that, in the continuum theory, we were able to write the Einstein-Hilbert action in a form where y derivatives only appeared linearly.

The failure of deconstruction is itself evidence that there may not be a consistent derivative interaction for a massive graviton. Nevertheless the next logical step is to perform a systematic search to search for such a kinetic interaction.

4.4 Systematic argument

Since deconstruction has failed to produce a ghost-free kinetic interaction, there are no obvious physically motivated candidates for a kinetic interaction. In light of this, it is necessary to do a brute force search through diﬀerent possibilities. In this section, we will perform such a search and give an argument that there are no ghost-free kinetic interactions beyond the Einstein-Hilbert term for a massive spin-2 ﬁeld.

Our method is built on the fact that in the decoupling limit, the BD ghost manifests itself as an Ostrogradsky instability. We will work with massive gravity with a ﬂat reference metric, ηµν. However as we have discussed in Section 2.7 this can be viewed as a consistent decoupling limit of bi-gravity.

109 4.4.1 St¨uckelberg criteria

In the next two subsections we will outline the theoretical underpinnings of our

method. The bottom line is summarized in the algorithm we give in Section 4.4.3,

which we then apply in the remaining parts of this section.

In the decoupling limit we can ask that the equations of motion for all the helicities

h, B, π be second order, when we use the non-linear St¨uckelberg decomposition.

In this section we connect this to the standard form of the St¨uckelberg procedure.

We can deﬁne the non-linear metric perturbation for a ﬁxed background metric ηµν

Hµν ≡ gµν − ηµν. (4.26)

Then introducing the St¨uckelberg ﬁelds through gµν

α β gµν → g˜µν = ∂µφ ∂νφ gαβ(φ(x)), (4.27)

and perturbing

S hµν gµν = ηµν + MPl α α α α B ∂ π Φ = x + + 2 , (4.28) mMPl m MPl

110 we can write the metric ﬂuctuation as

α β S ∂µΦ ∂νΦ hαβ(Φ) ∂(µBν) ∂µ∂νπ Hµν = + + 2 2 (4.29) MPl mMPl m MPl α α ∂µB ∂µ∂ π ∂νBα ∂ν∂απ + + 2 + 2 . mMPl m MPl mMPl m MPl

In fact we can simplify the analysis, in the decoupling limit. In the above decompo-

sition it is suﬃcient to replace

α β S ∂µΦ ∂νΦ hαβ(Φ) → hµν(x). (4.30)

The reason is that we will be focusing on interactions that are at most linear in h.

Thus we can write this interaction (in the decoupling limit) as

(d.l.) S a α β µν Lder = hαβ(Φ )∂µΦ ∂νΦ Y µν 1 µ σν µν σ = hµν(x) Y + 3 (2∂σ∂ πY − ∂σ(Y )∂ π) + ··· Λ3 S, µν = hµν(x)Y , (4.31)

where in the second line we have Taylor expanded h(x+∂∂π) and integrated by parts,

S 1 and in the last line we have deﬁned Y = Y + 3 ∂(∂πY ) + ··· . Λ3 The condition that the equations of motion be second order is equivalent to de-

manding that Y must vanish. This, in turn, implies Y S must vanish to the order we

are working since the relationship between Y and Y S is invertible. Furthermore there

are no interactions at higher order in hS.

111 The bottom line is that we can make the replacement

hµν ∂(µBν) ∂µ∂νπ Hµν = + + 2 2 (4.32) MPl mMPl m MPl α α ∂µB ∂µ∂ π ∂νBα ∂ν∂απ + + 2 + 2 , mMPl m MPl mMPl m MPl and ask for second order equations in the decoupling limit.

4.4.2 Decoupling limits

Having introduced the St¨uckelberg ﬁelds, we now turn to the equations of motion.

Before going into the details it is useful to look at the schematic form of the interactions.

The general interaction is an inﬁnite series in powers of H which we write sym- bolically as

p p+1 Lder = ∂H∂HH + ∂H∂HH + ··· , (4.33)

where each term in this expression refers to a sum over every possible tensor contraction of that form, with arbitrary coefficients in front of each term. At order Hn, there are interactions on the Stückelberg fields of the form

(n) 2 n−2 Lder ∼ ΛderH ∂H∂H

2nh+nB −2 4−3(nh+nB +nπ) 2 nh nB 2 nπ ∼ m Λ3 (∂ )h (∂B) (∂ π) . (4.34)

112 We will focus on interactions that come in at a scale where there is at most one power

of h, so that we can consistently take the background to be Minkowski, rather than

ηµν + hµν.

To organize the interactions, it is convenient (though not necessary) to take a scaling limit so that all interactions occur at the scale Λ3. This procedure will not

change the number of degrees of freedom (although we will not be able to directly

infer information such as the mass of a ghost from this scaling limit). We choose the

scaling of Λder so that the leading non-trivial interactions occur at Λ3.

First we have the free Lagrangian

2 LFP ∼ h∂ h. (4.35)

Since this will be the leading order piece, it should involve second order equations of

motion under a linear St¨uckelberg decomposition.

Then we have interactions ∼ ∂2hn for n > 2. These interactions could either be

a higher order term in a non-linear completion, or the beginning of a new non-linear

completion. First, scaling Λder ∼ Λ3 in Eq. 4.34, we ﬁnd interactions

2 nπ 2 2 ∂ π L 2 1/3 ∼ Λ ∂ . (4.36) der,(m MPl) 3 3 Λ3

The equations of motion will automatically be higher than second order just by count-

ing derivatives and ﬁelds. Demanding all these interactions have vanishing equations

of motion will constrain the parameters of the theory.

113 2 1/3 Next, scaling Λder ∼ (mMPl) , we ﬁnd the interactions

2 nπ 2 nπ 2 ∂ π 2 2 ∂ π L 2 1/3 ∼ Λ3h ∂ , (∂B) ∂ . (4.37) der,(mMPl) 3 2 Λ3 Λ3

Again, imposing that the equations of motion are second order constrains the parameters.

5 1/6 Finally, scaling Λder ∼ (mMPl) we ﬁnd the interactions

2 nπ 2 nπ 1 2 ∂ π 1 2 3 ∂ π L 5 1/6 ∼ ∂ h∂B , ∂ (∂B) . (4.38) der,(mMPl) 5 3 4 3 Λ3 Λ3 Λ3 Λ3

After cancelling these interactions we are left with interactions at O(h2). In that case it is less clear how to use the higher derivative requirement to identify a ghost mode.

4.4.3 Algorithm

The bottom line of the preceding two sections is that we will implement the following algorithm using Mathematica:

• We will construct the most general Lagrangians with two derivatives and arbi-

trary powers of Hµν (up to total derivatives and combinations that vanish in 4

space-time dimensions) of the schematic form

L = ∂H∂H + ∂H∂HH + ∂H∂HH2 + ··· , (4.39)

4 where all indices are contracted with ηµν. It is suﬃcient to work to O(H ) as

114 we describe below.

• We will introduce the St¨uckelberg ﬁelds through H directly as in Eq. 4.32

hµν 2∂(µBν) ∂µ∂νπ Hµν = + + 2 2 (4.40) MPl mMPl m MPl α α ∂µB ∂µ∂ π ∂νBα ∂ν∂απ + + 2 + 2 . mMPl m MPl mMPl m MPl

• We focus on the interactions that are linear in h so that the background is ηµν

and we can identify the helicity degrees of freedom with hµν,Bµ, π.

• We demand all equations of motion be second order. Demanding that any

terms higher than second order vanish identically will put constraints on the

parameters implicitly contained in Eq. 4.39.

• We solve all constraints for the parameters at a given order to determine the

parameters that are ghost-free in the decoupling limit.

Any interaction that survives this check has a ﬂat-space decoupling limit where it is ghost-free. It still may be that the ghost appears beyond the decoupling limit, but nevertheless the interaction will have passed a highly non-trivial check. Indeed, for the mass term, avoiding a ghost in the Λ3 decoupling limit is fully suﬃcient to avoid the BD ghost non-linearly.

As we will now show, the only interaction that survives this check is the Einstein-

Hilbert term.

115 4.4.4 Quadratic order

The most general ansatz (after integrations by parts) is

(2) 2 2 µν µν 2 Lg.f. = a1(∂µHρσ) + a2(∂µH) + a3∂µH ∂νH + a4(∂µH ) . (4.41)

Plugging in the linear St¨uckelberg decomposition and demanding that all equations

of motion be second order, we are led (up to an overall normalization a) to the

Fierz-Pauli Lagrangian

1 1 1 1 L(2) = a (∂ H )2 − (∂ H)2 + ∂ Hµν∂ H − (∂ Hµν)2 . (4.42) a 4 µ ρσ 4 µ 2 µ ν 2 µ

4.4.5 Cubic order

Next we consider the interactions at cubic order. We consider the Lagrangian made of

the ghost-free Lagrangian at quadratic order, plus the most general cubic Lagrangian

with three powers of H and two derivatives L(2) = L + L(3) where L(3) has 14 g.f.,Λ3 a gen gen parameters (after integrating by parts and removing total derivative combinations)

(3) µν ρσ µν σ µν Lgen = b1H ∂µH ∂νHρσ + b2H ∂µH ν∂σH + b3H ∂µH∂νH

µν σ µ µν σ ρ + b4H ∂σH∂ Hµν + b5H∂µH∂ H + b6H ∂ρHµσ∂ H ν

σ µν µν ρ σ µ νσ + b7H∂µHνσ∂ H + b8H ∂ρHµσ∂ H µ + b9H∂µHνσ∂ H

µν σ ρ µν ρ σ µν σ + b10H ∂σH µ∂ρH ν + b11H ∂µH ν∂σH ρ + b12H∂µH ∂σH ν

µν ρ σ µ ν + b13H ∂ Hµν∂ Hρσ + b14H∂ H∂ Hµν. (4.43)

116 Then introducing the non-linear St¨uckelberg decomposition we can demand that all

equations of motion be second order. For example the equations of motion contain

the term

(3) δLgen 4 h i µ ν σ ⊃ − 5 a + 4b3 + 2b7 + 2b9 ∂µ∂ν∂σh∂ ∂ ∂ π. (4.44) δπ Λ3

In order for the equations of motion to be second order, the parameters must satisfy

a + 4b3 + 2b7 + 2b9 = 0. (4.45)

Solving all such constraints we are led to a three parameter family of Lagrangians that is ghost-free in the Λ3 decoupling limit

L(3) = a L(2) + L(3) + b L(3) + b L(3) g.f.,Λ3 a a 1 b1 2 b2

µν αβ = −2aH Eµν Hαβ

µν ρσ µν σ µν + b1H ∂µH ∂νHρσ + b2H ∂µH ν∂σH − b1H ∂µH∂νH 1 a + (2b − b ) Hµν∂ H∂σH + −b + b + H∂ H∂µH 1 2 σ µν 1 2 2 2 µ 1 + (5b − b − a) Hµν∂ H ∂σHρ + b + b H∂ H ∂σHµν 1 2 ρ µσ ν 1 2 2 µ νσ 1 1 + (−2b + b ) Hµν∂ H ∂ρHσ + b − b − a H∂ H ∂µHνσ 1 2 ρ µσ ν 1 2 2 2 µ νσ

µν σ ρ µν ρ σ + (−3b1 + a) H ∂σH µ∂ρH ν − b2H ∂µH ν∂σH ρ 1 + −3b + b + a H∂ Hµν∂ Hσ + (−2b + b ) Hµν∂ρH ∂σH 1 2 2 µ σ ν 1 2 µν ρσ

µ ν + (2b1 − b2 − a) H∂ H∂ Hµν. (4.46)

117 It is useful to introduce the short-hand notation

L(3) = L(3) , (4.47) a g.f.,Λ3 b1=b2=0 and similarly for L(3) . Conceptually, L(3) is part of the non-linear completion of L(2). b1,b2 a a

(3) (2) (3) La is a sum of cubic interactions such that the combination La + La is ghost-free.

Interestingly, the deconstruction terms may be written as linear combinations of the Lagrangians which are ghost-free to cubic order in the Λ3 decoupling limit

√ 1 L = −gR = [L − L − 4L ] EH 4 a b1 b3 √ 1 3 11 L = −gK Gµν = − L − L − L KG µν 4 a 2 b1 2 b2

√ ∗ µµ0νν0 (der) ∗ 0 0 LKK R = −gKµνKµ ν R = Lb1 + 4Lb2 = L3 . (4.48)

As an important check, the candidate non-linear interaction in Eq. 4.2 is one of the terms we have identiﬁed by our consistency conditions at cubic order. This can

4d be seen directly since to cubic order the deconstruction ansatz SKK?R is identical to

(der) L3 .

We also have identiﬁed a potentially new non-linear completion of Einstein-Hilbert,

(2) consistent with the deconstruction ansatz. Starting from La , there is a two parameter family of possible non-linear completions arising at cubic order.

118 4.4.6 Quartic order

At quartic order, we conisder a Lagrangian made of the ghost-free kinetic terms up to cubic order, plus the most general two derivative interaction at quartic order

L = a L(2) + L(3) + b L(3) + b L(3) + L(4) . (4.49) a a 1 b1 2 b2 gen

(4) Lgen has 38 parameters after integrating by parts and removing total derivatives

(4) αβ γ δ γ αβ δ γ αβ Lgen = c1H H α∂βH ∂γHδ + c2H αH ∂γH β∂δH + c3H αH ∂βH∂γH

βγ αβ γδ αβ γδ + c4HH ∂βH∂γH + c5H H ∂γH α∂δHβ + c6H H ∂βH α∂δHγ

αβ γδ αβ γδ βγ δ + c7H H ∂βHαγ∂δH + c8H H ∂γHαβ∂δH + c9HH ∂βH ∂γHδ

βγ δ γ αβ δ βγ δ + c10HH ∂γH β∂δH + c11H αH ∂δH∂ Hβγ + c12HH ∂δH∂ Hβγ

αβ δ 2 α βγ δ + c13HαβH ∂δH∂ H + c14H ∂αH∂ H + c15HH ∂δH β∂H γ

αβ γδ γ αβ δ βγ δ + c16H H ∂βHαγ∂H δ + c17H αH ∂γH β∂H δ + c18HH ∂γH β∂H δ

αβ γδ 2 γδ γ αβ δ + c19HαβH ∂γH ∂H δ + c20H ∂γH ∂H δ + c21H αH ∂ Hβγ∂H δ

βγ δ αβ δ 2 δ + c22HH ∂ Hβγ∂H δ + c23HαβH ∂ H∂H δ + c24H ∂ H∂H δ

αβ γδ αβ γδ αβ γδ + c25H H ∂δHγ∂ Hαβ + c26H H ∂Hγδ∂ Hαβ + c27H H ∂δHβ∂ Hαγ

αβ γδ γ αβ δ βγ δ + c28H H ∂Hβδ∂ Hαγ + c29H αH ∂γHδ∂ H β + c30HH ∂γHδ∂ H β

γ αβ δ βγ δ γ αβ δ + c31H αH ∂δHγ∂ H β + c32HH ∂δHγ∂ H β + c33H αH ∂Hγδ∂ H β

βγ δ αβ γδ 2 γδ + c34HH ∂Hγδ∂ H β + c35HαβH ∂δHγ∂ H + c36H ∂δHγ∂ H

αβ γδ 2 γδ + c37HαβH ∂Hγδ∂ H + c38H ∂Hγδ∂ H . (4.50)

119 Table 4.1: Parametric solution for Lagrangian of form 4.50 to avoid a ghost in the decoupling limit up to quartic order.

1 c3 = −c1 c21 = b1 − 4 b2 − 2c1 + c2 1 1 c4 = 4 (−a + 9b1 − 2b2 + 4c1) c22 = 2 (a − b1 + 4c1 − 2c2) 1 1 c5 = 2 (−a + 7b1 − 2b2 + 4c1) c23 = 4 (a − b1 + 4c1 − 2c2) 1 1 c6 = 2 (a − 7b1 + 2b2 − 4c1) c24 = 4 (−b1 + b2 − 4c1 + 2c2) 1 3 c7 = 4 (−2a + 18b1 − 5b2 + 8c1) c25 = −a + 5b1 − 2 b2 + c2 1 1 c8 = 2 (a − 7b1 + 2b2 − 4c1) c26 = 4 (a − 3b1 + b2 + 4c1 − 2c2) 1 1 c9 = 4 (a − 9b1 + 2b2 − 4c1) c27 = 4 (−a + 7b1 − 2b2 + 16c1 − 6c2) 1 c10 = a − 5b1 + b2 − c2 c28 = 4 (−a + 7b1 − 2b2 − 4c1 + 2c2) 1 1 9 1 c11 = 4 (−4b1 + b2 + 8c1 − 4c2) c29 = − 4 a − 4 b1 + b2 − 6c1 + 2 c2 1 1 c12 = 2 (−a + b1 − 4c1 + 2c2) c30 = 4 (−2a + 18b1 − 5b2 + 16c1) 1 c13 = 8 (−a + b1 − 4c1 + 2c2) c31 = 2c1 − c2 1 1 c14 = 8 (b1 − b2 + 4c1 − 2c2) c32 = 4 (a − 11b1 + 3b2 − 4c1 + 4c2) 1 c15 = 4 (−3a + 13b1 − 3b2 − 4c1) c33 = −2c1 + c2 1 1 c16 = 4 (5a − 35b1 + 10b2 − 16c1 + 2c2) c34 = 2 (a − b1 + 4c1 − 2c2) 1 1 c17 = 4 (a + 9b1 − 4b2 + 24c1 − 6c2) c35 = 8 (−3a + 17b1 − 5b2 + 4c1 + 2c2) 1 1 c18 = 4 (−2a + 2b1 + b2 − 16c1 + 4c2) c36 = 8 (3a − 19b1 + 4b2 − 4c1 − 2c2) 1 1 c19 = 8 (a − 15b1 + 5b2 − 12c1 + 2c2) c37 = 8 (a − b1 + 4c1 − 2c2) 1 1 c20 = 8 (−3a + 21b1 − 6b2 + 12c1 − 2c2) c38 = 8 (−b1 + b2 − 4c1 + 2c2)

If we introduce Stückelberg fields and perform the decoupling limit in Eq. 4.37, there are five interactions in D space-time dimensions that have second order equations of motion. However, in D = 4 there is only a five parameter family of solutions, because one linear combination of the Lagrangians vanishes identically

(4) (4) Lc1 + 4Lc2 ≡ 0. (4.51)

120 c2 Then redeﬁning c1 − 4 → c1 the Lagrangian has 4 parameters

L(4) = a L(2) + L(3) + L(4) g.f.,Λ3 a a a +b L(3) + L(4) + b L(3) + L(4) + c L(4). (4.52) 1 b1 b1 2 b2 b2 1 c1

We see that there are potential non-linear completions for Lb1 and Lb2 and La to quartic order, as well as one new parameter of an interaction that starts at quartic order.

The Einstein-Hilbert term, as well as the deconstruction terms, may be expressed in terms of these Lagrangians

1 L = [L − L − 4L ] EH 4 a b1 b2 1 3 11 15 L = − L − L − L − L KG 4 a 2 b1 2 b2 16 c1 1 L ∗ = L + 4L − L . (4.53) KK R b1 b2 8 c1

There is also a fourth parameter for which there is no obvious non-linear completion.

However, this is irrelevant, as utlimately this theory has a ghost. The bottom line of the analysis, to this stage, is that we have indeed found several candidate kinetic interactions for a massive spin-2 ﬁeld, using the decoupling limit in Eq. 4.37. We do not know not yet know whether these interactions may be non-linearly continued to higher order. It may also be that the interactions contain a ghost that appears only beyond the decoupling limit that we have considered so far.

There are diﬀerent ways to proceed from this point.

121 One method is to perform an ADM analysis on the interactions we have identiﬁed.

This analysis was performed in my work [61]. When working in a perturbative ADM

framework, one can show that for the three interactions that survive to quartic order

that are not the Einstein-Hilbert term, there are time derivatives on the shift. This

signals that there is a new active degree of freedom beyond the ﬁve for a massive

graviton, since as we have seen in a healthy theory of massive gravity the shift appears

as an auxiliary variable which may be integrated out.

A second method (which I performed during work on [63]) is to perform a diﬀerent

decoupling limit using the scaling in Eq. 4.38. The equations of motion should still

be second order in this decoupling limit, because all interactions at that scale contain

one power of h and so it is consitent to treat the ﬁelds as living on a ﬁxed Minkowski background. If we perform the analysis with this scaling, then the only interaction that gives rise to second order equations of motion is the Einstein-Hilbert term.

Using either of the above methods, in the end, we are left with the only ghost-free action for a massive graviton up to quartic order

1 L(4) = L = [L − L − 4L ] . (4.54) g.f EH 4 a b1 b2

This is the central result of this chapter.

4.4.7 Extension to higher order

We have shown that the only possible non-linear completion of the spin-2 kinetic interactions, up to quartic order, is the Einstein-Hilbert term. In fact, this is suﬃcient

122 to argue that the only kinetic interaction for a massive graviton, which propagates

ﬁve degrees of freedom, is the Einstein-Hilbert term.

Let us suppose that there was such a kinetic interaction. We will refer to it as

L(der). Then we may perturb this interaction around a Minkowski background as bn

L(der)(H ) = Λ2 L(der) + sub-leading terms , (4.55) bn µν der n

where all indices (which are implicit) are raised and lowered with respect to ηµν. Here,

L(der) is an interaction ∼ ∂2Hn that starts at order n. Its non-linear completion is bn

(der) Lnew , which is not necessarily order n.

In order for L(der) to propagate ﬁve degrees of freedom at the fully non-linear level, bn

(der) a necessary condition is that Ln gives rise to second order equations of motion when acting on the Stückelberg fields, after the linear Stückelberg decomposition has been performed. This is because the linear Stückelberg procedure, applied to the leading order part of a kinetic interaction, identifies the physical degrees of freedom. Bµ gives

the helicity-1 modes, and π gives the helicity-0 mode.

As we argued in Section 4.2, (see also [92]), in four dimensions, the only inter-

actions which are ghost-free when the linear St¨uckelberg procedure is performed are

(der) (der) (der) L2 and L3 . There is no other candidate leading order interaction Ln with n ≥ 4. In the previous section we have shown that the only non-linear completion of

(der) (der) L2 is the Einstein-Hilbert term. We also showed that L3 does not admit a non- linear completion, by showing that this interaction could not be extended to quartic order. As a result, the only consistent kinetic interaction for a massive graviton,

123 which propagates ﬁve degrees of freeom, is given by the Einstein-Hilbert term.

4.5 Relationship to the coupling to matter

4.5.1 Eﬀective vielbein coupling

Recently there has been a great deal of interest in coupling matter ﬁelds to both

metrics gµν and ηµν simultaneously. There are several reasons for considering this kind

of coupling. First, quantum corrections which will generically induce such couplings

[89,97]. Second, these couplings have potential applications [89,97,98] in cosmology.

Additionally, matter couplings to two metric simultaneously can be generated through

deconstruction by including matter ﬁelds coupled to gravity before discretizing.

There is a close relationship between generalized kinetic interactions and gener-

alized matter couplings, much as there is a close relationship between Einstein and

Jordan frame in scalar tensor theories. This is particularly true when the matter is

coupled to a single ‘eﬀective’ metric built out of gµν and ηµν

(eﬀ) (eﬀ) gµν = gµν [g, η]. (4.56)

In this case, we can write the full action for gravity and matter in the form

(eﬀ) S = SEH [g] + Sm[g, η] + Smatt[g [g, η], ψi], (4.57)

where SEH is the Einstein-Hilbert action, Sm are the ghost-free potential interactions,

124 (eff) and where Smatt[g ψi] denotes a generic matter coupling involving matter fields ψi and the effective metric. We can now do a field redefinition so that the matter is minimally coupled to only one metric. In practice this may be implemented by doing

(eff) a field redefinition gµν → gµν[g , η], in which case the action is

(eff) (eff) (eff) S = SEH [g[g , η]] + Sm[g[g , η], η] + Smatt[g , ψi]. (4.58)

In this form we see that generically that the mass term acting on g(eﬀ) does not have the ghost-free form. This will lead to a BD ghost.

There is a clever choice of coupling where the form of the mass term is still a ghost-free mass term after doing this field redefinition. This corresponds to coupling to an effective metric

α (eﬀ) 2 p −1 2 gµν [g, f] = α gµν + 2αβgµα g f + β fµν. (4.59) ν

The reason that this eﬀective metric is special can be seen most easily by writing it in terms of an eﬀective vielbein

eff (eff),a (eff),b gµν = ηabeµ eν , (4.60)

where

(eﬀ),a a a eµ = αeµ + βδµ. (4.61)

125 Then, because the ghost-free mass terms are polynomial in the vielbein, and because switching to the effective vielbein is a linear field redefinition, the mass terms written in terms of the effective vielbein will still be of the ghost-free (polynomial-in-vielbein) form.

It was shown that this coupling is ghost-free in the natural decoupling limit where

2 1/3 we scale α, β ∼ O(1) and keep Λ3 = (m MPl) ﬁxed in [89]. In fact this is the only such coupling that obeys a weak equivalence principle [99].

However, this coupling this provides a puzzle as this should lead to a kinetic term which is ghost-free in the decoupling limit

p −g(eﬀ)R[g(eﬀ)], (4.62)

but we have seen that there are no ghost-free interactions in the decoupling limit.

I provided a resolution to this puzzle in [62] where I showed that the interaction in the original decoupling limit actually vanishes, and that there was a diﬀerent decoupling limit in which the ghost was manifest. In this section we will see how this works working with the kinetic interaction.

4.5.2 Eﬀective metric

We can introduce Stückelberg fields through fµν. Let us define the matrix

1 1 Mµν ≡ 2 ∂µ∂νπ + ∂µBν. (4.63) m MPl mMPl

126 If we neglect the vectors then M is a symmetric matrix. However in general M is not

symmetric, the antisymmetric part is Mµν − Mνµ = 2Fµν = 2(∂µBν − ∂νBµ).

Writing

αβ gµν = ηµν, fµν = ηµν + Mµν + Mνµ + Mµαη Mβν, (4.64)

p −1 µ 2 we can expand the matrix ( g f)ν to O(M ) as

µ p −1 µ 1 µ µ g f = δν + (M ν + Mν ) ν 2 1 − M µ M λ + M µM λ + M µM λ − 3M µ M λ + ··· .(4.65) 8 λ ν λ ν λ ν λ ν

Using this expansion, the eﬀective metric takes the form

α (eﬀ) ρ σ Fµα∂ ∂νπ gµν = ∂µY ∂νY ηρσ + 2αβ 3 2 + ··· , (4.66) m MPl

where

µ µ µ µ B ∂ π Y = (α + β) x + β + 2 . (4.67) mMPl m MPl

In this form, it is clear that the eﬀective metric has zero curvature, up to corrections of

order αβ. This explains why studies of the matter coupling found that the equations

of motion were second order in the decoupling limit. The coupling between matter and

π is removable with a ﬁeld redeﬁnition (a Galileon duality transformation [100,101])

in this case.

127 In ‘Einstein frame,’ in which the eﬀective metric arises in the kinetic term, the

eﬀective metric is ﬂat and so the Einstein-Hilbert term vanishes on this combination.

The form of the eﬀective metric guarantees that there are no higher derivative oper-

ators acting purely on π. Working to higher order in fields to keep the vector fields, we find an operator

2 p (eﬀ) (eﬀ) (αβ) α µνρσ MPl −g R ⊃ 3 (Fµα∂ ∂νπ) E hρσ. (4.68) MPlm

This is in fact precisely one of the operators that can be of the form of Eq. 4.38.

This shows explicitly the connection between ghosts seen directly through the matter

coupling, and ghosts seen through the kinetic interaction.

An amusing point is that the Ostrogradsky instability here does not arise as higher

derivatives acting on a scalar, which is the original BD ghost problem. The problem

in this case comes from higher derivatives acting on a vector ﬁeld.

4.6 Outlook

The bottom line of this chapter is quite remarkable. In four space-time dimensions

the kinetic interactions for a massive spin-2 ﬁeld (with Lorentz invariant interactions)

must take the form of the Einstein-Hilbert term. This is remarkable since there is no

symmetry requirement that this be the case. We have derived this result purely on

the basis of stability requirements.

The results are very powerful and have implications for other extensions of massive

128 gravity. For example, the ghost in the linear eﬀective vielbein matter coupling which was discovered in [89] can be seen from the perspective of the kinetic interactions as

I described in Section 4.5.

In performing this analysis, there are several assumptions that have been made that we will now review. First, we have assumed that the interactions are local and

Lorentz invariant, and that there exists a stable, Poincar´einvariant vacuum solution.

These assumptions are typically required to have a well-deﬁned notion of a spin-2

ﬁeld. Nevertheless, breaking these assumptions may allow for more general kinetic terms, for example recent work has established that a Lorentz-violating theory of massive gravity may have modiﬁcations to the kinetic structure, which propagate two degrees of freedom [102].

We have also assumed that the theory admits a formulation in the metric language. We have considered interactions which violate this assumption in [85], but those interactions always come with more degrees of freedom and thus they are not appropriate for a massive spin-2 ﬁeld.

Finally, it is worth pointing out that we can still make sense of the new kinetic interactions at low energies, by treating them perturbatively. At low energies when it is consistent to neglect any ghost instabilities, it is still possible to include modiﬁed kinetic interactions. However, in that case we must be careful that the ghost does not acquire a small mass on backgrounds which exhibit the Vainshtein regime.

129 Chapter 5

Galileon radiation from binary

systems

The inspiral of compact binary star systems due to the emission of gravitational ra-

diation is an important observational test of General Relativity.1 A compact binary star system consists of two compact objects (for example, two neutron stars) in orbit around a common center. Because the stars are accelerating, the system emits gravitational radiation. The ﬁrst example of such a system is the binary pulsar system

PSR B1913+16 discovered by Hulse and Taylor in [14, 15]. Since then, many other inspiralling binary systems have been measured, for a review see [16].

In GR it is possible to treat inspiralling binary systems with analytic techniques because there is a natural small parameter in which to expand, the orbital velocity v

(in units where c = 1). For the Hulse-Taylor system, the orbital velocity is ∼ 10−3.

From the point of view of eﬀective ﬁeld theory, the existence of this small parameter

1This chapter is based on my work in [63] with Claudia de Rham and Andrew J. Tolley.

130 is fundamentally due to a separation of two dimensionful scales: the size of the orbit r¯ and the orbital frequency ΩP . The velocity v = ΩP r¯ is a small quantity, because

−1 r¯ ΩP . As a result, the long wavelength physics associated with the gravitational radiation is, to first approximation, insensitive to the details of the objects making up the system. An effective field theory approach to calculating gravitational wave emission was developed in [103].

At leading order in the post-Newtonian expansion, the time averaged power of radiation into gravitational waves (in GR) is given by the Peters-Mathews formula

[104]

32 G4 m2m2 (m + m ) 73 37 hP i = 1 2 1 2 1 + e2 + e4 , (5.1) 5 c5 a5(1 − e2)7/2 24 96 where e is the eccentricity. The power emission can then be related to the spin-down rate of the binary system, which is observable.

In this chapter I will discuss progress towards ﬁnding an analogue of the Peters-

Mathews formula in massive gravity. This is complicated by the fact that in order to compute the power consistently, we should take into account the strong coupling effects responsible for Vainshtein screening. This apparently prevents a simple, analytic treatment.

In [64], an approximation scheme was developed to handle the time dependence of the pulsar system. The key idea is that there is a new length scale in the problem,

131 the Vainshtein radius r?. For a realistic system,

−1 r¯ ΩP r?. (5.2)

−6 As an example, for the Hulse-Taylor system, we have ΩP r? ∼ 10 , taking m ∼

−33 −1 10 eV. Physically, we expect that the separation of scales between r? and ΩP will allow us to compute the radiated emission using eﬀective ﬁeld theory techniques.

Such a procedure was carried out in the case of the cubic Galileon.

In this chapter I will apply this formalism to study the emission from binary systems including all the Galileon interactions. We will find that the results depend strongly on the strength of the quartic Galileon interaction. Therefore it will be useful to develop a formalism where the Galileons enter with different scales, in order to see how this difference emerges.

This formalism is also of interest from a theoretical point of view because it involves treating the Vainshtein mechanism in a time-dependent situation. This pushes the theory beyond the case of a static, spherically symmetric setup we considered in

Section 2.6.

The rest of this chapter is organized as follows. In Section 5.1 I will discuss the notion of Galileons with diﬀerent scales. In Section 5.2 I will describe the approximation scheme that we will use for computing the power emission in a generic Galileon.

In Section 5.3 I will give a derivation for the power emission as a sum over multipoles. We can then apply this formula to diﬀerent cases of the Galileon. In Section

5.4 I will review the results of [64] which computed the power emission for the pure

132 cubic Galileon. In Section 5.5, we will apply the formalism to a case where the cubic

and quartic Galileon interactions enter at the same scale. We will ﬁnd that the ap-

proximation scheme developed for the cubic Galileon theory breaks down for realistic

binary systems. However we will explain this, and work through the details of the

case when the two objects in the binary system have very diﬀerent masses. In Section

5.6 I will describe another regime in which perturbation theroy is valid. I conclude

with a summary and outlook in Section 5.7.

5.1 Galileons with diﬀerent scales

We will compute the Galileon radiation in a general Galileon theory. Our starting

point will be the Galileon action given in Eq. 2.5.3, which as we have seen is related

to the decoupling limit of massive gravity. The Galileon action is

Z 5 (n) ! 3 3 X cnL(Gal) 1 S = d4x − (∂π)2 − + πT . (5.3) 4 4 Λ3(n−2) 2M n=3 Pl

2 1/3 In massive gravity, Λ = Λ3 = (m MPl) . However, since our results are phrased in

terms of Galileons, we will allow this scale to be arbitrary. The bounds we ﬁnd will

be most directly phrased in terms of Λ, but can be converted into a bound on m.

In this chapter I will be interested in considering various hierarchies of the coef-

ﬁcients cn. For example of particular interest is the case of the pure cubic Galileon c4 = 0, and a regime where the quartic Galileon interactions are ‘small’ c4 c3. As

133 a result it is useful to deﬁne three new scales

3(n−2) ˜ 3(n−2) Λ Λn ≡ , n = 3, 4, 5. (5.4) cn

In terms of these new scales, the action is

Z 5 (n) ! 3 3 X L(Gal) 1 S = d4x − (∂π)2 − + πT . (5.5) 4 4 ˜ 3(n−2) 2M n=3 Λn Pl

5.2 Perturbations around a spherically symmetric

background

5.2.1 Keplerian source

To leading order in the gravitational coupling, we can neglect the back-reaction of the radiation emission on the pulsar’s orbit. As a result, for the purposes of computing the emitted radiation, we can treat the binary system as an external source which drives the Galileon ﬁeld. The stress-energy for the binary system, ignoring relativistic corrections, is given by

(3) Kepler (3) Kepler Tµν = M1δ (~x − ~x1 (t)) + M2δ (~x − ~x2 (t)) δµ,0δν,0, (5.6)

134 where M1 (M2) is the mass of the ﬁrst (second) objects, and where we will take the objects to follow Keplerian (elliptical) orbits2

2 Kepler M2,1 r¯(1 − e) Kepler π Kepler r1,2 = , θi = , ϕi = ΩP t + πδi,2, (5.7) Mtot 1 + e cos ΩP t 2

where e is the ellipticity of the orbit.

In principle, one should solve the exact equations of motion for the Galileon in

the presence of this source. However it is impossible to solve the equations of motion

in this case, because the Galileon equations are highly non-linear and the source is

time-dependent. An approximation scheme is needed to make progress.

The central idea to make analytic progress is to treat the departure from a static

and spherically symmetric source perturbatively. The basic assumption behind this

split is that the monopole of the background is largely responsible for the Vainshtein

screening.

Explicitly, we split the source in to a static, spherically symmetric background,

and a time-dependent perturbation

µ ¯µ µ Tν = Tν (r) + δTν (~x,t), (5.8)

where

¯µ (3) µ 0 Tν ≡ Mtotδ (~x)δ0 δν. (5.9)

2 Actually saying ϕ(t) = ΩP t is not quite correct for elliptical orbits. However this approximation will suﬃce for the level of precision we want.

135 Here we have deﬁned the total mass as Mtot = M1 + M2, and

2 µ X (3) Kepler (3) µ 0 δTν = Miδ (~x − ~xi (t)) + Mtotδ (~x)δ0 δν. (5.10) i=1

After doing this, we will split the Galileon π into a spherically symmetric background plus fluctuations, π =π ¯(r) + p2/3φ. Written in this form, the source T¯ creates a background (spherically symmetric) profile for the Galileon,π ¯(r). The fluctuations

φ are then directly sourced by the time-dependence in δT , and power is radiated into the modes of φ. However there is also the a Vainshtein screening effect due to the background. The fluctuations receive a large rescaling of their kinetic term, as described in Section 2.6.1. For example, if the quartic Galileon dominates, then the kinetic term for the fluctuations is ∼ (∂2π¯)2(∂φ)2, and at small distances when ∂2π¯ is large, the kinetic term for the fluctuations is large as well, so that the canonically normalized fluctuation φˆ = φ/(∂2π¯) is only weakly coupled to the source.

We expect this approximation scheme to be valid on eﬀective ﬁeld theory grounds when there is a large hierarchy of scales between the size of the systemr ¯ and the

−1 orbital frequency ΩP . This holds for realistic systems where ΩP r¯ 1. Because the spherical symmetry is only broken at a very large energy scaler ¯−1, we would generically expect spherical symmetry to be a good approximation to the background. More pragmatically, we may make any split we desire between background and perturbations so long as perturbation theory remains valid.

136 5.2.2 Background Galileon ﬁeld

As we have described, T¯ creates a spherically background proﬁleπ ¯(r). The proﬁle is determined from an algebraic equation for E =π ¯0

E 2 E 2 2 E 3 M + + = tot . (5.11) ˜ 3 ˜ 6 3 r 3Λ3 r Λ4 r 12πMPlr

˜ ˜ Since we allow for Λ3 and Λ4 to be very diﬀerent, there are two relevant dimensionful quantities

(M/M )1/3 r ≡ Pl , n = 3, 4, (5.12) ?,n ˜ Λn which we refer to as cubic and quartic strong coupling radii (or Vainshtein radii), respectively.

Note that the quintic Galileon does not enter into the equations of motion for the background. The reason for this is that in D space-time dimensions, there are only

D + 1 Galileon interactions that are relevant. By considering a time-independent background, the equations of motion for the background essentially live in a D − 1 dimensional space, and the quintic Galileon becomes a total derivative.

Focusing on the non-vanishing parts of the equations of motion, this background takes a simple form in diﬀerent spatial regimes.

• The weakly coupled regime, r r?,3, r?,4. At distances from the source

larger than either Vainshtein radius, the background takes the usual Newtonian

137 form

M /M 1 E(r r ) = tot Pl . (5.13) ?,3 12π r2

• The cubic Galileon regime, r?,4 r r?,3. This regime only exists if

r?,3 r?,4. The background behaves as if only a cubic Galileon is present

1/2 (Mtot/MPl) ˜ 3/2 E(r?,4 r r?,3) = √ Λ . (5.14) 8πr 3

• The quartic Galileon regime, r r?,4. This regime always dominates at

small enough distances, so long as r?,4 is non-zero. The background has the

form

(M /M )1/3 E(r r ) = tot Pl Λ˜ 2. (5.15) ?,4 (24π)1/3 4

There is no need to consider a range of parameters when r?,3 < r?,4. At small distances, the quartic Galileon always dominates the background, because it involves more derivatives on π in the action.

5.2.3 Perturbations for Galileon

We now study perturbations around this background

r2 π =π ¯(r) + φ(~x,t). (5.16) 3

138 To quadratic order, the action for φ is given by

Z (2) 4 1 µν φδT S [φ] = d x − Z ∂µφ∂νφ + √ 2 6MPl Z 1 1 1 φ 4 tt 2 rr 2 ΩΩ 2 √ = d x Z (r)(∂tφ) − Z (r)(∂φ) − 2 Z (∂Ωφ) + δT . 2 2 2r 6MPl (5.17)

The background-dependent matrix Zµν is called the eﬀective metric3. Its components are given in terms of the background quantities by

4 E(r) 6 E(r)2 Zrr(r) ≡ 1 + + (5.18) ˜ 3 ˜ 6 2 3Λ3 r Λ4 r 1 d 2 E(r) 18 E(r)2 24 E(r)3 Ztt(r) ≡ r3 1 + + + (5.19) 2 3 ˜ 6 2 ˜ 9 3 3r dr Λ3 r Λ4 r Λ5 r 1 d 4 E(r) 6 E(r)2 ZΩΩ(r) ≡ r2 1 + + . (5.20) ˜ 3 ˜ 6 2 2r dr 3Λ3 r Λ4 r

The equations of motion for φ are then given by (in Cartesian coordinates)

˜ µν δT φ = ∂µ (Z ∂νφ) = −√ . (5.21) 6MPl

˜ The operator is called the eﬀective d’Alembertian operator, which takes the following form for a spherically symmetric background

1 1 ˜φ = −Ztt(r)∂2 + ∂ r2Zrr(r)∂ + ZΩΩ(r)∇2 φ, (5.22) t r2 r r r2 Ω

2 where ∇Ω is the Laplacian on a sphere. 3Technically the true eﬀective metric is related to this by a conformal rescaling.

139 Rather than giving an explicit closed form expression for the eﬀective d’Alembertian

it is more useful to give an approximate form in diﬀerent regimes.

Weakly coupled region: r r?,3 . In this regime, the background forπ ¯ is small

2 ˜ µν µν so that ∂ π¯ Λ3,4 and the eﬀective metric is approximately Minkowski, Z ≈ η .

As a result, the eﬀective d’Alembertian takes the standard Minkowski form

2 1 ˜ φ = −∂2φ + ∂2φ + ∂ φ + ∂2 φ . (5.23) t r r r r2 Ω

Cubic Galileon region: r?,4 r r?,3. In this regime, the eﬀective d’Alembertian

is given by4

r 512 r 3/2 2 1 ˜ φ = ?,3 −3∂2φ + 4∂2φ + ∂ φ + ∂2 φ . (5.24) 9π r t r r r r2 Ω

Quartic Galileon region: r r?,4 Finally, in this regime the eﬀective d’Alembertian is given by

!6 128 × 31/3 Λ˜ r 2 1 k ˆ φ = 4 ?,4 − ∂2φ + ∂2φ + Ω ∂2 φ , (5.25) 2/3 ˜ 2 t r 2 Ω π Λ3 r cr r?,4

where the speed of sound of the radial ﬂuctuations cr is given by

!−1/2 4 Λ˜ 12 c = 1 − c 4 , (5.26) r 5 ˜ 3 ˜ 9 9 Λ3Λ5

4 1/3 This assumes Λ˜ 5 > (Λ˜ 4/Λ˜ 3) Λ˜ 4. If this is violated then the quintic Galileon contributes to Ztt, this case is beyond the scope of this chapter.

140 and the coeﬃcient kΩ is given by

 !6 π2/3 27 Λ˜ k = 1 − 3 . (5.27) Ω 1/3  ˜  1728 × 3 2 Λ4

Here we note the important feature that the coeﬃcient of the angular derivative

−2 −2 is r?,4, not r . In other words, the angular derivative is not suppressed at large r.

This will have important consequences as we see in Section 5.5.

5.3 Formula for the power emission

Given the equations of motion,

˜ δT φ = −√ , (5.28) 6MPl

we can compute the energy radiated into φ. We can now write a formal expression

for the power emission for a generic eﬀective d’Alembertian of the form in Eq. 5.22

as a sum over multipoles.

We deﬁne the eﬀective action

Z 1 4 Seﬀ = √ d xφF (x)δT (x), (5.29) 6MPl

where

Z 4 0 0 0 φF (x) = d x GF (x, x )δT (x ), (5.30)

141 is a solution of the equations of motion

˜ δT φF = √ . (5.31) 6MPl

0 Here, GF (x, x ) is the Feynman propagator, which is a solution to the equation

˜ 0 4 0 GF (x, x ) = iδ (x − x ), (5.32)

with in-out boundary conditions.

The Feynman propagator can be written in terms of the Wightman function

0 0 + 0 0 − 0 GF (x, x ) = θ(t − t )W (x, x ) + θ(t − t)W (x, x ) , (5.33)

where

Z ∞ + 0 X ∗ 0 ∗ 0 0 −iω(t−t0) W (x, x ) = dω u`ω(r)u`ω(r )Y`m(θ, ϕ)Y`m(θ , ϕ )e , (5.34) `m 0

−iωt where u`mω(r, Ω, t) = u`ω(r)Y`m(Ω)e are a complete set of eigenmodes for the eﬀective d’Alembertian.

The power is computed by the time averaged energy emission

dE Z ∞ P = − = dω f(ω), (5.35) dt 0

142 where f(ω) is related to the eﬀective action by

2 Z ∞ ImSeﬀ = dωf(ω). (5.36) TP 0

It is useful to deﬁne the moments

∞ Z TP Z 1 X 0 M = einΩP t dt d3xu (r)Y (θ, ϕ)e−inΩP t δT (~x0, t0), (5.37) `mn T `n `m P n=−∞ 0

where u`n ≡ u`,ω=nΩP . Taking the Fourier transform

∞ X inΩP t M`m = M`mne , (5.38) n=−∞

we have that

0 Z TP Z t 1 X iω(t−t0) ? 0 f(ω) = 2 Re e M`m(t)M`m(t ) , (5.39) 3M TP Pl `m 0 −∞ leading to

∞ ∞ ` π X X X hP i = nΩ |M |2. (5.40) 3M 2 P `mn Pl n=0 `=0 m=−`

From this formula one can compute the power emission given the correctly normalized mode functions.

However, it is not necessary to construct the solutions to the exact eﬀective d’Alembertian. Instead we can construct the mode function separately in the weakly

143 coupled regime and in the strongly coupled regime(s). We ﬁx the normalization of

the mode asymptotically by matching to the properly normalized modes for a free

ﬁeld, and then match the mode functions at the boundaries between the weak and strong coupling regimes.

We now apply these techniques to the Galileon.

5.4 Power emission for the cubic Galileon

The formalism in the previous section was applied to the cubic Galileon in [64]. Here we brieﬂy recall the main results before turning to the quartic Galileon, so that we

˜ may compare our results. In this section we therefore take Λ4,5 → ∞. Since there is only one strong coupling scale and one Vainshtein radius in this case, we deﬁne

˜ Λ ≡ Λ3, and we use r? = r?,3 to refer to the Vainshtein radius.

The correctly normalized mode functions are given by

√ ! 9π 1/4 1 r 1/4 3 u (r) = J ? ωr , (5.41) ` √ ν` 128 r? r? 2

? where the index ν` is determined by the boundary condition that the modes are

? ? regular at the origin. Explicitly, ν` = (2` + 1)/4 for ` > 0, and ν` = −1/4 for ` = 0.

To compute the source moments, we work at small distances where ωr 1. In this

144 regime, the mode functions are approximately given as

√ ν?/2 1/4 1/4 ! ` 9π 1 1 r ? 3 3 ν` 2 u`(r) ≈ ? √ (ωr) 1 − ? (ωr) . 128 Γ(1 + ν` ) r? r? 4 16 (1 + ν` ) (5.42)

The dominant contributions to the power come from two sources:

• First, there is monopole radiation ` = 0. In GR, the monopole radiation van-

ishes exactly due to Birkhoﬀ’s theorem. In a Galileon theory, the monopole

radiation also vanishes at zeroth order in relativistic corrections because of en-

ergy conservation (the monopole moment is time-independent). However, there

is a non-trivial time-dependent monopole moment coming from relativistic cor-

rections. This sources a monopole radiation component not present in GR.

• Second, there is quadrupole ` = 2 radiation. This is alredy present in GR. It

comes from the fact that the source has a time-dependent quadrupole moment.

Similarly in the present case, the time-dependent quadrupole moment of the

source leads to quadrupole radiation for the Galileon.

The dipole radiation was also computed. However this is suppressed relative to the monopole and quadrupole due to conservation of momentum.

The power in the monopole and quadrupole was computed for arbitrary eccentricity. For realistic systems the quadrupole was the dominant contribution to the power. For comparison with our results in the next sections, here I report the result

145 for the quadrupole. The quadrupole radiation is

2 3 2 ∞ 5λ (ΩP r¯) MQ X hP i = Ω2 |IQ(e)|2, (5.43) 32 (Ω r )3/2 M 2 P n P ?,3 Pl n=0

where the constant λ is deﬁned as λ = 39/8π1/42−17/4Γ(9/4)−1 ≈ 0.2, and where the

eﬀective quadrupole mass is given by

√ √ M M ( M + M ) M = 1 2 1 2 . (5.44) Q M 3/2

Q The integrals In (e) are given by

7/4 Z 2 −i(n−2)x Q 2 3/2 n e In (e) = (1 − e ) π 3/2 dx, (5.45) 2π 0 (1 + e cos x)

where e is the eccentricity, and can be computed numerically as a function of e.

This result is to be compared with the Peters-Mathews formula in Eq. 5.1. In the cubic Galileon, the ratio between the power emission in the cubic Galileon and in GR is given by

Pgal 1 ∼ 3/2 , (5.46) PGR (ΩP r?)

where q is a factor that depends on the masses and eccentricities of the system (for

Hulse-Taylor q ≈ 0.08), and v = ΩP r¯ is the velocity of the system.

3/2 The crucial point is the factor of (ΩP r?) 1 which represents a suppression of

power emission in the Galileon as opposed to GR, due to the Vainshtein mechanism.

146 It is interesting to compare the Vainshtein suppression for a force between two

objects to the Vainshtein suppression for the power emission

Vainshtein Suppression r¯ 3/2 ∼ Static Force, r? (Cubic Galileon)

Ω−1 3/2 ∼ P Power Emission. (5.47) r?

The Vainshtein suppression is less eﬀective in the time-dependent case, as can be seen above from the fact that ΩP r¯ 1. Based on this result, in [64] a bound

Λ < 10−9 eV (corresponding to m < 10−27 eV) was placed by comparison to observed binary pulsars.

5.5 Power emission for the general Galileon

In this section we will consider the case when all Galileon strong coupling scales are

˜ ˜ equal, Λ ≡ Λ3 = Λ4, so again there is only one Vainshtein radius r? ≡ r?,3 = r?,4.

Unlike the previous case, the approximation scheme will break down. We will explore why this occurs. However, in Section 5.5.5 we will consider a regime when the masses of the binary system are very different, where perturbation theory still applies. In that case, we find that the Vainshtein screening is fully effective.

147 5.5.1 Mode functions

Given the form of the eﬀective d’Alembertian in the quartic Galileon strong coupling regime in Eq. 5.25, the unnormalized mode functions are given by

π u =u ¯ sin ω r − δ , (5.48) `n `n `n `,0 2 where

2 1 2 `(` + 1) ω`n = 2 (nΩP ) − kΩ 2 , (5.49) cr r?

and whereu ¯`n is a normalization constant. The phase shift of the monopole of −π/2 compared to the other modes arises from the requirement that the modes be analytic at the point r = 0.

The normalization constantu ¯`n is ﬁxed by matching to the weakly coupled mode functions at asymptotically large distances. The normalization will depend on whether

−1 −1/2 ` is larger or smaller than `crit = cr kΩ ΩP r?. Explicitly, the normalization constant is given by

1 u¯`n = √ , ` < n`crit, πωr? e−`2 = √ , ` > n`crit. (5.50) πωr?

148 where ω = nΩP . This is well approximated with a step function

1 u¯`n ≈ √ θ(n`crit − `), (5.51) πωr? where θ(x) = 0 for x < 0, and θ(x) = 1 for x > 0, so the correctly normalized modes in the strong coupling regime are given by

1 π un` ≈ √ θ(n`crit − `) sin ω`nr − δ`,0 . (5.52) πωr? 2

Strangely, the mode functions do not fall oﬀ with r within the quartic Galileon strong

coupling regime. In other words, as the wave spreads out from the origin, its amplitude

does not decay. The short distance behavior of this mode function does not depend

strongly on `. Because of this, diﬀerent ` modes will contribute equally to the power.

5.5.2 General form for the power

Starting from Eq. 5.40, the power has the form

∞ ∞ ` 2 π X X X θ(n`crit − `) hP i = nΩ √ (5.53) 3M 2 P πnΩ r Pl n=0 `=1 m=−` P ? 2 1 Z TP Z 3 −inΩP t × dt d x e sin(nΩP r)Y`m(θ, φ)δT (~x,t) , TP 0

where the perturbed source δT (~x,t) is given as in in (5.10).

149 To leading order in ΩP r¯ = v, the power radiated by the system is given by

∞ M2 Ω2 r¯2 X X π 2 hP i = red P θ(n` − `)n2Y , 0 (5.54) 3M 2 r2 crit `m 2 Pl ? n=0 `m 2 m Z TP 2 1 + (−1) −i(n−m)Ω t 1 − e × dt e P , TP 0 1 + e cos ΩP t

−1 −1 −1 where the reduced mass Mred is deﬁned by Mred ≡ M1 + M2 , and where we use the short-hand

∞ ` X X X ≡ . (5.55) `m `=0 m=−`

The integral can be done analytically

n−m 1 Z TP e−i(n−m)ΩP t 2π e dt = (−1)n−m √ √ , (5.56) 2 2 TP 0 1 + e cos ΩP t 1 − e 1 + 1 − e for n − m ≥ 0, and as a result the ﬁnal expression for the power is

2 2 2 8π Mred r¯ 2 hP i = ΩP Se , (5.57) 3 MPl r?

with the sum Se given by

∞ n` ` crit 2 2(n−m) X X X 2 π 2 e 2 mπ Se = n Y`m , 0 1 − e √ cos . 2 2 2 n=0 `=0 m=−` 1 + 1 − e

The ﬁrst multipole that has nonzero radiation is the quadrupole ` = 2 (as expected, to zeroth order in relativistic corrections, the monopole ` = 0 and the dipole ` = 1

150 radiation vanish because of energy and momentum conservation, repsectively).

Typically in ordinary radiation problems on ﬂat space-time, the multipole expansion becomes an expansion in powers of the orbital velocity v = ΩP r. However, for the power emitted in the present case, we see that there is no suppression factor as higher order multipoles are included. All modes with ` < n`crit contribute to this expression with equal strength.

5.5.3 Power in quadrupole

Despite the fact that the power is a sum over multipole moments, in this section to get a clearer picture we focus on the power in just the ` = 2 mode and in just one harmonic (n = 2). At small r the mode function becomes

nΩP r un2 ≈ √ , (5.58) πnΩP r? so the power emission becomes

2 2 Mred (ΩP r¯) 2 hP i ≈ 2 ΩP . (5.59) MPl (ΩP r?)

This expression Eq. 5.59 has many features in common with the power emission by the cubic Galileon in equation 5.43. In particular there is a Vainshtein suppression

−2 ∼ (ΩP r?) . The diﬀerence in power between the cubic and quartic Galileon cases

(3/2 for the cubic vs 2 for the quartic) matches the expectation from the static force.

Because the screening is more eﬀective for the quartic Galileon, the power in this

151 mode is even smaller than the cubic Galileon result.

5.5.4 Breakdown of perturbation theory

However, in the previous section we have only considered the expression for a single

multipole. The sum over multipoles leads to a divergent answer because there is no

suppression at higher values of `. To understand this divergence we will regulate expression of the power.

We do this by regulating the sum over ` by only summing over multipoles below some cutoﬀ ` < L. Physically this is justiﬁed because it is impossible to know the angular resolution of our source to arbitrary precision. Since `crit 1 for any reasonable system we will take L < `crit.

We may then compute the regulated power by taking the expression in Eq. 5.57, and cutting oﬀ the sum over L. Explicitly,

L L ` 1 X X X π 2 L4 S (L) = n2Y δ ≈ , (5.60) 0 π2 `m 2 n,m 48π2 n=0 `=1 m=−`

where we have used the approximation, valid for large m (which is satisﬁed by most

terms in the sum)

π 1 π Y ( , 0) ≈ cos (` + n) . (5.61) `m 2 π 2

The power depends sharply on the cut-oﬀ. Our perturbative calculation has relied

on the assumption that the solution to the linearized equations of motion is a good

152 approximation to the solution of the fully non-linear equations of motion. We now revisit that assumption.

The fully non-linear non-linear equation of motions are given by

1 π + ( π)2 − (∂ ∂ π)2 Λ3 µ ν 1 3 2 3 T + 6 (π) − 3 (∂µ∂νπ) + 2 (∂µ∂νπ) = − . (5.62) Λ 3MPl

To construct a perturbative solution, we split the Galileon ﬁeld as

r2 π =π ¯ + φ(1) + φ(2) + ··· , (5.63) 3 whereπ ¯(r) solves Eq. 5.11. The linearized equation obeyed by the perturbations has the form

˜ (n) (n) (1) (n−1) φ = J [δT, φ , ··· , φ ], (5.64)

where J (n) is an effective source which depends on the time-dependent quadrupole moment, δT , as well as the lower order fluctuations. Before giving the explicit form for J, it is useful to make some general comments. For example, the first order

ﬂuctuation is sourced directly by δT , while the second order ﬂuctuation is sourced by

φ(1) itself.

153 To solve these equations for φ(1) and φ(2) we use the retarded Green’s function5

Z (n) 4 0 0 (n) 0 φ (x) = d x GR(x, x )J (x ), (5.65)

where the retarded Green’s function is given by

0 0 0 GR(x, x ) = −iθ(t − t ) 0 [φ(x), φ(x )] 0 (5.66) Z = −θ(t − t0) dω sin ω(t − t0) (5.67)

L ` X X 0 ∗ 0 0 × u`ω(r)u`ω(r )Y`m(θ, ϕ)Y`m(θ , ϕ ) . `=0 m=−`

Note that we have regulated the retarded Green’s function to have ` < L, consistent with the philosophy of this section.

We now turn to constructing the solution for the ﬁeld. In the WKB regime, the

first order field fluctuation φ(1) is then given by

Z 0 (1) 4 0 0 δT (x ) φ (x) = − d x GR(x, x )√ (5.68) 6MPl L ` 2πMredr¯ X X = √ θ(m`crit − `) (5.69) 2 6MPlr? `=1 m=0 π mπ ×Y (θ, 0) Y , 0 sin mΩ r sin(mΩ t − mϕ) cos2 . `m `m 2 P P 2

There are several key features of this expression. First, there are special points xmax on the radial light cone, deﬁned by r = t = (2k + 1)π/2ΩP for integer k, and

(1) θ = π/2 and ϕ = 0, where the ﬁeld φ diverges as L → ∞. At these points xmax,

5We use the retarded Green’s function, not the Feynman Green’s function, because we would like to compare the physical values of the real-valued ﬁelds.

154 sin(ΩP r) sin(mΩP t − mϕ) = 1 and it is easy to estimate the sum with a cutoﬀ L,

explicitly we have6

2 (1) 1 2πMredr¯ L 4 Mred r¯ 2 φ (xmax,L) ≈ √ = √ L . (5.70) 2 2 2 4 6MPlr? π 9π 6 MPl r?

The second key feature, evident from the above expression, is that the ﬁrst order

ﬂuctuation does not fall oﬀ with increasing r. In other words, as the wave spreads radially outward, its amplitude does not damp out.

We now turn to solving for the second order ﬂuctuation φ(2). To do this, it is ﬁrst

necessary to compute the eﬀective source J (2).

3 h 2 2 J (2)[φ(2)] = π¯ φ(1) − 2 π¯ ∂ ∂ φ(1) Λ3 µ ν

µ ν (1) (1) ν λ (1) µ (1)i −∂µ∂νπ∂¯ ∂ φ φ + 2∂µ∂ π∂¯ ν∂ φ ∂λ∂ φ

1 h 2 2 i + φ(1) − ∂ ∂ φ(1) . (5.71) Λ3 µ ν

In principle one can then integrate the source explicitly using the retarded Green’s

function in Eq. 5.66. However, it is equivalent and more practical to instead make a

WKB ansatz for φ(2)

(2) φ (~x,t) ∼ A(r)B(θ, ϕ) cos(nΩP t + Pt) cos(nΩP r + Pr) , (5.72)

where A(r) is a slowly-varying function of r (varies over distances much bigger than

−1 ΩP ) and Pt,r are irrelevant phases.

6 1 The factor of 4 comes because about half of the multipoles contribute to this sum.

155 The solution for φ(2) can then be obtained, in the WKB regime, by inserting this ansatz into the equations of motion, and matching with the eﬀective source J (2).

In the limit L → ∞, we would find again a diverging expression on the radial light cone, however keeping a fixed cutoff L we find

2 2 6 (2) r 2 2πMredr¯ 2 L φ (xmax,L) = √ Ω . (5.73) 3 2 P 4 r? Λ 6MPlr? 121π

(1) (2) We can then explicitly compare φ and φ at xmax

√ (2) φ (r?; xmax,L) 3 6π Mred 1 ΩP 4 (1) = v 2 L (5.74) φ (r?; xmax,L) 121 MPl (Λr?) Λ r 3 3 Mred 4 = vΩP r?L (5.75) 968π 2 Mtot

4 ≈ 0.1 × vΩP r?L . (5.76)

From this comparison, we see that perturbation theory is trustworthy whenever the

cutoﬀ L satisﬁes

−1/4 Mred L . vΩP r? . (5.77) Mtot

If we only include modes below the cutoﬀ L, then the power is given by

2 2 2 8π Mred r¯ 2 hP i = 2 ΩP S0(L) (5.78) 3 MPl r? 2 2 4 Mred v 2 4 = 2 2 ΩP L (5.79) 27 MPl (ΩP r?) 4 121 M M v √ red tot 2 = 2 3 ΩP . (5.80) 27 3 6π MPl (ΩP r?)

156 We may now apply this formula to realistic systems to compute the power emitted into modes below this cutoﬀ. However, this will not be relevant for binary pulsar systems. Using Hulse-Taylor parameters, the upper bound of L is less than 1. As a result, perturbation theory is never valid for any range of angular momentum, for a binary system like the Hulse-Taylor pulsar.

5.5.5 Hierarchy of masses

Despite the fact that perturbation theory does not converge for a realistic binary system, there is a regime where this calculational scheme is under perturbative control.

The perturbative calculation ultimately failed because there was no cost in energy to excite high ` modes which are highly sensitive to any departure from spherical symmetry. If there is a large hierarchy between the two masses in the system (M1

M2), the spherical symmetry will be stronger. This will not be relevant for a binary pulsar system, but we can use it to compute the emission of (say) the earth-moon system and verify that the emitted power is small in this case, both in the sense that the answer is convergent and in the sense that the Vainshtein screening mechanism is active.

If we take M1 M2 then

Mred M2 ≈ 1 if M1 M2. (5.81) Mtot M1

Taking the expression for the power in Eq. 5.78, and using the maximum value of L

157 −1/4 for which perturbation theory is still valid, L = (vΩP r?M2/M1) we have

4 121 M M v √ 1 2 2 hP i ≈ 2 3 ΩP . (5.82) 27 3 6π MPl (ΩP r?)

This expression has several key features. First, the power emission exhibits a Vain-

−3 shtein suppression ∼ (ΩP r?) . In fact the Vainshtein suppression is even stronger than might be expected by analogy to the static and spherically symmetric case.

For the quartic Galileon, the force between two objects separated by a distancer ¯

2 is (¯r/r?) , and from this we would have naively inferred a Vainshtein suprression

−2 ∼ (ΩP r?) for the power emission. Second, the power emission is proportional to the smaller mass M2, and therefore in the limit M2 → 0 we recover zero power emission appropriate for a static, spherically symmetric mass. This expression therefore passes several important consistency checks that are not a priori obvious from the divergent expression in Eq. 5.57.

It is interesting to estimate this expression numerically for an explicit example.

The Earth-Moon system is an example of a binary system where one mass is much

−2 larger than the other. For the Earth-Moon system, MMoon/MEarth ≈ 10 . From the power emission we can compute the spin down rate of the binary system using the time derivative of Kepler’s law

r¯ dE r¯˙ = − NR , (5.83) ENR dt

158 where

2 1/3 1 M1M2 ΩP ENR = 2/3 2 MPl. (5.84) (8π) MPl MtotMPl

Comparing the result for r¯˙ using the quartic Galileon with the Peters-Mathews for-

10 mula Eq. 5.1 for zero eccentricity, we ﬁnd thatr ˙Gal/r˙GR ∼ 10 . While this is larger than in GR, it is still small in absolute terms, contributing to a spin-down rate of

10−6 cm/year. More to the point, the estimate in this section relies on trusting perturbation theory with the cut-oﬀ in L given in Eq. 5.77, and it may be that a fuller analysis shows that perturbation theory breaks down even earlier. Future work is needed to rigorously establish the power emission for the quartic Galileon.

5.6 Hierarchy between two strong coupling scales

The approximation scheme that worked for the cubic Galileon has failed for a quartic

Galileon at full strength. We can however ﬁnd a regime where the quartic Galileon is present, and where perturbation theory holds, by considering an intermediate case between these regions the quartic Galileon only becomes relevant at very small distances.

−1 More precisely, we assume a hierarchy of scales r?,4 ΩP r?,3. As a ﬁducial value

−2 in what follows to check our approximations, we will take ΩP r?,4 = 10 .

In the cubic Galileon regime, the mode functions are given by Eq. 5.41, and in the quartic Galileon regime the mode functions are given by Eq. 5.48. We normalize the cubic Galileon mode function by matching to the mode function in the weakly coupled

159 regime. We then match the cubic and quartic mode functions and their derivatives

at r = r?,4, the boundary between the cubic and quartic Galileon regimes. The result

of this matching calculation is that the normalization constantu ¯`n for the quartic

Galileon is given by

1/4 √ !ν` 1/4 1 + 2` 9π 3 (nΩ r )ν`−1 r u¯ = P ?,4 ?,4 (5.85) `n 2 + ` 128 4 Γ(ν + 1) 3/4 ` r?,3 r 1/4 1 ν`−1 ?,4 = β`(nΩP r?,4) √ , (5.86) r?,3 r?,3

√ ν` 1+2` 3 9π 1/4 −1 where ν` = 2` + 1 for all `, and where β` = 2+` 4 128 Γ(ν` + 1) is dimen-

sionless. Crucially, β` falls oﬀ with increasing `, so that the multipole expansion now corresponds to a convergent series. This is the key property that is needed for the power emission to converge.

5.6.1 Power emission

Using the correctly normalized mode functions, we can now compute the power by using Eq. 5.40. Explicilty we ﬁnd that

π Ω rr ∞ P ?,4 X X 2ν`−1 2 2(ν`−1) hP i = 2 n β` (ΩP r?,4) (5.87) 3M r?,3 r?,3 Pl n=0 `m 2 1 Z TP Z −inΩP t 3 × dt e d x sin(ω`nr)Y`m(θ, ϕ)δT (x, t) . TP 0

160 When ` n`crit we may approximate ω`n ≈ nΩP . Since ΩP r¯ 1, we may use the

approximation sin(nΩP r) ≈ nΩP r, which leads to

2 p ∞ πM ΩP r?,4 X X 2(ν −1) π hP i = Ω2 red v2 n2ν`+1β2 (Ω r ) ` Y ( , 0)2δ . P 3M 2 3/2 ` P ?,4 `m 2 m,n Pl (ΩP r?,3) n=0 `m (5.88)

To check the validity of perturbation theory at large `, let us focus on the sum over n

∞ X π S(`) ≡ n2ν`+1β2(Ω r )2(ν`−1)Y ( , 0)2 . (5.89) ` P ?,4 `n 2 n=0

In this sum, the sum over multipoles is an expansion in powers of the small parameter

ΩP r?,4. This corresponds to an orbital velocity with an eﬀective orbital size r?,4. It is

as if the quartic Galileon has the eﬀect of increasing the eﬀective orbit of the source.

At large `, the sum S(`) scales with ` as

√ !` e 3 S(`) ∼ Ω r `−1/2 , (5.90) 2 P ?,4 where e is not the eccentricity but rather the base of the natural logarithm, and √ where we have used Stirling’s approximation Γ(z) ∼ 2πzz−1/2e−z. Thus for small

ΩP r?,4, the sum over multipoles is a convergent series with higher order terms in

` being exponentially suppressed. Numerically we ﬁnd that for the ﬁducial choice

−2 −3 ΩP r?,4 = 10 that S(4)/S(2) ≈ 10 , veryifying that the perturbative series is well

deﬁned because there is a suppression at higher multipoles and moments.

161 5.6.2 Quadrupole radiation

Focusing the power emitted into the Galileon quadrupole ` = 2 by the binary system, the sums in Eq. 5.88 are given by

∞ 2 π X X π 2 π π 2 β2 n7/2Y , 0 δ = β2 × 27/2Y , 0 ≈ 0.1 , (5.91) 3 2 2m 2 m,n 3 2 22 2 n=0 m=−2 and so the power in the quadrupole is

2 Mred (ΩP r?,4) 2 2 hP i = 0.1 × 3/2 v ΩP . (5.92) MPl (ΩP r?,3)

This result is almost identical to the cubic Galileon result in Eq. 5.43, but with one power of velocity replaced with ΩP r?,4. In particular, from this expression we can see that there is an active Vainshtein mechanism.

5.7 Outlook

In this section we have considered the radiation from binary systems in Galileon theories. We developed a formalism to handle a general Galileon theory with arbitrary interaction scales. Then we discussed how to compute the power emitted by a binary system in this framework. We reviewed how this was applied to the case of the cubic

Galileon in [64], before applying it to the general case.

For the general Galileon where all interaction scales are comparable, we found that perturbation theory diverges. The divergence of perturbation theory is ultimately tied

162 to the fact that the eﬀective metric in the quartic Galileon strong coupling regime takes the form

µ ν 2 2 2 2 Zµνdx dx ∝ −dt + dr + r?,4dΩ , (5.93)

where r?,4 is a constant. As a result of this constant factor, the Galileon fluctuations propagate on an approximately one dimensional metric, and the mode function does not fall off at large distances. This creates a situation where the first order perturbation acts as a large source for the second order perturbation, ultimately leading our perturbative approach to break down.

We identiﬁed several regions where perturbation theory is under control. Within these regions it is possible to obtain sensible results. By studying the radiation from a system with a large hierarchy of masses (such as the Earth-Moon system), we con-

ﬁrmed that within this regime the power remains ﬁnite, and that there is a Vainshtein suppression is present. We have also seen that in the limit where the system becomes static and spherically symmetric, the power emission smoothly approaches zero.

We also studied the case of an intermediate Galileon, where the strong coupling regime for the quartic Galileon occurs only at very small distances compared to the strong coupling regime for the cubic Galileon. In this case, we obtained a formula very similar to the one obtained for the cubic Galileon in isolation.

As a result, in all cases where a perturbative treatment is possible and under control in Galileon theories, the results are ﬁnite and suppressed relative to GR. This suggests that the Vainshtein mechanism is operative even in time-dependent situation.

163 Future work on computing the power emission in the quartic Galileon case would require a better approximation scheme. For example, this could involve including some breaking of spherical symmetry in the background to avoid the problems we have described above. It would also be interesting to study this problem numerically.

164 Chapter 6

Conclusions

6.1 Summary

In this dissertation I have studied ghost-free massive gravity in a wide variety of contexts. After presenting an introduction to the theory in Chapter 2 focusing on techniques and concepts that I would use later, I presented several important results that have been obtained during my dissertation work.

In Chapter 3 I gave an explicit derivation of the potential interactions of ghost-free massive gravity using Dimensional Deconstruction. This is an important development as it was not previously known whether Deconstruction could be applied to gravity to generate a ghost-free theory. The key insight was to apply a discretization of the derivative that acts linearly on the vielbein. This result is quite natural as the vielbein has nice properties as a choice of variable in a gravitational theory, for example fermions can couple to gravity through the vielbein but not the metric. We also showed how several features of ghost-free massive gravity, such as the strong coupling

165 scale, can be interpreted as arising from ﬁve dimensional General Relativity.

In Chapter 4, I studied kinetic interactions for a massive graviton. I have shown that the only consistent ghost-free theory kinetic term for a massive graviton is given by the Einstein-Hilbert term. This result was not obvious in advance because the mass term breaks diﬀeomorphism invariance and there is no symmetry reason to privelege the Einstein-Hilbert term. These results ﬁt with other work in the literature showing how the structure of General Relativity emerges from consistency requirements. I also applied the techniques from this work to study non-minimal matter couplings, which have been considered in the literature.

In Chapter 5, I discussed a method to compute the power emission in a binary star system in Galileon theories, accounting for the Vainshtein mechanism. I developed a formalism to compute the power in a set-up with arbitrary Galileon interactions.

For the quartic Galileon, I discussed how the perturbative methods that previously worked for the cubic Galileon broke down, and explained this phenomenon in terms of the eﬀective metric for the Galileon perturbations. I also studied several cases where the perturbative method is under control. In all cases where perturbation theory can be trusted, the Vainshtein mechanism has been shown to be active. This work connects Galileon theory with a real physical system. It also gives an analytic way to extend previous known results of the Vainshtein mechanism into time dependent regimes.

166 6.2 Outlook

There are several open questions raised by this work.

First, while we have given an argument that any kinetic term for a massive graviton beyond the Einstein-Hilbert term introduces new degrees of freedom, we have not learned about the nature of the degrees of freedom. There may be special choices of kinetic term for which ghost instabilities have a large mass in interesting backgrounds.

For example, while the non-minimal eﬀective vielbein matter coupling which we discussed in Section 4.5 exhibits a ghost, the ghost does not appear perturbatively around

FLRW solutions and so still could be useful for building models in cosmology. It may be interesting to explore this further.

Second, it would be very interesting to extend the gravitational wave emission in binary pulsar systems in the full decoupling limit of massive gravity. A crucial step along that path will likely involve computing the power emission for the quartic

Galileon. There may be analytic approaches, for example using a variational ansatz, that allow for tractable analytic progress to be made. The full decoupling limit of massive gravity also contains a coupling to matter which we did not consider. It could be interesting to understand the eﬀect of this coupling in the cubic Galileon where the analytic approach currently appears more tractable.

There are many outstanding issues in the ﬁeld that my work can be relevant for.

First, the Vainshtein mechanism is not understood in all cases, even within the decoupling limit. In order to develop observational tests of massive gravity in cosmology, it is important to understand these eﬀects in more detail. For example, the

167 Vainshtein mechanism should play a crucial role in cosmic structure formation. How- ever, the static and spherically symmetric case will not be a good ﬁrst approximation.

There has been progress in developing numerical codes to calculate these eﬀects [105].

However another interesting avenue is to study the decoupling limit around cosmological solutions. For example, the decoupling limit around cosmological solutions in interesting extensions to massive gravity, such as the quasi-dilaton, are only known in special cases [106]. It may be that by understanding these decoupling limits we can develop an understanding of how the Vainshtein mechanism enters into structure formation.

Second, lessons learned from massive gravity are actively being applied in other areas of ongoing research. Recent work on the eﬀective ﬁeld theory of dark energy

[107, 108] has focused on trying to ﬁnd the most general class of consistent scalar- tensor theories. These results rely on methods to avoid an Ostrogradsky instability, using insights that were discovered in the context of massive gravity [72]. I have worked on making this connection more explicit [109].

As I began this dissertation, I noted that there were many notions of gravity.

Massive gravity builds upon these ideas and adds to them in a non-trivial way. Taking the possibility of a graviton mass seriously opens up new and interesting framework in which to questions to ask about gravity. By exploring these theories we can hope to learn more about the nature of gravity itself.

168 Appendix A

Lovelock interactions

The following interactions are the most general covariant actions built out of a metric

gµν which lead to second order equations of motion for the metric [110]

Z √ S = (M )D−2n dDx −gεµ1···µD ε Rν1ν2 ··· Rν2n−1ν2n δν2n+1 ··· δνD D,n D,n ν1···νD µ1µ2 µ2n−1µ2n µ2n+1 µD Z D−2n D √ n D−2n ∼ MD,n d x −gεεR δ , (A.1)

where MD,n is a scale with units of energy, and n is a positive integer. In D dimensions this is non-vanishing only if n < D/2. In odd spacetime dimensions there are (D−1)/2 non-trivial Lovelocks, and in even spacetime dimensions there are (D − 2)/2. For

D = 4 there is only one non-trivial Lovelock combination, the Einstein-Hilbert term.

Since these interactions involve higher derivatives at low energies (if all MD,n ∼

MPl then for E < MPl) the higher order (n > 1) the Lovelock interactions are subdominant compared to the Einstein-Hilbert and cosmological constant pieces.

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