Faculty of Physics and AstronomyInstitute for Theoretical Physics

Linearised nonsymmetric metricperturbations in cosmology

byWessel Valkenburg

M.Sc. ThesisJune 2006

Supervisor: Dr. Tomislav Prokopec

Instituut voor Theoretische FysicaUniversiteit Utrecht

MinnaertgebouwLeuvenlaan 43584 CE UtrechtThe Netherlands

Abstract

We consider the generation and evolution of quantum fluctuations of a massive nonsym-metric gravitational field (B-field) from inflationary epoch to matter era in the simplest vari-ant of the nonsymmetric theory of gravitation (NGT), which consists of a gauge kinetic termand a mass term. We observe that quite generically a nonsymmetric metric field with mass,µ ' 0.03(HI/1013 GeV)4 eV, is a good dark matter candidate, where HI denotes the inflationaryscale. The most prominent feature of this dark matter is a peak in power at a comoving momen-tum scale, k ∼

õH0/(1 + zeq)1/4, where zeq is the redshift at equality and H0 is the Hubble

parameter today. Today this corresponds to the Earth-Sun distance.

Contents

Introduction 7I General relativity and Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7II Modern cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7III Nonsymmetric Theory of Gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . 8IV The layout of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1 NGT geometry 111.1 The metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2 Einstein-Hilbert action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.1 The connection in terms of the metric . . . . . . . . . . . . . . . . . . . . . 141.3 The metric: symmetric and antisymmetric components . . . . . . . . . . . . . . . . 15

1.3.1 The connection expanded . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.4 Effective action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.5 Dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.5.1 Massless free KR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.5.2 Massive free antisymmetric tensor field . . . . . . . . . . . . . . . . . . . . . 201.5.3 Coupled massive antisymmetric tensor field . . . . . . . . . . . . . . . . . . 221.5.4 Coupled massless Kalb-Ramond . . . . . . . . . . . . . . . . . . . . . . . . 22

2 Instabilities of NGT 232.1 Field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2 FLRW Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3 Schwarzschild background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4 Geometry and instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Canonical quantisation 313.1 Scalar field quantisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Antisymmetric tensor field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2.1 Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.2 Field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3 Antisymmetric tensor field quantisation . . . . . . . . . . . . . . . . . . . . . . . . 353.3.1 Straightforward quantisation . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3.2 Gauge invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3.3 Constrained quantisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4 Vacuum states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.4.1 Vacuum in Minkowsky space . . . . . . . . . . . . . . . . . . . . . . . . . . 433.4.2 Ambiguity in curved space . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.4.3 Conformal vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.4.4 Bunch-Davies vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3

CONTENTS CONTENTS

4 Quantum fluctuations 474.1 Quantities of interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.1.1 Massive or massless . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.1.2 Power spectrum of density fluctuations . . . . . . . . . . . . . . . . . . . . . 48

4.2 Dynamics during inflation era . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2.1 General solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.2.3 The spectrum of the density fluctuations . . . . . . . . . . . . . . . . . . . . 51

4.3 Dynamics during radiation era . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.3.1 General solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.3.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.3.3 The spectrum of energy density fluctuations . . . . . . . . . . . . . . . . . . 57

4.4 Dynamics during matter era . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.4.1 General solution and boundary conditions . . . . . . . . . . . . . . . . . . . 614.4.2 The spectrum of energy density fluctuations . . . . . . . . . . . . . . . . . . 64

4.5 Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5 Dark-matter energy density 695.1 Energy density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.2 Hot or cold Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6 Conclusion 73

A The standard model of Cosmology 75A.1 Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75A.2 Horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80A.3 Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

B Massive antisymmetric field action 85B.1 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

B.1.1 Covariant way of writing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85B.1.2 Rewriting the EOM as ~E and ~B . . . . . . . . . . . . . . . . . . . . . . . . 86

B.2 Tµν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

B.2.1 Covariant notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87B.2.2 Rewriting T0

0 as ~E and ~B . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88B.2.3 Ti

j as a function of BL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

C Special functions 91C.1 Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

C.1.1 General solutions and their Wronskians . . . . . . . . . . . . . . . . . . . . 91C.1.2 Rewriting the equation to the desired form . . . . . . . . . . . . . . . . . . 92C.1.3 Asymptotic expansions of the Bessel functions . . . . . . . . . . . . . . . . 92C.1.4 Recursion relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92C.1.5 Taylor expansions of the Hankel functions . . . . . . . . . . . . . . . . . . . 93C.1.6 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

C.2 Confluent Hypergeometric functions . . . . . . . . . . . . . . . . . . . . . . . . . . 94C.2.1 General solutions and their Wronskians . . . . . . . . . . . . . . . . . . . . 94C.2.2 The Whittaker function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95C.2.3 Rewriting the equation to the desired form . . . . . . . . . . . . . . . . . . 95C.2.4 Large k asymptote . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96C.2.5 k=0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Acknowledgements 99

Bibliography 101

4

Notation

Greek indices µ, ν etc. run over all spacetime coordinates, i.e. from 0 to 3, with x0 the timecoordinate.

Latin indices i, j, k etc. run over all spatial coordinates, i.e. from 1 to 3.

Repeated indices are summed over.

The Minkowsky metric ηµν is chosen as diagonal(+,−,−,−), and therefore δij = −ηij .

Two distinguished squares of the partial derivative are used: the d’Alembertian = 1√−g∂µ (

√−ggµν∂ν)

and the ’ordinary’ square ∂2 = ηµν∂µ∂ν . This distinction is made since in conformal spacetimesgµν = a(η)2ηµν . In that case gµν∂µ∂ν = a(η)−2∂2, which is comfortable in rewriting equations.

The fully antisymmetric Levi-Civita symbol ε is defined in both three and four dimensions, εijk

(with ε123 = +1) and εµνρσ (with ε0123 = +1), where the indices run as defined above. TheLevi-Civita tensor, which in curved space differs from the symbol, is not given another notationas all additional curvature terms will be written explicitly.

For a vector ~V superscripts T and L denote the transversal and longitudinal parts of this vector,where T is not to be confused with the transposition operation on a vector or matrix. There willnot, however, be much confusion about this throughout this thesis, since matrix manipulationis always done in index notation. In this notation transposing simply means switching indices,Mµν →Mνµ, and no T will be used.

Complex conjugation is denoted by a superscript ∗.

Natural units are used throughout the thesis. I.e. ~ = c = GN = 1.

5

NOTATION

6

Introduction

I General relativity and Cosmology

In the beginning of the twentieth century, Einstein developed the general theory of relativity [14],in which he identified gravity with the curvature of space. In the framework of this theory, one isable to derive an equation of state for the Universe, which allowed an initially surprising result toexist, namely the fact that the vacuum might be an expanding medium. Since this was in contrastwith Einsteins expectations of the Universe, he introduced a constant in this equation of state,the cosmological constant Λ, to create a static Universe without expansion, an Einstein-de SitterUniverse.

About a decade later, looking at distant galaxies, Hubble [20] found that the spectrum of lightcoming from these galaxies was redshifted with respect to the spectrum of the stars in our owngalaxy. He found that the larger the distance between the Sun and a galaxy is, the heavier thespectrum is redshifted. This was the first indication that the Universe might be expanding afterall, giving rise to redshift of photons, depending on the distance to the observer.

Hubble’s observations made Einstein call his cosmological constant ‘the biggest blunder of hislife’, since there was no reason at all to presume the Universe to be static.

These events can be thought of as the foundations of the present theory of cosmology, theorigin and evolution of the Universe.

II Modern cosmology

Extrapolating back in time the notion of an expanding Universe, it is not very strange to believethat the Universe once was very compact, with all matter compressed in the small volume ofthe young Universe. Matter might have been so dense, that it was in thermal equilibrium withradiation, and hence optically thick, like our own sun. As an expanding gas cools down, the mattercontent of the Universe became optically thin, at once allowing all light to freely emerge frommassive particles (decoupling). The Universe today is still filled with these ancient photons, calledthe ’Cosmic Microwave Background Radiation’ (CMBR). After the CMBR had been predicted byGamow and Alpher [16, 2], it has been observed, and its spectrum coincides with that of a blackbody with a temperature of 2.725 ± 0.002 K [5].

This fluid of photons is presently the most ancient source of information in the Universe, sincebefore these photons were born, the Universe was optically thick and all information was washedout by thermal interactions. Hence, many cosmological theories can be tested by calculating theirprediction of the shape of the CMBR and see whether it suits observations. That is, we found avery old picture of the Universe, and now we try to reconstruct that picture with various theories,trying to say something about the Universe before and just after decoupling.

The CMBR can be measured all over the sky. Doing so, the map of the CMBR shows only smallperturbations in the measured temperature, denoted by δT

T . From these temperature fluctuations,the density fluctuation of the massive matter can be calculated, which typically is [37],(

δρ

ρ

)dec

= const× δT

T. 8× (10−5). (II.1)

7

III Nonsymmetric Theory of Gravitation INTRODUCTION

This means that just after the last scattering, the Universe was close to homogeneity. Yet thesesmall perturbations can, due to gravitational instability, lead to the structure of the Universetoday.

The equation of state of the Universe depends on the contents of the Universe. Via thismechanism, the cosmological constant has made its comeback into the equation of state of theUniverse, allowing us to fine tune the theory to comply with present observations. Next to thecosmological constant1, an extensive amount of massive matter has to be present in the Universe,in order to fit the observations. An estimate of observed matter (stars, earth, we) will give anumber that is too small to give the Universe the equation of state it obeys today. Hence, morematter is needed: dark matter (DM).

Dark matter has been needed not only by cosmologists. In the field of astrophysics, classicalgravitation in combination with the observed amount of matter has some shortcomings. Matterin the outer regions of galaxies, appears to be rotating around the galaxy with too high a speed.Ordinarily, one would expect these stars to fly away due to a centrifugal force, out of the galaxy,however observations of galaxies and calculations of their ages, imply that this matter is in a stableorbit. This means that either the force of gravity on large distance scales is larger then we know,or the amount of matter in the outer regions of a galaxy is larger than we know.

The same effect is observed within clusters of galaxies, where the motion of galaxies relativeto each other is out of equilibrium in the classical theory involving gravity and observed matter.

Many sources of dark matter have been suggested and some of them were ruled out. Examplesare relic neutrinos, axions, Goldstone bosons, many different supersymmetric particles, gravitino’s,monopoles and other topological defects, and so on. In this thesis we will find another newcandidate: the nonsymmetric metric field.

III Nonsymmetric Theory of Gravitation

The theory of nonsymmetric gravitation (NGT) finds its origin in 1925, when Einstein first triedto unify gravity and electrodynamics in a geometrical way [15]. This unification stems from theresemblance of a nonsymmetric admixture to the general-relativistic metric and the antisymmetricfield strength tensor in electromagnetism. This unification was without success, primarily becausethe Lorentz force law is not correctly reproduced. However, this still leaves open the possibility ofadding a nonsymmetric term to the metric.

In 1979 Moffat proposed a new theory of gravitation [31], using the same antisymmetric ad-mixture in a gravitational way, with no reference to electromagnetism (EM). Since then NGT hasbeen subject to a revived research, testing its viability as a geometrical theory, as well as tryingto use it to solve the mystery of dark matter or the unappealing singularities in general relativity(GR) [35, 33, 34, 12, 10, 11].

If one allows nonsymmetric terms to appear in the metric, one is led to

gµνgαν = gνµg

να = δαµ (6= gµνg

να) . (III.1)

If we assume the nonsymmetric term to be small, the metric can be decomposed in a symmetricbackground and an infinite series of combinations of symmetric and antisymmetric terms,

gµν = Gµν +Bµν + αBµαBα

ν + βBαβBαβGµν +O

(B3). (III.2)

Expressing NGT up to first order in Bµν as in equation (III.2), it is tempting to point at theresemblance not only with EM, but also with string theory [12], as the extension is nothing buta non-symmetric term in the general relativistic metric. However, there is no correlation betweenNGT and string theory [36].

The geometrical implications of NGT for cosmology have been briefly investigated [33, 32], butthe energy density in stable modes of NGT and its consequences for the matter content of thestandard model of cosmology were as yet unexplored.

1The cosmological constant is often denoted as dark energy, DE, not to be mistaken for the dutch coffee brandDouwe Egberts, which in fact is also a source of Dark Energy.

8

INTRODUCTION IV The layout of this thesis

IV The layout of this thesis

The aim of this thesis is to investigate the nonsymmetric extension to general relativity in theframework of cosmology. By means of this investigation we are given a view on the current statuscosmological research and it leads us to new results.

I will begin with the linearisation of the action for antisymmetric perturbations on a cosmo-logically justified background. From this action I derive the equations of motion for the theory.As the title of this thesis suggests, I restrict to linear order. Although this leaves open certainquestions, our hands are well filled when treating just the linear order.

With the linearised equations at hand I will discuss the stabilities of the theory. In thisdiscussion I show that NGT in its full shape is not stable. However, I also derive what the actionmust look like to be stable after all. The resulting action is slightly different from NGT.

In a geometric theory the mass term is naturally induced by the cosmological constant, inwhich case [22],

µ2 ' 2Λ(1− 2α+ 8β) , (IV.1)

where α and β are defined by the decomposition of the metric tensor in equation (III.2). Yet thereis no reason to assume that the mass term is fully of geometric origin, and thus the relation (IV.1)needs not to hold in general.

Equation (IV.1) is also significant because it implies that, if the B field is of a geometric origin,then it would be unnatural to assume that its mass vanishes. Indeed the current observations,based on the luminosity-redshift relation of distant supernovae Ia [39, 43, 44], suggest that thecosmological term today is of the order, Λ ∼ 10−84 GeV2. To maintain generality we assume inthis work that the mass µ is unspecified and study its cosmological implications.

Applying to the linearised equations the presently assumed boundary conditions of the Uni-verse, we are led to a specific amplitude of quantum fluctuations for the derived stable action, bymeans of canonical quantisation. These quantum fluctuations may become observable as classicalexcitations when they cross outside the Hubble radius and subsequently evolve from inflation erato the radiation era; this is similar to the scalar metric perturbations which freeze in during in-flation. Evolving the antisymmetric metric fluctuations up to different timescales in the historyof the Universe, I calculate the expected energy density of the fluctuations, depending on fewparameters. My main finding is that the B field with a mass of the order,

µ ' 0.03(1013 GeV

HI

)4

eV (IV.2)

which corresponds to a lengthscale, µ−1 ' 7 × 10−8(HI/1013 GeV

)4 m, is a good dark mattercandidate.

Since the B field is produced in inflation and does not couple to the matter fields, its spectrumis highly nonthermal. Indeed, we find that the spectral power is peaked at a comoving momentum,k '

õH0/(1 + zeq)1/4, where zeq is the redshift at matter-radiation equality, and corresponds to

a physical scale at structure formation (z ∼ 10), k−1phys ∼ 2× 107 km. This peak is generated as a

consequence of a different nature of the vacuum states in inflation and radiation era. This is themain feature by which this dark matter can be distinguished from other dark matter candidates,which typically obey a thermal statistic.

Another important feature are Fourier space pressure oscillations, which occur after the secondHubble crossing. Although we find that the pressure of the B-field drops to zero before the decou-pling of the cosmic photon fluid, the Fourier pressure components exhibit significant oscillations.These oscillations may have a potentially observable impact on the gravitational potentials, andthey are thus the second distinct feature of our dark matter candidate. The physical significanceof these spectral pressure oscillations should be further investigated.

A brief overview of the standard model of cosmology is given in appendix A.

9

IV The layout of this thesis INTRODUCTION

10

Chapter 1

NGT geometry

1.1 The metric

In general relativity, the infinitesimal line element is defined as

ds2 = dxµdxµ. (1.1.1)

The metric tensor (gµν) is the entity that maps a vector to its dual vector,

dxµ = gµνdxν , (1.1.2)

and is often addressed as the metric, although officially the metric is the infinitesimal line element.When a nonsymmetric metric is allowed, the choice of equation (1.1.2) is not unique anymore.

I could have also written gνµ in stead of gµν , which will lead to the same physics, but may giveminus signs in several places in the mathematics describing the physics.

By virtue of equation (1.1.2), ds2 can be written as

ds2 = dxµdxµ

= gµνdxνdxµ

= gµνgανdxαdx

µ (1.1.3)

where I let the inverse metric, gµν , be the map from dual vectors to vectors by the ordering ofindices such that

dxµ = gµνdxν . (1.1.4)

From equation (1.1.3) we find that the inverse metric is defined such that

gµνgαν = gνµg

να = δαµ (6= gµνg

να) . (1.1.5)

Due to curvature, the partial derivative (∂µ) no longer transforms as a vector, but the covariantderivative,

∇µVν = ∂µVν − ΓλνµVλ, (1.1.6)

does. The quantity Γλνµ is called the connection, and its contents is to be defined later. At this

moment it is sufficient to understand that the connection takes account exactly for the differencebetween the partial derivative in a new frame and the partial derivative transformed to that frame.

The change of a vector Vν when it is parallel transported along a direction xµ is then given bythe quantity ∇µVν .

The Riemann tensor is defined as the quantity that embodies the curvature. In this sense,curvature is quantified by the rotation of a vector, when it is parallel transported from one pointto another along two different paths. The rotation is then calculated by

[∇µ,∇ν ]V ρ = RρλµνV

λ.

11

1.2 Einstein-Hilbert action CHAPTER 1 NGT GEOMETRY

V

y

x x

y

V’ V’’

Figure 1.1: Without the presence of torsion, the vector V is transported along paths x− y and y − x, totransform to V ′ and V ′′ respectively.

V’V’’

x

y

x

y

V

Figure 1.2: In the presence of torsion, the vector V is again transported along paths x− y and y − x, totransform to V ′ and V ′′ respectively, with a translation between the endpoints.

See also figure 1.1.When torsion is present, a vector is not only rotated but translated as well when parallel

transported [45]. In other words, the paths x−y and y−x do not close as illustrated in figure 1.2.In order to appropriately calculated the rotation of a parallel transported vector, one has to

close the parallelogram, to find

[∇µ,∇ν ]V ρ = RρλµνV

λ − 2Γλ[µν]∇λV

ρ. (1.1.7)

The expression for the Riemann tensor now reads

Rρλµν = ∂µΓρ

λν − ΓσλµΓρ

σν − ∂νΓρλµ + Γσ

λνΓρσµ. (1.1.8)

Note that, since torsion may be present, the ordering of indices in the definition of the covariantderivative, and hence in the Riemann tensor, is important.

1.2 Einstein-Hilbert action

A geometric theory is defined by an action which is constructed from covariant objects builtfrom the connection and the metric on the spacetime manifold [12]. For a symmetric metric,

12

CHAPTER 1 NGT GEOMETRY 1.2 Einstein-Hilbert action

the connection is usually fully defined in terms of the metric, by an equation in which one doesnot have to keep track of the ordering of the lower indices of the connection symbols and ofboth indices of the metric. When the connection is not a free parameter and is solely defined interms of the metric, literature speaks of second order formalism. However, when a nonsymmetriccomponent is allowed for the metric, the ordering of the indices is not trivial any longer. One wayof deriving the connection in terms of the metric is in Palatini form (also addressed as first orderformalism). In this formalism, both the metric and the connection coefficients are considered to befree parameters. The action based on which I define the geometry, is the Einstein-Hilbert actionincluding the cosmological constant,

SEH =∫d4x

√−g(R− 2Λ), (1.2.1)

where I choose to work in units were 1/(16πGN ) = 1. In this equation the Ricci scalar, R, isdefined as

R = Rλµλνg

µν . (1.2.2)

The field equation for the metric is as usual given by

1√−G

δ

δgµν(SEH + SM ) = 0

= Rµν − 12gµνR+ Λgµν − 8πTµν . (1.2.3)

The action of any present matter is denoted by SM . Equation (1.2.3) is called Einstein’s equationand is the most fundamental equation in general relativity, since it determines fully how the metricis determined by any present matter.

The field equation for the connection now reads

δ

δΓαβγ

S = 0 (1.2.4)

which becomes

(−g)− 12 ∂α

√−ggβγ − (−g)− 1

2 ∂λ

√−ggβλδγ

α −Γγαλg

βλ −Γβναg

νγ + Γλαλg

βγ + Γβλσg

λσδγα = 0. (1.2.5)

Looking at the trace of this equation, α = γ, we find

(−g)− 12 ∂α

√−ggβα +

2D − 1

Γαgβα + Γβ

λσgλσ = 0, (1.2.6)

whereΓα ≡ Γλ

[αλ], (1.2.7)

following the notation of Ref. [12]. Using this result, equation (1.2.5) becomes,

(−g)− 12 ∂α

√−ggβγ + Γβ

λαgλγ + Γγ

αλgβλ − 2Γαg

βγ − Γλλαg

βγ +2

D − 1Γλg

βλδγα =

∂αgβγ − 1

2gβγgλσ∂αg

λσ + Γβλαg

λγ + Γγαλg

βλ − Γαgβγ − 1

2Γλαλg

βγ − 12Γλ

λαgβγ +

2D − 1

Γλgβλδγ

α =

∂αgβγ + Γβ

λαgλγ + Γγ

αλgβλ − 1

2gβγgλσ

(∂αg

λσ + Γλµαg

µσ + Γσαµg

λµ)

−Γαgβγ +

2D − 1

Γλgβλδγ

α = 0.

If we now substituteΓα

βγ ≡ Γαβγ +

2D − 1

δαβ Γγ , (1.2.8)

13

1.2 Einstein-Hilbert action CHAPTER 1 NGT GEOMETRY

we arrive at(δβλδ

γσ − 1

2gβγgλσ

)(∂αg

λσ + Γλµαg

µσ + Γσαµg

λµ)

+2− 2D − 1

Γλgβλδγ

α +−2 +D + 1− (D − 1)

D − 1Γαg

βγ

=(δβλδ

γσ − 1

2gβγgλσ

)(∂αg

λσ + Γλµαg

µσ + Γσαµg

λµ)

= 0, (1.2.9)

which implies∂αg

λσ + Γλµαg

µσ + Γσαµg

λµ = 0. (1.2.10)

Using the fact that Γλ[µλ] = 0, we can write,

Rµν (Γ) = Rµν(Γ)− 2D − 1

∂[µ Γν]. (1.2.11)

With this result we can now rewrite the action as

S =∫d4x

√−gR(Γ)− 2

D − 1∂[µ Γν]

. (1.2.12)

From this equation we can see that, if the connection is treated as a free parameter, it has only onedegree of freedom, represented by Γµ. All other degrees of the connection are again fully definedin terms of the metric by equation (1.2.10).

Since we now have an expression which defines the nonsymmetric connection in terms of thenonsymmetric metric and an action in which the degree of freedom in the connection is isolatedinto a single term, we can continue in second order formalism by simply setting

Γµ = 0

Γαβγ = Γα

βγ .

1.2.1 The connection in terms of the metric

Equation (1.2.10) defines the connection in terms of the inverse metric, gµν . Using equation (III.1),we find

gλβgρσ

(∂αg

λσ + Γλµαg

µσ + Γσαµg

λµ)

=

gλβ∂αδλρ − gλβg

λσ∂αgρσ + gλβgρσΓλµαg

µσ + gλβgρσΓσαµg

λµ =

−∂αgρβ + gλβΓλρα + gρσΓσ

αβ = 0, (1.2.13)

defining the connection in terms of the metric.In the symmetric theory the usual way to solve the connection in terms of the metric is by

subtracting two even permutations of equation (1.2.13) of its original form. With the nonsymmetrictheory the same can be done, to find(

∂αgρβ − gλβΓλρα − gρλΓλ

αβ

)−(∂ρgβα − gλαΓλ

βρ − gβλΓλρα

)−(∂βgαρ − gλρΓλ

αβ − gαλΓλβρ

)=

∂αgρβ − ∂ρgβα − ∂βgαρ + Γλβρ (gλα + gαλ)− Γλ

ρα (gλβ − gβλ)− Γλαβ (gρλ − gλρ) ,

(1.2.14)

yielding,g(λα)Γλ

βρ = 12 (−∂αgρβ + ∂ρgβα + ∂βgαρ) + g[λβ]Γλ

ρα + g[ρλ]Γλαβ . (1.2.15)

This is the best we can do, when treating the full metric. Note that when a symmetric metricis inserted, the connection turns out to be the Christoffel symbol, also known as the Levi-Civitaconnection.

14

CHAPTER 1 NGT GEOMETRY 1.3 The metric: symmetric and antisymmetric components

1.3 The metric: symmetric and antisymmetric components

Equation (1.2.15) can be solved iteratively when the metric can be expanded in a symmetric back-ground metric and an antisymmetric perturbation. As general relativity deals with a symmetricmetric by definition, and has been tested up to great accuracy, it is very likely that if the met-ric contains a nonsymmetric component, it will appear in terms of an antisymmetric expansion.Hence, throughout the rest of this thesis, I will assume the following form of the metric,

gµν = Gµν +Bµν + αBµαBα

ν + βBαβBαβGµν +O

(B3). (1.3.1)

Here Gµν denotes the symmetric background metric and Bµν denotes the antisymmetric per-turbation. Using the fact that the background metric has a known inverse,

GµνGµρ = δρ

ν , (1.3.2)

and the fact that we may expand for small antisymmetries, we can set up the inverse of the fullmetric,

gµνgµρ = δρ

ν , (1.3.3)

which becomes,

gµν = Gµν +Bµν + (1− α)BµρB

ρν − βBαβBαβGµν +O

(B3). (1.3.4)

Here Bµν is defined asBµν = GµρGνσBρσ. (1.3.5)

Having the metric and its inverse, the determinant can be expanded as

δ[ln detGµν ] = δ[Tr lnGµν ]

δ[detGµν ]detGµν

= Gµνδ[Gµν ]− 12G

µνδ[Gνρ]Gρσδ[Gσµ]

= −αBµνBµν +DβBµνB

µν + 12BµνB

µν +O(B3), (1.3.6)

leading to √−g =

√−G

[1 + 1

2

(12 − α+Dβ

)BµνB

µν]+O

(B4), (1.3.7)

where D = 4 is the number of dimensions. Here I defined δGµν = gµν − Gµν . I expanded upto second order in δGµν , since that term contains both first and second order terms in Bµν , andwe want to expand up to second order in Bµν . Note that all third orders terms in Bµν vanish inequation (1.3.7) due to symmetry reasons.

1.3.1 The connection expanded

Now, with the given expansion of the metric, equation (1.2.15) can be solved iteratively. Up tolowest order in Bµν we retrieve the original Christoffel connection,

λβρ

, with respect to the

background metric,

Sλ(0)

βρ =

λβρ

= 1

2Gαλ (−∂αGρβ + ∂ρGβα + ∂βGαρ) . (1.3.8)

Up to first order in Bµν we find,

Aλ(1)

βρ = 12G

αλ (−∂αBρβ + ∂ρBβα + ∂βBαρ) +BσβSσ(0)

ρα Gαλ +BρσSσ(0)

αβ Gαλ

= 12G

αλ(−∇αBρβ +∇ρBβα +∇βBαρ

)(1.3.9)

The connection can be split in two series, one containing components that are symmetric in thetwo lower indices (Sλ(n)

βρ ), and one containing components that are antisymmetric in the two lower

15

1.4 Effective action CHAPTER 1 NGT GEOMETRY

indices (Aλ(2n+1)

βρ ). In the last line of equation (1.3.9), I denoted the covariant derivative withrespect to the background connection by ∇µ. Throughout the rest of this thesis, an overlinedentity denotes that entity, with all connection symbols within it evaluated with respect to thebackground.

Up to quadratic order in Bµν , we find

Aλ(0)

βρ =Aλ(2)

βρ = Sλ(1)

βρ = 0, (1.3.10)

Sλ(2)

βρ =− αBλσBσγSγ(0)

ρβ − βBγσBγσSλ(0)

ρβ +GλαBσβAσ(1)

ρα +GλαBρσAσ(1)

αβ

+ 12G

αλ [−∂α (αBρσBσ

β + βBσγBσγGρβ) + ∂ρ (αBβσB

σα + βBσγB

σγGβα)

+∂β (αBασBσ

ρ + βBσγBσγGαρ)] . (1.3.11)

Of course all solutions obey the condition

Γλ[αλ] = 0. (1.3.12)

1.4 Effective action

The full action describing our geometry remains equation (1.2.1),

S =∫d4x

√−g(R− 2Λ). (1.4.1)

But, now that we chose to work in second order formalism, we found the connection in terms ofthe metric, henceforth equation (1.4.1) can be written in a more explicit form as a function of thebackground metric, its connection and the antisymmetric perturbation. Later on in this thesis wewill calculate the linearised vacuum fluctuations of the antisymmetric perturbation. In order todo so, we need the effective action to be of order BµνB

µν .The integrant in the action can be split in terms of different order in Bµν ,

S =∫d4x

√−g(R− 2Λ)

=∫d4x

√−G

[1 + 1

2

(12 − α+Dβ

)BµνB

µν] [gαβRαβ(Γ(g))− 2Λ

]=∫d4x

√−G

R− 2Λ +

(12 − α+Dβ

)ΛBµνB

µν + 12

(12 − α+Dβ

)BµνB

µνR

+GαβR(1)αβ +GαβR

(2)αβ + g(1)αβR

(1)αβ + g(2)αβRαβ

. (1.4.2)

Here the superscript (n) denotes the order of Bµν in that specific quantity. In the following, themass term can be written as (

12 − α+Dβ

)Λ =

14µ2. (1.4.3)

Let us now take a closer look at terms R(n). For a given value of n ≥ 1, one can write

R(n)µν =R(n)α

µαν

=∂αΓα(n)

µν − ∂νΓα(n)

µα −n∑

i=0

Γλ(i)

µα Γα(n−i)

λν +n∑

i=0

Γλ(i)

µν Γα(n−i)

λα

=∇αΓα(n)

µν −∇νΓα(n)

µα −n−1∑i=1

Γλ(i)

µα Γα(n−i)

λν +n−1∑i=1

Γλ(i)

µν Γα(n−i)

λα . (1.4.4)

16

CHAPTER 1 NGT GEOMETRY 1.4 Effective action

Combining this with the fact that for any vector Vν we have

Gµν∇µVν = ∇µVµ

= ∂µVµ + Γ

µ

µρVρ

= ∂µVµ + 1

2VρGσλ∂ρGσλ

= ∂µVµ + V ρ(−G)−

12 ∂ρ

√−G

= (−G)−12 ∂µ

(√−GV µ

), (1.4.5)

we arrive at√−G

[GαβR

(1)αβ +GαβR

(2)αβ

]= −GαβAλ(1)

ασ Aσ(1)

λβ +GαβAλ(1)

αβ Aσ(1)

λσ + total derivative. (1.4.6)

From this derivation we can conclude that the quantity Sλ(2)

βρ only contributes in by means of a

total derivative. For Aλ(1)

βρ we may write

Aλ(1)

βρ = 12G

αλ(−∇αBρβ +∇ρBβα +∇βBαρ

)= 1

2Fλ

ρβ −∇λBρβ , (1.4.7)

in which we recognise the antisymmetric field strength of an antisymmetric 2-form, analogous tothe Kalb-Ramond field [25],

Fµνρ = ∂µBνρ + ∂νBρµ + ∂ρBµν

= ∇µBνρ +∇νBρµ +∇ρBµν . (1.4.8)

Then equation (1.4.6) turns into

GαβR(1)αβ +GαβR

(2)αβ =− 1

4FµνρF

µνρ + 12 (∇λ

Bσα)Fσαλ − 12 (∇σ

Bαλ)Fλ

σα

+ (∇λBσα)∇σ

Bαλ + 0 +

1√−G

total derivative

=112FµνρF

µνρ − (∇λBσα)∇σ

Bαλ +

1√−G

total derivative. (1.4.9)

Here I made use of the fact that

FµνρFµνρ = 3

(∇µBνρ

)Fµνρ. (1.4.10)

Likewise we have

g(1)αβR(1)αβ = Bαβ

(∇λ

[12F

λβα −∇

λBβα

]+∇β∇

λBλα

)= Bαβ

(− 1

2∇λ∇λBβα + 1

2∇λ∇βBαλ + 1

2∇λ∇αBλ

β +∇β∇λBλα

)= Bαβ

(12∇λ∇

λBαβ −∇λ∇βB

λα +∇β∇

λBλα

)= Bαβ

(12∇λ∇

λBαβ +

[∇β ,∇λ

]Bλ

α

)= Bαβ

(12∇λ∇

λBαβ +R

λσβλB

σα −R

σαβλB

λσ

)(1.4.11)

17

1.4 Effective action CHAPTER 1 NGT GEOMETRY

If we now combine equations (1.4.9) and (1.4.11), this results in

GαβR(1)αβ +GαβR

(2)αβ + g(1)αβR

(1)αβ =

112FµνρF

µνρ − (∇λBσα)∇σ

Bαλ −R

λσλβB

σαB

αβ

−Rσ

αβλBλ

σBαβ + 1

2Bαβ∇λ∇

λBαβ +

1√−G

total derivative

=112FµνρF

µνρ −Rσ

αβλBλ

σBαβ −R

λσλβB

σαB

αβ

+ 12∇λ

(Bαβ∇λ

Bαβ

)− 1

2

(∇λB

αβ)∇λ

Bαβ

− 12 (∇λ

Bσα)∇σBα

λ − 12 (∇λ

Bσα)∇αBλ

σ +1√−G

total derivative

=112FµνρF

µνρ −Rσ

αβλBλ

σBαβ −R

λσλβB

σαB

αβ

− 16FαβγF

αβγ +1√−G

total derivative (1.4.12)

Again I made use of equalities (1.1.7) and (1.4.10).Plugging these results in equation (1.4.2), we get

S =∫d4x

√−G

R− 2Λ +

14µ2BµνB

µν + 12

(12 − α+ (D − 2)β

)BµνB

µνR

− 112FµνρF

µνρ −RµανβBµνBαβ − αBα

ρBρβRαβ

+∫d4x total derivative. (1.4.13)

In a number of the steps taken, I have extracted all total derivative terms. Any term that is equalto a total derivative has no contribution to the theory and can be neglected. That is the case,since all that is important is the variation of the action with respect to various parameters. In theEuler-Lagrange formalism, the variation of a parameter can be chosen to vanish at plus and minusinfinity. If such a variation appears in a total derivative term, the contribution of that term tothe evaluated action will be linear in the variation at plus and minus infinity, and hence be equalto zero.

All single terms in the action, (1.4.13), are scalar by nature on their own. This means thatin a general action, these terms might be added or subtracted by arbitrary amount, apart fromtheir geometrical (Einstein-Hilbert action) origin. The action in its particular form given in equa-tion (1.4.13) is the special case when the full geometrical action is given by the Einstein-Hilbertaction solely. There is no reason why that has to be the only possibility, therefore I present thefinal form of the general effective action, up to quadratic order in antisymmetric perturbations as

Seff =∫d4x

√−G

R− 2Λ− 1

12FµνρFµνρ +

(14µ

2 + θ1R)BµνB

µν

+θ2RµανβBµνBαβ + θ3B

αρB

ρβRαβ

. (1.4.14)

18

CHAPTER 1 NGT GEOMETRY 1.5 Dualities

1.5 Dualities

For different choices of the (coupling) constants µ and θn, the theory of the antisymmetric field ona symmetric background is the dual of different other theories. Looking at the dual of a theory,often gives insight in certain behaviour of the theory, since the actual physics must be the same.A general property of the physics, as for example the number of degrees of freedom, must beexhibited in both theories. The content of this section is mainly based on Refs. [25, 19, 30, 4, 3,47, 48, 27, 18, 29].

1.5.1 Massless free KR

Setting µ = θn = 0 leads us to the uncoupled (hence ‘free’) Kalb-Ramond action,

SKR =∫d4xLKR =

∫d4x

√−G

− 1

12FµνρF

µνρ

. (1.5.1)

In order to prove the quantum equivalence of dual theories and to get the same result as algebraicapproach [48], I choose to calculate the dual theory in path integral formulation. The path integralcalculates the partition function, which is known from statistical physics,

Z[J ] =∫

[DF ][DB] exp−∫d4x

√−G

(112FµνρF

µνρ − 12Fµνρ∇[µBνρ] −

e

2BµνJ

µν

),

(1.5.2)where Jµν is the source for the Kalb-Ramond field, and e is its coupling constant. I will work infirst order formalism, this time meaning that the field strength is not a priori defined as a functionof Bµν but will be defined so by the field equations.

The B-integral can be evaluated,∫[DB] exp

−1

2

∫d4x

√−GBνρ (∂µF

µνρ − eJνρ)

= constant× δ(∂µ

√−GFµνρ −

√−GeJνρ

).

(1.5.3)Here I made use of partial integration, the antisymmetry of F , and the equality

∇[µBνρ] = ∂[µBνρ]. (1.5.4)

When evaluating the F -integral, the delta function has the same role as in ordinary analysis,although it now states a differential equation to be solved in stead of an ordinary equation. Thereader may verify that this differential equation represents the field equations, that would be foundif the action were varied with respect to B.

The homogeneous solution is determined by

∂µ

√−Guµνρ

1 = 0,

which has the solution

uµνρ1 =

1√−G

εµνρσ∂σφ. (1.5.5)

In this solutions the explicit quantity 1√−G

εµνρσ is a tensor density, such that

u1µνρ = −√−Gεµνρσ∂

σφ. (1.5.6)

Now we find that φ is a residual freedom in F , which still has to be integrated over in the partitionfunction. This is the case since the delta function gave a restriction on the derivative of F , whichmeans that the whole class of functions F that are not of the form of (1.5.5) are now ruled out.

19

1.5 Dualities CHAPTER 1 NGT GEOMETRY

The inhomogeneous solution is given by

1√−G

∂µ

√−Guµνρ

2 = eJνρ

= ∇µuµνρ2 , (1.5.7)

solved by means of a Green function,

uµνρ2 = ∇[µ(x)

∫d4y√−G(y)G(x, y)Jνρ](y)

= ∇[µ 1Jνρ]. (1.5.8)

The Green function is for this equation defined by

∇σ(x)∇σ(x)G(x, y) = G(x)−

14G(y)−

14 δ4(x− y), (1.5.9)

which gives a complicated retarded Green function, thoroughly explained in Ref. [40].Hence the field strength is given by

Fµνρ =1√−G

εµνρσ∂σφ+∇[µ 1Jνρ]. (1.5.10)

With this solution the partition function becomes

Z[J ] =∫

[Dφ] exp−∫d4x

√−G

[112Fµνρ(φ, J)Fµνρ(φ, J)

]

=∫

[Dφ] exp

−∫d4x

√−G

[1√−G

εµνρσ∂σφ+∇[µ 1Jνρ]

]2

=∫

[Dφ] exp−∫d4x

√−G

[−∂σφ∂

σφ+(∇[µ 1

Jνρ]

)∇[µ

1Jνρ]

+1√−G

1J[νρ∇µ]

√−Gεµνρσ∂σφ−

1√−G

1J[νρ∇µ]

√−Gεµνρσ∂σφ

].

(1.5.11)

Hence we find that the massless free Kalb-Ramond action in curved space is dual to the scalarfield action

Sφ =∫d4x

√−G

[∂σφ∂

σφ+(∇[µ 1

Jνρ]

)∇[µ

1Jνρ]

]. (1.5.12)

With this action, we can conclude that the massless, free Kalb-Ramond field has only one degreeof freedom. In other words, the antisymmetric perturbation on the metric has for the chosenparameters, µ = θn = 0, just one degree of freedom.

Note that in the presence of sources, the action (1.5.12) contains nonlocal source terms.

1.5.2 Massive free antisymmetric tensor field

When we allow for a mass term to be nonzero but constant, we can choose µ 6= 0 and θn = 0,giving us the action of a massive uncoupled antisymmetric tensor field,

SmKR =∫d4xLmKR =

∫d4x

√−G

− 1

12FµνρF

µνρ + 14µ

2BµνBµν

. (1.5.13)

20

CHAPTER 1 NGT GEOMETRY 1.5 Dualities

Then we find the partition function to be,

Z[J ] =∫

[DF ][DB] exp−∫d4x

√−G

(− 1

12FµνρF

µνρ +12Fµνρ∂[µBνρ]

+ 14µ

2BµνBµν +

e

2BµνJ

µν)

. (1.5.14)

Rewriting this equation as

Z[J ] =∫

[DF ][DB] exp−∫d4x

√−G

(− 1

12FµνρF

µνρ − 12

1√−G

B[νρ∂µ]

√−GFµνρ

+ 14µ

2BµνBµν +

e

2BµνJ

µν)

=∫

[DF ][DB] exp−∫d4x

√−G

[− 1

12FµνρF

µνρ+

(12µBµν −

12µ

1√−G

∂σ√−GFσµν +

e

2µJµν

)2

− 14µ2

(1√−G

∂σ√−GFσµν − eJµν

)2]

(1.5.15)

we see that the B-field only appears in a rescaled, squared form. That is, it only contributes asGaussian term to the distribution function, which only adds a constant and hence is irrelevant.In this case the rescaling with a constant factor µ does not alter the irrelevance.

If we identify

Fµνρ =√−G√−G

εµνρσεσαβγFαβγ

=√−GεµνρσµA

σ (1.5.16)

we see that F can be written is an antisymmetric multiplication of a one-form. This is true becausethe new defined quantity Aσ transforms as a vector. Implicitly I have already chosen a convenientrescaling of that particular one-form with a factor of µ. Imposing this rewriting on the partitionfunction, we find,

Z[J ] =∫

[DA] exp−∫d4x

√−G

[−1

2µ2AνA

ν

−GαµGβν 14µ2

(µ√−G

∂σ(−G)εσµνρAρ − eJµν

)(µ√−G

∂λ(−G)ελαβξAξ − eJαβ

)]

=∫

[DA] exp−∫d4x

√−G

[−1

2µ2AνA

ν

−GαµGβν 14µ2

(µ√−G

∇σ√−GεσµνρAρ − eJµν

)(µ√−G

∇λ√−GελαβξAξ − eJαβ

)]

=∫

[DA] exp−∫d4x

√−G

[−1

2µ2AνA

ν − 14FρσF

ρσ +e2

4µ2JµνJ

µν

], (1.5.17)

where

Fρσ = ∇ρAσ −∇σAρ (1.5.18)

In this partition function we find that the free theory, containing only the kinetical Kalb-Ramondterm and a mass term, is dual to a (massive) Proca theory, which has 3 degrees of freedom. The

21

1.5 Dualities CHAPTER 1 NGT GEOMETRY

number of degrees of freedom can be easily counted when one remembers that Maxwell’s theory,represented by the massless Proca lagrangian, has 2 degrees of freedom and one then takes intoaccount that the mass term breaks the gauge invariance, adding one degree of freedom. Hence thepresence of a mass term induces a change in spin from spin 0 (1 degree of freedom) to spin 1 (3degrees of freedom).

Again, the reader may verify that the completion of the square in B and the evaluation of theB-integral led to the same solution as the one that would have been obtained, if the action werevaried with respect to B, and if the field equations were solved and imposed on the effective actionrespectively.

1.5.3 Coupled massive antisymmetric tensor field

When we set all parameters nonzero, the partition function has the form

Z[J ] =∫

[DF ][DB] exp−∫d4x

√−G

[− 1

12FµνρF

µνρ +(

14µ2 + θ1R

)BµνB

µν

+θ2RµανβBµνBαβ + θ3B

αρB

ρβRαβ +12Fµνρ∂[µBνρ] +

e

2BµνJ

µν

]. (1.5.19)

It seems difficult to evaluate the B-integral, because B is not any longer simply squared but alsointertwined with the curvature. However, the action may still be varied with respect to B, leadingto a condition on F which may be solved. The field equation reads

2(

14µ2 + θ1R

)Bµν + 2θ2Rµ

αν

βBαβ + 2θ3Bα

µRαν +

12√−G

∂σ

√−GF σµν +

e

2Jµν = 0.

(1.5.20)

Equation (1.5.20) can be solved by

Fσµν(x) =εµνρσ µ√−G

Aσ +4√−G

∂[σ

∫d4y

√−GG(x− y)

[(14µ2 + θ1R

)Bµν]

+θ2Rµα

ν]βB

αβ + θ3Bα

µRαν +

e

4Jµν

]. (1.5.21)

When this is inserted in the partition function, equation (1.5.19), the dual theory is a fact. Givensolution (1.5.21), the conclusion can be drawn, that in the full theory the dual field is againa massive Proca field. Hence, the number of degrees of freedom is three. But as we can see,the solution contains integrated curvature terms. This means that the dual theory has becomenonlocal in the curvature. This is in no way an improvement with respect to the original theory.

1.5.4 Coupled massless Kalb-Ramond

If the constant mass term is omitted, but the curvature couplings are maintained, the dual theorybecomes

Fσµν(x) =εµνρσ µ√−G

Aσ +4√−G

∂[σ

∫d4y

√−GG(x− y)

[(θ1R

)Bµν]

+θ2Rµα

νβB

αβ + θ3Bα

µRαν] +

e

4Jµν

]. (1.5.22)

since effectively nothing changes with respect to the massive coupled version of the theory, only aconstant shift in the effective mass, i.e. the term multiplying B2. Hence, the theory still has threedegrees of freedom.

22

Chapter 2

Instabilities of NGT

In this chapter I will investigate the full Lagrangian derived in chapter 1, equation (1.4.14). Iwill show the field equations for NGT in a Friedmann-Lemaıtre-Robertson-Walker background(FLRW) as well as in a Schwarzschild background. From those field equations, we will find thatthe theory is not stable for any choice of parameters. The contents of this chapter are based onReference [22].

2.1 Field equations

The action that was derived in chapter 1 reads

Seff =∫d4x

√−G

R− 2Λ− 1

12FµνρFµνρ +

(14µ

2 + θ1R)BµνB

µν

+θ2RµανβBµνBαβ + θ3B

αρB

ρβRαβ

. (2.1.1)

By varying this action we find for Bµν ,√−G 1

2∂ρ

(√−GF ρµν

)+(

12µ

2 + 2θ1R)Bµν − θ2B

αβRµ

αν

β − θ3

(BναR

µα +BαµR

να

)= 0,

(2.1.2)

whereas the Einstein equations remain

Rµν − 12RGµν − ΛGµν = 0. (2.1.3)

Thus we find that the Einstein equation is independent of B, at least up to linear order. Themetric is fully defined by the choice of background, and is not influenced by NGT at first order.

2.2 FLRW Background

For distinguished independent components of Bµν the field equation can be solved, during differentstages in the history of the Universe. What I will do now, is pick one component, and discuss thesolution in a FLRW background. In the next section I will discuss the another component in aSchwarzschild background.

The FLRW metric in a space with zero spatial curvature is defined as

Gµν = a(η)2ηµν , (2.2.1)

where ηµν is the usual Minkowsky metric and η denotes conformal time, dt = a(η)dη. With thismetric we have

√−G = a(η)4, (2.2.2)

23

2.2 FLRW Background CHAPTER 2 INSTABILITIES OF NGT

and all curvature terms(R)

can be expressed in terms of a(η), a(η)′ and H = a(η)′/a(η).During radiation era we have,

a(η) = HIη with ( 1HI

< η < ηe), (2.2.3)

where HI = 1013 GeV, whose meaning is explained in chapter A.When we take the divergence of equation (2.1.2), we have

Xηij∂iB0j = 0, (2.2.4)

η00∂0 (XBj0) + Yηik∂kBji = 0, (2.2.5)

using the notation of Ref. [22], with parameters

X =a(η)−2[(12θ1 + 2θ3)H2 + (12θ1 − 2θ2 + 4θ3)H′ − 1

2µ2a(η)2

], (2.2.6)

Y =a(η)−2[(12θ1 − 2θ2 + 4θ3)H2 + (12θ1 + 2θ3)H′ − 1

2µ2a(η)2

]. (2.2.7)

The equations of motion in the FLRW-background become

a(η)−2ηρσ∂ρFσ0i + 2XBj0 = 0, (2.2.8)

a(η)−2 [ηρσ∂ρFσij −HF0ij ] + 2YBij = 0. (2.2.9)

If we write ~E = B0i, we find

a(η)−2[∂2 ~E + ∂0

Y−1∂0

(X ~E

)]− 2XE = 0. (2.2.10)

Note that this equation becomes singular when Y = 0.We can immediately drop the vector notation, as we see that the field equation does not depend

on the direction of ~E. That is, we can write ~E = E ~n, where ~n is some norm vector.The field equation can be written in a more convenient shape, if we rescale E,

E =√YX

E, (2.2.11)[∂20 +

YXηij∂i∂j +M2

eff

]E =0, (2.2.12)

where I defined the effective mass as

M2eff = −2Ya(η)2 +

Y ′′

2Y− 3

4

(Y ′

Y

)2

. (2.2.13)

If we transform to momentum space, by means of a Fourier transform1, this equation becomes[∂20 +

YXk2 +M2

eff

]E =

[∂20 +

µ2H2I η

4 + 4(θ2 − θ3)µ2H2

I η4 − 4(θ2 − θ3)

k2 +M2eff

]E =0. (2.2.14)

The rescaling of E in equation (2.2.11), is only regular if both X 6= 0 6= Y. Together withthe above mentioned singularity of the unrescaled field equation, during the radiation era thiscondition becomes

Y =4(θ3 − θ2) + µ2H2I η

4 6= 0, (2.2.15)

X =4(θ3 − θ2)− µ2H2I η

4 6= 0. (2.2.16)

1For an extensive treatment of the process of canonical quantisation, whilst performing a transformation tomomentum space, the reader is referred to chapter 3.

24

CHAPTER 2 INSTABILITIES OF NGT 2.3 Schwarzschild background

Assuming that these conditions hold, we find that in field equation (2.2.14), a regions exists where

YXk2 +M2

eff < 0. (2.2.17)

The solution to this equations consists of one exponentially growing mode and one exponentiallydecaying mode. Both solutions have unphysical asymptotes, filling the Universe with an infiniteamount of energy. This is for instance to be expected at the beginning of radiation era, when

YX

=µ2

H2I

+ 4(θ2 − θ3)µ2

H2I− 4(θ2 − θ3)

'−1, (2.2.18)

for large k, if µ |θ2 − θ3|HI . As the action is geometric of origin, the mass is expected to be ofthe same order of magnitude as the cosmological constant, for which we have today

µ2 ∼ Λ ≤ 10−84 GeV. (2.2.19)

Unless |θ2 − θ3| is very small, conditions (2.2.15) and (2.2.16) are satisfied, and the field willcontradict present observations.

However, for special choices of parameters, the evolution of the field may still be withouttrouble. For instance, when θ2 = θ3, we have Y/X = 1. Then the field equation has stablesolutions, [

∂20 + k2 +M2

eff

]E = 0. (2.2.20)

A next option that might influence the scenario, is the case when X goes to zero, since then therescaling becomes singular and one could expect the field thereby to be driven to zero always.However, from the unrescaled field equation,

a(η)−2[∂2 ~E + ∂0

Y−1∂0

(X ~E

)]− 2XE = 0, (2.2.21)

we find that nothing special happens at X → 0.Next consider Y → 0. Then the field equation becomes singular. However, at that time the

rescaling (2.2.11) still holds, and the field dynamically becomes zero-valued. Thus, this localsingularity does not change the (in)stability of the solutions.

Another configuration is the choice θ1 = −θ2/6. In that case X = Y = −µ2/2, and the fieldequations turn into,

∂ρFρ0i + µ2Bj0 =0, (2.2.22)

∂ρFρij −a(η)′

a(η)3F0ij + µ2Bij =0, (2.2.23)

which is perfectly stable. In fact, as we will see in chapter 4, these field equations are exactly thesame as the ones that are retrieved in the choice θ1 = θ2 = θ3 = 0.

As a result we must conclude that the action, (2.1.1), only becomes stable on a FLRWbackground for the choice θ2 = θ3 or the choice θ2 = θ3 = 6θ1.

2.3 Schwarzschild background

Next to FLRW-metric which is a good approximation for the evolution of the Universe on largescales, the Schwarzschild metric is a good approximation for all (by approximation) sphericallysymmetric objects, i.e. stars etcetera.

25

2.3 Schwarzschild background CHAPTER 2 INSTABILITIES OF NGT

The Schwarzschild metric for a spherically symmetric object is, in spherical coordinates andfor r > 2Ms, given by

ds2 =(

1− 2Ms

r

)dt2 −

(1− 2Ms

r

)−1

dr2 − r2dΩ2. (2.3.1)

Here dΩ2 = dϑ2 + sinϑ dφ2. For this metric we have the following quantities:√−G =r2 sinϑ, (2.3.2)

Rµν =0, (2.3.3)

R =0. (2.3.4)

Therefore the equation of motion becomes

12∂ρ

(√−GF ρµν

)+√−G

(µ2

2Bµν − 2θ2BαβRµ

αν

β

)= 0. (2.3.5)

Note that the coupling of Bµν to gravity only depends on θ2, which is the only one of the threeinitial parameters θn which is equal to one by definition, if the action is chosen as the linearisationof a purely geometric quantity, the Einstein-Hilbert action.

The divergence of equation (2.3.5) leads to the consistency condition

µ2

2∂µB

µν − 2θ2∂µBαβRµ

αν

β + µ2B1ν − 4θ2rBαβR1

αν

β (2.3.6)

+µ2

2B2ν cotϑ− 2θ2BαβR2

αν

β cotϑ = 0. (2.3.7)

The next thing to do is write these equations in terms of the explicit components of the Riemanntensor. The most convenient step appears to be to look at all four equations separately, 0 ≤ ν ≥ 3,

T B10 +RB20 cotϑ+ S∂1B10 +R∂iB

i0 =0, (2.3.8)

RB21 cotϑ− S∂0B10 +R∂iB

i1 =0, (2.3.9)

QB12 −R∂0B20 −R∂1B

21 − S∂3B23 =0, (2.3.10)

QB13 + SB23 cotϑ−R∂0B30 −R∂1B

31 − S∂2B32 =0. (2.3.11)

Here I again have used the notation from Ref. [22], where

Q ≡ µ2

r+

2θ2Ms

r4, (2.3.12)

R ≡ µ2

2− 2θ2Ms

r3, (2.3.13)

S ≡ µ2

2+

4θ2Ms

r3, (2.3.14)

T ≡ µ2

r− 4θ2Ms

r4. (2.3.15)

With the explicit field equations for the different components of Bµν , we can see that a thefollowing rescaling is useful,

Bµν → BµνRS

(2.3.16)

26

CHAPTER 2 INSTABILITIES OF NGT 2.3 Schwarzschild background

Then the consistency conditions become

∂µBµν +

QRB1ν + B2ν cotϑ = 0. (2.3.17)

The field equation, equation (2.3.5), in terms of the Riemann components becomes

∂ρFρµν +

2rF 1µν + F 2µν cotϑ+ 4δµ

[0δν1] (S −R)B01 + 4δµ

[2δν3] (S −R)B23 + 2RBµν = 0.

(2.3.18)

This time I will focus on the component B01. The rescaled field equation for this componentbecomes

RS(G22 cotϑ∂2 +G22∂2∂2 +G33∂3∂3 + 2S

)B01 (2.3.19)

+G00(∂0B

12)

cotϑ+G00∂0

(∂2B

12 + ∂3B13)

(2.3.20)

+ cotϑG11∂1B20 +G11∂1

(∂2B

20 + ∂3B20)

= 0. (2.3.21)

Combining the consistency condition, (2.3.17) and the field equation (2.3.21), we arrive at[G00∂0∂0 +G11∂1∂1 +

RS(G22∂2∂2 +G22 cotϑ∂2 +G33∂3∂3

)(2.3.22)

+2R+G11∂1

(QR

)+G11Q

R∂1

]B01 = 0. (2.3.23)

Now we have to perform a last rescaling,

B01 = λB01 = −√r√

4θ2Ms − µ2r3

8θ2Ms + µ2r3B01, (2.3.24)

where

λ =√r

4θ2Ms − µ2r3. (2.3.25)

The final field equation now reads,[∂20 −

(r − 2Ms)2

r2∂21 −

r − 2Ms

r

RSL2 +M2

eff

]B01 = 0, (2.3.26)

with the effective mass Meff and the spatial operator L2,

M2eff ≡

(1− 2Ms

r

)(2R−G11∂2

1λ), (2.3.27)

L2 ≡ 1r2

(∂22 + cotϑ∂2 +

1sin2 ϑ

∂23

). (2.3.28)

The solution for B01 can be written in terms of spherical harmonics,

B01 =∑l,m

blm(t, r)Ylm(ϑ, φ), (2.3.29)

27

2.4 Geometry and instability CHAPTER 2 INSTABILITIES OF NGT

such that the operator L2 gives

L2B01 = − l(l + 1)r2

. (2.3.30)

In this case l is the multipole moment. Then the equation of motion becomes[∂20 −

(r − 2Ms)2

r2∂2

r +r − 2Ms

r

RSl(l + 1)r2

+M2eff

]blm(t, r) = 0. (2.3.31)

Effectively this differential equation is of the form (∂20+C)b = 0, where the number C is determined

by the value of time t and the multipole moment l. Depending on the sign of R/S, the solutionof the field is in rough approximation either a periodical function of an exponentially growingor decaying function of time. Hence, if R/S is negative, the field grows without bounds in itslarge-multipole components.

The sign of R/S,

RS

=µ2r3 − 4θ2Ms

µ2r3 + 8θ2Ms, (2.3.32)

is negative when

r <

(4θ2Ms

µ2

) 13

for θ2 > 0, (2.3.33)

and

r <

(−8θ2Ms

µ2

) 13

for θ2 < 0. (2.3.34)

Even though we have restricted r > Ms, it is impossible to prevent the exponential growth fromoccurring anywhere in the Universe. One can always imagine an object with small enough Ms

such that the conditions (2.3.33) and (2.3.34) hold; the mass of an object causing NGT to explodeis determined by

Ms = 2GNm < r <

(λ |θ2|Ms

µ2

) 13

, (2.3.35)

such that

m2 <λ |θ2|m4

Pl

µ2, (2.3.36)

where λ = 4 if θ2 > 0 and λ = 8 for θ2 < 0. Thus, even if the mass of NGT, µ, is of the order ofthe Planck mass, which is far above the expected value, the maximum mass of the object sourcingthe Schwarzschild metric is also of the order of the Planck mass, which for a classical object (e.g.a bunch of particles) is not extreme. Of course in that case the object must be a black hole inorder to let the Schwarzschild metric still be the correct solution at r ∼Ms, but there is no reasonto expect that NGT is stable everywhere in the Universe.

This time, as a result we must conclude that the action, (2.1.1), only becomes stable on aSchwarzschild background for the choice θ2 = 0.

2.4 Geometry and instability

As we find from the Schwarzschild background, we have to build the action such that θ2 = 0.On the other hand we have from the FLRW background the condition that θ2 = θ3. The only

28

CHAPTER 2 INSTABILITIES OF NGT 2.4 Geometry and instability

parameter that survives the quest for a stable action is θ1. But, if one straightforwardly calculatesthe Einstein-Hilbert action with the antisymmetric perturbation, as defined in chapter 1, theaction has a fixed value of θ2 = 1. That means that the Einstein-Hilbert action can not lead to astable linearised nonsymmetric theory of gravity.

One could try to construct an action out of several geometric quantities, and then try to finda configuration from which automatically θ2 = 0. I will not show the proof here, however it issufficient to say that an action which leads to θ2 = 0 also lacks a kinetic term (∝ Fµνρ)for theB-field [22]. Without this term, there is no dynamical theory for the B-field.

Another option is to try to find higher order terms in B to solve the instabilities. In this ThesisI restrict myself to the linear order theory, however in Ref. [22] it is shown that also higher orderterms do not solve the problem.

In its strict form, as it was proposed in the previous chapter, NGT is not a stable theory. Ifone however may find an adjustment to NGT that is stable, its action must obviously be at leastsimilar to the stable action

Seff =∫d4x

√−G

R− 2Λ− 1

12FµνρFµνρ +

(14µ

2 + θ1R)BµνB

µν. (2.4.1)

Therefore, we do not have to quit here, and I will continue the calculation of quantum fluctuations,starting from a stable action, inspired on NGT.

29

2.4 Geometry and instability CHAPTER 2 INSTABILITIES OF NGT

30

Chapter 3

Canonical quantisation

In the previous chapter the action is derived in a classical field theory, without really botheringabout quantising the theory, although I did investigate the dual theories in the path-integralformalism. In this chapter I show how the theory can be quantised in the framework of canonicalquantisation.

3.1 Scalar field quantisation

First I will give a brief sketch of the canonical quantisation of a scalar field, in order to illustratethe process of quantisation. For a free massive scalar field, the action is

S =∫d4x

√−G

12(∂µφ)(∂µφ)−m2φ2

. (3.1.1)

The field equation for this field is given by

δS

δφ=0

=δ

δφ

∫d4x a(η)4

12a(η)−2ηµα(∂µφ)(∂αφ)−m2φ2

=δ

δφ

∫d4x

−1

2ηµαφ∂µ

(a(η)2∂αφ

)− a(η)4m2φ2

=− a(η)2∂2

0φ− 2a(η)′a(η)∂0φ+ a(η)2δij∂i∂jφ− a(η)4m2φ = 0. (3.1.2)

This is equivalent to

∂20φ+

2a(η)′

a(η)∂0φ− δij∂i∂jφ+m2a(η)2φ = 0. (3.1.3)

The canonical momentum of this field is defined by

p =δS

δ∂0φ

=a(η)2∂0φ. (3.1.4)

Then canonical quantisation is achieved by imposing the equal time commutation relation[p(η, ~x), φ(η, ~x′)

]= −iδ3(x− x′). (3.1.5)

31

3.1 Scalar field quantisation CHAPTER 3 CANONICAL QUANTISATION

Commutators of different fields vanish at all times.Now one can transform to momentum space and write the operators in terms of creation and

annihilation operators,

φ(η, ~x) =∫

d3k

(2π)3[ei~k·~xφ(η,~k)a~k

+ e−i~k·~xφ∗(η,~k)a†~k

], (3.1.6)

p(η, ~x) =∫

d3k

(2π)3[ei~k·~xp(η,~k)a~k

− e−i~k·~xp∗(η,~k)a†~k

]. (3.1.7)

(3.1.8)

This decomposition is strictly speaking only exact for the noninteracting scalar theory. Then thecommutation relation (3.1.5) transforms to[

a~k, a†~k′

]= (2π)3δ3(~k − ~k′), (3.1.9)

together with the functional relation

p(η, k)φ∗(η, k)− p∗(η, k)φ(η, k) = i, (3.1.10)

which is, using the definition of the canonical momentum (3.1.4), equivalent to

φ∗∂0φ− φ∂0φ∗ = ia(η)−2. (3.1.11)

Here we recogniseW [φ(η), φ∗(η)] ≡ φ(η)∂0φ

∗(η)− φ∗(η)∂0φ(η), (3.1.12)

which is called the Wronskian of two functions. If the Wronskian of two functions is nonzero, andboth functions solve the same differential equation, then the functions are linearly independentsolutions.

If we now write φ = ψ/a(η), we find that this is equivalent to

ψ∗∂0ψ − ψ∂0ψ∗ = i. (3.1.13)

In order to have consistency, this must be in agreement with the field equation. If we apply thesame rescaling of φ to the field equation in momentum space, such that ∂i → iki, we have[

∂20 + k2 +m2a(η)2 − a(η)′′

a(η)

]ψ = 0, (3.1.14)

since ψ and ψ∗ are linearly independent by virtue of equation (3.1.13). If we now take the timederivative of equation (3.1.13), we find

∂0 [ψ∗∂0ψ − ψ∂0ψ∗] =∂0ψ

∗∂0ψ − ∂0ψ∂0ψ∗ + ψ∗∂2

0ψ − ψ∂20ψ

∗

=F [k,m, η] |ψ|2 − F [k,m, η] |ψ|2

=0. (3.1.15)

In the last line F [k,m, η] = k2 +m2a(η)2 − a(η)′′/a(η). Equation (3.1.15) confirms that ψ∗∂0ψ−ψ∂0ψ

∗ is indeed equal to a constant. This constant then must be equal to i in order to have thecorrect normalisation for canonical quantisation, given in equation (3.1.5).

To conclude we can say that any quantum state of the scalar field is given by a linear com-bination of the solutions to the field equations. If we write the solutions as ϕ1 and ϕ2, then thefield is in momentum space defined by

φ(η, k) = αϕ1 + βϕ2. (3.1.16)

The factors α and β multiplying the distinct solutions, define the state in which the field is, andthe factors always have to be normalised in order to have

φ∗∂0φ− φ∂0φ∗ = ia(η)−2. (3.1.17)

32

CHAPTER 3 CANONICAL QUANTISATION 3.2 Antisymmetric tensor field

3.2 Antisymmetric tensor field

3.2.1 Action

In chapter 2, the resulting action is

Seff =∫d4x

√−G

R− 2Λ− 1

12FµνρFµνρ +

(14µ

2 + θ1R)BµνB

µν. (3.2.1)

whereFµνρ = ∂µBνρ + ∂νBρµ + ∂ρBµν (3.2.2)

is the field strength associated with the antisymmetric Kalb-Ramond field Bµν [25]. Note thatFµνρ is symmetric under even permutations of its indices, and antisymmetric under uneven per-mutations. This action can be split into two parts:

S = SEH + SmKR, (3.2.3)

where

SEH =∫d4xLEH =

∫d4x

√−G

R− 2Λ

, (3.2.4)

SmKR =∫d4xLmKR =

∫d4x

√−G

− 1

12FµνρFµνρ + ( 1

4µ2 + θ1R)BµνB

µν. (3.2.5)

SEH is the classical Einstein-Hilbert action including the cosmological constant and SmKR de-notesthe action containing the Kalb-Ramond kinetical term and a mass term, without curvaturecoupling. L denotes the Lagrangian density associated with a given action S.

The action contains two free parameters which are unknown. The first is the mass term µ,the second is the coupling of Bµν to the Ricci scalar. For simplicity I will now continue with theaction where θ1 = 0. As the field B is not coupled to gravity in that case, I will refer to it as afree antisymmetric field.

3.2.2 Field equations

For the free antisymmetric field, we have the following action,

S =∫d4x

√−G

R− 2Λ− 1

12FµνρF

µνρ +14µ2BµνB

µν

. (3.2.6)

Varying this action with respect to Bµν and demanding its variation to be equal to zero,

δ

δBµνS = 0, (3.2.7)

gives the field equations:

a(η)2∂ρFρµν −2a′

aF0µν + µ2a(η)2Bµν = 0, (3.2.8)

Multiplying equation (3.2.8) by a(η)−2 and taking its divergence, we also find

∂µBµν = 0. (3.2.9)

This is a consistency constraint. Taking the divergence of the field equations is classically nothingspecial, however in a quantised scenario this is only as simple as expressed above as long as thefield equations are linear in the Bµν-field.

33

3.2 Antisymmetric tensor field CHAPTER 3 CANONICAL QUANTISATION

For a derivation of the field equations and the consistency term, the reader is referred toappendix B, section B.1.1.

When Bµν is rewritten as

B0i = −Bi0 = Ei (3.2.10)

Bij = −Bji = −εijkBk, (3.2.11)

field equation (3.2.8) results in:

EL(x) = 0, (3.2.12)(∂2 + µ2a(η)2

)~ET (x) = 0, (3.2.13)(

∂2 − 2a′(η)a(η)

∂0 + µ2a(η)2)BL(x) = 0, (3.2.14)

∂0~ET − ~O× ~BT (x) = 0. (3.2.15)

An explicit derivation of these equations can be found in appendix B, section B.1.2.In four spacetime dimensions an antisymmetric tensor field initially has six degrees of freedom.

In our notation these are contained in the three dimensional vectors ~E and ~B. From equations(3.2.12–3.2.15) one can conclude that the theory possesses only three physical degrees of freedom,one longitudinal (L) and two transversal (T ) modes, since ~EL = 0 and ~BT is defined by ~ET via(3.2.15). This confirms the statement I made in Chapter 1, section 1.5 where I calculated the dualfield of this theory.

Rescaling

When the longitudinal mode, BL(x), is written as

BL = a(η)BL(x), (3.2.16)

equation (3.2.15) becomes (∂2 − 2a′(η)

a(η)∂0 + µ2a(η)2

)a(η)BL(η, k) =

a(η)∂2BL + 2a′(η)∂0BL + BLa′′(η)− 2a′(η)∂0B

L−

BL 2a′(η)2

a(η)+(µ2a(η)2

)a(η)BL =

a(η)(∂2 + µ2a(η)2 +

a′′(η)a(η)− 2a′(η)2

a(η)2

)BL = 0. (3.2.17)

Massless limit

When the mass is set equal to zero, equation (3.2.8) becomes

∂ρa(η)−2Fρµν = 0, (3.2.18)

which is solved in Chapter 1 bya(η)−2Fρµν = ερµνσ∂

σφ, (3.2.19)

indeed leaving only one degree of freedom, as stated in that same chapter. This field is related toBµν by

6∂[µBνρ] = a(η)2εµνρσ∂σφ,

(3.2.20)

34

CHAPTER 3 CANONICAL QUANTISATION 3.3 Antisymmetric tensor field quantisation

From this equation one can see that all B0i = Ei are constant in time, since ε00µν = 0. By virtueof equation (3.2.15), the transversal modes of Bij = −εijkBk are cancelled as well, leaving BL

as the only surviving mode in the massless limit. The field equations then define BL, leaving allother terms unphysical. One remark has to be made about condition (3.2.9). This condition is infact multiplied by a factor of µ2, making it degenerate in the massless limit, hence invalid. Yet,the massless action reveals a gauge invariance,

Bµν → Bµν + (∂µVν − ∂νVµ),

Fµνρ → Fµνρ, (3.2.21)

for an arbitrary vector Vµ. This gauge invariance has four degrees of freedom, allowing us toimpose four constraints on the theory. This can be equation (3.2.9), such that the massive theoryand the massless theory should coincide in the massless limit.

3.3 Antisymmetric tensor field quantisation

The field equations restrict the behaviour of the field, both classically and in quantum theory.In the following I present a common method for quantising a field, giving a measure for thequantum fluctuations of the field in any state, which together with the field equation defines thefull evolution of the field as a quantum operator.

3.3.1 Straightforward quantisation

For the given fields Bµν , canonical momenta Πµν can be defined by the variational derivative ofthe action with respect to ∂0Bµν [13, 50], as

Πµν ≡ δS

δ∂0Bµν=

δ

δ∂0Bµν

∫d4x

√−G

R− 2Λ− 1

12FαβγF

αβγ +14µ2BαβB

αβ

=δ

δ∂0Bµν

∫d4x

√−G

− 1

12FαβγF

αβγ

= −1

4δ

δ∂0Bµν

∫d4x

√−G

(∂αBβγ + ∂βBγα + ∂γBαβ)(∂αBβγ)

= −1

4δ

δ∂0Bµν

∫d4x

√−G

(∂ρBστ + ∂σBτρ + ∂τBρσ)(∂αBβγ)a(η)−6ηραησβητγ

= −a(η)

−2

2∂0Bρσ + ∂ρBσ0 + ∂σB0ρ ηρµησν . (3.3.1)

Quantisation of the field would now be easily achieved if we could turn Bµν and Πµν intooperators and impose the equal-time commutation relations, without yet imposing the constraints,[

Παβ(η, ~x), Bµν(η, ~y)]

= − i2(δα

µδβν − δβ

µδαν )δ3(~x− ~y), (3.3.2)[

Παβ(η, ~x), Πµν(η, ~y)]

= 0, (3.3.3)[Bαβ(η, ~x), Bµν(η, ~y)

]= 0. (3.3.4)

Now a problem arises. Because of the definition of Πµν and the antisymmetry of Bµν we find theidentity Π0i = 0. This leads to the conclusion that our action cannot be canonically quantised [25].Π0i = 0, means that the action does not depend on ∂0

~E, such that this field is not dynamical.

35

3.3 Antisymmetric tensor field quantisation CHAPTER 3 CANONICAL QUANTISATION

More inconsistency

If we ignore the zero valued canonical momenta, we can just continue quantisation procedure,by imposing only commutation relations on the nonzero momenta and their conjugate spatialoperators. When transforming to momentum space, the next thing to do [6, 50] is to express thefields in terms of creation and annihilation operators (ak and a†k) at the same time,

Bµν(x) =∑P

∫d3k

(2π)3ei~k·~xQP

(η,~k)εPµν(~k)aP

~k+ e−i~k·~xQ∗P

(η,~k)ε∗Pµν (~k)aP†

~k

,(3.3.5)

where I have also split the expression in the distinctly polarised degrees of freedom, which maybecome useful as we see in the field equations. The index P runs all six degrees of freedom. Thatis, two transversal polarisations and one longitudinal for both vectors ~E and ~B. The polarisationtensor is made up of polarisation vectors, and it is not an actual tensor, merely a useful tool,

εPBij (~k) = − εijsε

PBs (~k), (3.3.6)

εPE0i (~k) = −εPE

i0 (~k) = εPE

k (~k), (3.3.7)

where PE and PB denote only the electric or magnetic degrees of freedom (polarisations) re-spectively. In order to maintain the correct decomposition for equations (3.3.5) and (3.3.29), wenecessarily define,

εPEij = εPB

0i = εPBi0 = 0. (3.3.8)

This means that the ij-components of Bµν do not involve any of the electrical degrees of freedomof course, and the 0i-components do not contain any magnetic degrees of freedom. The threespatial polarisation vectors ε are constructed such that

kiεTi (~k) = 0, (3.3.9)

where T is either one of the two transversal polarisations, and

kiεLi (~k) = |k| , (3.3.10)

where L denotes the longitudinal polarisation. These vectors satisfy

3∑P=1

εPi ε

Pj = δij , (3.3.11)

εPi ε

P ′

i = δP,P ′. (3.3.12)

Note that for these actual polarisation vectors there is no distinction between ~E and ~B. Then thepolarisation tensor satisfies,

δαβεPµα(~k)εP′

ρβ(~k) = δP,P ′δµρ, (3.3.13)

6∑P=1

εPij(~k)ε∗Plm (~k) = εijkεlmk

= (δilδjm − δimδjl) , (3.3.14)

6∑P=1

εPi0(~k)ε∗Pj0 (~k) = δij , (3.3.15)

6∑P=1

εPi0(~k)ε∗Pjk (~k) = 0. (3.3.16)

36

CHAPTER 3 CANONICAL QUANTISATION 3.3 Antisymmetric tensor field quantisation

Using the definition of the polarisation tensor, we can make the same decomposition of themomentum space mode functions as I did for Bµν(x) in equations (3.2.10) and (3.2.11). Thisbecomes,

QTE (η,~k)εTE0i (η,~k) = −QTE (η,~k)εTE

i0 = ETi (η,~k), (3.3.17)

QLE (η,~k)εLE0i (η,~k) = −QLE (η,~k)εLE

i0 = ELi (η,~k), (3.3.18)

QTB (η,~k)εTBij (η,~k) = −εijkB

Ti (η,~k), (3.3.19)

QLB (η,~k)εLBij (η,~k) = −εijkB

Li (η,~k), (3.3.20)

(3.3.21)

Let us go back to the field equations and take a closer look at, for example, equation (3.2.14). Wefind that (

∂2 − 2a′(η)a(η)

∂0 + µ2a(η)2)BL(x) =

(∂2 − 2a′(η)

a(η)∂0 + µ2a(η)2

)∫d3k

(2π)3eikx

BL(η, k)a L

k +BL∗(η,−k)a†L−k

=

∫d3k

(2π)3eikx

(∂20 + k2 − 2a′(η)

a(η)∂0 + µ2a(η)2

)BL(η, k)a L

k +

(∂20 + k2 − 2a′(η)

a(η)∂0 + µ2a(η)2

)BL∗(η,−k)a†L

−k

= 0.(3.3.22)

If we assume that Bµν and B∗µν are linearly independent solutions, equation (3.3.22) becomes(∂20 + k2 − 2a′(η)

a(η)∂0 + µ2a(η)2

)BL(η, k) = 0. (3.3.23)

As we saw in the quantisation of the scalar field, the momentum space solution and its complexconjugate were linearly independent because of the demand of the commutation relations in mo-mentum space. As we will see later on, this is also the case for Bµν , such that the assumption

of linear independence is correct. We find that B(η,~k)

is only a function of η and k =√‖ ~k ‖.

Note that this is an explicit prove that in the case of linear field equations, the field equations leadonly to a differential equation for the mode functions which are not operators themselves. As thefield equations reduce to a differential equation, taking the divergence of this differential equationis also straightforwardly allowed, as was already mentioned in the context of equation (3.2.9).

The steps taken in equation (3.3.22) can of course be taken for all components of Bµν . Theresult is that the replacements ∂i → iki and Bµν(x) → Bµν(η, k) are made in the field equations,

EL(η, k) = 0, (3.3.24)(∂20 + k2 + µ2a(η)2

)~ET (η, k) = 0, (3.3.25)(

∂20 + k2 − 2a′(η)

a(η)∂0 + µ2a(η)2

)BL(η, k) = 0, (3.3.26)

∂0ET1 + ikBT2(η, k) = 0, (3.3.27)

∂0ET2 − ikBT1(η, k) = 0. (3.3.28)

The explicit evaluation of the indices in the last two equations indicates that ~ET and ~BT areorthogonal to each other within the transversal plane.

37

3.3 Antisymmetric tensor field quantisation CHAPTER 3 CANONICAL QUANTISATION

Now let us get back to the process of quantising the distinct polarisations. The canonicalmomentum operator can be written as

Πij(x) = −a(η)−2

2ηaiηbj

∑P

∫d3k

(2π)3ei~k·~x

(∂0Q

P εPab(~k) + ikaQP εPb0(~k) + ikbQ

P εP0a(~k))aP

~k

− hermitian conjugate, (3.3.29)

because now we have ∂j → ikj . If I insert equations (3.3.5) and (3.3.29) in the commutationrelations (3.3.2–3.3.4), we find that these relations imply[

a†P~k, a†P

′

~k′

]= 0, (3.3.30)[

a P~k, a P ′

~k′

]= 0, (3.3.31)[

a P~k, a†P

′

~k′

]= (2π)3δP,P ′

δ3(~k − ~k′), (3.3.32)

and∑P

[−a(η)−2ηmaηnb

2QP (η, k)εPij(~k)

×

(∂0Q∗P (η, k)ε∗P

ab (~k)− ikaQ∗P (η, k)ε∗P

b0 (~k)− ikbQ∗P (η, k)ε∗P

0a (η, k)

+a(η)−2ηmaηnb

2Q∗P (η, k)ε∗P

ij (~k)

×

(∂0QP (η, k)εPab(~k) + ikaQ

P (η, k)εPb0(~k) + ikbQP (η, k)εP0a(~k)

]=i

2(δmi δ

nj − δm

j δni

).

(3.3.33)

Together with equations (3.3.13–3.3.16), this leads to the condition

QP ′(η,~k)∂0Q

∗P(η,~k)−Q∗P ′

(η,~k)∂0Q

P(η,~k)

=ia(η)2, for P = P ′, (3.3.34)

which has become independent of the polarisation. Note that in the derivation above, Q(η,~k)

isjust a function in momentum space, not an operator.

However, when we look at the field equations defining the transversal components,(∂20 + k2 + µ2a(η)2

)~ET (η, k) = 0, (3.3.35)

∂0ET1 + ikBT2(η, k) = 0, (3.3.36)

∂0ET2 − ikBT1(η, k) = 0, (3.3.37)

we find that,

BT1∗∂0BT1 −BT1∂0B

T1∗ =∂0ET2∗∂2

0ET2 − ∂0E

T2∂20E

T2∗

=−(k2 + µ2a(η)2

) ET2∂0E

T2∗ − ET2∗∂0ET2, (3.3.38)

such that

W [ET2 , ET2∗] =W [BT1 , BT1∗]k2 + µ2a(η)2

=ia(η)2

k2 + µ2a(η)2

6=constant. (3.3.39)

38

CHAPTER 3 CANONICAL QUANTISATION 3.3 Antisymmetric tensor field quantisation

This is in contradiction with the field equation for ET , which, like for the rescaled scalar field,implies W [ET2 , ET2∗] = constant.

3.3.2 Gauge invariance

On basis of a count a degrees of freedom, one finds that the path integral integrates over toomany paths. The massless theory has eventually one degree of freedom, where the path integralintegrates over all six components of Bµν independently. The paths which determine the resultingpartition function are determined by the field equations, which in some manner restrict the numberof paths, however the field equations may still contain degeneracy. In the massless case the actionhas a gauge freedom,

Bµν → Bµν + ∂µVν − ∂νVµ, (3.3.40)

Fµνρ → Fµνρ. (3.3.41)

Under this transformation the action is left unchanged, where the field does change. The path in-tegral is supposed to take account for all the physical paths the action contains. When we howevercalculate the path integral by means of a functional integral over all possible field configurations,we find that one particular path will have an infinite contribution from all field configurations thatthrough the gauge invariance resemble the same path. Of course this path should only contributeto the partition function once. This gauge freedom can be fixed by imposing constraints on thesystem. Effectively this means one is decreasing the number of degrees of freedom in the pathintegral. One choice is for example,

∂µBµν = 0. (3.3.42)

In the massive case, this constraint follows directly from the classical equations of motion. Thereis no freedom to choose a different constraint, as the massive theory has no gauge invariance.

This constraint on the field amplitudes turns out to be too strong for the quantised theory, asit leads to inconsistent canonical momenta. However we do need a constraint, as the path integralin principle runs over six degrees of freedom, where the classical theory shows that only three ofthem are physical.

Now what we can do is insert the constraint in a more relaxed way in the path integral, suchthat the quantum theory ’knows’ about the constraints and does not develop degeneracies. Thequestion is: what is the correct constraint?

Based on the classical equations, we expect the constraint to be similar to ∂µBµν = 0 andwe therefore expect it not to interfere with the longitudinal magnetic component. With thisknowledge, let us now make a comparison with electromagnetism. In electromagnetism the actionis given by

SEM =− 14

∫d4x

√−G (∇µAν −∇νAµ) (∇µAν −∇νAµ) . (3.3.43)

This action has a gauge invariance, Aµ → Aµ + ∂µλ, which can be fixed by subtracting a term(∇µA

µ)2. This term follows from an analysis of the photon propagator which shows that it is onlythis term that leads to inconsistencies [24]. For the antisymmetric tensor field the same analysiscan be done, leading to a fixing term [21],

Sfix =∫d4x

√−G−λ

4Gνα (∇µB

µν)∇ρBρα. (3.3.44)

It should be understood that λ is a Lagrange multiplier, which should be integrated over in thepath integral. As λ appears unconstrained in all field equations, it can be chosen to be any valueduring the evaluation of the theory. A useful choice for the parameter is λ = 2, similar to theFeynman ’gauge’ in the theory of vector fields. For simplicity I omit the analysis of the propagatorsof the antisymmetric tensor field, and will continue with the result.

39

3.3 Antisymmetric tensor field quantisation CHAPTER 3 CANONICAL QUANTISATION

This choice of gauge fixing has in fact been proposed by Kalb and Ramond [25], as theyinvestigated the massless antisymmetric tensor field. They proposed two methods. The firstmethod is to ignore all zero canonical momenta and only impose commutation relations for thenonzero momenta and fix the gauge freedom (present for the massless theory) by ∂µBµν = 0. Thismethod is in fact the one we followed above, except that we now have a mass term. It is thismethod that leads to inconsistency in the massive theory.

The second method they proposed is still to fix the gauge by setting ∂µBµν = 0 and to add aterm to the Lagrangian: the Fermi term, which is exactly the term we are now adding based on apropagator analysis. This term renders all canonical momenta nonzero a priori. However, becauseof the mass term, the theory has lost the gauge freedom which demanded fixing ∂µBµν = 0. Wewill let go of this additional constraint1 and only insert the Fermi term in the action.

3.3.3 Constrained quantisation

Now that the action is ’gauge’ fixed2, we have the new field equations,

∂ρ

√−GFµνρ +

√−Gµ2Bµν + 2

√−GGδ[νGµ]α∂α

Gβδ√−G

∂ρ

√−GBρβ = 0. (3.3.45)

We can again take the divergence of the field equation; we now find[∂µ∂µ +

µ2

2

]∂ρBρν = 0, (3.3.46)

which different from the classical consistency term which was given by ∂µBµν = 0.In the fixed action the field equations become[

∂20 −

2a(η)′

a(η)∂0 − ∂2

i + a(η)2µ2

]BP

i =2a(η)′

a(η)(∇× ~EP )i, (3.3.47)

[∂20 −

2a(η)′

a(η)∂0 − ∂2

i + a(η)2µ2

]EP

i =− 2a(η)′

a(η)(∇× ~BP )i, (3.3.48)

such that we have for the longitudinal modes,[∂20 −

2a(η)′

a(η)∂0 + k2 + a(η)2µ2

]BL =0, (3.3.49)

[∂20 −

2a(η)′

a(η)∂0 + k2 + a(η)2µ2

]EL =0. (3.3.50)

The consistency constraint reads,[∂20 − ∂2

i +µ2

2

](a(η)−2∂0

~E − a(η)−2~∇× ~B)

=0, (3.3.51)

such that[∂20 −

4a(η)2

a(η)∂0 + k2 +

µ2a(η2)2

+ 6(a(η)′

a(η)

)2

− 2a(η)′′

a(η)

](∂0E

T − ikεTT ′BT ′

)=0, (3.3.52)

[∂20 −

4a(η)2

a(η)∂0 + k2 +

µ2a(η2)2

+ 6(a(η)′

a(η)

)2

− 2a(η)′′

a(η)

] (ikEL

)=0. (3.3.53)

1As previously mentioned, the constraint ∂µBµν = 0 is a direct consequence of the classical massive action. Inthe gauge fixed action it does not follow from the field equations, and could be imposed manually. This is not tobe done since the theory has no gauge invariance which would allow us to do so.

2The theory has not got any gauge freedom, however it did have degenerate canonical momenta. The additionalterm does repair the action, and thus can be regarded as similar to gauge fixing.

40

CHAPTER 3 CANONICAL QUANTISATION 3.3 Antisymmetric tensor field quantisation

It must be understood that from now on repeated polarisation indices are summed over, as inthe Einstein convention, where previously polarisation sums were explicitly written. The twodimensional antisymmetric Levi-Civita symbol is defined as εT1T2 = −εT2T1 = 1 and εT1T1 =εT2T2 = 0.

Before considering the canonical momenta and the commutation relations, let us first inves-tigate the implications of the field equations and the consistency constraint. The longitudinalB-mode is left unaltered by the Fermi term. The longitudinal E-mode was set equal to zero bythe consistency constraint in the classical theory, however now it seems only constrained by twodifferential equations. If we subtract equations (3.3.50) and (3.3.53), we have,

[∂0 +K(a)]EL =0, (3.3.54)

where

K(a) =− 3a(η)′

a(η)+a(η)′′

a(η)′+µ2a(η)2

4a(η)′. (3.3.55)

When we hit this equation from the left with ∂0 and subtract it again from equation (3.3.50), wehave [(

K(a) +2a(η)′

a(η)

)∂0 − k2 − µ2a(η)2 +K(a)′

]EL = 0, (3.3.56)

which after insertion of equation (3.3.54) leads to[−(K(a) +

2a(η)′

a(η)

)K(a)− k2 + µ2a(η)2 +K(a)′

]EL = 0. (3.3.57)

This last line does not contain any operators, hence it is a simple multiplication. As it has to betrue for any value of k, this last line implies that the consistency constraint and the field equationfor EL only share one solution, which is

EL =0. (3.3.58)

The transversal modes obey the field equations

DBT =2a(η)′

a(η)ikεTT ′

ET ′, (3.3.59)

DET =− 2a(η)′

a(η)ikεTT ′

BT ′. (3.3.60)

with the differential operator,

D =[∂20 −

2a(η)′

a(η)∂0 + k2 + a(η)2µ2

]. (3.3.61)

By taking rotations of these differential equations and inserting the results in one another, wefind [

D

(a(η)

2a(η)′D

)− 2a(η)′

a(η)k2

]BT =0, (3.3.62)

[D

(a(η)

2a(η)′D

)− 2a(η)′

a(η)k2

]ET =0, (3.3.63)

such that ET and BT solve the same fourth order differential equation. This does not yet implythat these four degrees of freedom are constrained in any way, since a fourth order differentialequation may give rise to four linearly independent solutions. If we however also impose the

41

3.3 Antisymmetric tensor field quantisation CHAPTER 3 CANONICAL QUANTISATION

consistency condidion, equation (3.3.52), we can reduce the order of the differential operatorsacting on ET by using,

∂30E

T =∂0

(∂20E

T),

=∂0

(2a(η)′

a(η)∂0 − k2 − µ2a(η)2

)ET + ∂0

(2a(η)′

a(η)ikεTT ′

BT ′). (3.3.64)

Repeating the same step we can eventually reduce equation (3.3.52) to a relation between ET

and BT ′which is of first order in differential operators. As a first order differential operator

can have at most one solution, this does in fact directly constrain ET in terms of BT ′. Hence

the initial four transversal degrees of freedom are reduced to two degrees of freedom by theconsistency condition. This analysis leads to no statement about the size of the solution space forthe transversal modes obeying both the field equations and the consistency condition, however forthis moment the conclusion that there are only three true degrees of freedom in the ’gauge fixed’theory is sufficient.

Now the canonical momenta are

Πij =√−G2

F 0ij , (3.3.65)

Π0i = −Πi0 =−Gijησρ∂σBρj . (3.3.66)

In momentum space and in terms of creation an annihilation operators this becomes,

Πij(x) = −a(η)−2

2ηaiηbj

∑P

∫d3k

(2π)3ei~k·~x

(∂0Q

P εPab(~k) + ikaQP εPb0(~k) + ikbQ

P εP0a(~k))aP

~k

− hermitian conjugate, (3.3.67)

Π0i(x) = −a(η)−2ηim∑P

∫d3k

(2π)3ei~k·~x

(∂0Q

P εP0m(~k)− ikjQP εPjm(~k)

)− hermitian conjugate

.

(3.3.68)

As we can see for the ij-components nothing has changed, however for the 0i components now wefind from the equal time commutation relations,∑

P

[−a(η)−2QP (η, k)εP0j(~k)

(∂0Q

∗P (η, k)ε∗P0m(~k)− ikaQ

∗P (η, k)ε∗Pam(~k)

+a(η)−2Q∗P (η, k)ε∗P

0j (~k)

(∂0QP (η, k)εP0m(~k) + ikaQ

P (η, k)εPam(~k)]

=i

2δmj . (3.3.69)

Since the polarisation pseudo tensors still obey properties (3.3.13–3.3.16), we now find for E andB, independent of their spatial polarisation,

QP ′(η,~k)∂0Q

∗P(η,~k)−Q∗P ′

(η,~k)∂0Q

P(η,~k)

=ia(η)2, for P = P ′ = PB . (3.3.70)

QP ′(η,~k)∂0Q

∗P(η,~k)−Q∗P ′

(η,~k)∂0Q

P(η,~k)

=ia(η)2

2, for P = P ′ = PE . (3.3.71)

Luckily, now this condition is in accordance with the field equations (3.3.47–3.3.50). Althoughthese Wronskian relations may not be consistent with the consistency condition, we did makeprogress. Previously, we had field equations and consistency conditions that allowed for a nonemptyset of solutions for ET . We also had degenerate canonical momenta. The nondegenerate canonicalmomenta led to inconsistencies with the field equations. This time we do not have degeneratemomenta, and the Wronskian conditions are consistent with the set of solutions to the new fieldequations combined with the new consistency condition, as some of these sets only contain thezero valued function regardless of the commutation relations.

42

CHAPTER 3 CANONICAL QUANTISATION 3.4 Vacuum states

3.4 Vacuum states

The background metric in the derived expansion of NGT, can be any general relativistic metric.In the case of cosmology, the most general metric to choose is the Friedmann-Lemaıtre-Robertson-Walker metric (FLRW), which is in general given by

gµν = diagonal(1,−a(t)2,−a(t)2,−a(t)2). (3.4.1)

3.4.1 Vacuum in Minkowsky space

For any canonically quantised field in Minkowsky space, the vacuum state is the state that isannihilated by any annihilation operator a~k

. That is,

a~k|0〉 = 0. (3.4.2)

Let us for instance look at the massless scalar field in flat space, with the Lagrangian density,

L = 12η

µν∂νφ∂µφ, (3.4.3)

where ηµν is the Minkowski metric. The scalar field obeys the field equation,

∂2φ(η, k) = 0. (3.4.4)

One set of solutions is

u~k(t, ~x) = ei~k·~x−ikt (3.4.5)

u∗~k(t, ~x) = e−i~k·~x+ikt. (3.4.6)

The question then is, what combination of both solutions to choose as the vacuum state. Toanswer that question, I will make use of the concept of Killing vectors. For a given metric, aKilling vector points in a certain direction. For a given path in spacetime, the component of thispath, which points in the direction of the Killing vector, is conserved. In other words, Killingvectors imply conserved quantities associated with the motion of free particles [9].

The vacuum state of a field should be the ground state of that field. Since the ground state isthe state of lowest energy, the field will not evolve beyond that state without external influencesand with a time independent Hamiltonian. This means that the ground state must be completelyconserved in time. If the ground state is not directed along the timelike Killing vector, then thecomponents not directed along this vector will not be conserved in time. These extra componentswould radiate away in some manner, leading to a lower energy state. This is in contradiction withthe definition of the ground state. If a state is directed along a Killing vector, it is an eigenfunctionof that vector.

In flat space, ∂t is the timelike Killing vector. To define the ground state, we demand that

∂tφ(t,~k) = σφ(t,~k), (3.4.7)

with σ being any constant.Now we find, that for the ground state I can choose solely either u~k or u∗~k.The functions u~k and u∗~k are orthogonal, but also satisfy the equality

u∗−~k(t, ~x) = u~k(t, ~x). (3.4.8)

Hence, all k-modes of uk are covered when only one of both solutions is integrated over all k.The field φ(t, ~x) may be expanded as

φ(t, ~x) =∑~k

[u~k(t, ~x)a~k

+(u~k(t, ~x)a~k

)†]

=∑~k

ei~k·~x[eikta~k

+ e−ikta†−~k

](3.4.9)

43

3.4 Vacuum states CHAPTER 3 CANONICAL QUANTISATION

Here k =∥∥∥~k∥∥∥. I have made the usual choice to define the modes (3.4.5) to have positive frequency

when∂tu~k(t, ~x) = −iku~k(t, ~x), (3.4.10)

with k > 0.

3.4.2 Ambiguity in curved space

Now moving back to curved space, we can recognise that the positive frequency choice is ambiguousin curved space. In flat space, ∂t is a Killing vector, and we defined the positive frequencymodes to be eigenfunctions of this Killing vector. In an arbitrary curved spacetime, ∂t is notautomatically a Killing vector, and there may not be a timelike Killing vector at all. This meansone cannot by definition uniquely define a vacuum solution, in agreement with equation (3.4.7).Hence different bases may be chosen in which to express the solutions to the field equations. I.e.,linear combinations of uk and u∗k can be used as a new basis. In that case, the new solutions uk

and u∗k can be expressed in terms of the old solutions as

u~k = α~k~k′u~k′ + β~k~k′u∗~k′. (3.4.11)

Likewise the new creation and annihilation operators can now be expressed in terms of the oldones as

a~k= α~k~k′a~k′

− β∗~k~k′a†~k′. (3.4.12)

Equations (3.4.11) and (3.4.12) are known as the Bogoliubov transformations [7, 6].With these new defined operators, a new vacuum is also defined, since

a~k

∣∣0⟩ = 0. (3.4.13)

That this is not the same vacuum as |0〉 follows from

a~k|0〉 = −β∗~k~k′

a†~k′|0〉 = −β∗~k~k′

∣∣1~k′

⟩6= 0. (3.4.14)

A vacuum in one frame, contains particles in another frame. The vacua of both frames may notbe conserved in time, as there may not be a timelike Killing vector. Only when a timelike Killing-vector field exists, the positive frequency modes (eigenfunctions of the timelike Killing vector withpositive eigenvalues) can be defined as in equation (3.4.10). Then the ground state is uniquelydefined.

3.4.3 Conformal vacuum

Special cases exist for which one can define a vacuum state in curved space. Let us for instancelook at the massless conformally coupled scalar field, with the Lagrangian density,

L = 12

√−g∂µφ∂µφ−

112√−gRφ2, (3.4.15)

where R denotes the Ricci scalar. The scalar field obeys the field equation,

φ(η, k) +16Rφ = 0. (3.4.16)

A conformal transformation of the metric is defined as,

gµν →gµν = Ω(x)2gµν . (3.4.17)

Under such a transformation the Ricci scalar transforms as,

R→R = Ω(x)2R+ 6Ω(x)−3√−gΩ(x), (3.4.18)

44

CHAPTER 3 CANONICAL QUANTISATION 3.4 Vacuum states

and the field equation transforms as

φ(η, k) +16Rφ = 0 → (3.4.19)

φ+16Rφ =0, (3.4.20)

= Ω(x)−3

[ +

16R

]Ω(x)φ =0, (3.4.21)

such that φ = Ω(x)−1φ and the field equation is invariant under conformal transformations.Now if we start in a FLRW-Universe, with metric gµν = a(η)2ηµν , and choose the conformal

transformation such that Ω(x)2 = a(η)−2, we find in the conformal coordinate system

φ(η, k) +16Rφ = 0 → (3.4.22)

∂2φ = 0, (3.4.23)

since for the Minkowsky metric we have R = 0 and Γρµν = 0.

In general for conformal spacetimes we can always perform a conformal transformation whichtakes us to the flat Minkowsky frame in which R = 0 and Γρ

µν = 0.Equation (3.4.23) shows that the field equation for the conformally coupled scalar reduces to

that of a massless scalar in flat space, under a conformal transformation. As a consequence themodes of the conformally coupled field contain the flat space solution. We already know that inflat space we have the timelike Killing vector ∂t, and have the well defined flat space vacuum statefor the scalar field.

In that way is is natural to choose the vacuum state in the conformal spavetime in accordancewith its flat space equivalent. We find that the vacuum state in the FLRW-metric must be givenby,

φvac = a(η)−1φvac. (3.4.24)

In general we find that the Killing vector in flat spacetime is identified with a conformal Killingvector in curved (conformal) spacetime. A conformal Killing vector no longer implies a conservedquantity for any geodesic, but it still does for null geodesics [49]. A massless field evolves along nullgeodesics. Therefore the vacuum state in any frame can be defined to be the state that containsonly positive-frequency modes in the comoving frame. For example, a field component ψ, notnecessarily a scalar field, whose field equation under transformation (3.4.17) transforms to

∂2 (a(η)ψ) = 0, (3.4.25)

has in the flat frame the solutions given in equation (3.4.5) and its complex conjugate. Hence, inconformally flat space, that is gµν = a(η)2ηµν , its vacuum state can be written as

ψ(η, ~x) = a(η)−1∑

k

ei~k·~x[eikηa~k

+ e−ikta†−~k

]. (3.4.26)

The behaviour of the field under transformation (3.4.17) is of course defined by nature of the field,as it may be the component of a tensor or vector field.

3.4.4 Bunch-Davies vacuum

Now let us move on to a less simple example, the minimally coupled massless scalar field. TheLagrangian density is given by

L = 12

√−g∂µφ∂µφ. (3.4.27)

45

3.4 Vacuum states CHAPTER 3 CANONICAL QUANTISATION

From this Lagrangian density we find the field equation,

φ = 0. (3.4.28)

Apparently the minimally coupled massless scalar breaks conformal invariance.If we quantise the field and transform to momentum space, the field equation becomes,[

∂20 +

2a(η)′

a(η)∂0 + k2

]φ~k(η) = 0. (3.4.29)

Now in the special case of a quasi-de Sitter Universe, with the metric

gµν = a(η)ηµν , with a(η) = −1HIη , (3.4.30)

where HI is the Hubble constant during inflation and −∞ < η < −1/HI , the field equationbecomes, [

∂20 −

2η∂0 + k2

]a(η)φ~k(η) = 0. (3.4.31)

The solution to this equation is given in terms of Hankel functions (see Appendix C.1),

φk(η) =a(η)−1

√2k

[α

(1− i

kη

)e−ikη + β

(1 +

i

kη

)eikη

], (3.4.32)

where |α| − |β| = 1 satisfies the canonical commutation relations for this action.For each mode with a comoving momentum ~k, there exists a time η such that k/a(η)

ggH. In that case k/(HIa(η)) = kη 1. Hence for each a time exists when the solution iseffectively reduced to

φk(η) =a(η)−1

√2k

[αe−ikη + βeikη

]. (3.4.33)

If we compare this to the conformal vacuum, defined in the previous subsection in equation (3.4.26),we see that the observer must effectively observe the conformal vacuum. In that case the choice ofparameters has to be α = 1 and β = 0. This is what is called the Bunch-Davies vacuum[6, 8, 42]:a field reduces to the conformal vacuum in the asymptotic limit (early time) of the spacetime.

46

Chapter 4

Quantum fluctuations

In this chapter I solve the field equations for the antisymmetric tensor field whose action is

S =∫d4x

√−G

R− 2Λ− 1

12FµνρF

µνρ +14µ2BµνB

µν

− λ

4(∇µB

µν)∇ρBρν . (4.0.1)

I impose boundary conditions, such that the field is in its vacuum state at the beginning of inflation,which in this analysis is treated as being the beginning of time. Statements about the period beforeinflation are currently highly hypothetical, such that we cannot make any general assumption otherthan we do now. In the vacuum state, the field can produce quantum fluctuations, which, due toinflation and due to the sudden transition from the inflationary era to radiation era and after thatto matter era, cause the field to get to an excited state. The goal of this chapter is to calculatethe state in which the field is during matter era, as then the decoupling of the photon fluid occurs,which is very likely to have an imprint of the Bµν stress-energy structure.

4.1 Quantities of interest

4.1.1 Massive or massless

In the next section I will solve the field equations and define the vacuum state for the field, inaccordance with the definition of the vacuum from the previous chapter.

A result of the following sections is that the momentum modes for which energy density of thetheory is significantly excited out of the vacuum state, are modes that crossed the Hubble radiusduring inflation, outwards. The Hubble parameter during inflation is denoted by HI , which is ingeneral expected to be of the orderHI ∼ 1013 GeV. The Hubble radius is then given byRH = H−1

I ,and a mode crosses the horizon when its momentum shrinks below kphys = k/a(η) = HI . In theprevious chapter I explained how in the case of a Bunch-Davies vacuum every mode once musthave been larger than HI somewhere during inflation. We will see in the following that the vacuumstate for Bµν is not the Bunch-Davies vacuum, but we can at least conclude that the modes ofinterest will satisfy kphys µ during inflation. This is the case since from a geometrical point ofview, the mass is expected to be of the order of the cosmological constant, which obeys Λ HI .

This implies that during inflation we are only looking at highly relativistic modes. The impli-cation of the modes being highly relativistic is that the theory effectively behaves as its masslesslimit during inflation.

As we saw in the previous sections, the massless theory only possesses one degree of freedom,the longitudinal B-mode. When the modes of interest are highly relativistic, and thus the theorybehaves massless, an observer can not distinguish between the massless and the massive theory.This is a strong argument to state that the amount of energy in the transversal modes of the fieldwhich start subhubble during inflation, has to be negligible. Note that we will not investigate thisstatement here. Further research is needed on this statement. For now we will do with the givenassumption.

47

4.1 Quantities of interest CHAPTER 4 QUANTUM FLUCTUATIONS

Throughout the rest of this section we will only be interested in the longitudinal B-modes, asthese are believed to be the only modes that contribute to a measurable result. Note that thegauge fixing term, added to be able to quantise the theory, does not contain BL. Hence now thetheory is quantised properly and we know which modes to evolve, the gauge fixing term can beignored.

4.1.2 Power spectrum of density fluctuations

As one of the goals of the following calculations is to check whether the energy density of NGT,after an evolution through the history of the Universe, is viable for containing an extensive amountof energy, we are interested in its energy density. The energy density is given by T 0

0 .In fact from the action of this theory we see that the antisymmetric tensor theory does not

couple to anything but gravity, which it does via its stress-energy tensor. That is the quantity wecalculate in this chapter.

From the action (4.0.1) we find without the gauge fixing term,

T00(η, ~x) =

12a(η)6

(∂iBj(η, ~x))

2 +(∂0~B(η, ~x) +∇× ~E(η, ~x)

)2

+ a(η)2µ2(~E(η, ~x)2 + ~B(η, ~x)2

). (4.1.1)

For the explicit derivation of the stress-energy tensor from the action and the rewriting in E andB, the reader is again referred to Appendix B.2.

When I write T 00 in terms of the fields in their quantised form in momentum space (equa-

tion (3.3.5)), T 00 has become an operator, with an expectation value given by

〈ρNGT (η)〉 =⟨T0

0NGT (η)

⟩= 〈0|T0

0NGT (η, ~x) |0〉

=∫

d3k

(2π)31

2a(η)6

k2 |B(η, k)|2 +

∣∣∣∂0~B(η, k) + ik ~E(η, k)

∣∣∣2+ a(η)2µ2

(∣∣∣ ~E(η, k)∣∣∣2 +

∣∣∣ ~B(η, k)∣∣∣2)

=∫k2dk

2π2T0

0(η, k), (4.1.2)

where the vacuum state |0〉 is to be defined as in the previous section. The last line defines thequantity T0

0(η, k), and 〈. . .〉 denotes the averaging procedure. Usually in this procedure one hasto subtract the energy density of the background vacuum, as the energy density proposed here hasan ultraviolet divergence. As we however are only interested in a certain band in the spectrum,the modes that leave the vacuum state sometime, we will not deal with this divergence and simplyignore the vacuum energy density.

It is in general not yet understood what is the correct vacuum energy solution, as we measuretoday a tiny cosmological constant, in contradiction with the theoretical vacuum fluctuations. Onthe other hand, the hypothetical vacuum fluctuations for scalar perturbations in the metric whichare pushed outside the horizon during inflation and are excited out of the vacuum, do fit thepresently measured CMB perturbations.

The penultimate step in the derivation shown in equation (4.1.2) follows from the fact that for

48

CHAPTER 4 QUANTUM FLUCTUATIONS 4.2 Dynamics during inflation era

any quantised field φ(x),

⟨φ2⟩

= 〈0|(∫

d3k

(2π)3ei~k·~x

φ~k

(η)a~k+ φ∗

−~k(η)a†

−~k

)2

|0〉

= 〈0|∫ ∫

d3k d3k′

(2π)6ei(~k+~k′)·~x

φ~k

(η)a~k+ φ∗

−~k(η)a†

−~k

φ~k′

(η)a~k′+ φ∗

−~k′(η)a†

−~k′

|0〉

= 〈0|∫ ∫

d3k d3k′

(2π)3δ3(~k + ~k′)φ~k

φ∗−~k′

a~ka†−~k′

|0〉

=∫

d3k

(2π)3∣∣φ~k

∣∣2 ⟨1~k

∣∣1~k

⟩=∫k2 dk

2π2|φk|2 . (4.1.3)

Of course from the third to the fourth line I made use of 〈k| −k′〉 = (2π)3δ3(~k + ~k′).To say anything about the physics at a given moment in time, one has to distinguish between

different wavelengths on subhorizon and superhorizon scales. As the spectrum of wavelengthsextends from zero to infinity, the future analysis of distribution over wavelengths will be done ina logarithmically scaled frame of reference. Therefore it will turn out convenient to express theenergy density in a spectrum, defined as the energy ’per decade in k’: the power spectrum P, suchthat

ρNGT =∫d(ln k)PNGT (η, k)

=∫dk

kPNGT (η, k). (4.1.4)

The result is that we will, for different episodes in history, calculate the power spectrum, given by

PNGT =k3

2π2T0

0NGT (η, k). (4.1.5)

We can split the power spectrum in a transversal and a longitudinal contribution. The expres-sion for the longitudinal B-modes becomes,

PLNGT =

k3

4π2a(η)4

∣∣∣∣∂0BL(k, η) +

a′

aBL(k, η)

∣∣∣∣2 + (k2 + a2µ2)∣∣∣BL(k, η)

∣∣∣2 . (4.1.6)

(4.1.7)

4.2 Dynamics during inflation era

4.2.1 General solution

Since during inflation the scale factor is defined as

a(η) =−1HIη

(with −∞ < η < −1HI

), (4.2.1)

where HI denotes the Hubble constant during inflation, the field equation for BL becomes(∂20 + k2 +

2η∂0 +

µ2

H2I η

2

)BL

inf (η, k) = 0, (4.2.2)

49

4.2 Dynamics during inflation era CHAPTER 4 QUANTUM FLUCTUATIONS

or, (∂20 + k2 +

µ2

H2I η

2

)BL

inf (η, k) = 0, (4.2.3)

where we still have BL = BL/a(η). Apparently, for the specific parametrisation of inflation, inwhich explicit curvature terms vanish in equation (4.2.3), besides the one in the mass term, wehave, [

∂2 + µ2a(η)2]BL =0

=[ +

16R+ µ2a(η)2

]BL

=a(η)3[ +

16R+ µ2a(η)2

]BL

a(η). (4.2.4)

If we compare this with the conformal vacuum, defined in the previous chapter, we find thatBL/a(η) obeys the field equation for the conformally coupled scalar field, during inflation. Again, and R are to be evaluated in the flat comoving frame.

Rewriting the general differential equation solved by Bessel functions1, we find that the equa-tion

u′′ +1− 2αz

u′ +(β2 +

α2 − ν2

z2

)u = 0 (4.2.5)

is solved byu = zαZν(βz) (4.2.6)

where Zν is a Bessel function of the first, second or third kind, u′ = dudz etc.

Setting α = 12 , β2 = k2, ν2 = 1

4 −µ2

H2I

we find that, using (4.2.3) and (3.2.16),

BLinf (η, k) =

αinfB√ηZ

(1)

12

r1− 4µ2

H2I

(kη) +βinf

B√ηZ

(2)

12

r1− 4µ2

H2I

(kη). (4.2.7)

The solution is written as a linear combination of Z(1)ν and Z

(2)ν . It is yet left undefined which

set of Bessel functions is chosen. Distinct combinations of two of the Bessel functions of the first,second and third kind form a full basis for the space of solutions. Which combinations are apossible choice can be read in Appendix C.1. In the next section I will show that the boundaryconditions imply a preferred choice for the basis. The coefficients αinf

B and βinfB are as well to be

set by the boundary conditions.

4.2.2 Boundary conditions

The first condition is given by the canonical quantisation condition on the wronskian, equa-tion (3.3.70), the second condition is the demand that at the beginning of the inflation era, i.e. atη → −∞, the field is in its ground state [13].

Taking a look at the field equations at this early stage of the Universe, we find that equa-tions (4.2.3)2 reduces to nothing but an harmonic oscillator equation(

∂20 + k2

)BL

inf (η, k) = 0. (4.2.8)

The harmonic oscillator equations lead to the general solution,

BLinf (η, k) = C3e

ikη + C4e−ikη, (4.2.9)

1For a summary of properties of Bessel functions, see appendix C, section C.1.2Note that we could as well take the limit in equation (4.2.2), but then the term 2

η∂0 is not equal to zero when

η → −∞. However, the right limit would still be achieved.

50

CHAPTER 4 QUANTUM FLUCTUATIONS 4.2 Dynamics during inflation era

When we compare this result with equation (4.2.7), and using the asymptotic expansions for thedifferent, optional, Bessel functions3, we find that Bessel Functions of the third kind (often calledHankel functions), converge to harmonic oscillator solutions for η → −∞. Hence,

BLinf (η, k) ' αinf

B

√2

πkη2ei

„kη−π

4

r1− 4µ2

H2I

−π4

«+ βinf

B

√2

πkη2e−i

„kη−π

4

r1− 4µ2

H2I

−π4

«

' i

√2πη2

(−αinf

B eikη + βinfB e−ikη

)(4.2.10)

Now it becomes clear why I chose to rescale B: at the beginning of inflation its field equationreduces to the plane wave equation. As a consequence we find that during inflation when thefield is effectively massless, B/a(η)2 solves the conformally-coupled-scalar field equation. Henceusing the definition of the conformal vacuum as explained in the previous chapter, I choose theconformal vacuum as the vacuum state for B.

The boundary conditions now impose the following choice of parameters,

C3 = 0 = −iαinfB

√2πk,

C4 =1√2k

= iβinfB

√2πk.

Thus finally the desired vacuum fluctuations in inflation are found,

BLinf (η, k) =− i

√π

4H2I ηH

(2)

12

r1− 4µ2

H2I

(kη). (4.2.11)

I could have chosen to express the solutions as combinations of Bessel functions of the firstkind, or of the first and second kind. If I then would have applied the procedure of matching to theground state at the beginning of inflation, I would eventually have found the same solution. Thatis, the parameters C3 and C4 would be equivalent to a Bogoliubov transformation; a reordering ofthe basis such that we were again left with the Hankel functions. The reason for this conclusionis the fact that the vacuum state is defined uniquely by this procedure.

4.2.3 The spectrum of the density fluctuations

In section 4.1.2 the power spectrum of density fluctuations is given by equation (4.1.6). We findfor the longitudinal spectrum,

PNGT (η, k) =k3

16π2a(η)4

∣∣∣∣∣∣ ∂∂η√ηH(2)

12

r1− 4µ2

H2I

(kη) +1√ηH

(2)

12

r1− 4µ2

H2I

(kη)

∣∣∣∣∣∣2

+(k2 +

µ2

H2I η

2

) ∣∣∣∣∣∣√ηH(2)

12

r1− 4µ2

H2I

(kη)

∣∣∣∣∣∣2 . (4.2.12)

In figure 4.1 we see the power spectrum. The ultraviolet momenta are redshifted like a rela-tivistic particle field in vacuum. The running parameter is kη, which is during inflation equal tokη = −kphys/HI . Hence kη = −1 is the moment of crossing the Hubble radius outwards for amode k.

On very subhubble scales the spectrum is a power law spectrum, P ∼ k4, which is the conformalvacuum. For a given wavelength, the amplitude changes ∼ a(η)−4 on these scales, hence the field

3See appendix C, section C.1, subsection C.1.3 for a summary of properties of Bessel functions.

51

4.2 Dynamics during inflation era CHAPTER 4 QUANTUM FLUCTUATIONS

-1000 -100 -10 -1 -0.1k Η

1

10

100

a HΗ L4

k4

P NGT

ΜHI

=1

ΜHI

< È k Η È

Figure 4.1: The spectrum during inflation, plotted for several values of µHI

. Only when a very large mass

is added, the spectrum (solid line) differs from the relativistic spectrum (dashed line).

behaves as relativistic matter. However on superhubble scales, we see that the scaling of theenergy density for the massless field (dashed line) in a mode k switches to ∼ a(η)2. The modes areantidamped on superhubble scales. This process is similar to the freeze in of scalar field fluctuationsand gravitational wave fluctuations on superhubble scales during inflation, however the energydensity per mode of the minimally coupled scalar field becomes constant on superhubble scales,whereas for the antisymmetric tensor field it proceeds in damping with a factor a(η)−2. The readermust not be mislead by the rescaling of the plotted spectrum with a factor a(η)4/k4 = 1/(HIηk)4.The spectrum is in fact ultraviolet divergent and infrared safe. The ultraviolet spectrum is believedat least to stop being correct at the Planck scale, where physics may not be a simple continuationof known physics.

From this figure it becomes clear that although BL couples conformally, its stress-energy tensorbreaks conformal invariance.

As an illustration, the spectrum is also plotted for an extremely high mass, µHI

= 1 (solidline). For such a field ultraviolet modes are overproduced with respect to the massless conformalvacuum. One can see that for small k, the spectrum then starts scaling as nonrelativistic matter,i.e. ∼ a(η)−3. If the mass would be that big, the ground state chosen at η → −∞ is still the correctvacuum state as any physical momentum exceeds the mass at that time, however the transversalmodes may become important already early during inflation. Hence, the extremely massive graphin this context is of physical meaning, but cannot be taken in account without further investigationof the transversal modes. It is however illustrative for our small-mass limit. Throughout the restof this thesis I will assume the field to remain massless up to the beginning of the radiation era.

52

CHAPTER 4 QUANTUM FLUCTUATIONS 4.3 Dynamics during radiation era

4.3 Dynamics during radiation era

4.3.1 General solution

In radiation era the scale factor is

a(η) = HIη with ( 1HI

< η < ηe), (4.3.1)

where ηe denotes the conformal time at a sudden radiation-to-matter transition.The field equation for BL in radiation era is given by(

∂20 + k2 − 2

η∂0 + µ2H2

I η2

)BL

rad(η, k) = 0. (4.3.2)

This equation can be identified with

d2y(x)dx2

+A0x+

B0

x

dy(x)dx

+A1x

2 +B1 +C1

x2

y(x) = 0, (4.3.3)

which is solved by a form of the confluent hypergeometric function (see appendix C, section C.2) ,

y(x) = x−Ae−Bx2

1F1[a; b;Cx2] (4.3.4)

given the parameters,

A0 = (4B − 2C) = 0,

B0 = (2A+ 2b− 1) = −2,

A1 = (4B2 − 4BC) = µ2H2I ,

B1 = −2CA+ 2B(2b− 1) + 4AB + 2B − 4aC = k2,

C1 = A(A+ 2b− 2) = 0.

As a result of some algebra, we are led to the conclusion that equation (4.3.2) yields two solutions,writing y(η) instead of BL(η, k),

y1(η) = ei2 µHIη2

1F1

[− ik

2 + µHI

4µHI;−1

2;−iµHIη

2

]=

√ηM ik2

4µHI, 3

4

(−iµHIη

2)

(4.3.5)

y2(η) = η3ei2 µHIη2

1F1

[− ik

2 + 5µHI

4µHI;52;−iµHIη

2

]=

√ηM ik2

4µHI,− 3

4

(−iµHIη

2), (4.3.6)

where Mn,m(x) is the Whittaker function.The Wronskian for these solutions is

W [y1(η), y2(η)] = 3iµHIη2. (4.3.7)

The Wronskian is nonzero for all η during radiation era. Therefore equations (4.3.5) and (4.3.6)are linearly independent solutions to equation (4.3.2).

Despite of a nonzero Wronskian, we cannot yet be sure that we have chosen the right basis inthe space of solutions. The reader will notice from the previous section, that in inflation the fieldequations were solved by Bessel functions, and I chose the Hankel functions as a basis. That basis

53

4.3 Dynamics during radiation era CHAPTER 4 QUANTUM FLUCTUATIONS

was oriented in the same way as the basis at η → −∞, hence boundary conditions were easilysatisfied. What happened, is that the basis of solutions converged to the commonly used basisof solutions to the field equations for µa(η) → 0. If we look at the field equations at this time,we see that in the same limit, this equation again reduces to an equation that is solved by Besselfunctions: (

∂20 + k2 − 2

η∂0

)BL

rad(η, k) = 0. (4.3.8)

with the solutionsu = η

32Z 3

2(kη). (4.3.9)

The configuration of solutions that gives the ground state, demands some explanation. Al-though I do not let radiation era extend infinitely far back in time, we may choose the Bunch-Davies vacuum with the positive frequency modes as a hypothetical vacuum.

Now we may try to rotate the basis of the solution space for the full equation, such that itreduces to the basis of Hankel functions for µa(η) → 0, because these define the ground state.Since at this time η is finite, we are not taking the limit of η → −∞, but the limit of µ→ 0. If Imake the substitutions ζ = k2

4µHIand x = 1

4k2η2 in equations (4.3.5) and (4.3.6),

√ηM ik2

4µHI,± 3

4

(−iµHIη

2)

=√ηMiζ,± 3

4

(x

iζ

), (4.3.10)

it is clear that we must expand for large ζ. Since this expansion is not a usual one, it is calculatedexplicitly in appendix C, subsection C.2.4. The expansion is,

Miζ,± 34

(x

iζ

)=

x12

(iζ)12±

34Γ(

1± 32

)[(1− 1

2x

iζ

)J± 3

2

(2x

12

)+ix

2ζJ± 3

2+2

(2x

12

)]+O

(ζ−

52∓

34

),

(4.3.11)or, in terms of the physical parameters:

M ik2

4µHI,± 3

4

(−iµHIη

2)

=

√−iµHIη2

(4µHI

ik2

)± 34

Γ(

1± 32

)[(1 +

iµHIη2

2

)J± 3

2(kη) +

iµHIη2

2J± 3

2+2 (kη)]

+O

((µ

HI

) 52±

34). (4.3.12)

In appendix C, subsection C.1.1, I explain

H(1),(2)ν (z) = (1± i cot νπ) Jν(z)∓ i

sin νπJ−ν(z),

(4.3.13)

where the upper sign of every ± or ∓ will give H(1)ν (z), and the lower sign will give H(2)

ν (z). Hence,

M(1)34

[ζ; kη] = eiπ4

√πHI

4µ

[ζ−

34

Γ(

52

)Miζ, 34

(−ik

2η2

4ζ

)+ i

ζ34

Γ(− 1

2

)Miζ,− 34

(−ik

2η2

4ζ

)],(4.3.14)

M(2)34

[ζ; kη] = eiπ4

√πHI

4µ

[ζ−

34

Γ(

52

)Miζ, 34

(−ik

2η2

4ζ

)− i

ζ34

Γ(− 1

2

)Miζ,− 34

(−ik

2η2

4ζ

)],(4.3.15)

54

CHAPTER 4 QUANTUM FLUCTUATIONS 4.3 Dynamics during radiation era

reduce to√

π4 a(η)H

(1)32

(kη) and√

π4 a(η)H

(2)32

(kη) respectively, for ζ →∞. In this case the Wron-skian is

W [√ηM

(1)34,√ηM

(2)34

] = ia(η)2. (4.3.16)

Now we can write the most general solution for equation (4.3.2) as

BLrad(η, k) = αrad

B

√ηM

(1)34

[k2

4µHI; kη]

+ βradB

√ηM

(2)34

[k2

4µHI; kη]. (4.3.17)

Note that from the definitions of M (1),(2)34

we find for their complex conjugates

M(1)∗34

[ζ; kη] = e−iπ4

√πHI

4µ

[ζ−

34

Γ(

52

)M−iζ, 34

(ik2η2

4ζ

)− i

ζ34

Γ(− 1

2

)M−iζ,− 34

(ik2η2

4ζ

)]

= e−iπ4

√πHI

4µ

[ζ−

34

Γ(

52

)e− iπ2 Miζ, 3

4

(−ik

2η2

4ζ

)− i

ζ34

Γ(− 1

2

)e− iπ2 Miζ,− 3

4

(−ik

2η2

4ζ

)]

= −e iπ4

√πHI

4µ

[ζ−

34

Γ(

52

)Miζ, 34

(−ik

2η2

4ζ

)− i

ζ34

Γ(− 1

2

)Miζ,− 34

(−ik

2η2

4ζ

)]

= −M (2)34

[ζ; kη], (4.3.18)

M(2)∗34

[ζ; kη] = −M (1)34

[ζ; kη]. (4.3.19)

Thus we can see

W [BLrad(η, k), B

L∗rad(η, k)] =

(|αrad

B |2 − |βradB |2

)ia(η)2, (4.3.20)

which tells us that the condition|αrad

B |2 − |βradB |2 = 1, (4.3.21)

will make the theory canonically quantised. The ground state of the field would be given by thechoice of parameters αrad

B = 1 and βradB = 0.

4.3.2 Boundary conditions

The only boundary condition is that the field Bµν(x) evolves continuously up to its first derivativethrough time. This means matching the solutions for the inflation and the radiation equationsand their first time derivatives,

BLinf

(−1HI

, k

)= BL

rad

(1HI

, k

), (4.3.22)

∂0BLinf (η, k)

∣∣η=− 1

HI

= ∂0BLrad (η, k)

∣∣η= 1

HI

. (4.3.23)

When I write this explicitly, it is convenient to introduce a shorthand notation,

BLinf (η, k) = −i

√π

4H2I ηH

(2)

12

r1− 4µ2

H2I

(kη)

≡ C3H,

BLrad(η, k) = αrad

B

√ηM

(1)34

[k2

4µHI; kη]

+ βradB

√ηM

(2)34

[k2

4µHI; kη]

≡ αradB M1 + βrad

B M2,

f ′(η) = ∂0f(η).

55

4.3 Dynamics during radiation era CHAPTER 4 QUANTUM FLUCTUATIONS

Then the matching leads to

C3H∣∣∣η=− 1

HI

= αradB M1

∣∣η= 1

HI

+ βradB M2

∣∣η= 1

HI

(4.3.24)

C3H′∣∣∣η=− 1

HI

= αradB M ′

1

∣∣η= 1

HI

+ βradB M ′

2

∣∣η= 1

HI

. (4.3.25)

Multiplying equation (4.3.24) by M ′2 and equation (4.3.25) by M2, gives, as a result of some more

algebra,

αradB =

C3

M1M ′2 −M2M ′

1

HM ′

2 −M2H′

=C3

W [M1,M2]

HM ′

2 −M2H′

= −√

π

4H2I

HM ′

2 −M2H′. (4.3.26)

Likewise we findβrad

B =√

π

4H2I

HM ′

1 −M1H′. (4.3.27)

With these expressions for αradB and βrad

B , the field satisfies the boundary conditions. It willalso satisfy equation (3.3.70), the condition that the field is canonically quantised. That this isthe case, can be understood when we realise that, since the field and its derivative are continuousat the border of the inflation and radiation era, the Wronskian is continuous as well. The pa-rameters αrad

B and βradB are just numbers, hence time independent. Therefore Wronskian satisfies

equation (3.3.70) and αradB and βrad

B by definition satisfy equation (4.3.21).

0.01 0.02 0.05 0.1 0.2 0.5 1

kHI

0.3

0.5

0.7

1

È Β rad ÈÈ Α rad È

Figure 4.2: The relative value of αradB and βrad

B , for µHI

< 0.1. For regular values of k, αradB and βrad

B areof equal size.

I can simplify expressions once we remember that the value of HI is rather large with respectto the mass µ, given that µ has its origin from the cosmological constant Λ and HI ∼ 1013 GeV.

56

CHAPTER 4 QUANTUM FLUCTUATIONS 4.3 Dynamics during radiation era

Since12

√1− 4µ2

H2I

=12− µ2

H2I

+O(µ4

H4I

), (4.3.28)

µHI

only appears in H as second and higher order terms (see appendix C, section C.1, subsec-tion C.1.5). With this in mind, I can write

1√ηH

(2)

12

r1− 4µ2

H2I

(kη)

∣∣∣∣∣∣η=−1

HI

=1√ηH

(2)12

(kη)∣∣∣η=−1

HI

+O(µ2

H2I

)

= −i√

2HI

πkei k

HI +O(µ2

H2I

), (4.3.29)

∂0

1√ηH

(2)

12

r1− 4µ2

H2I

(kη)

∣∣∣∣∣∣η=−1

HI

=

√2H4

I

πk

(−i− k

HI

)ei k

HI +O(µ2

H2I

). (4.3.30)

For the expansion of the reordered Whittaker functions M (1) and M (2) I will of course makeuse of equation (4.3.12). This quantity and its time derivative may be evaluated at η = 1

HI. After

more simple algebra the result for αradB and βrad

B up to lowest order in µHI

is

αradB = −1

2H2

I

k2+O

(µ

HI

)(4.3.31)

βradB = −1

2H2

I

k2

[1− 2i

k

HI− 2

k2

H2I

]e2i k

HI +O(µ

HI

). (4.3.32)

From this we can see that for superhubble modes at the end of inflation,

βradB ' αrad

B = −12H2

I

k2+O

(µ

HI

). (4.3.33)

Now I have calculated the configuration of the field during the radiation era. As the reader mayremember, I explained that the ground state is given by the configuration αrad

B = 1 and βradB = 0.

By the matching of the solutions in the inflation era and the radiation are, another configurationis given. Hence, we can conclude that particles have been created, by the mismatching of theground states in the different eras.

4.3.3 The spectrum of energy density fluctuations

Now I again use equation (4.1.6) to calculate the spectrum,

PNGT (η, k) =k3

4π2a(η)4

∣∣∣∣∂0BL(η, k) +

a(η)′

a(η)BL(η, k)

∣∣∣∣2 + (k2 + a(η)2µ2)∣∣∣BL(η, k)

∣∣∣2 .(4.3.34)

This time we find that there is an extra dimensionless parameter in which to express the spectrum,k

HI. Both dimensionless parameters, k

HIand µ

HI, always appear together, as

k2

4µHI=

14k2

H2I

HI

µ=

14ξ,

iµHIη2 = ik2η2ξ2.

57

4.3 Dynamics during radiation era CHAPTER 4 QUANTUM FLUCTUATIONS

When I was discussing the solution of inflation, I plotted a renormalised spectrum, a(η)4

k4 PNGT (kη).

If I take the leading order of the matching coefficients αradB and βrad

B (i.e. H2I

k2 ) in equations (4.3.31)and (4.3.32) outside of the brackets in the expression for PNGT (kη), we find that I can now plota(η)4

H4IPNGT (kη).

For a constant value of k, in the limit where η → 0 the mass term vanishes and the solutionof the massless field equation, equation (4.3.9), determines the behaviour of the spectrum. Thespectrum becomes,

PNGT (η, k) =k3

4π2a(η)4|αrad(k)|2

(kη)21 + 2(kη)2 − cos 2kη − 2kη sin 2kη

+O

(µ2

H2I

), (4.3.35)

such that the leading order in the region for small kη is

PNGT (η, k) ∼ k3

4π2a(η)4|αrad(k)|2

(kη)2

+O

(kη,

µ2

H2I

), (4.3.36)

which if of the order a(η)−2. Here I made use of the fact that for small k the matching coefficients,equations (4.3.31) and (4.3.32), are equal: αrad ∼ βrad.

In the limit of η →∞, the field is still well defined. As the mass term becomes dominant, thefield equation becomes equivalent to setting k → 0. In appendix C, section C.2.5, we have

M0,m(ix) = 22mΓ[m+ 1]√xe−

imπ2 Jm

(−x

2

), (4.3.37)

such that the power spectrum has a massive asymptote, using the asymptotic expansion of theBessel function given in appendix C, section C.1.3,

PNGT (η, k) ∝a(η)−3, (4.3.38)

which is the scaling of nonrelativistic matter.Before we can say anything about the meaning of some value of kη, it is necessary to note the

following. Since during the radiation era a(η) = HIη, the Hubble constant H(η) is defined as

H(η) ≡ a(η)′

a(η)2=

1HIη2

. (4.3.39)

This implies that that the Hubble radius is given by

RH ≡ H(η)−1 = HIη2. (4.3.40)

Since the wave vector k is equal to the inverse of the wavelength, k represents a superhubble modeif kRH < 1 and a subhubble mode if kRH > 1. From equation (4.3.40) we find that

kRH = kηa(η), (4.3.41)

such that kη is a superhubble mode if kη < a(η)−1, and kη is a subhubble mode if kη > a(η)−1.Regarding the size of the scale factor during radiation era, which is smaller than the scale factortoday which is normalised to unity, it is justified to say that k is on a superhubble scale if kη < 1,and to say that k is on a subhubble scale if kη 1. The spectrum in the area of kη ∼ 1 showsthe behaviour of the modes at horizon crossing.

In figure 4.3 the spectrum is plotted for several values of ξ. Because of the appearance of theparameter k

HI, figures 4.3 (b–d) can be only interpreted as functions of η, for a specific value of

µk2 . The plot gives the evolution of one mode of the spectrum (the evolution of the share of energycontained in one mode k).

Figure 4.3 (d) is plotted to illustrate that for an extremely massive field, any mode will evolveas a(η)−3. That means that non-relativistic modes may not change their behaviour outside theHubble radius and will not feel that they cross the horizon.

58

CHAPTER 4 QUANTUM FLUCTUATIONS 4.3 Dynamics during radiation era

HcL

0.1 0.2 0.5 1 2 5 10k Η0.01

0.02

0.05

0.1

0.2

0.5

1

2

16 Π2

a HΗ L4

HI

4P NGT

ΜHI

HI2

k2

= 1

HdL

0.1 0.2 0.5 1 2 5 10k Η0.01

0.02

0.05

0.1

0.2

0.5

1

2

16 Π2

a HΗ L4

HI

4P NGT

ΜHI

HI2

k2

= 10

HaL

0.1 0.2 0.5 1 2 5 10k Η0.01

0.02

0.05

0.1

0.2

0.5

1

2

16 Π2

a HΗ L4

HI

4P NGT

ΜHI

HI2

k2

<< 0.1

HbL

0.1 0.2 0.5 1 2 5 10k Η0.01

0.02

0.05

0.1

0.2

0.5

1

2

16 Π2

a HΗ L4

HI

4P NGT

ΜHI

HI2

k2

= 0.1

Figure 4.3: The spectrum during radiation era, plotted for several values ofH2

Ik2

µHI

= ξ. In figure (a) we

see the massless spectrum, which grows as (kη)2 for kη < 1 and as (kη)0 for (kη) > 1. The larger themass with respect to the momentum, the earlier the (kη)0-asymptote turns into a (kη)1-asymptote, evenbeyond kη < 1.

In figure 4.3 (a) we can see that the spectrum doesn’t change with decreasing µ any longeras soon as ξ < 0.1. For the massless case, and for modes for which k2

H2I< µ

HI, we find that

the spectrum PNGT ∝ a(η)−2 and PNGT ∝ k2 on superhubble scales. That is an infrared-savespectrum [42].

On subhubble scales on the other hand, the spectrum reduces to PNGT ∝ a(η)−4 and PNGT ∝k0, with an oscillatory behaviour, decaying as a(η)−5k−1. The scaling with a(η)−4 is just thebehaviour of relativistic matter (massive case) or a radiative field (massless case). The spectrumis reduced with respect to inflation, where we had PNGT, subhubble ∝ k4. We can conclude thatthe particles that are created during the inflation-radiation transition, obey a converging energydensity.

Now let us compare figures 4.3 (a–c). What we see is that for kη 1, i.e. for superhubblescales, the energy in one mode still evolves as a(η)−2 and k2. In these scales the spectrum doesnot differ from the massless spectrum. On subhubble scales (kη 1), a mode will evolve as amassive non-relativistic field, i.e. as a(η)−3. The higher the mass, or the smaller the value of k fora given mode, the earlier will the mode evolve as a massive non-relativistic field. Hence we canconclude: the more massive the field, the faster it switches from a(η)−2 behaviour on superhubblescales to non-relativistic-matter behaviour on subhubble scales. Figure 4.3 (b) shows that there isa transition region between massless behaviour and massive behaviour during the crossover from

59

4.3 Dynamics during radiation era CHAPTER 4 QUANTUM FLUCTUATIONS

1 2 5 10 20 50 100k Η

0.05

0.1

0.5

1

5

10

16 Π2

a HΗ L4

HI

4P NGT

Μ HI Η2

= 10 3

Μ HI Η2

= 10

Μ HI Η2

= 10-2

Hk Η L2

Figure 4.4: The spectrum during radiation era, plotted for several values of µHIη2. For early timesor, equivalently, small mass, the spectrum is that of the massless theory. In that case it is a copy offigure 4.3 (a). As soon as the mass term enters the horizon, the peak of the spectrum lies at k =

õHI

and from the peak on, the spectrum grows as k−1 back to the relativistic spectrum.

superhubble to subhubble scales, in which the spectrum at first behaves as that of relativisticmatter before adjusting to the spectrum of non-relativistic matter.

The evolution of the full spectrum as a function of k is sketched in figure 4.4. The small andlarge k behaviour can be easiest described with respect to the massless theory. In the masslesscase, the spectrum depends only on kη, which leads to the spectrum in figure 4.3 (a). As forsuperhubble kη this figure grows as (kη)2, the spectrum grows as k2, and consequently as k0 onsubhubble scales.

As long as NGT scales as a relativistic field, the super-horizon spectrum (k2η2 < 1) is of theorder ∼ k2/a2. These modes start scaling as non-relativistic matter all together as soon as (inthe field equation) the Hubble constant shrinks below the mass, µ > H(η). That is the case whena(η) =

√HI/µ, such that

Psuper(k) ∼neH

4I (kη)2

16π2a(η)4→ neH

4I k

2

16π2a(η)31

H32I

õ. (4.3.42)

For sub-horizon modes, the field starts scaling massive as soon as (in the field equation) the wavevector shrinks below the mass, µ > k/a(η), when a(η) = k/µ, such that

Psub(k) ∼neH

4I

16π2a(η)4→ neH

4I

16π2a(η)3µ

k(4.3.43)

for H(η) < ka(η) < µ. The peak of the massive spectrum lies at the mode of which the physical

momentum is equal to the mass when it crosses inside the horizon, that is, k2 = a(η)2H(η)2 =a(η)2µ2,

kpeak =µa(ηNR)

=√HIµ. (4.3.44)

We can conclude that on scales which correspond to momenta that remained subhubble duringinflation, the power spectrum in approximation still contains only fluctuations of a conformal

60

CHAPTER 4 QUANTUM FLUCTUATIONS 4.4 Dynamics during matter era

Region Effective field equation Scale

I(∂2

X + 1− 6X2

)BL

mat(X) = 0 X2 4kηe

ξ

II(∂2

X + 1− 6X2 + ξ2

16kηeX4)BL

mat(X) = 0 X2 ∼ 4kηe

ξ

III(∂2

X + ξ2

16k2η2eX4)BL

mat(X) = 0 X2 4kηe

ξ and X3 4√

6kηe

ξ

Table 4.1: The effective field equations for different values of X ≡ kη ≡ k(ηe + η). Note that theseexpressions are written in terms of the previously defined parameter ξ ≡ µHI

k2 .

vacuum. For momenta that were superhubble at the end of inflation particles have been created.This is the result we aimed at, allowing us to focus only on modes which were superhubble at theend of inflation.

4.4 Dynamics during matter era

In matter era the scale factor is defined as

a(η) =HI

4ηe(η + ηe)2. (4.4.1)

The field equation for BL in radiation is given by(∂20 + k2 − 6

(η + ηe)2 + µ2 H

2I

16η2e

(η + ηe)4)BL

mat(η, k) = 0. (4.4.2)

4.4.1 General solution and boundary conditions

When I write η = η + ηe, this equation becomes(∂2

η + k2 − 6η2

+ µ2 H2I

16η2e

η4

)BL

mat(η, k) = 0. (4.4.3)

This differential equation contains a term of the order ∼ η4. Therefore we can not solve itanalytically. What we can do is look at several approximations for the field equation, in differentregions of values of k, µ and η. The field equation can be rewritten as(

∂2X + 1− 6

X2+ µ2 H2

I

16η2ek

6X4

)BL

mat(X) = 0, (4.4.4)

where X ≡ kη. For different regions of X, different terms dominate this equation. I devide thesolution in three regions, one region in which the mass term can be neglected, one region in whichthe mass term is the dominant term, and an intermediate region. The effective field equations arestated in table 4.1. Region I is the region which is equivalent to the massless theory. Region IIwill have to be solved numerically, and the solution in region III can be approximated analytically.Note that for region III there are 2 separate conditions on the size of kη, since the mass term hasto dominate two distinct terms, both 1 and 6

η2 .

61

4.4 Dynamics during matter era CHAPTER 4 QUANTUM FLUCTUATIONS

Region I

When the mass term vanishes, the resulting homogeneous equation is solved by

u0(k, η) = η12H

(1),(2)52

(kη), (4.4.5)

where C1 is an arbitrary constant. Using the previously discussed Bessel equation, I write thegeneral massless solution as

BLmat(η, k) = αmat

B η12H

(1)52

(kη) + βmatB η

12H

(2)52

(kη). (4.4.6)

Boundary conditions for region I

Performing the same method of continuously matching both field solutions at the radiation-to-matter-era transition, we find for the coefficients,

αmatB =

1W [H1,H2]

αrad

B W [M1,H2] + βradB W [M2,H2]

, (4.4.7)

βmatB =

−1W [H1,H2]

αrad

B W [M1,H1] + βradB W [M2,H1]

, (4.4.8)

where I again use the shorthand notation introduced in section 4.3, with

H1 = a(η)η12H

(1)52

(kη),

H2 = a(η)η12H

(2)52

(kη).

By definition the coefficients αmatB and βmat

B satisfy the canonical-quantisation condition, equa-tion (3.3.70),

W [BL(η, k), B∗L(η, k)] = ia(η)2,

like during inflation and radiation era.Calculating the coefficients α and β explicitly, we retrieve the same result as in Ref. [42], up

to a normalisation factor in the general solutions,

αmatB = αrad

B

(4i+

2kηe

− i

2k2η2e

)e−ikηe + βrad

B

i

2k2η2e

e−3ikηe , (4.4.9)

βmatB = αrad

B

−i2k2η2

e

e3ikηe + βradB

(−4i+

2kηe

+i

2k2η2e

)eikηe . (4.4.10)

Region II

The field equation for region II will be used to numerically connect region I to region III. Asthe field has to be continuous at the radiation-to-matter transition, the boundary conditions forthe numerical continuation4 of the field in region II are the value of the field and its derivative atthe time of η = ηe, which we know of course exactly.

Region II, as mentioned above, and resembled by the equations in table 4.1, has a lowerboundary given by

X2 =kηe

ξ. (4.4.11)

4I have used a standard procedure for solving differential equations numerically in Mathematica 5.1/5.2. For anexplicit explanation of the procedure see Ref. [52].

62

CHAPTER 4 QUANTUM FLUCTUATIONS 4.4 Dynamics during matter era

For the upper bound some care has to be taken. Both conditions X2 4kηe

ξ and X3 4√

6kηe

ξ

are equivalent at kηe

ξ = 32 . Hence I demand as an upper bound

X2 = 8kηe

ξfor

kηe

ξ<

32,

X3 = 8√

6kηe

ξfor

kηe

ξ>

32. (4.4.12)

A physical conclusion can be drawn from these bounds. If we take a look at figure 4.3 again,we notice that for large values of ξ, the effect of the mass term becomes dominant earlier in thespectrum. Both conditions on the upper bound of region II lead to a similar effect in matter era:for larger values of ξ, the switch to region III is allowed earlier.

The effective field equations given in table 4.1 are valid for the given values of kη. But in orderto calculate the actual value of the field, the solutions are matched to the field at radiation-to-matter equality. Hence, to give a valid matching coefficient using the effective solution in a givenregion, the boundaries (4.4.11) and(4.4.12) have to be satisfied at the time of equality. If a givenmode lays in region II at the time of equality, it will most certainly grow into region III as timeevolves, since the mass term will only grow bigger and bigger as the scale factor grows. This leadsto the actual bounds for region II when k is fixed,

1ξ< kηe(η) <

8ξ, (4.4.13)

and when η is fixed,

kηe 14µHIη

2e ,

µHIη2e 4

√6. (4.4.14)

Note that the distinction between the two conditions on the upper boundary of region II,for kηe

ξ < 32 and for kηe

ξ > 32 , may be neglected during the numerical calculations. It is more

reasonable to deal with the upper bound of region II as to be the ‘ultra violet’5 bound for regionIII. This is the case as region III is a very rough estimate of the field equation, and regionII may be calculated up to high precision, only increasing the numerical effort of the calculatingdevice. As the lower bound of region III is set by kηe(η) > 8

ξ , both conditions on the value ofkηe on region III (given in table 4.1), are automatically met.

Region III

The field equation in region III becomes(∂2

η + ω(η)2)BL

mat(η, k) = 0, (4.4.15)

where ω2 = µ2 H2I

16η2eη4 = ξ2

16(kηe)2 . If ω′

ω < ω and ω′′

ω < ω2, which is the case for the demandedvalues of kη in this region, we can approximate the general solution by

u1 =1√

2ω(η)e±i

Rdη ω(η), (4.4.16)

where ∫dη ω(η) =

µ2H2I

12η2e

η3. (4.4.17)

5Of course region III itself is in the far infrared, but the region where it connects to region II is relatively violetwith respect to region III.

63

4.4 Dynamics during matter era CHAPTER 4 QUANTUM FLUCTUATIONS

In that case

∂20u1 =

−ω(η)2 +

34

(ω(η)′

ω(η)

)2

− ω(η)′′

2ω(η)

u1

' −ω(η)2u1, (4.4.18)

andW [u1, u

∗1] ' ∓i. (4.4.19)

Now these solutions may in their turn be matched to their values at the radiation-to-matterequality.

Mathematically the calculations for regions II and III are not interesting, since it is just an-other repeating of the process of continuously matching functions that solve continuously matchedequations. Therefore I will not present the full solutions of the matching coefficients, as I willpresent the behaviour the spectrum, depending on the different parameters.

4.4.2 The spectrum of energy density fluctuations

Let me restate expression (4.1.6) to calculate the spectrum,

PNGT (η, k) =k3

4π2a(η)4

∣∣∣∣∂0BL(η, k) +

a(η)′

a(η)BL(η, k)

∣∣∣∣2 + (k2 + a(η)2µ2)∣∣∣BL(η, k)

∣∣∣2 .(4.4.20)

The spectrum as a function of k for a given time in history is shown in figure 4.6. The evolutionof a particular mode k is illustrated in figure 4.5.

As I did for the spectrum during radiation era, we can calculate the asymptotes of the spectrumduring matter era. For µ and k small enough, one finds that αmat = β∗mat, such that for the(massless) early-time infrared region, kη → 0, i.e. region I, for small k the spectrum becomes ofthe order

PNGT (η, k) =k3

4π2a(η)42|αmat|2

π(kη)49 + 4(kη)2 + 2(kη)4 + (−9 + 14(kη)2) cos 2kη + 2kη(−9 + (2kη)2) sin 2kη

∼ k3

4π2a(η)4(kη)4

+O

((kη)4,

µ

HI

)∼ a(η)−2. (4.4.21)

Here I assumed that k is small enough to set αmat = βmat, defined in equations (4.4.9) and (4.4.10).The late-time asymptote, region III, gives the following spectrum,

PNGT (η, k) =k4

4π2a(η)4

∣∣∣∣√ ηe

µHI(kη)2(αmate

iµHI6ηe

(kη)3 + βmat

)∣∣∣∣2

+116

∣∣∣∣∣√ηe(kη)2

µHI

(αmate

iµHI6ηe

(kη)3 + βmat

)∣∣∣∣∣2

∼ k4

4π2a(η)4ηe

µHI

(kη)2

+O

((kη)−1,

µ2

H2I

)∼ ηe

µHIa(η)−3. (4.4.22)

64

CHAPTER 4 QUANTUM FLUCTUATIONS 4.4 Dynamics during matter era

HaL

0.02 0.05 0.1 0.2 0.5k Η

0.0001

0.001

0.01

0.1

1

10

16 Π2 a HΗ L

4

HI

4P NGT

Ξ = 100000.

Ξ = 1000.

Ξ = 10.

HbL

2 5 10 20 50k Η

1

10

100

16 Π2 a HΗ L

4

HI

4P NGT

Ξ = 10

Ξ = 0.1

Ξ = 0.001

Figure 4.5: The spectrum during matter era, plotted for several values of kηe. Figures (a) and (b) representkηe = 0.1 and kηe = 1, respectively.

Now let us remember from chapter A that the scale factor and conformal time with respect tothe situation at ηe, when z = 3230± 200 lead to,

ae

a(η)=z + 1ze + 1

=η2

4η2e

, (4.4.23)

such that

η = 2ηe

√ze + 1z + 1

. (4.4.24)

During matter era the Hubble radius is given by

RH =a(η)2

a′(η)=HI

8ηeη3. (4.4.25)

Hence again, a mode k is on superhubble scales if

kη < 12a(η)

−1, (4.4.26)

and on subhubble scales if

kη > 12a(η)

−1. (4.4.27)

Thus again we can say that a mode k is on superhubble scales if kη < 1 and on subhubble scalesif kη 1, such that the region where kη 1 shows the behaviour of the field at horizon crossing,and the field on the large-kη asymptote of figure 4.6 well before it is on subhubble scales.

A discussion of the matter-era spectra and their evolution is similar to that of radiation era.For very small mass the spectrum remains relativistic, and hence will not differ from the actualmassless theory. The massless spectrum develops a peak on kphys = H(η). The spectrum formasses which were still superhubble at radiation-to-matter equality, again shows a peak on themode for which kphys = H(ηcross) = µ, that is k ∼ (µHI/ηeq)1/3. If the spectrum already wasmassive at the equality, the peak remains at k ∼

õHI . The transition between the values of

the peak lies at the mass for which the spectrum becomes massive just at the equality, such thatηeq ∼ (µHI)−1/2.

One final remark is that the reader must not forget that figure 4.6 show the spectrum rescaledwith a factor a(η)4, such that the highest graph in figure 4.6(a) for instance, is magnified with afactor of about 32304.

65

4.5 Pressure CHAPTER 4 QUANTUM FLUCTUATIONS

HcL

0.01 0.1 1 10 100k Η

0.001

0.1

10

1000

16 Π2 a HΗ L

4

HI

4P NGT

z = 1z = 10z = 1089z = 3230

HdL

0.01 0.1 1 10 100k Η

0.00001

0.001

0.1

10

1000

16 Π2 a HΗ L

4

HI

4P NGT

z = 1z = 10z = 1089z = 3230

HaL

0.01 0.1 1 10 100k Η

0.001

0.1

10

1000

16 Π2

a HΗ L4

HI

4P NGT

z = 1z = 10z = 1089z = 3230

HbL

0.01 0.1 1 10 100k Η

0.001

0.1

10

16 Π2

a HΗ L4

HI

4P NGT

z = 1z = 10z = 1089z = 3230

Figure 4.6: The spectrum during matter era, plotted for several values of µHη2e . Figures (a)–(d) represent

µHη2e = 10−6, 10−2, 1 and 10, respectively.

4.5 Pressure

Similar to the energy density, the pressure of the theory can be calculated as the pressure is givenby the spatial components of the diagonal of the stress-energy tensor,

Tii = −3P. (4.5.1)

If isotropy is assumed, that is kikj = 1

3δij |k|2 for i = j, the expression for the ’pressure per mode’

P becomes,

P (η, k) =−1

24π2a(η)4

[∣∣∣∣∂0BL(η, k) +

a(η)′

a(η)BL(η, k)

∣∣∣∣2

−3k2∣∣∣BL(η, k)

∣∣∣2 − µ2a(η)2∣∣∣BL(η, k)

∣∣∣2] . (4.5.2)

The physical pressure can then be calculated by

P (η) =∫dk

kP (η, k). (4.5.3)

When the field is dominated by relativistic modes, during radiation era this leads to,

P (η, k) =1

4π2a(η)4

[1− cos 2kη

(kη)2− 2 sin 2kη

kη+

23

+43

cos 2kη], (4.5.4)

66

CHAPTER 4 QUANTUM FLUCTUATIONS 4.5 Pressure

10 20 30 40 50k Η

-0.2

0.2

0.4

0.6

0.8

1

a HΗ L4

HI

4P NGT

Μ HI Η2

= 0.01

0.5 1 1.5 2Log@k Η D

-4

-2

2

4

a HΗ L4

HI

4P NGT

Μ HI Η2

= 103

Figure 4.7: The ’pressure per mode’ for both relativistic and non-relativistic modes during radiation era,for µHIη2 = 10−2 and µHIη2 = 103 respectively. Note that the x-axis is linear for the relativistic pressuremodes and logarithmic for the nonrelativistic pressure modes, where the y-axis is linear for both figures.For superhubble scales the pressure scales as − 1

3ρ for both regions, whereas on subhubble scales is averages

around one third of the energy density for the relativistic modes, and around zero for nonrelativistic modes.

such that

P (η) =1

4π2a(η)4

[13

log a(η) +13

log 2 +13γE −

12

+16a2

ina(η)2 +

sin 2a(η)3a(η)

+O(a4

ina(η)4, a(η)−2

)]. (4.5.5)

Covariant conservation of the stress-energy tensor implies

ρ′ + 3a(η)′

a(η)(ρ+ P ) = 0. (4.5.6)

Comparing expressions (5.1.6) and (4.5.5), we find that energy conservation is indeed obeyed orderby order in η.

On superhubble scales, kη 1, the power spectrum and the pressure per mode are dominatedby the first terms in relations (4.4.21) and (4.5.2). Thus one finds that for superhubble modes,both relativistic and non-relativistic, we have P (η, k) ' − 1

3P(η, k), in accordance with the scalingof ρ ∝ a(η)−2.

During inflation and during matter era similar results are found for the relativistic modes,satisfying energy conservation.

This integration must be performed for the massive modes as well. However, for the same rea-sons as for the energy density, the pressure for the massive modes can only be either approximatedor integrated numerically. I do not perform here a complete analysis of the pressure. Instead, Iillustrate the pressure for one specific choice of parameters, and comment on how the pressurebehaves for a general choice of parameters.

In figure 4.7 the ’pressure per mode’ is illustrated for both an effectively massless antisymmetricfield and a massive antisymmetric field. On superhubble scales for relativistic modes, that iskη 1, the pressure per mode behaves as mentioned above as P (η, k) = − 1

3P(η, k). On subhubblescales the relativistic pressure per mode oscillates around ∼ 1

3P(η, k). On both superhubble andsubhubble scales the pressure per mode becomes zero for the massive modes, conserving energywith a density scaling as ρ ∝ a(η)−3.

During inflation and matter era the behaviour of the pressure is consistent with covariantenergy conservation as well. The result is that we have zero pressure at the time of decouplingwhen the field is massive. This is in agreement with the demands for cold dark matter.

67

4.5 Pressure CHAPTER 4 QUANTUM FLUCTUATIONS

68

Chapter 5

Dark-matter energy density

5.1 Energy density

As the mass term in the field equations grows with the scale factor, at a certain moment theenergy density of the field switches from being dominated by relativistic modes to domination bynon-relativistic modes, as previously explained.

When this switch in domination occurs, the energy density of the field starts growing withrespect to the total energy density of the Universe during radiation era.

If we assume that inflation can be described by a scalar inflationary model, such that

H2I =

ρφ

3M2P

, (5.1.1)

where MP = (8πGN )−12 ' 2.4× 1018 GeV is the reduced Planck mass, the energy density of the

radiation field at the end of inflation equals

ρrad(ηE) = 3H2IM

2P . (5.1.2)

The energy density of the nonsymmetric metric field is given by

ρ(η) =∫dk

kPNGT(η, k). (5.1.3)

The only modes that contribute to the actual energy-density spectrum are the modes that areexcited out of the vacuum state. As we saw in radiation era, those are the modes which weresuperhubble at the end of inflation. The smallest momentum is given kphys = HI at the beginningof inflation, the largest momentum by the same equation at the end of inflation. As the variableover which is integrated is kη, the minimum and maximum value are given by aina(η) and aEaη

respectively. Thus the energy density of NGT becomes

ρNGT =∫ aEHI

ainHI

dk

kPNGT

=H4

I

4π2a(η)4

[− 1

4x2+ log x+

cos 2x4x2

+sin 2x

2x− ci2x

]aEa(η)

aina(η)

. (5.1.4)

where x = kη, and with

PNGT(η, k) =H4

I

8π2a(η)4

[(1 +

12

1(kη)2

)− 1

21

(kη)2cos 2kη − sin 2kη

kη

]. (5.1.5)

69

5.1 Energy density CHAPTER 5 DARK-MATTER ENERGY DENSITY

From equation (4.3.1) we have aE = 1, such that we may expand for small ain,

ρNGT =H4

I

4π2a(η)4

[log a(η)− 1

2+ log 2 + γE

+12a2

ina(η)2 +O

(a(η)−2

)+O

(a3

in

)]. (5.1.6)

Here γE denotes the Euler constant. Now we see that the energy density contains a subhubbleterm scaling as a(η)−4, a term scaling as a(η)−4 log a(η) representing the continuous filling up ofthe density on horizon scales, and a small term aina(η)−2 for the superhubble energy density. Weassume the scale of inflation to be large enough to have aina(η) < 1, such that the expansion isindeed allowed.

When the energy density is dominated by non-relativistic modes, the calculation of the energydensity is less trivial. As the field is given in terms of M ik2

4µHI,± 3

4(−iµHIη

2), the integral over all

fourier modes is too complicated to be evaluated exactly. However, we do know the asymptoticbehaviour of the field in the infrared and the ultraviolet regions separately.

In the infrared the leading order of the spectrum is ∝ (kη)2. In the ultraviolet the spectrumbecomes that of the massless theory.

This information together with a closer look at the field equation,(∂20 + k2 − 2

η2+ µ2H2

I η2

)BL

rad(η, k) = 0. (5.1.7)

allows a simple approximation. Note that during radiation era the term ∝ η−2 representsa(η)2H(η)2, where H(η) is the Hubble parameter.

The approximation is the following: each of the second, third and fourth term in equa-tion (5.1.7) has its own region in which it defines the behaviour of the field, but for each value of kthe mass eventually will dominate, leading to a scaling of a(η)−3. Thus the energy density duringthe period of massive behaviour can be given in terms of the energy density during the period ofmassless behaviour.

The superhorizon modes all together are the first ones to start scaling as non-relativistic matter,as soon as the Hubble constant shrinks about below the mass, µ '

√2H(η). That is the case

when a(η) '√HI/µ, such that

PsuperH(k, η) ' PmasslesssuperH (k, η) → a(η)

õ

HIPmassless

superH (k, η). (5.1.8)

For sub-horizon modes, the field starts scaling massive as soon as the wave vector shrinks belowthe mass, µ > k/a(η), when a(η) ' k/µ, such that

PsubH(k, η) ' PmasslesssubH (k, η) → µa(η)

kPmassless

subH (k, η) (5.1.9)

for H(η) < ka(η) < µ. The peak of the massive spectrum lies at the mode of which the physical

momentum is equal to the mass when it crosses inside the horizon, that is, k2 = a(η)2H(η)2 =a(η)2µ2,

kpeak =µa(ηNR)

=√HIµ, (5.1.10)

where ηNR denotes the moment in time of switching from relativistic to non-relativistic for themode kpeak.

70

CHAPTER 5 DARK-MATTER ENERGY DENSITY 5.1 Energy density

Now we can approximate the non-relativistic energy density by

ρNGT =∫ √

µHI

ainHI

dk

ka(η)

õ

HIPmassless

superH (k, η)

+∫ µa(η)

õHI

dkµ a(η)k2

PmasslesssubH (k, η)

+∫ aEHI

µa(η)

dk

kPmassless

subH (k, η)

=H4

I

8π2a(η)3

õ

HI

[B − 2a(η)−1 log

µa(η)HI

− 2√

µ

HI

+O(a2

ina(η)2, a(η)−2, (

√µHIη)−2

)]. (5.1.11)

In this expansion, B is the leading order, which can be determined by a numerical calculation of theenergy density. In our approximation with a sudden relativistic-to-non-relativistic transition, wefind B ' −2

14 +2+2

34 log 2+2

34 γE ' 1.35. The third integral term only contributes if µa(η) < HI .

Of course, when µa(η) > HI , the upper bound of the second integral becomes aEHI , and thelogarithmic term in the energy density becomes nonexistent. This logarithmic term represents thecontribution of the enhanced and relativistic modes. This term damps quickly, and is neglectedin the following. We assume that the scale of inflation is large enough to let aina(η) 1 always,and we already made use of the fact that during the non-relativistic behaviour of the theory√µHIη > 1.

Taking account only of the leading order, we find for the NGT-to-radiation ratio when the fieldhas become massive,

ρNGT

ρrad=ε a(η)

õ

HIfor ε =

H2IB

3πm2P

, (5.1.12)

where m2P = 8πM2

P is the Planck mass squared.In order for the energy density to fully take account of the dark-matter energy density, it has

to satisfy

ρNGT

ρrad= 1 at η = ηe. (5.1.13)

Hence,

ε aeq

õ

HI= 1 (5.1.14)

This gives a rough approximation for the mass,

µ ' HI

ε2a2eq

' Heq

ε2

' 2.8× 10−2

(1013 GeV

HI

)4

eV , (5.1.15)

or

µ−1 ' 7× 10−8

(HI

1013 GeV

)4

m. (5.1.16)

In the last line we used for the Hubble parameter at radiation-matter equality, Heq = (zeq +1)3/2H0, H0 ' 1.4× 10−42 GeV, zeq = 3230± 200 and the scale factor aeq/a0 = 1/(zeq + 1).

71

5.2 Hot or cold Dark Matter CHAPTER 5 DARK-MATTER ENERGY DENSITY

5.2 Hot or cold Dark Matter

Now the question is: is this cold or hot Dark Matter? In order to answer that question, wemust know the momentum at which the spectrum peaks. In the previous chapter we saw thatfor a normalisation in which the scale factor at the end of inflation is equal to unity, we hadkpeak =

õHI . We can now renormalise to a0 = 1, where a0 is the scale factor today. We need

to express scale factors in a normalisation independent way, that is,

a(η1)a0

=a(η1)a0

, (5.2.1)

where η1 denotes conformal time at the inflation-to-radiation-era transition, and a is normalisedsuch that a0 = 1 and a is normalised such that a(η1) = 1. We have

a(η1)aeq

=η1ηeq

=

√aeq′/a2

eq√a(η1)′/a(η)12

=√Heq

HI. (5.2.2)

In the new normalisation the comoving momentum of the peak in power spectrum reads

kpeak =a(η1)a0

õHI

=aeq

õHeq

=õH0

(1 + zeq)14. (5.2.3)

This correspond to a physical momentum today of

k−1peak =2× 107 km ∼ O(AU). (5.2.4)

The result is that the peak in power spectrum lies at a length scale which is about 1017 times aslarge as the inverse-mass length. If the peak where located just a few orders of magnitude beyondthe mass scale, then the matter would have been warm dark matter. This is for example the casefor the thermal spectrum of background neutrinos, where we find,

Ekin,ν ∼⟨mνv

2

2

⟩∼ kBTν , (5.2.5)

Pν

ρν∼Ekin,ν

mνc2> O

(10−4

), (5.2.6)

where I take for the largest neutrino mass a rough upper limit of mν . 1 eV, and Tν is the back-ground neutrino temperature today. Hence, background neutrinos still have a nonzero pressure asthey have a thermal spectrum with a small yet nonzero temperature.

For the nonthermal spectrum of the antisymmetric tensor theory, we find,

Ekin,B ' (~kpeak)2

2mB, (5.2.7)

PB

ρB∼ Ekin,B

mBc2∼ 10−34. (5.2.8)

The antisymmetric tensor field fluctuations are indeed very cold in comparison with the ‘warm’background neutrinos. The length scale of the peak in power spectrum indicates that this mattermay be involved in earlier structure formation then other Dark Matter candidates.

72

Chapter 6

Conclusion

On geometrical grounds there is no reason why a nonsymmetric term should not be added to themetric of general relativity. Moffat et al. have used the nonsymmetric admixture to alter Einstein’sgravity in a geometrical way. The alteration of Einstein’s gravity is inspired by the shortcomings ofthe combination of general relativity and the visible matter content of the Universe. An importantaim of the nonsymmetric theory of gravitation is to provide a solution for the observed state ofthe Universe without adding the ’magical’ dark matter.

What we have seen in this thesis, is that NGT as a geometrical theory does not contain stablesolutions in the framework of cosmology. We then proceeded by abstracting from the geometricalaction all terms containing the antisymmetric admixture which do not lead to instability. In thatway we obtained an action which contains the terms which must at least be present in a stableand dynamical theory of a nonsymmetric metric. That is a reasonable procedure, since a priorithe statement that a nonsymmetric admixture to the metric is allowed, still holds. However, nowwe know that the statement must be accompanied by the notion that the admixture appears inthe action in a non-geometrical way.

After making the theory suited for canonical quantisation, we considered the cosmology of amassive antisymmetric tensor field whose dynamics is given by the action

S =∫d4x

√−G

R− 2Λ− 1

12FµνρF

µνρ +14µ2BµνB

µν

. (6.0.1)

In this form without sources, the massless theory (µ = 0 is dual to the Kalb-Ramond pseudoscalar. The cosmology of the Kalb-Ramond scalar has been extensively investigated. If the theorycouples to sources, the dual scalar field contains non-local source terms. Dualisation represents auseful tool to study the theory only when there are no sources. The same goes for massive NGT(µ 6= 0), which is in no sense dual to the massive Kalb-Ramond axion, which has equally well beenextensively investigated in cosmology. One can show show that massive NGT is dual to a massivevector field, and not to a massive scalar field. The argument that sources transform to nonlocalterms supports our investigation of massive NGT as it is. As such, this investigation has not beendone before and the results are new.

We followed the evolution of the vacuum fluctuations generated during an inflationary epoch,through inflation, radiation and matter era. We find that the antisymmetric field with a mass,µ = 3 × 10−2

(1013 GeV/HI

)4 eV, results in the right energy density today to account for thedark matter of the Universe. This then implies that below the corresponding length scale, µ−1 '0.1 µm

(HI/1013 GeV

)4, the strength of the gravitational coupling may change, as it has beenargued in Refs. [35, 34]. This scale is about two orders of magnitude below the present experimentalbound, which is of the order 10µm [51]. Note that the mass scale (5.1.15) depends strongly onthe scale of inflation, such that, if the gravitational force law below the scale of 0.1 µm remainsunchanged, would imply that, either the B field is nondynamical, or the inflationary scale is lowerthan 1013 GeV. Note that, in contrast to gravitons, the antisymmetric tensor field begins scaling

73

CHAPTER 6 CONCLUSION

as non-relativistic matter during radiation era, making it potentially the most sensitive probe ofthe inflationary scale.

If we now get back to NGT, we see that a theory that initially was hoped to solve the needfor dark matter in the Universe by altering gravity, is itself a very good candidate to be the darkmatter without the ability to change gravity on scales necessary to get rid of dark matter.

The peak in the energy density power spectrum, generated as a consequence of a breakdownof conformal invariance in radiation era, corresponds to a comoving momentum scale of the order,k '

õH0/(1 + zeq)1/4, where zeq ' 3230 is the redshift at matter-radiation equality. Hence

the most prominent matter density perturbations at a redshift z = 10, when structure formationbegins, occur at a scale, k−1

phys ∼ 2× 107 km (corresponding to a wavelength, λphys ∼ 1× 108 km,which is (co-)incidentally the Earth-Sun distance), which may boost early structure formation onthese scales. We therefore expect that, when compared to other CDM models, this type of colddark matter may induce an earlier structure formation. Even though the mass of the antisymmetricfield is quite small (of the order of the heaviest neutrino mass, mν ∼ 0.1 eV), the antisymmetrictensor field dark matter is neither hot nor warm. Indeed, since the antisymmetric tensor field doesnot couple to matter fields, it cannot thermalise and its spectrum remains primordial, and thushighly non-thermal. Hence, in spite of its small mass the field pressure remains small, such thatstructure formation on small scales does not get washed out.

Although the pressure converges to zero as the field becomes dominated by non-relativisticmodes, the pressure per mode in Fourier space shows a characteristic oscillatory behaviour shownin figure 4.7. These oscillations may influence cosmological perturbations in a manner analogousto Sakharov oscillations [1], which in turn may produce an observable imprint in the cosmicbackground photon fluid. This question deserves further investigation.

74

Appendix A

The standard model of Cosmology

This chapter gives the reader a brief introduction in Cosmology. Its goal is not to derive the fullevolution of the Universe, as it is to explain the terms and eras discussed in this thesis in chapter 4.The contents are mainly based on References [28, 41, 9].

A.1 Spacetime

In general relativity the infinitesimal line element contains the curvature of a spacetime, as oneneeds to deal with the curvature of space when one is calculating distances. As usual, this can bewritten as

ds2 = dxµdxµ, (A.1.1)

where the metric tensor (gµν) is the entity that maps a vector to its dual vector:

dxµ = gµνdxν . (A.1.2)

The metric tensor is often addressed as the metric, although officially the metric is the infinites-imal line element. In general relativity the metric tensor is by definition a symmetric tensor.Equation (A.1.2) defines how the metric (tensor) raises and lowers indices.

Due to curvature, the partial derivative (∂µ) no longer transforms as a vector, but the covariantderivative,

∇µVν = ∂µVν − ΓλνµVλ, (A.1.3)

does. The quantity Γλνµ is called the connection.

The change of a vector Vν when it is parallel transported along a direction xµ is then given bythe quantity ∇µVν .

The Riemann tensor is defined as the quantity that resembles the curvature. In this sense,curvature is quantified by the rotation of a vector, when it is parallel transported from one pointto another along two different paths. The rotation is then calculated by

[∇µ,∇ν ]V ρ = RρλµνV

λ.

The Riemann tensor reads

Rρλµν = ∂µΓρ

λν − ΓσλµΓρ

σν − ∂νΓρλµ + Γσ

λνΓρσµ. (A.1.4)

As the covariant derivative of a p-form gives us a p+1-form, we find the condition not onlythat the derivative transforms as a vector itself, but also the condition of metric compatibility,

gµν∇αTνρσ = ∇αTµ

ρσ. (A.1.5)

That is,∇αgµν = 0. (A.1.6)

75

A.1 Spacetime CHAPTER A THE STANDARD MODEL OF COSMOLOGY

When we take the sum of three even permutations of this equation, we are led to(∂αgρβ − gλβΓλ

ρα − gρλΓλαβ

)−(∂ρgβα − gλαΓλ

βρ − gβλΓλρα

)−(∂βgαρ − gλρΓλ

αβ − gαλΓλβρ

)=

∂αgρβ − ∂ρgβα − ∂βgαρ + Γλβρ (gλα + gαλ)− Γλ

ρα (gλβ − gβλ)− Γλαβ (gρλ − gλρ) ,

(A.1.7)

yielding,Γλ

βρ = gλα 12 (−∂αgρβ + ∂ρgβα + ∂βgαρ) . (A.1.8)

Hence the metric defines the connection and the Riemann tensor, thereby containing all informa-tion about the spacetime manifold.

By requiring that in the weak field nonrelativistic limit Newton’s theory of gravitation isrecovered, Einstein derived the following equations, referred to as Einstein’s equations,

Rµν − 12gµνR =

8πGN

c4Tµν . (A.1.9)

In my units in which c = 1 and 8πGN = 1, this simplifies to

Rµν − 12gµνR = Tµν . (A.1.10)

Here Tµν is the stress energy tensor. Note that the stress energy tensor has a mass dimensionof four, whereas the Ricci quantities have a dimension of two and the metric is dimensionless.Equation (A.1.10) would be dimensionally inconsistent, if we would not keep track of the numberof Newtons constants that are actually set to unity. Now we have a spacetime, given shape by itsenergy content.

The Universe and geometry

The Copernican Principle states that the Universe is the same everywhere and therefore that thereis no reason to expect our solar system to be in the centre of the Universe. On small scales thisis far from the truth, for example compare the centre of the sun with an ordinary living room.However, on large scales it is a good assumption, as we can see when we gaze at the stars. Withinthis principle, it is a good assumption that the laws of nature are the same everywhere. Add tothat the fact that the Universe appears isotropic from our point of view, and we are led to theconclusion that the Universe is homogeneous. This is a conclusion which has consequences forthe space dependence of our description of space and time in general relativity, leading us to thegeneral metric,

ds2 = dt2 − a(t)2(dx2 + dy2 + dz2), (A.1.11)

orgµν(x) = diag[1,−a(t),−a(t),−a(t)]. (A.1.12)

This is the Friedmann-Lemaıtre-Robertson-Walker metric (FLRW metric). The quantity a(t)is called the scale factor, as it determines the rescaling of spatial sections of the Universe withtime. Solving Einstein’s equations for this metric, leaves no static solution for the Universe ifit has nonzero matter content. Believing that the Universe was static, Einstein introduced thecosmological constant to establish a static solution. Then Einstein’s equations became,

Rµν − 12gµνR+ gµνΛ = Tµν . (A.1.13)

Einstein’s equations for the FLRW metric then become

H2 ≡(a

a

)2

=13ρ+

Λ3

(A.1.14)

a

a= −4

3(ρ+ 3p) +

Λ3, (A.1.15)

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CHAPTER A THE STANDARD MODEL OF COSMOLOGY A.1 Spacetime

often called the FRLW equations. The parameter H is called the Hubble parameter.At a later time, Einstein called this the biggest blunder of his life. Hubble had discovered

profound evidence that the Universe was not static but expanding, and Einstein spoiled the op-portunity of predicting the expansion. This expansion is represented by increasing redshift formore distant objects in the Universe, hence the previously mentioned Hubble parameter. Yet,present observations of distant supernovae [43, 39, 44] indicate that the Universe is accelerating.This can only be the case either if the active gravitational energy density is negative, where theactive energy density is

ρact = ρ+ 3P, (A.1.16)

where ρ is the energy density and P is the pressure of the matter, or if the cosmological constant ispositive. Since we have not yet seen any matter with negative gravitational mass in laboratories,we assume the latter condition.

When we denote the Hubble parameter, aa , for today’s Universe by H0, we can transform

equations (A.1.14) and (A.1.15) to

1 = Ωm + ΩΛ (A.1.17)

q0 = 12Ωm − ΩΛ, (A.1.18)

where we at the moment (still) neglect spatial curvature in the Universe. Here I have assumedthe matter content of the Universe to be dominated by pressureless particles, both relativistic andnonrelativistic, ρ ' ρm and pm = 0. Now we have

Ωm =ρm

3H20

, (A.1.19)

ΩΛ =Λ

3H20

, (A.1.20)

q0 = − a(t0)a0

. (A.1.21)

According to the previously mentioned measurements, the acceleration of the expansion isgiven by q0 ≈ −0.6, such that we are led to [5],

Ωm ≈ 0.27, (A.1.22)

ΩΛ ≈ 0.73. (A.1.23)

Conformal flatness

In general relativity, flat space is characterised by the Minkowsky metric, ηµν =diagonal(+,−,−,−).Any curved spacetime is said to be conformally flat if it can be written as

gµν = Ω(x)2ηµν . (A.1.24)

In the case of the FLRW-metric, equation (A.1.12), the metric can be written as in equation (A.1.24)by the substitution dη = dt/a(t),

ds2 = dt2 − a(x)2(dx2 + dy2 + dz2

)= a(x)2

(dη2 − dx2 − dy2 − dz2

). (A.1.25)

This is usually written asgµν = a(η)2ηµν , (A.1.26)

where I substituted t in a(t) by η = η(t). The parameter η is called the conformal time coordinate.

77

A.1 Spacetime CHAPTER A THE STANDARD MODEL OF COSMOLOGY

When a spacetime is conformally flat, one can perform a transformation to co-moving coordi-nates, to get

gµν → Ω(x)−2gµν = gµ′ν′ = ηµ′ν′ . (A.1.27)

In the case of the FLRW-metric this means that x = (η, ~x) → x = (η, ~x) = (a(η)η, a(η)~x). Theuse of conformal flatness is that in this spacetime some theories will possess conformal invariance.This then leads to a natural generalisation from properties the theory would have in flat space tothe properties is has in a conformal flat space.

Spatial curvature

The WMAP project [5], next to the results cited in subsection A.1, gives a measure for the totalenergy density, which is given by

Ωtot = 1.02± 0.02. (A.1.28)

Previously I defined Ωtot to be equal to one, as I chose a spatially flat metric. If one allows anisotropic curvature in the metric, the metric becomes, in spherical coordinates,

ds2 = dt2 − a(t)1− kr2

dr2 − a(t)2r2(dθ2 + sin2(θ)dφ2). (A.1.29)

Then the FRLW equations become

1 = Ωm + ΩΛ −k

H20

(A.1.30)

q0 = 12Ωm − ΩΛ. (A.1.31)

Often this is recast as

Ωtot = 1− Ωk, (A.1.32)

Ωk =k

H20

. (A.1.33)

The quantity Ωk is taken apart from Ωtot, since it does merely resembles an energy density con-tribution but it is actually not an energy density.

Due to the WMAP results, we can not exclude spatial curvature in the full theory. However therole of spatial curvature is much magnitude smaller then the other contributions to the equationof state, ΩΛ and Ωmat, in this thesis it is therefore neglected.

Matter

WMAP also gives a constraint on the amount of baryonic matter in the Universe,

Ωb = 0.044± 0.004, (A.1.34)

Ωb

Ωm= 0.17± 0.01. (A.1.35)

Some other densities are known, the density background neutrino’s and background photons, theorigin of which will be explained below,

Ωγ ≈ 0.000043± 0.004, (A.1.36)

Ων < 0.015, (A.1.37)

Adding all known matter densities, we are left with a lack of

Ωdm = Ωm − Ωγ,ν,b ' 0.23± 0.04. (A.1.38)

This matter causing this extra matter density is called dark matter, since it apparently does notinteract with electromagnetic radiation. If it would, we would have seen it by now.

78

CHAPTER A THE STANDARD MODEL OF COSMOLOGY A.1 Spacetime

A thermal history of our spacetime

Extrapolating back our present knowledge of the Universe, leads us to the theory in which theUniverse originates from a singularity in spacetime, when all space was reduced to a minimal size:the Big Bang. What happened before the Universe reached its minimal size is as yet subjectto thorough investigation, however whether we live in a bouncing Universe or a Universe whichresides from a singularity is not of importance to this thesis. We know that the energy content ofthe Universe is made almost homogeneous during inflation, such that all matter is in its vacuumstate during inflation.

With present theories we could just naively extrapolate back as far as to the Planck epoch (t ∼10−43s and T ∼ 1019GeV), where quantum corrections to general relativity become important.Combining what we know of field theory at ultra high densities, we know that if such a BigBang happened, the Universe shortly after the Big Bang consisted of a rapidly cooling plasma ofrelativistic particles such as quarks, leptons, gauge bosons, Higgs bosons and so on. This plasmawould then be in near thermal equilibrium, due to the interactions between all present particles.All relativistic matter energy density scales with the scale factor as

ρ =ρ(t0)a4

, (A.1.39)

hence this era is denoted as the radiation dominated era, or radiation era. Equation (A.1.14) tellsus that in this era the scale factor behaves as

a ∼ 1a,

a(t) ∼ t12 . (A.1.40)

As the Universe expands and the plasma cools down, interaction lengths increase, leading todecoupling of the various fluids in the Universe. Apart from gravitons, the first fluid to decouple,which we should be able to measure today, is that of neutrinos, whose interaction is the weakestof present known particles. The decoupling happens at a temperature of about T ' 1MeV, ata time t ' 1 s. This leaves the Universe with two distinct fluids with their own temperature,the fluid of neutrinos and the fluid of other massive matter and photons. The difference intemperature of present background neutrinos and background photons stems from the fact thatafter the decoupling of neutrinos, the photon temperature increases again due to recombinationprocesses.

As the baryonic matter and the dark matter cools down, it becomes less relativistic, and startsscaling as nonrelativistic matter,

ρdm,b =ρdm,b(t0)

a3. (A.1.41)

Eventually the radiative energy density drops below the non-relativistic matter energy density.Therefore this era is called the matter era. Equation (A.1.14) tells us that in this era the scalefactor behaves as

a ∼ 1√a,

a(t) ∼ t23 . (A.1.42)

Cooling down further, massive matter starts combining to protons and neutrons we know today.Next another decoupling occurs: the decoupling of photons from electrons. This is the last

moment at which the primordial photons scatter off electrons. The last scattering happens at atemperature of T ' 0.26eV, at a time t ' 8×1012s. The fluid of photons, which can be treated asinteraction free, remains present today. Now it has a black-body spectrum with a temperature of2.725± 0.002 ' 2.35× 10−4eV [5], and is the source of oldest information about the Universe wecan measure directly, since al earlier information is wiped out by scattering in the thermal plasma.It is detected as cosmic microwave background radiation, CMBR.

79

A.2 Horizons CHAPTER A THE STANDARD MODEL OF COSMOLOGY

Structure formation

If the Universe would be filled with a homogeneous distribution of energy density, the Universewould be in perfect gravitational equilibrium. However, something must have caused the Universeto roll of this instable equilibrium, to form the structures present today. If we look at the CMBR, itappears homogeneous on large scales, but on small scales it shows small inhomogeneities, indicatingsmall inhomogeneities in the energy density at the time of decoupling. Computational models showthat these inhomogeneities are good enough for causing structure formation with the characteristicsit shows today [23, 26, 38]. The question remains, what caused these inhomogeneities in theCMBR?

A.2 Horizons

For a photon the infinitesimal line element is equal to zero, ds2 = 0. In spherical coordinates,with θ = φ = 0, this implies

dt2 = a(t)2dr2, (A.2.1)

such that the time it takes for a photon to travel from the coordinate r = 0 to the coordinater = r0 is determined by ∫ t

0

dt′

a(t′)=∫ r0

0

dr. (A.2.2)

On the other hand, the distance between these coordinates is given by

d(t) =∫ r0

0

√grrdr = a(t)r0 (k = 0). (A.2.3)

Thus we find for the distance a photon can travel in a time t,

d = a(t)∫ t

0

dt′

a(t′). (A.2.4)

Writing in terms of conformal time η, we have,

dH(t) = a(η(t))∫ η(t)

η(0)

dη′ = a(η(t))η(t)− η(0). (A.2.5)

A more useful expression would be the horizon distance in terms of known parameters. Insertingdt = dt

dada = 1ada, we have

dH = a

∫ a

0

da′

a′a′. (A.2.6)

By virtue of the FRLW-equations and the scalings of the different energy densities, we have

a2 = Ωγa20

a2+ Ωm

a0

a. (A.2.7)

so we can write, if we set a0 = 1,

dH(a) =a

H0

∫ a

0

da′√Ωγ + Ωma′

. (A.2.8)

In a matter dominated Universe, Ωm = 1 and Ωγ = 0, a =(

tt0

) 23

and a0 = 1, we find

dH(a) =2a

32

H0, (A.2.9)

80

CHAPTER A THE STANDARD MODEL OF COSMOLOGY A.2 Horizons

such that, if we assume the Universe has been matter dominated always, today (a0 = 1) theparticle horizon is

dH =2H0

= 2RH(t0). (A.2.10)

Here I introduced the quantityRH ≡ H−1, (A.2.11)

called the Hubble radius. The assumption that the Universe has been matter dominated is justifiedfor the calculation of the particle horizon, since the matter-radiation energy density equalityhappened before t ∼ 1012s, while the age of the Universe today is estimated at t0 ∼ 4.3 × 1017s.Changing the scale factor for the first 1012s, to be the radiation era scale factor, would hardlychange the result.

At the end of radiation era, the particle horizon was

dH(aeq) =1Heq

a2

a2eq

, (A.2.12)

where it is constructed such that it matches to the scale factor at the end of radiation era. Henceat that time the particle horizon is equal to the Hubble radius,

dH(aeq) = RH(aeq) = H−1eq . (A.2.13)

The scale factor and redshift

As mentioned earlier, the expansion of the Universe is resembled by a redshift of emitted photons.The simplest way of understanding, is by looking at the photons as being stretched along theirway. The further away a galaxy is from us, the more its photons are redshifted before they reachus. As in a curved space no unambiguous notion of universal distance and time exists, it is mostaccurate to express the distance to such a galaxy purely in terms of its redshift.

As the reader may remember, the age of the information we receive from an object, increaseswith distance. This is a somewhat dangerous statement, since there is no universal notion of time.Actually I should say, the larger the distance to an object, the younger the Universe is at the timeof transmission of the information from the spacetime position of that object. As the Universe isyounger, its scale factor is smaller.

The redshift z is given by the change in wavelength,

λ0

λ1≡ 1 + z, (A.2.14)

where λ0 is the laboratory wavelength, and λ1 is the measured wavelength. Since we know whichprocess causes the redshift, we can say

a(t0)a(t1)

=λ0

λ1≡ 1 + z, (A.2.15)

giving us for a measurement today (t = t0 and a0 = 1),

a(t1) =1

z + 1, (A.2.16)

i.e. the scale factor at an observed coordinate is a factor 1+ z smaller than the scale factor today.

Conformal time and redshift

When we assume that the Universe is still matter dominated today, we can find a relation betweenthe conformal time coordinate η and the redshift of a given observation, from equation (A.2.15).

81

A.2 Horizons CHAPTER A THE STANDARD MODEL OF COSMOLOGY

First let us express the scale factor in radiation and matter era in terms of conformal time, usingdη = dt/a(t),

arad(η) = C1η (A.2.17)

amat(η) =C1

4ηe(η + ηe)2. (A.2.18)

Here rad and mat denote the quantity during the radiation era and matter era respectively. Theconstant C1 at this moment is an arbitrary scaling, which shall be defined in a narrower sense lateron. The conformal time at the moment of transition from radiation era to matter era is denotedby ηe, by virtue of which I have constructed the scaling of the scale factor during the matter erasuch that the scale factor and its first derivative match continuously at the radiation-to-mattertransition.

With this knowledge of the scale factor, we can rewrite equation (A.2.15),

a(η0)a(η1)

=z1 + 1z0 + 1

=η20

η21

, (A.2.19)

where η = η+ηe. Substituting today’s redshift, z=0, and choosing today’s conformal time, η0 = 1,we have,

η1 =√

1z1 + 1

. (A.2.20)

Horizon problem

The WMAP measurements tell us the redshifts for equality and decoupling,

zeq = 3233, (A.2.21)

zdec = 1089. (A.2.22)

Hence we find

aeq =a0

3234, (A.2.23)

adec =a0

1090. (A.2.24)

Thus, as one would expect, the comoving particle horizon at the time of decoupling was

dH(tdec) =dH(t0)1090

53≈ dH(t0)× 10−5. (A.2.25)

Now take a look at the sky. The CMBR spectrum from two opposite directions hardly differs,up to the dipole caused by earth’s speed in the CMBR rest frame. But the two regions wherethese signals come from, lie today at a distance of twice the particle horizon from each other. And,as equation (A.2.25) tells us, these were not in causal contact at the moment the spectrum of theCMBR was imprinted. More important, extrapolating back up to the spacetime singularity (thebig bang), we find that these regions have never been in causal contact. How can it be then, thatthese regions show the same initial conditions and laws of physics? This problem is addressed asthe horizon problem.

This problem could be solved if a mechanism were found which lifts the particle horizon abovethe Hubble radius, equations (A.2.13) and (A.2.13). Such a mechanism exists: inflation.

82

CHAPTER A THE STANDARD MODEL OF COSMOLOGY A.3 Inflation

A.3 Inflation

To solve the horizon problem, we have to inflate the particle horizon with respect to the hubbleradius. The particle horizon will grow as long as time is running, but the Hubble radius isdetermined by the behaviour of the scale factor. Several expressions for the scale factor obey thecondition that the Hubble radius increases slowly with respect to the particle horizon, the mostelegant of which is that of a constant Hubble radius,

RH = H−1I = constant. (A.3.1)

Hence we find,

a

a= HI

a = aineHIt. (A.3.2)

Here ain is the initial scale factor. This solution is called de Sitter inflation, since it can beparametrised as a solution of a quasi de Sitter space. The particle horizon becomes

dH = a(t)∫ t

0

dt′

a(t′)

= eHIt

∫ t

0

dt′

eHIt′

=1HI

(eHIt − 1

)= RH

(eHIt − 1

). (A.3.3)

Hence, if HI is such that HIt > 1 the particle horizon grows larger than the Hubble radius.

Other problems solved

Inflation is not only a successful model for solving the horizon problem, it also solves an impressivelist of problems.

• It gives a reason for why the Universe is so big and old.

• It gives large scale homogeneity and isotropy a causal nature.

• It creates a Universe that is as flat as ours from less flat initial conditions.

• It dilutes any cosmic relics that may exist from phase transitions.

• And, it gives raise to CMBR perturbations.

Quantum fluctuations

The last of these is a consequence of solution to the horizon problem. As it is in fact the phe-nomenon I investigate for the nonsymmetric metric field, I will explain the concept in more detail.Quantum fields fluctuate. On average we have a theory in equilibrium, for example in its classi-cal limit. In order to fluctuate around an equilibrium, distributed over space, the field at pointx must ’know’ what is happening at point y. As information travels with the speed of light, afluctuation at point y induces a fluctuation at point x, such that on average the field is still inits equilibrium. The averaging procedure not only averages over space, but over time as well, justlike the amplitude of all the points on the string of a violin is on average equal to zero. Nowwhat happens if a quantum fluctuation at a specific scale grows out of the Hubble radius due to

83

A.3 Inflation CHAPTER A THE STANDARD MODEL OF COSMOLOGY

inflation? Remember that the information from an event happening in radiation era or matterera, can not travel beyond the Hubble radius. In terms of the violin string, what happens if thevibrating string suddenly cant see or feel its other end? I could try to give some explanation usinga metaphor, but that would always go at the cost of giving wrong explanations for certain details.What must be clear, is that vacuum fluctuations are influenced by the Hubble scale.

The quantum field behaves as follows: as the wavelength of a fluctuation mode grows outsidethe horizon, this often influences the field equation in such a manner that the time evolution ofsuch a mode changes. For both scalar and tensor perturbations the two-point correlations stopdecaying once they grow superhubble. This process is called ’freezing in’. The process gives rise toenhanced energy densities on superhubble scales. Eventually these quantum fluctuations becomebig enough to be measured as classical stochastic energy fluctuations in the field.

As the mode re-enters the horizon at later times, when the scale of the fluctuation modedoes not grow faster than the Hubble radius (in radiation or matter era), the mode will againstart evolving as a subhubble fluctuation. However, due to the period at which the fluctuationwas superhubble, the energy density contains is excited with respect to the subhubble vacuumstate. This give rise to measurable energy density fluctuations. These fluctuations have influenceon the whole state of the Universe through Einsteins equations. The fluctuations create smallinhomogeneities in the spacetime, seeding the gravitational potentials, creating an inhomogeneousenergy density. These affect the CMBR homogeneity, where on subhubble scales one will mainlysee ’ordinary’ quantum fluctuations, and on superhubble scales one will identify the freezed-inenergy density fluctuations from earlier eras.

84

Appendix B

Massive antisymmetric field action

B.1 Equations of motion

B.1.1 Covariant way of writing

With the following action,

S =∫d4x

√−G

R− 2Λ− 1

12FµνρF

µνρ +14µ2BµνB

µν

, (B.1.1)

the field equations for Bµν can be derived by varying this action with respect to δBµν and de-manding this variation to be equal to 0:δ

δBµνS = 0

=δ

δBµν

∫d4x

√−G

R− 2Λ− 1

12FαβγF

αβγ +14µ2BαβB

αβ

=∫d4x

√−G

− 1

12∂

∂BµνFαβγF

αβγ +12µ2Bαβ

∂

∂BµνBαβ

=∫d4x

√−G

− 1

12∂

∂BµνFρστFαβγg

ραgσβgτγ +12µ2gραgσβBαβ

∂

∂BµνBρσ

=∫d4x a(η)4

− 3

12a(η)−6ηραησβητγ ∂

∂Bµν(∂ρBστ )Fαβγ +

12µ2a(η)−4ηραησβBαβ

∂

∂BµνBρσ

=∫d4x

− 6

12ηραησβητγa(η)−2Fαβγ

∂

∂Bµν∂ρBστ +

12µ2ηραησβBαβ

∂

∂BµνBρσ

=∫d4x

+

12ηραησβητγ∂ρ

(a(η)−2Fαβγ

) ∂

∂BµνBστ +

12µ2ηραησβBαβ

∂

∂BµνBρσ

=

12ηραηµβηνγ∂ρ

(a(η)−2Fαβγ

)+

12µ2ηµαηνβBαβ . (B.1.2)

In the step from line 7 to line 8 in this derivation we used partial integration, where the residingterm a(η)4Fαβγ δ

δBµνBβγ |∞∞ is equal to zero, since we demand the variation of the field to vanish

at infinity. From line 8 to 9 the equality δBβγ

δBµν= (ηβ

µηγν − ηγ

µηβν−)δ(n)(x − x′) leads to evaluating

the integral. The fact that, due to its (anti)symmetries

FµνρFµνρ =

∂µBνρ + ∂νBρµ + ∂ρBµνFµνρ = 3∂µ(Bνρ)Fµνρ, (B.1.3)

85

B.1 Equations of motion CHAPTER B MASSIVE ANTISYMMETRIC FIELD ACTION

and∂

∂BµνFµνρF

µνρ =

a(η)−6ηραησβητγ

(Fρστ

∂

∂BµνFαβγ + Fαβγ

∂

∂BµνFρστ

)=

6a(η)−6ηραησβητγFρστ∂

∂Bµν∂αBβγ , (B.1.4)

was used as well.Equation (B.1.2) gives:

ηρα∂ρ

(a(η)−2Fαµν

)+ µ2Bµν = (B.1.5)

∂ρFρµν −2a′

a3F0µν + µ2Bµν =

a(η)2∂ρFρµν −2a′(η)a(η)

F0µν + µ2a(η)2Bµν = 0. (B.1.6)

Acting on equation (B.1.5) with ηµλ∂λ gives the following consistency equation (again using thefact that Fµνρ is antisymmetric under even permutations),

ηλµηρα∂λ∂ρ

(a(η)−2Fαµν

)+ µ2ηλµ∂λBµν =

0 + µ2∂µBµν = 0. (B.1.7)

B.1.2 Rewriting the EOM as ~E and ~B

It appears that when Bµν is rewritten as

B0i = −Bi0 = Ei (B.1.8)

Bij = −Bji = −εijkBk, (B.1.9)

the equations of motion (EOM, i.e. field equations (B.1.6)) will become the following, for (ρ, σ) =(i, j):

∂0F0ij − ∂kFkij −2a′(η)a(η)

F0ij + µ2a(η)2Bij =

−εijk∂20Bk + ∂0∂jEi − ∂0∂iEj + εijl∂

2kBl + εkil∂j∂kBl − εkjl∂i∂kBl +

2a′(η)a(η)

εijk∂0Bk −2a′(η)a(η)

∂jEi +2a′(η)a(η)

∂iEj − µ2a(η)2εijkBk =

~εij

−∂2 ~B − ∂0

(~O× ~E

)+ ~O× ~O× ~B +

2a′

a∂0~B +

2a′

a

(~O× ~E

)− µ2a(η)2 ~B

= 0, (B.1.10)

and for (ρ, σ) = (0, i):

− ∂kFk0i −2a′(η)a(η)

F00i + µ2a(η)2B0i =

−∂k (∂kB0i + ∂iBk0 − ∂0Bki) + 0 + µ2a(η)2B0i =

−O2 ~E + ~O(~O · ~E

)+ ∂0

(~O× ~B

)+ µ2a(η)2 ~E = 0. (B.1.11)

86

CHAPTER B MASSIVE ANTISYMMETRIC FIELD ACTION B.2 Tµν

Likewise the consistency equation (B.1.7) is rewritten, for ν = 0 to

∂µBµ0 =

−∂iBi0 =

−~O · ~E = 0, (B.1.12)

and for ν = i to

∂µBµi =

∂0B0i − ∂jBji =

∂0~E − ~O× ~B = 0. (B.1.13)

Combining equations (B.1.10)–(B.1.13) leads to the final resulting equations of motion, fullydefining Bµν :

EL = 0 (B.1.14)(∂2 + µ2a(η)2

)~ET = 0 (B.1.15)(

∂2 − 2a′(η)a(η)

∂0 + µ2a(η)2)BL = 0 (B.1.16)

∂0~ET − ~O× ~BT = 0. (B.1.17)

Here L denotes the longitudinal modes of the fields, and T denotes the transversal modes. Notethat the longitudinal mode BL contains one degree of freedom, where an equation for transversalmodes is in fact an equation for two fields two transverse degrees of freedom. Hence the vectorsigns.

In calculating the above, we made use of the following identities,

∂iBij = −∂iεijkBk = (~O× ~B)j (B.1.18)

∂iB0j + ∂jBi0 = ∂iEj − ∂jEi = εijk(~O× ~E)k. (B.1.19)

B.2 Tµν

B.2.1 Covariant notation

The stress-energy tensor Tµν is defined by the following equality:

2√−G

δS

δGµν= Rµν −

12GµνR+GµνΛ− Tµν = 0, (B.2.1)

where

2√−G

δSEH

δGµν= Rµν −

12GµνR+GµνΛ (B.2.2)

−2√−G

δSmKB

δGµν= Tµν . (B.2.3)

87

B.2 Tµν CHAPTER B MASSIVE ANTISYMMETRIC FIELD ACTION

Therefore

−√−GTµν = 2

δ

δGµν

∫d4x

√−G

− 1

12FλσρF

λσρ +14µ2BσρB

σρ

= 2∫d4x

[− 1

12FλσρF

λσρ +14µ2BσρB

σρ

∂

∂Gµν

√−G

+√−G ∂

∂Gµν

− 1

12FλσρF

λσρ +14µ2BσρB

σρ

]

= 2∫d4x

[− 1

12FλσρF

λσρ +14µ2BσρB

σρ

−12

√−GGαβ

∂

∂GµνGαβ

+√−G

− 3

12FασρFβ

σρ ∂

∂GµνGαβ +

24µ2BασBβ

σ ∂

∂GµνGαβ

]

= 2

124FλσρF

λσρ − 18µ2BσρB

σρ

√−GGµν

+√−G

−1

4FµσρFν

σρ +12µ2BµσBν

σ

(B.2.4)

hence

T00 = −2

124FλσρF

λσρδ00 −18µ2BσρB

σρδ00 −14F0σρF

0σρ +12µ2B0σB

0σ

= −2

124FijkF

ijk − 18µ2BσρB

σρ − 18F0jkF

0jk +12µ2B0iB

0i

, (B.2.5)

and

Tij = −2

124FλσρF

λσρδji −

18µ2BσρB

σρδji −

14FjσρF

iσρ +12µ2BjσB

iσ

, (B.2.6)

where Tµν = GνλTµλ, GµλG

λν = δνµ and of course δ00 = 1.

B.2.2 Rewriting T00 as ~E and ~B

Continuing from equation (B.2.5), we insert

FijkFijk = −6a(η)−6 (∂iBj)

2, (B.2.7)

F0jkF0jk = 2a(η)−6

(∂0~B +∇× ~E

)2

, (B.2.8)

BρσBρσ = 2a(η)−4

(− ~E2 + ~B2

), (B.2.9)

B0σB0σ = −a(η)−4 ~E2. (B.2.10)

Then we have,

T00 =

a(η)−4

2

a(η)−2

4(∂iBj)

2 +a(η)−2

4

(∂0~B +∇× ~E

)2

+µ2

4

(~E2 + ~B2

)

=1

2a(η)6

(∂iBj)

2 +(∂0~B +∇× ~E

)2

+ a(η)2µ2(~E2 + ~B2

)(B.2.11)

88

CHAPTER B MASSIVE ANTISYMMETRIC FIELD ACTION B.2 Tµν

B.2.3 Tij as a function of BL

Next to the identities (B.2.7)–(B.2.10), we can write in momentum space,

FiµνFjµν = a(η)−6δjm

[−2kikm

∣∣BL(η, k)∣∣2 + 2

∣∣∂0BL(η, k)

∣∣ δim − 2kikm

k2

∣∣∂0BL(η, k)

∣∣] ,(B.2.12)

BiµBjµ = a(η)−4δjm

[∣∣BL(η, k)∣∣2 δim − kikm

k2

∣∣BL(η, k)∣∣2)] . (B.2.13)

Here I made use of Bi = ( ~BL + ~BT )i such that when we neglect BT , we have Bi = ki(~k · ~B)/|k|.If we assume isotropy, i.e. kik

j = 13δ

ji for i = j, we find

Tij = a(η)−6

[− 1

12k2∣∣BL(η, k)

∣∣2 − 112

∣∣∂0BL(η, k)

∣∣2 + 112µ

2a(η)2∣∣BL(η, k)

∣∣2]= − 1

3T00 + 1

6µ2a(η)2

∣∣BL(η, k)∣∣2 . (B.2.14)

89

B.2 Tµν CHAPTER B MASSIVE ANTISYMMETRIC FIELD ACTION

90

Appendix C

Special functions

C.1 Bessel functions

This section will give a short description of the characteristics of Bessel functions, abstracted from[17]. A Bessel function is the solution of the differential equation

d2Fν

dz2+

1z

dFν

dz+(

1− ν2

z2

)Fν = 0. (C.1.1)

C.1.1 General solutions and their Wronskians

The most general solution is

Jν(z) =zν

2ν

∞∑k=0

(−1)k z2k

22kk!Γ(ν + k + 1)(| arg z| < π), (C.1.2)

where Jν is called the Bessel function of the first kind.When two distinct functions form a solution to the same differential equation, there is a quantity

called the Wronskian, defined as

W [f1(z), f2(z)] = f1(z)d

dzf2(z)− f2(z)

d

dzf1(z). (C.1.3)

If the Wronskian of two solutions is nonzero, the solutions are linearly independent. In the caseof Jν and J−ν the Wronskian is

W [Jν(z), J−ν(z)] =2πz

sin νπ. (C.1.4)

Hence for noninteger ν, Jν and J−ν form a complete set of solutions. Throughout the literaturedifferent combinations of both functions are used, namely the Bessel function of the second kind

Nν(z) =1

sin νπcos νπJν(z)− J−ν(z) (for nonintegral ν, and | arg z| < π), (C.1.5)

and the Bessel Functions of the third kind, also called the Hankel’s functions

H(1)ν (z) = Jν(z) + iNν(z) (C.1.6)

H(2)ν (z) = Jν(z)− iNν(z). (C.1.7)

The following Wronskians can be calculated,

W [Jν(z), Nν(z)] =2πz

(C.1.8)

W[H(1)

ν (z),H(2)ν (z)

]=

−4iπz

. (C.1.9)

91

C.1 Bessel functions CHAPTER C SPECIAL FUNCTIONS

C.1.2 Rewriting the equation to the desired form

When in equation (C.1.1) Fν(z) is replaced by z−αzαFν(z), after some algebra the followingidentity is found,

d2z−αzαFν

dz2+

1z

dz−αzαFν

dz+(

1− ν2

z2

)z−αzαFν = (C.1.10)

z−α

[d2

dz2+

1− 2αz

d

dz+(

1 +α2 − ν2

z2

)]zαFν(z) = 0. (C.1.11)

Likewise, when equation (C.1.11) is multiplied from the left with zα, and afterwards z is rewrittenas βz, we find [

d2

d(βz)2+

1− 2α(βz)

d

d(βz)+(

1 +α2 − ν2

(βz)2

)](βz)αFν(βz) = (C.1.12)

βα−2

[d2

dz2+

1− 2αz

d

dz+(β2 +

α2 − ν2

z2

)]zαFν(βz) = 0. (C.1.13)

C.1.3 Asymptotic expansions of the Bessel functions

For large values of |z| we have [17],

H(1)ν (z) =

√2πz

ei(z−π2 ν−π

4 )

i

[n−1∑k=0

(−1)k

(2iz)k

Γ(ν + k + 1

2

)k!Γ

(ν − k + 1

2

) + θ1(−1)n

(2iz)n

Γ(ν + n+ 1

2

)k!Γ

(ν − n+ 1

2

)] , (C.1.14)

H(2)ν (z) =

√2πz

e−i(z−π2 ν−π

4 )

−i

[n−1∑k=0

1(2iz)k

Γ(ν + k + 1

2

)k!Γ

(ν − k + 1

2

) + θ21

(2iz)n

Γ(ν + n+ 1

2

)k!Γ

(ν − n+ 1

2

)] , (C.1.15)

J±ν(z) =

√2πz

cos(z ∓ π

2ν − π

4

)[n−1∑k=0

(−1)k

(2z)2k

Γ(ν + 2k + 1

2

)(2k)!Γ

(ν − 2k + 1

2

)]

−sin(z ∓ π

2ν − π

4

)[n−1∑k=0

(−1)k

(2z)2k+1

Γ(ν + 2k + 3

2

)(2k + 1)!Γ

(ν − 2k − 1

2

)] , (C.1.16)

(where Re ν > −12, | arg z| < π).

For z real, ν real, and n+ 12 > |ν|,

|θ1| < 1, and |θ2| < 1. (C.1.17)

C.1.4 Recursion relations

More algebra leads to the following recursion relation,

Fν−1(x)− Fν(x)+1 = 2∂xFν(x), (C.1.18)

where Fν(x) denotes any Bessel function. The derivation of this relation is omitted since it wouldnot lead to further understanding of the topic.

92

CHAPTER C SPECIAL FUNCTIONS C.1 Bessel functions

C.1.5 Taylor expansions of the Hankel functions

The Bessel functions are defined as a power series, so by definition we have

H(1),(2)ν (z) = Jν(z)± iNν(z)

= (1± i cot νπ) Jν(z)∓ i

sin νπJ−ν(z)

= (1± i cot νπ)zν

2ν

∞∑k=0

(−1)k z2k

22kk!Γ(ν + k + 1)∓ i

sin νπz−ν

2−ν

∞∑k=0

(−1)k z2k

22kk!Γ(−ν + k + 1),

(C.1.19)

where the upper sign of every ± or ∓ will give H(1)ν (z), and the lower sign will give H(2)

ν (z).

In chapter 4 we are facing ν → 12 + ε2. If we expand up to first order in ε, we will need [17]

1Γ(x+ 1)

=∞∑

k=0

dkxk, (C.1.20)

where d0 = 1 and d1 = γE , and γE = 0.577215664 . . . is Euler’s constant. This gives,

1Γ(1 + ε2 + y)

=1

Γ(1 + y)+O(ε2). (C.1.21)

Also,

sin (12

+ ε2)π = 1− πε2 +O(ε4)

1sin ( 1

2 + ε2)π=

11− πε2

+O(ε4) = 1 + πε2 +O(ε4)

cos (12

+ ε2)π = −πε2 +O(ε4)

cot (12

+ ε2)π =cos(1 + ε2)πsin(1 + ε2)π

= (−πε2)(1 + πε2) +O(ε4) = −πε2 +O(ε4)

x12+ε2 = x

12 (1 + ε2 lnx) +O(ε4) = x

12 +O(ε2).

Then H(1),(2)12+ε2

(z) becomes

H(1),(2)12+ε2

(z) =√z

2

∞∑k=0

(−1)k z2k

22kk!Γ( 32 + k)

∓ i

√2z

∞∑k=0

(−1)k z2k

22kk!Γ( 12 + k)

+O(ε2)

= H(1),(2)12

(z) +O(ε2)

=

√2πz

e±iz

±i+O(ε2). (C.1.22)

93

C.2 Confluent Hypergeometric functions CHAPTER C SPECIAL FUNCTIONS

C.1.6 Special cases

Here some Bessel functions for special values of ν are given, as used throughout the thesis:

J 12(z) =

√2πz

sin z

J− 12(z) =

√2πz

cos z

J 32(z) =

√2πz

(sin zz

− cos z)

J− 32(z) =

√2πz

(− sin z − cos z

z

)H

(1)12

(z) =

√2πz

eiz

i

H(2)12

(z) =

√2πz

e−iz

−i

C.2 Confluent Hypergeometric functions

This section will give a short description of the characteristics of confluent hypergeometric func-tions, abstracted from [46]. The confluent hypergeometric equation reads

xd2y(x)dx2

+ (b− x)dy(x)dx

− ay(x) = 0. (C.2.1)

C.2.1 General solutions and their Wronskians

Equation (C.2.1) is solved by the confluent hypergeometric function,

y1 = 1F1[a; b;x] ≡∞∑

n=0

(a)nxn

(b)nn!, (C.2.2)

where(a)n = a(a+ 1)(a+ 2) . . . (a+ n).

A second solution to can be found,

y2 = x1−b1F1[1 + s− b; 2− b;x]. (C.2.3)

The radius of convergence (rC) of 1F1[a; b;x] can be calculated by

r−1C = lim

n→∞

(a)n+1

(b)n+1(n+ 1)!(b)nn!(a)n

= 0, (C.2.4)

which tells us that 1F1[a; b;x] is an analytic function for every a and x. As a result from equa-tion (C.2.2), 1F1[a; b;x] is not an analytic function when b is equal to a negative integer or zero.However, since

limb→1−n

1F1[a, b, x]Γ(b)

=(a)nx

n

n! 1F1[a+ n; 1 + n;x], (C.2.5)

the modified solution 1F1[a, b, x]Γ(b)

is an analytic function for all a, b and x.

94

CHAPTER C SPECIAL FUNCTIONS C.2 Confluent Hypergeometric functions

The derivative of 1F1[a; b;x] is easily calculated,

d

dx1F1[a; b;x] =

∞∑n=1

a(a+ 1)n−1nxn−1

b(b+ 1)n−1n(n− 1)!=

=a

b1F1[a+ 1; b+ 1;x]. (C.2.6)

Using this definition and the definition of the Wronskian (C.1.3), it can be found that

W [y1(x), y2(x)] = (1− b)exx−b. (C.2.7)

Hence, y1 and y2 are linearly independent solutions of equation (C.2.2) as long as b 6= 1.

C.2.2 The Whittaker function

When we make the following substitution in equation (C.2.2),

y = e12

Rdx(1− b

x )z = x−12 be

12 xz, (C.2.8)

we are left with the equation

d2z(x)dx2

+(−1

4+k

x+

14 −m2

x2

)z(x) = 0, (C.2.9)

where m = 12 (b− 1) and k = 1

2b− a. This is then solved by

z = x12 be−

12 xy1, (C.2.10)

with a = 12 +m− k and b = 1 + 2m. Hence

z1 = Mk,m(x) = x12+me−

12 x

1F1[12

+m− k; 1 + 2m;x] (C.2.11)

z2 = Mk,−m(x), (C.2.12)

andW [z1(x), z2(x)] = −2m. (C.2.13)

The function Mk,m(x) is called the Whittaker function. Again, for equation (C.2.5),Mk,m(x)

Γ(1 + 2m)is

an analytic function for all k, m and x. Throughout the calculations in this thesis some dependentforms of the Whittaker functions are encountered:

M−k,m(−x) = e−iπ2 Mk,m(x), (C.2.14)

M−k,−m(−x) = e−iπ2 Mk,−m(x), (C.2.15)

if x 6= 0. These can be found in [17].

C.2.3 Rewriting the equation to the desired form

Making some more substitutions, which we will not explicitly give because they do not lead tofurther understanding, we can find that

d2y(x)dx2

+A0x+

B0

x

dy(x)dx

+A1x

2 +B1 +C1

x2

y(x) = 0 (C.2.16)

is solved byy(x) = x−Ae−Bx2

1F1[a; b;Cx2] (C.2.17)

95

C.2 Confluent Hypergeometric functions CHAPTER C SPECIAL FUNCTIONS

where

A0 = (4B − 2C),

B0 = (2A+ 2b− 1),

A1 = (4B2 − 4BC),

B1 = −2CA+ 2B(2b− 1) + 4AB + 2B − 4aC,

C1 = A(A+ 2b− 2).

C.2.4 Large k asymptote

In chapter 4.3 we encounter Whittaker function with an index k of the following form,

Mia,m

( xia

)(a, k,m, x ∈ R), (C.2.18)

where a becomes very large. Expanding in the parameter 1a can now be done be taking the

limit a→∞ and maintaining first order corrections. We may write the confluent hypergeometricfunction as Barnes’s integral [46],[17],

Γ(a)Γ(b) 1F1[a; b; z] =

12πi

∫ c+i∞

c−i∞

Γ(−s)Γ(a+ s)Γ(b+ s)

(−z)sds, (C.2.19)

provided that

| arg(−z)| < π, b 6= 0,−1,−2, . . . , c ∈ R. (C.2.20)

96

CHAPTER C SPECIAL FUNCTIONS C.2 Confluent Hypergeometric functions

Then the expansion becomes,

(ia)12+mMia,m

( xia

)= (ia)

12+m

( xia

) 12+m

e−12

xia 1F1[

12

+m− ia; 1 + 2m;x

ia]

= x12+me−

12

xia

Γ(1 + 2m)Γ( 1

2 +m− ia)1

2πi

∫ c+i∞

c−i∞

Γ(−s)Γ( 12 +m− ia+ s)

Γ(1 + 2m+ s)

(− x

ia

)s

ds

= x12+me−

12

xia

12πi

∫ c+i∞

c−i∞

Γ(−s)Γ(1 + 2m)Γ(1 + 2m+ s)

xse−s ln(−ia) Γ( 12 +m− ia+ s)Γ( 1

2 +m− ia)ds

= x12+m

(1− 1

2x

ia

)1

2πi

∫ c+i∞

c−i∞

Γ(−s)Γ(1 + 2m)Γ(1 + 2m+ s)

xs

(1− s2 + 2ms

2ia

)ds+O

(1a2

)

= x12

(1− 1

2x

ia

)1

2πi

∫ +i∞

−i∞

Γ(−s)Γ(1 + 2m)Γ(1 + 2m+ s)

(x

12

)2s+2m

ds

− x122m2ia

12πi

∫ +i∞

−i∞

Γ(−s+ 1)Γ(1 + 2m)Γ(1 + 2m+ s)

(x

12

)2s+2m

ds

− x12

12πi

∫ +i∞

−i∞

Γ(−s)Γ(1 + 2m)Γ(1 + 2m+ s)

(x

12

)2s+2m (−s)2

2iads+O

(1a2

)

=(

1− 12x

ia

)x

12 Γ(1 + 2m)J2m

(2x

12

)− x

322m− 1

2ia1

2πi

∫ +i∞

−i∞

Γ(−s+ 1)Γ(1 + 2m)Γ(1 + 2m+ 1 + s− 1)

(x

12

)2(s−1)+2m+1

ds

− x32

12ia

12πi

∫ +i∞

−i∞

Γ(−s+ 2)Γ(1 + 2m)Γ(1 + 2m+ 2 + s− 2)

(x

12

)2(s−2)+2m+2

ds+O(

1a2

)= x

12 Γ(1 + 2m)

[(1− x

2ia

)J2m

(2x

12

)− x

2ia

(2m− 1)J2m+1

(2x

12

)+ J2m+2

(2x

12

)]+O

(1a2

)(C.2.21)

From step three to four in this derivation we used the equalities

Γ(z + s)Γ(z)

e−s ln z = exp [ln Γ (z + s)− ln Γ(z)− s ln z]

= exp[(z + s− 1

2

)ln(z + s)− (z + s)−

(z + s− 1

2

)ln(z) + z +

112

(1

z + s− 1z

)+O

(1z2

)]

= exp[(z + s− 1

2

)ln(z + s

z

)− s+

112

(1z

(1− s

z

)− 1z

)+O

(1z2

)]

= exp[(z + s− 1

2

)(s

z− s2

2z2

)− s+O

(1z2

)]

= exp[s− s+

s2 − s

2z+O

(1z2

)]

= 1 +s2 − s

2z+O

(1z2

), (C.2.22)

97

C.2 Confluent Hypergeometric functions CHAPTER C SPECIAL FUNCTIONS

from which follows that

Γ( 12 +m− ia+ s)Γ( 1

2 +m− ia)e−s ln(−ia) =

(1− s2 − s

2ia

)es ln( 1

2+m−ia)e−s ln(−ia) +O(

1a2

)

=(

1− s2 − s

2ia

)es ln(1− 1+2m

2ia ) +O(

1a2

)

= 1− s2 + 2ms2ia

+O(

1a2

)(C.2.23)

and

s2Γ(s) = sΓ(s+ 1)

= (s+ 1− 1)Γ(s+ 1)

= Γ(s+ 2)− Γ(s+ 1). (C.2.24)

C.2.5 k=0

When the parameter k reaches zero, the Whittaker equation, equation (C.2.9), becomes

d2z(x)dx2

+(−1

4+

14 −m2

x2

)z(x) = 0. (C.2.25)

This equation is solved by a Bessel function, such that we find

M0,m(x) = 22mΓ[m+ 1]√xIm

(x2

), (C.2.26)

and for imaginary argument,

M0,m(ix) = 22mΓ[m+ 1]√xe−

imπ2 Jm

(−x

2

), (C.2.27)

where the last equation holds for −π < arg ix ≤ π2 .

98

Acknowledgements

I would like to thank a number of persons for their contribution to my road to a Masters degreein Physics. First of all I thank Tomislav Prokopec for being an enthusiastic, patient and energeticsupervisor who investigated an extensive amount of time in my education and for bringing me intocontact with Julien Lesgourgues, giving me the grand opportunity to extend my life in Cosmology.I thank all the fellow students who simultaneously worked on their Masters thesis for their usefuland useless discussions on work related and other topics. Among this group I would especially liketo mention Gijs van den Oord and Tristan Dupree for their patience during the daily contest fornot-having-to-make-by-yourself coffee. I also thank my parents and brothers for stimulating mychoice for Theoretical Physics, but also of course for the eighteen years in advance of starting atthe University of Utrecht. Next to these people I could actually thank everybody around me forall their humour, patience, hate, love and coffee. I do not want to burn my fingers by extendingthe list of names at the risk of not mentioning others. Hence, I really thank you all.

99

Acknowledgements

100

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