graviton two-point function inside the cosmological

UNIVERSIDADE FEDERAL DO PARÁ
INSTITUTO DE CIÊNCIAS EXATAS E NATURAIS
PROGRAMA DE PÓS-GRADUAÇÃO EM FÍSICA
Masters Dissertation
GRAVITON TWO-POINT FUNCTION
INSIDE THE COSMOLOGICAL HORIZON
OF DE SITTER SPACETIME
Rafael Pinto Bernar
Advisor: Prof. Dr. Luı́s Carlos Bassalo Crispino
Belém-Pará
Graviton two-point function inside the cosmological horizon of de Sitter spacetime
Rafael Pinto Bernar
Dissertation presented to the Programa de Pós-Graduação
em Fı́sica of the Universidade Federal do Pará (PPGFUFPA) as part of the requirements needed to obtain the
Master Degree in Physics.
Advisor: Prof. Dr. Luı́s Carlos Bassalo Crispino
Examiners
Prof. Dr. Luı́s Carlos Bassalo Crispino (Advisor)
Prof. Dr. Valdir Barbosa Bezerra (Outside member)
Prof. Dr. Ednilton Santos de Oliveira (Inside member)
Prof. Dr. João Vital da Cunha Junior (Substitute)
Belém-Pará
i
Abstract
Perturbation theory applied to Einstein’s gravitation can give satisfactory results to
many problems in general relativity and quantum field theory. Perturbation theory allows
to find approximate solutions that slightly deviate from a known exact solution. Einstein’s
equation is linearized so that we establish a linear theory for general relativity. By quantizing gravitational perturbations, we obtain a linear theory for a tensorial field of spin 2:
the graviton. Due to its relevance to the inflationary cosmology, the study of phenomena
in de Sitter spacetime has increased. Quantum gravitational perturbations in de Sitter
spacetime can have an impact in the evolution of an inflationary universe, which seems
to be the case of our own Universe according to recent observations. The properties of
the graviton two-point function are especially important within this context. Due to the
gauge invariance of the linearized theory, divergences in the graviton two-point function
can be gauge artifacts, that is, they can be non physical. Particularly, infrared divergences in the graviton two-point function, if they are physical, may act as mechanisms
to break the de Sitter symmetry, thus leading to the failing of perturbative approach in
which the background is non-dynamical. As an example, some authors state that these
infrared divergences could create a time-dependent cosmological constant. In this dissertation, we study gravitational perturbations in curved background spacetimes. Using a
gauge-invariant formalism for gravitational perturbations in background spacetimes with
special symmetries, particularly spherical symmetry, we compute the graviton two-point
function in the static patch of de Sitter spacetime. We analyze its properties, including
its behavior in the infrared limit. The static patch of de Sitter spacetime represents the
region accessible to an inertial observer in de Sitter spacetime. In the Bunch-Davies like
vacuum state, which represents a thermal bath with temperature H/2π, H being the
Hubble constant, the two-point function we have found is finite in the infrared limit. The
two-point function in the static patch has the additional property of time-translation invariance.
Keywords: De Sitter spacetime, gravitational perturbations, graviton twopoint function, infrared divergence.
Programa de Pós-Graduação em Fı́sica - UFPA
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Resumo
A teoria de perturbações aplicada à gravitação de Einstein pode gerar resultados
satisfatórios para muitos problemas em relatividade geral e teoria quântica de campos.
A teoria de perturbações nos permite encontrar soluções aproximadas, que se desviam
pouco de uma solução exata. Desta forma, a equação de Einstein é “linearizada” e estabelecemos uma teoria linear para a relatividade geral. Ao quantizarmos as perturbações
gravitacionais, temos uma teoria linear para um campo tensorial de spin 2: o gráviton. Devido à sua relevância para o estudo de cosmologias inflacionárias, a análise de fenômenos
no espaço-tempo de De Sitter tem aumentado. Perturbações gravitacionais quânticas
no espaço-tempo de De Sitter podem ter impactado a evolução de um universo inflacionário, que parece ser o caso de nosso próprio Universo, de acordo com observações
recentes. As propriedades da função de dois pontos do gráviton são especialmente importantes neste contexto. Devido à invariância de calibre da teoria linearizada, divergências
na função de dois pontos do gráviton podem possuir um caráter não fı́sico. Particularmente, divergências no infravermelho podem atuar como mecanismos relacionados à
quebra da simetria de De Sitter e o método perturbativo perde sua validade. Alguns
autores relatam que estas divergências no infravermelho levam a uma constante consmológica dependente do tempo, por exemplo. Nesta dissertação, estudamos perturbações
gravitacionais em espaços-tempos de fundo curvos. Usando um formalism invariante de
calibre para perturbações gravitacionais em espaços-tempos de fundo com simetrias especiais, particularmente simetria esférica, calculamos a função de dois pontos do gráviton
na região estática do espaço-tempo de De Sitter. Analisamos suas propriedades, incluindo
seu comportamento no limite infravermelho. A região estática do espaço-tempo de De
Sitter é fisicamente relevante, pois representa a região acessı́vel a um observador inercial.
No estado de vácuo tipo Bunch-Davies, o qual é um estado térmico com temperatura
H/2π, em que H é a constante de Hubble, a função de dois pontos que encontramos é
finita no limite infravermelho. Além disso, esta função de dois pontos na região estática
é invariante por translações temporais.
Palavras-Chaves: Espaço-tempo de De Sitter, perturbações gravitacionais,
função de dois pontos do gráviton, divergência no infravermelho.
Programa de Pós-Graduação em Fı́sica - UFPA
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To the memory of all people who have faced “The Man”...
Programa de Pós-Graduação em Fı́sica - UFPA
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Big bang, big crunch, you know there is no free lunch.
Kneel down and pray, here comes your judgment day.
Bad Religion in the song Big Bang.
Programa de Pós-Graduação em Fı́sica - UFPA
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Acknowledgments
• To Prof. Luı́s Carlos Bassalo Crispino for the supervision.
• To Prof. Atsushi Higuchi for the help in understanding many topics in physics.
• To my fiancée Melina Gonçalves for not doubting me, even when I had doubts about
myself. You are my true love.
• To my greatest friend and brother, Lucas Bernar. There is a lot of bad guys around
there and you are the good one.
• To my family for the greatest support and comprehension. Lı́gia Bernar, Brazı́lia
Bernar, Renato Bernar and Nazaré Santa Brı́gida, the good guys.
• To all my friends at the graduate program and at the university for discussions about
the most diverse themes. Alvaro Coelho, Luiz Carlos Leite, Ygor Pará, Amanda
Almeida, Everton da Silva, Leandro Oliveira and Caio Macedo. You are true friends.
I hope we can continue to “cut reality” together for a long time.
• To the Grupo de Teoria Quântica de Campos em Espaços Curvos (GTQCEC) for
discussions in physics.
• To the Universidade Federal do Pará and to the PPGF-UFPA, for the opportunity
of doing Science in Brazil, particularly in Amazonia.
• To Conselho Nacional de Desenvolvimento Cientı́fico e Tecnológico (CNPq) and
IRSES (European Union) for the financial support.
Programa de Pós-Graduação em Fı́sica - UFPA
Contents
Foreword
9
Introduction
10
Conventions and notations
13
1 De Sitter spacetime geometry
1.1 De Sitter metric in global coordinates (Global patch) . . . . . . . . . . .
1.2 Conformally flat representation and the causal structure of de Sitter spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Static coordinate system (Static patch) . . . . . . . . . . . . . . . . . . .
1.4 Inflationary coordinates (Poincaré Patch) . . . . . . . . . . . . . . . . . .
2 Linearized Gravity
2.1 Gravitational perturbations in flat spacetimes . .
2.1.1 Weak field limit . . . . . . . . . . . . . . .
2.1.2 Gauge invariance . . . . . . . . . . . . . .
2.2 Gravitational perturbations in curved spacetimes
2.2.1 Einstein’s equation . . . . . . . . . . . . .
2.2.2 Gauge invariance . . . . . . . . . . . . . .
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3 Gauge-invariant formalism for perturbations
3.1 Properties of the background spacetime . . . . . . . . . . . . . . . .
3.2 Gravitational perturbations - three different types of perturbations .
3.2.1 Scalar perturbations . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Vector perturbations . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Tensor (rank-2) perturbations . . . . . . . . . . . . . . . . .
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4 Graviton two-point function in de Sitter spacetime
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4.1 Normalization of gravitational perturbations . . . . . . . . . . . . . . . . . 50
4.1.1 Normalization of the tensor-type modes . . . . . . . . . . . . . . . . 50
Programa de Pós-Graduação em Fı́sica - UFPA
CONTENTS
4.2
vii
4.1.2 Normalization of the vector-type modes . . . . . . . . . . . . . . . . 53
4.1.3 Normalization of the scalar-type modes . . . . . . . . . . . . . . . . 55
Infrared finite two-point function . . . . . . . . . . . . . . . . . . . . . . . 59
Conclusion and perspectives
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A Calculation of the inner product for the scalar-type modes
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Programa de Pós-Graduação em Fı́sica - UFPA
Foreword
The content of this dissertation is a result of the research developed during the Masters
of the candidate. The content of the dissertation is mainly related to a published article,
namely:
• R. P. Bernar, Luı́s C. B. Crispino and A. Higuchi, Infrared-finite graviton two-point
function in static de Sitter space, Phys. Rev. D 90, 024045 (2014).
Programa de Pós-Graduação em Fı́sica - UFPA
Introduction
What do we know of our own Universe? This is certainly not an easy question to
answer. And it is not a question that is asked only by physicists. Slightly different
related questions are: What do we physically know of our own Universe? What can we
measure about the Universe? Since the Universe is very big and complicated, most of our
measurements are indirect. In most of the cases we can only infer its properties from these
indirect observations, and attempt to make theories from these inferences. Our Universe
expanding in an accelerated rate is one of these inferences [1, 2].
We can assume a model for the expanding Universe, constructed using general relativity and quantum field theory. The most accepted theory for the Universe evolution is
that the Universe has suffered a short time burst of an extremely accelerated expansion
(exponential era of expansion), known as the inflationary period, caused by a hypothetical
field called inflaton [3–7]. This could answer some questions, regarding the formation of
structures and the flatness problem [6]. After this exponential expansion, our Universe
continues to expand in a less accelerated rate. The causes for this profile of expansion
will not be discussed here. This inflationary cosmology recently appears to have gained
further evidence from observation [8]. One of the key ingredients to this model is the
cosmological constant, a term in Einstein’s field equation which could be there for several
reasons - dark energy, zero point energy, or simply being a fundamental term.
This (positive) cosmological constant gives rise to a maximally symmetric vacuum
solution to Einstein’s equation called de Sitter spacetime [9,10]. This spacetime was used
in one of the first models for the Universe, the Einstein-de Sitter universe [11], which
is a static universe, later recognized as an unrealistic model. De Sitter spacetime have
some special properties, the main ones (related to cosmology) being its expanding (and
contracting) property (ies) and that it possesses a horizon, the cosmological horizon.
The expanding half of de Sitter spacetime is approximately the exponentially expanding
universe in the inflationary model and our Universe may be approaching the de Sitter
spacetime asymptotically. Hence, de Sitter spacetime can be a model for our Universe in
its early expanding phase and in its far future.
Due to the non-linearity of Einstein’s equation, there are relatively few known exact
solutions with physical interest. The difficulty in solving these equations makes us wonder
if someday we will have the necessary techniques to do it in a complete way. However,
there are methods that allow one to obtain satisfactory results within certain limits.
Perturbation theory applied to Einstein’s gravitation is one of these methods. In this
method we look for approximate solutions that have a small deviation from a known
Programa de Pós-Graduação em Fı́sica - UFPA
Introduction
11
exact solution. Einstein’s equations are then “linearized‘” so that we establish a linear
theory for general relativity.
The first one to use linearized gravity was Einstein himself, in his works developing
general relativity. Following that, several other works were published using the linear
theory, most of them about calculations correcting Newtonian gravity, but some studying
new effects - gravitational waves, for example. Almost all of the gravitational phenomena
experimentally observed fit in the linear limit of general relativity.
Particularly, the interest in these perturbations in de Sitter spacetime is increasing
recently, especially due to its relevance for the inflationary cosmology. Gravitational
perturbations analysis inside the cosmological horizon of de Sitter spacetime (static patch)
is quite relevant in view of the role of this region in the cosmological scenario. We can
quantize these gravitational perturbations, obtaining a theory for a tensor field of spin 2:
the graviton, which can have an impact in the evolution of an inflationary universe.
The properties of the graviton two-point function are especially important within this
context. In particular, the infrared (IR) properties of the graviton two-point function in de
Sitter spacetime have remained a source of controversies over the past 30 years. The main
source of these controversies is that the graviton mode functions natural to the spatiallyflat (or Poincaré) patch of de Sitter spacetime behave in a manner similar to those for
minimally-coupled massless scalar field [12], which allows no de Sitter-invariant vacuum
state because of IR divergences [13, 14]. Ford and Parker found that this similarity leads
to IR divergences in the graviton two-point function though they found no IR divergences
in the physical quantities they studied [12]. (In fact their work deals with a more general
Friedmann-Lemaı̂tre-Robertson-Walker spacetime.)
However, since linearized gravity has gauge invariance, it is important to determine
whether or not these IR divergences are a gauge artifact. Indeed, the IR divergences
and breaking of the de Sitter symmetry they cause in the free graviton theory have been
shown to be a gauge artifact in the sense that they can be gauged away if we allow
nonlocal gauge transformations [15, 16]. This point has recently been made clearer by
explicit construction of an IR-finite two-point function [17]. The authors of Ref. [17] also
pointed out that a local gauge transformation is sufficient to render the two-point function
finite in the infrared limit in a local region of the spacetime. It is also worth noting that
the two-point function of the linearized Weyl tensor computed using a de Sitter noninvariant propagator with an IR cutoff exhibits no IR divergences [18] and agrees with
the result [19] calculated using the covariant propagator [20, 21]. In fact, in a recent
gauge-invariant formulation of free gravitons [22] the Weyl tensor and graviton two-point
functions have been shown to be equivalent in de Sitter spacetime [23]. It has also been
argued recently [24] that there is a de Sitter-invariant Hadamard state for free gravitons
defined in a way similar to the scalar case [22].
Gravitational perturbations in de Sitter spacetime have been analyzed mainly in the
Poincaré patch for two reasons. Firstly, this patch is the most relevant to the inflationary
cosmology. Secondly, the graviton mode functions in this patch are the simplest. But
there have been some works using other patches. It has been known for some time that in
the global patch of de Sitter spacetime the free graviton field theory has no IR divergences
and that there is a de Sitter-invariant vacuum state [25] analogous to the Bunch-Davies
vacuum [26] for the scalar field theory (see also Ref. [27]). As a result there is an IR-finite
Programa de Pós-Graduação em Fı́sica - UFPA
Introduction
12
graviton two-point function in this patch [28]. An IR-finite graviton two-point function
has also been found in the hyperbolic patch of de Sitter spacetime [29]. However, there
has been little work on quantum gravitational perturbations in the static patch, which is
of physical importance because it represents the region causally accessible to an inertial
observer.
In this dissertation, one of our main goals is to use a gauge-invariant formalism developed by Kodama and Ishibashi [30] to study quantum gravitational perturbations in
the static patch of de Sitter spacetime. In particular, we demonstrate that there is an
IR-finite graviton two-point function in the Bunch-Davies-like state in this patch. We
emphasize that this two-point function is time-translation invariant unlike that in the
global patch [28]. Thus, if linearized gravity is treated as a thermal field theory inside the
cosmological horizon [31], then one finds no IR divergences or secular growth of the kind
encountered in the Poincaré patch [17]. Although it has been shown that IR divergences
are a gauge artifact in the sense mentioned above, it is useful to demonstrate explicitly
that there is an IR-finite and time-translation invariant graviton two-point function, since
there are objections to the existence of de Sitter invariant Bunch-Davies-like state in
de Sitter space [32, 33].
Outline of this dissertation
In this dissertation we study quantum gravitational perturbations inside the cosmological horizon of de Sitter spacetime in n + 2 dimensions. The main results presented
in this dissertation have been published in Ref. [34]. The remainder of this dissertation
is divided as follows. In Chapter 1, we introduce de Sitter spacetime, presenting some of
its properties. We emphasize general features of this spacetime, such as causal structure,
symmetries and the cosmological horizon.
In Chapter 2, we write the field equations for gravitational perturbations. We start
exhibiting gravitational perturbations in Minkowski spacetime, due to its relative simplicity. Then we generalize the results for curved spacetimes, particularly for de Sitter
spacetime.
In Chapter 3, we review the gauge-invariant formalism developed by Kodama and
Ishibashi [30]. This formalism consists in expanding the perturbations in terms of spherical
harmonic tensors, so that we can classify the gravitational perturbations in three types:
scalar, vector and (rank-2) tensor perturbations. We then obtain a master equation for
each type of perturbation. We exhibit these master equations for de Sitter spacetime
inside the cosmological horizon and find their solutions.
In Chapter 4, we discuss the quantization of the free graviton field and explain how
to construct its two-point function. We define a sympletic inner product, through which
we can distinguish different solutions. Also, we normalize the modes of each type of
perturbation, so that the construction of the two-point function becomes straightforward.
We analyze the behavior of the two-point function in the infrared limit and show that
there are no IR divergences for the two-point function constructed in this way. We end
this dissertation with our conclusions and perspectives.
Programa de Pós-Graduação em Fı́sica - UFPA
Conventions and notations
Here we specify to the reader the conventions used throughout this dissertation.
• The units are such that G = c = ~ = 1, unless otherwise stated.
• The metric signature is (− + +...+).
• The background spacetime will be de Sitter in n + 2 dimensions, with n ≥ 2. (We
will exhibit expressions in four dimensions explicitly as frequently as possible.) The
line element will take the form
ds2 = gµν dxµ dxν = −(1 − λr2 )dt2 +
dr2
+ r2 dσn2 ,
1 − λr2
(1)
where
dσn2 = γij (x)dxi dxj
(2)
is the line element on the n-sphere S n . The greek indices are used for spacetime
indices running from 0 to n + 1, the first latin indices a, b, c, . . . stand for the first
two coordinates (for example, t and r) and the i, j, k, . . . stand for S n coordinates.
We will be working inside the cosmological horizon in the so-called static coordinate
system (static patch). We shall put λ = 1 for simplicity. We shall use the notation
established in Refs. [30, 35], with the exception of quantities of the background
spacetime, for which we use greek indices. We define the line element of the twodimensional orbit space by
ds2orb = gab dxa dxb = −(1 − r2 )dt2 +
dr2
.
1 − r2
(3)
We denote the covariant derivatives compatible with the full metric represented
by the line element ds2 , the two-dimensional metric represented by ds2orb , and the
metric on S n represented with dσn2 , by ∇µ , Da and D̂i , respectively. The connection coefficients for ds2 , ds2orb and dσn2 are denoted by Γαµν , Γabc (t, r) and Γ̂ijk (x),
respectively.
Programa de Pós-Graduação em Fı́sica - UFPA
Chapter 1
De Sitter spacetime geometry
The de Sitter spacetime is a maximally symmetric solution of Einstein’s equation with
positive cosmological constant, found by Willem de Sitter in 1917 [9, 10]. Considered
to be a mistake by Einstein himself, there are evidences indicating the existence of this
cosmological constant, crucial to the inflationary cosmology [3–7], which recently appears
to have gained further evidence from observation [8]. In addition, current observations
indicate that our Universe is expanding in an accelerated rate and may approach de Sitter
space asymptotically [1, 2]. Physics in de Sitter space is also attracting attention because
of the dS/CFT correspondence [36].
De Sitter spacetime can be understood in a relatively simple way, due to its maximal
symmetry. It is very similar to Minkowski spacetime, but the presence of the curvature
demands additional attention for the physical interpretation.
In this chapter, we study de Sitter spacetime. We construct a d-dimensional de Sitter
spacetime and investigate several of its properties. We do this by considering different
coordinate systems. Firstly, in Sec. 1.1, we define a global coordinate system, where we
can analyze the topology and rotational symmetry of de Sitter spacetime. In Sec. 1.2,
using Penrose diagrams for de Sitter spacetime, we have a view of its causal structure. In
Sec. 1.3, we define the static coordinate system, which is of physical importance because
it represents the region causally accessible to an inertial observer. In this region, the
cosmological horizon has its utmost importance. Finally, in Sec. 1.4, we present the
Poincaré patch, with inflationary coordinates. It is in this patch that it can be shown
that de Sitter spacetime is a good model for an inflationary universe.
Einstein’s equation1 with cosmological constant [39],
1
Rµν − Rgµν + Λgµν = 8πTµν ,
2
(1.1)
admits particularly simple vacuum solutions (Tµν = 0). By simple we mean that they
are maximally symmetric, that is, they have all possible continuous symmetries. If the
1
We are assuming that the reader possesses some previous knowledge about general relativity. As
references to general relativity we suggest, for example, [11, 37, 38].
Programa de Pós-Graduação em Fı́sica - UFPA
1.1. DE SITTER METRIC IN GLOBAL COORDINATES (GLOBAL
PATCH)
15
cosmological constant Λ is zero, we have Minkowski spacetime as the most simple solution.
It is the standard spacetime in special relativity. If Λ > 0, the simpler solution is de Sitter
spacetime. And if Λ < 0, we have anti-de Sitter spacetime. Here, we won’t discuss anti-de
Sitter spacetime, but we direct the interested reader to Ref. [39].
We can construct a d-dimensional de Sitter spacetime, dSd , by means of a hypersurface
equation, embedded in a d + 1-dimensional Minkowski spacetime, Md+1 , that is
− (x0 )2 + (x1 )2 + ... + (xd )2 = ηAB xA xB = l2
A, B = 0, 1, 2, ..., d,
(1.2)
in which ηAB = diag(−1, 1, 1, ..., 1) is the metric on Md+1 . The hypersurface (1.2) is a
hyperboloid and l is called de Sitter radius. Note that this Minkowski spacetime has only
one timelike direction, like the usual Minkowski spacetime. This construction have the
advantage of revealing some symmetries and the topology of de Sitter spacetime, as we
will see.
d+1
The metric in dSd is the induced metric by the hyperboloid (1.2) in Mp
. Spatial
0
2
sections with x = constant define (d − 1)-dimensional spheres of radius l + (x0 )2 .
It follows that dSd topology is cilindrical, R × S d−1 , where S d−1 is the hypersphere of
dimension d − 1.
It is possible to show that the resulting metric is a vacuum solution of Einstein’s
equation (1.1) with positive cosmological constant, and that dSd has a constant positive
scalar curvature (see, for example, Ref. [40]). The cosmological constant is related to de
Sitter radius by
(d − 2)(d − 1)
.
(1.3)
Λ=
2l2
As we already stated, the spacetime dSd is maximally symmetric, what means that it
has d(d+1)
Killing vectors [37], which represent kinematical symmetries of the spacetime.
2
In the case of four dimensional Minkowski spacetime, this symmetry group constitutes
the Poincaré group, and it has a direct physical interpretation. In de Sitter’s case, the
symmetry group is called the de Sitter group and the analysis of its physical interpretation
has to be more carefully done.
1.1
De Sitter metric in global coordinates (Global
patch)
We can obtain the metric gµν of dSd , using the equation (1.2) to eliminate the spatial
coordinate xd from the metric of M(d+1) , so that we have
gµν = ηµν +
l2
xµ xν
.
− ηαβ xα xβ
Programa de Pós-Graduação em Fı́sica - UFPA
(1.4)
1.1. DE SITTER METRIC IN GLOBAL COORDINATES (GLOBAL
PATCH)
16
Here, ηµν = diag(−1, 1, 1, ..., 1) is the d-dimensional Minkowski metric. Note that the
greek indices run from 0 to d − 1, while A, B = 0, 1, ..., d. We can choose a convenient
global system of coordinates, defined by
x0 = l sinh
τ l
xα = lω α cosh
and
τ l
(α = 1, ..., d),
(1.5)
with −∞ < τ < ∞ being a timelike coordinate. This choice of coordinates must satisfy
the equation (1.2). For this purpose, we must impose
d
X
(ω α )2 = 1.
(1.6)
α=1
We note that this restriction is a parameterization of a (d − 1)-sphere. Therefore, we can
choose a set of (d − 1) angular variables θi so that the ω α ’s will be
ω 1 = cos θ1 ,
ω 2 = sin θ1 cos θ2 ,
..
.
ω d−2 = sin θ1 sin θ2 ... sin θd−3 cos θd−3 ,
ω d−1 = sin θ1 sin θ2 ... sin θd−2 cos θd−1 ,
ω d = sin θ1 sin θ2 ... sin θd−2 sin θd−1 ,
(1.7)
with 0 < θi < π for all θi except θd−1 that varies between 0 < θd−1 < 2π.
Inserting the equations (1.7) and (1.5) in (1.4), we obtain the line element written in
global coordinates (τ, [θi ]):
ds2 = −dτ 2 + l2 cosh2
τ l
2
dσd−1
,
(1.8)
2
in which dσd−1
is the line element of the hypersphere S d−1 , namely
2
2
dσd−1
= dθ12 + sin2 θ1 dθ22 + sin2 θ1 sin2 θ2 dθ32 + ... + sin2 θ1 ... sin2 θd−2 dθd−1
(1.9)
!
j−1
d−1
X
Y
=
sin2 θi dθj2 .
(1.10)
j=1
i=1
In four dimensions, we are going to have three angular coordinates, i.e., θ1 , θ2 and θ3
1.2. CONFORMALLY FLAT REPRESENTATION AND THE CAUSAL
STRUCTURE OF DE SITTER SPACETIME
17
and the ω α ’s take the form
ω1
ω2
ω3
ω4
=
=
=
=
cos θ1 ,
sin θ1 cos θ2 ,
sin θ1 sin θ2 cos θ3 ,
sin θ1 sin θ2 sin θ3 .
(1.11)
The line element dσ32 of the 3-sphere S 3 is
dσ32 = dθ12 + sin2 θ1 (dθ22 + sin2 θ2 dθ32 ).
(1.12)
The 3-sphere is a three dimensional space, like R3 , but it is closed, what means that it
has a finite volume.
In these global coordinates, a spatial section with fixed τ corresponds to a (d − 1)sphere with radius l cosh τl . When τ → −∞, its radius is infinitely large, it goes to a
minimum value when τ = 0 and then it grows back to infinity when τ → ∞. Hence we
say that the spatial sections are compact, except in the far past and far future [40, 41].
It is interesting to note that ∂/∂τ is not a Killing vector. The inexistence of this Killing
symmetry gives rise to many problems in the definition of energy and its conservation [40],
which poses a few obstacles in a quantum field theory in de Sitter spacetime. We note
that the vector ∂/∂θd−1 is a Killing vector in this coordinate system, related to azimuthal
symmetry.
There is another way to obtain the line element (1.8). We write a general form for the
de Sitter spacetime metric, with some appropriate symmetries, and then it is possible to
solve Einstein’s equation with positive cosmological constant. To the interested reader,
we suggest Ref. [40] for this development.
1.2
Conformally flat representation and the causal
structure of de Sitter spacetime
De Sitter spacetime admits a conformally flat representation [40]. The correspondent
coordinates to this representation are related to the global coordinate system (1.5) by
cosh τ /l =
1
,
cos(T /l)
(1.13)
with −π/2 < T /l < π/2. The line element (1.8), in these coordinates given by (1.13),
becomes
1
2
ds2 =
(−dT 2 + l2 dσd−1
).
(1.14)
cos2 Tl
1.2. CONFORMALLY FLAT REPRESENTATION AND THE CAUSAL
STRUCTURE OF DE SITTER SPACETIME
18
The advantage of this representation is in the discussion of de Sitter spacetime causal
structure. Null geodesics with respect to (1.14) are also null with respect to the line
element
2
2
2 T
ds2 = −dT 2 + l2 dσd−1
.
(1.15)
ds̃ = cos
l
That way, we can easily construct the Penrose diagram [11].
Noth Pole
South Pole
I
I
Figure 1 - Penrose Diagram of de Sitter Spacetime.
The Penrose diagram in Figure 1 contains all the information of the causal structure of
dSd , although the distances are highly distorted. Each point in the diagram represents a
(d−2)-sphere, except points in both left and right boundary vertical lines, which represent
the North pole (θ1 = 0) and the South pole (θ1 = π), respectively. A horizontal line
represents a (d − 1)-sphere. Light rays are lines that form an angle of 45◦ with horizontal
lines, as usual in diagrams of this kind.
Any null geodesic originates on I − and terminates on I + . These spacelike hypersurfaces I − and I + are called past and future null infinity, respectively. Although the Penrose
diagram covers all of the spacetime, no observer has access to the whole spacetime. An
observer in the South pole, for example, will never see an event beyond the diagonal line
connecting I − in the North pole to I + in the South pole. It is possible to show that this
holds for any observer [11].
O‘s world-line
Particle that has been
observed by O at P
Particle horizon for
Particle
world-lines
P
I
-
O at P
Particle that has not been
observed by O at P
Past light cone
of O at P
Figure 2 - Representation of the particle horizon in de Sitter spacetime.
1.2. CONFORMALLY FLAT REPRESENTATION AND THE CAUSAL
STRUCTURE OF DE SITTER SPACETIME
19
In Figure 2, we consider a family of particles, following timelike geodesics. If P is a
given event in the worldline of a particle O in this family, the past lightcone in P is the
set of events that can, in principle, be observed by O at that event P. Particles of this
family visible to O are those whose worldlines intersect the past lightcone associated to
P. There are particles whose worldlines don’t intersect this lightcone, so they are not
visible by O at that event P. This separation between visible and non-visible particles
forms a boundary, which we call particle horizon for O at P. If we consider this event
P as a point at I + , then the past lightcone forms a boundary between observable events
and events that could never be observed by O, as can be seen in Figure 3. This boundary
is called future event horizon of O. We note that the existence of these horizons is deeply
connected to the fact that the null infinities are spacelike. We oppose this situation to
what happens in Minkowski spacetime, for which all the timelike geodesics originate from
the same point, the past timelike infinity i− [37], as can be seen in Figure 4.
O‘s world-line
I+
Future event horizon of O
Past event horizon of O
r
Events O will never
be able to see
Events O will never be
able to influence
I-
Q‘s world-line
Figure 3 - Representation of the event horizon and accessible region to O in de Sitter
spacetime.
1.3. STATIC COORDINATE SYSTEM (STATIC PATCH)
20
Figure 4 - Penrose Diagram for Minkowski spacetime.
In the same way, the future lightcone at P bounds the set of events O can have
influence on. If we locate P at I − , this future lightcone is the boundary of events O may
be able to have an influence on in the whole spacetime as it is also represented in Figure
3. We denote this boundary as past event horizon of O. Figure 3 also shows the worldline
of a particle Q. As the particle Q crosses the future event horizon of O at event r, it will
not be observable by O anymore. This clearly shows that no single observer has access
to the entire de Sitter spacetime.
1.3
Static coordinate system (Static patch)
Let us now construct a coordinate system that exhibits an explicit timelike Killing
symmetry. We can decompose the equation (1.2) into two constraint equations, namely
−
and
x1
l
x0
l
2
+
2
xd
l
+ ... +
2
=1−
xd−1
l
2
=
r 2
l
r 2
(1.16)
.
(1.17)
l
p
The equation (1.16) represents an hyperbole of radius 1 − (r/l)2 , while the equation
(1.17) describes a (d − 2)-sphere of radius r/l. A coordinate system that satisfies both
1.3. STATIC COORDINATE SYSTEM (STATIC PATCH)
21
constraint equations is
r
r 2
x0
t
= − 1−
sinh ,
l
l
l
i
x
r i
=
ω , (i = 1, 2, ..., d − 1),
l
lr
r 2
xd
t
= − 1−
cosh .
l
l
l
(1.18)
We identify the additional parameter r as the radial coordinate of the static coordinate
system. In the coordinate system given by Eq. (1.18), the line element of de Sitter
spacetime takes the form
r 2 dr2
2
+ r2 dσd−2
,
ds = − 1 −
dt2 + l
1 − ( rl )2
2
(1.19)
2
in which dσd−2
is the line element of a (d − 2)-sphere S d−2 .
In four dimensions, the line element (1.19) has a familiar form
r 2 dr2
+ r2 (dθ2 + sin2 θdφ2 ),
ds = − 1 −
dt2 + l
1 − ( rl )2
2
(1.20)
where θ and φ are the usual polar and azimuthal angles, respectively.
The line element (1.20) is similar to the line element of Schwarzschild spacetime in
a static coordinate system [42]. In fact, both solutions belong to the same family of
solutions [35]. Hence, all the analysis made for the event horizon in Schwarzschild case is
also valid here. This reveals that r = l is an event horizon, which we call cosmological
horizon, since it is a boundary of what we can observe in the universe. We can verify that
this coordinate system is adapted to an observer sitting in r = 0. It becomes clear from
Figure 2 that every inertial observer in de Sitter spacetime has an event horizon, so it is
not a surprise that this phenomenon becomes evident in some coordinate system. As in
the Schwarzschild case, this covers only one quarter of the maximal extension of de Sitter
spacetime [43]. However, there is, at least, two differences between the Schwarzschild and
de Sitter spacetimes:
(i) In de Sitter spacetime, the observer is located inside the horizon, while in the case
of Schwarzschild spacetime, the observer is located in the spacelike infinity outside the
horizon.
(ii) In Schwarzschild spacetime, the interior region to the event horizon is spacelike and
the exterior, timelike. In de Sitter’s case, it is the other way around.
The static coordinate system presents a timelike Killing vector ∂/∂t. So, in this patch,
we have a time translation symmetry. However the Killing vector is not globally timelike,
what also happens in Schwarzschild’s solution. For example, the vector becomes null at
the cosmological horizon.
1.4. INFLATIONARY COORDINATES (POINCARÉ PATCH)
22
In this dissertation, we will investigate the properties of the graviton two-point function in the static patch of de Sitter spacetime. Inside the cosmological horizon of de
Sitter spacetime, the static and spherical symmetry properties allow us to use the gaugeinvariant formalism presented in Chapter 2.
1.4
Inflationary coordinates (Poincaré Patch)
We have a freedom to decompose the hypersurface equation (1.2). So we may choose
another decomposition of Eq. (1.2) in terms of new mappings, such as
−
and
x1
l
x0
l
2
+
2
xd
l
+ ... +
2
2
R
=1−
e2t/l
l
xd−1
l
2
2
R
=
e2t/l .
l
(1.21)
(1.22)
This new variable R is to be understood as a function of the new coordinates (t, X i ),
namely
d−1 2
1 2
X
X
2
.
(1.23)
+ ... +
R =
l
l
To implement the constraints above, we can write
x0
t
et/l
= sinh + R2
,
l
l
2
X i t/l
xi
=
e
(i = 1, 2, ..., d − 1),
l
l
xd
t
et/l
= − cosh + R2
.
l
l
2
(1.24)
(1.25)
(1.26)
The new coordinates (t, X i ) can take all the real values. Since we have that
− x0 + xd = −let/l < 0,
(1.27)
these coordinates cover only the upper half of dSd . The line element of de Sitter spacetime
will take a much simpler form using the coordinates (t, X i ), namely
ds2 = −dt2 + e2t/l δij dX i dX j ,
(1.28)
1.4. INFLATIONARY COORDINATES (POINCARÉ PATCH)
23
where δij is the Kronecker delta and i, j = 1, 2, ..., d − 1. In four dimensions the line
element is simply
ds2 = −dt2 + e2t/l (dx2 + dy 2 + dz 2 ).
(1.29)
This form of de Sitter line element describes a universe with flat spatial sections that
expands exponentially in time. Our Universe in the inflationary period is believed to
behave in a way similar to this patch of de Sitter spacetime. We call these inflationary
coordinates, for obvious reasons. Since the metric does not depend explicitly on the
spacelike coordinates X i in this system, there are translational and rotational symmetries,
just as in the case of Euclidean geometry or Minkowski spacetime. Translational symmetry
is an important property of the Poincaré group, so this region in these coordinates is
usually called the Poincaré patch. There are other coordinate systems that investigate
different properties of de Sitter spacetime such as the hyperbolic patch, which is foliated
by a (d − 1)-dimensional hyperbolic space. To the interested readers, we direct them to
Ref. [44].
Chapter 2
Linearized Gravity
In this chapter we study general gravitational perturbations in a curved background
spacetime, seeking solutions of the type:
g̃µν = gµν + hµν ,
|hµν | 1,
(2.1)
in which gµν is the metric of the background spacetime, a known exact solution. We can
expand Einstein’s equation in powers of hµν . In the zeroth order in hµν , the resulting
equations will be just Einstein’s equation for the background. The first order equations
(or linear equations) establish a theory for linearized gravity. In a coordinate system
in which Eq. (2.1) is valid, we can ignore higher order contributions to gravitational
effects. We say then that the field (the perturbed one) is weak and the deviation from the
exact solution is small. This linearized Einstein’s equation are sufficient to study many
problems in gravitation. Gravitational waves in vacuum, for instance, or the gravitational
field of a nearly spherical star. Many problems do not admit exact analytical solutions,
but perturbation theory applied to gravity can give satisfactory results.
Besides the geometrical point of view, the linearized gravity theory may be understood
as a classical theory of a tensorial field hµν propagating in a non-dynamic background
spacetime. This point of view is conceptually important in the extension of the theory
to a quantum field theory, where the notion of a non-massive gravitational interaction
particle (at the linear level, at least) arises. This non-massive particle of spin 2 is called
the graviton.
There are several formalisms related to gravitational perturbations, from which we can
cite, for example, the Newman-Penrose formalism applied to perturbations, which presents
itself as a powerful tool in dealing with perturbations in axially symmetric spacetimes
such as the Kerr family of solutions, the so called Teukolsky formalism [45]. Another
example, that is specific to spherically symmetric spacetimes, is the Regge-Wheeler gauge
formalism, which consists in expanding the perturbations in terms of harmonic functions
defined in the S 2 sphere [46]. The formalism we will present in Chapter 3 generalizes the
Regge-Wheeler gauge to spherically symmetric spacetimes with arbitrary dimensions.
In Sec. 2.1, we present the gravitational perturbations in a flat background, due to
2.1. GRAVITATIONAL PERTURBATIONS IN FLAT SPACETIMES
25
its simplicity. We also discuss the gauge invariance of the theory. In Sec. 2.2 we study
gravitational perturbations in curved spacetimes, since de Sitter spacetime has curvature.
2.1
Gravitational perturbations in flat spacetimes
It is natural to consider gravitational perturbations in a flat background spacetime,
which is the case of the usual Minkowski spacetime. Being mathematically and conceptually simple, Minkowski spacetime facilitates the comprehension of the theory and allows
us a better understanding of several phenomena described by perturbations. For instance,
many properties of gravitational waves can be studied if we analyze them as perturbations
in flat spacetime, that is, waves propagating in a flat background. As a matter of fact,
this is the formalism used in the setting of experimental devices in the search for gravitational waves [47]. Post-newtonian phenomena, which are corrections to the Newtonian
gravitational field, are also studied by this formalism.
The gravitational field we are dealing with is considered “weak”. Most of the times,
it is an excellent approximation to gravitational phenomena. This weak field limit is the
baseline for gravitational perturbation analysis for gravitation in flat spacetimes.
2.1.1
Weak field limit
We consider the metric to be close to the Minkowski metric, so that we can write
g̃µν = ηµν + hµν ,
|hµν | 1,
(2.2)
in which ηµν = diag(−1, +1, ..., +1) is the Minkowski metric in standard cartesian coordinates. Note that we are not fixing the number of dimensions here. Obviously, the
components of a tensor depend on the coordinate system used. So, what does it mean
to have |hµν | 1? It means that, in the physical situation in which we are interested,
there is at least one coordinate system (or reference frame) in which Eq. (2.2) is valid,
in a sufficiently large region of spacetime. We are, in a certain way, breaking the diffeormorphism invariance of general relativity. However, breaking this invariance is in general
the best way to get rid of spurious degrees of freedom and revealing the actual physical
content of a theory [47].
We will consider quantities at most at first order in hµν , since we are looking for a
linear approximation. Then, the background metric can be used to raise and lower indices
for first order quantities, since the corrections would be of higher order. In that way, we
can write
g̃ µν = η µν − hµν + O(h2 ),
(2.3)
with
hµν = η µα η νβ hαβ .
(2.4)
It is due to this fact that we can consider the linear theory of gravity as being the theory
2.1. GRAVITATIONAL PERTURBATIONS IN FLAT SPACETIMES
26
of a tensorial field, hµν , propagating in a flat background spacetime. Indeed, this theory
is invariant under Lorentz transformations, i.e., for transformations of the following type
xµ → x0µ = Λµ ν xν ,
(2.5)
where Λµ ν is the Lorentz transformation matrix. The Minkowski metric is invariant, while
the perturbation is covariant, since it transforms as
h0µν = Λα µ Λβ ν hαβ .
(2.6)
Therefore hµν is a tensor under Lorentz transformations. We have to be careful in not
spoiling the condition |hµν | 1 with these transformations. Usual orthogonal rotations
do not break this condition, while boosts can break it. We have to limit ourselves only
to boosts that do not spoil the condition |hµν | 1. Additionally, hµν is invariant under
constant translations, i.e. transformations of the type
xµ → x0µ = xµ + aµ ,
(2.7)
where aµ is any constant vector. Hence, the linearized theory is invariant under finite
Poincaré transformations (the group formed by translations and Lorentz transformations).
We will see later that there is an additional invariance under local infinitesimal transformations.
To find the perturbed (or linearized) Einstein’s equation, we start with the Christoffel
symbol (or connection), given by
1 ρλ
g (∂µ gνλ + ∂ν gλµ − ∂λ gµν )
2
1 ρλ
∼
η (∂µ hνλ + ∂ν hλµ − ∂λ hµν ).
=
2
Γρµν =
(2.8)
We note that we are using ∼
= to mean that we are calculating first order quantities. The
Christoffel symbol is a first order quantity in hµν , so the only contributions to first order
to the Ricci tensor come from derivatives of the connection, that is
Rµν = Rρ µρν = ∂ρ Γρµν − ∂ν Γρρµ + Γρρλ Γλµν − Γρνλ Γλρµ
1
∼
(∂σ ∂ν hσ µ + ∂σ ∂µ hσ ν − ∂µ ∂ν h − η λρ ∂λ ∂ρ hµν ),
=
2
(2.9)
where h = η µν hµν . Contracting the indices, we obtain the Ricci scalar:
R∼
= ∂µ ∂ν hµν − η µν ∂µ ∂ν h.
(2.10)
2.1. GRAVITATIONAL PERTURBATIONS IN FLAT SPACETIMES
27
Einstein’s tensor will then be
1
Gµν = Rµν − gµν R
2
1
∼
(∂σ ∂ν hσ µ + ∂σ ∂µ hσ ν − ∂µ ∂ν h − ∂ ρ ∂ρ hµν − ηµν ∂ρ ∂λ hρλ + ηµν ∂ ρ ∂ρ h).(2.11)
=
2
We note that, since we are in flat background spacetime, the background Einstein’s tensor
is zero, as well as all the other curvature related tensors. We simplify Eq. (2.11) by writing
1
h̄µν = hµν − ηµν h.
2
(2.12)
In terms of h̄µν , linearized Einstein’s equation become
η λρ ∂λ ∂ρ h̄µν + ηµν ∂ ρ ∂ σ h̄ρσ − ∂ ρ ∂ν h̄ρµ − ∂ ρ ∂µ h̄ρν = −16πTµν .
(2.13)
The tensor Tµν is the energy-momentum tensor of matter and we are assuming it to be
weak, in the sense that its contribution to the curvature of the background spacetime can
be neglected. The energy-momentum tensor must be computed considering only quantities of order zero in hµν . We can show that the energy-momentum tensor is conserved in
the background spacetime. In fact, differentiating Eq. (2.13), we have
η λρ ∂ µ ∂λ ∂ρ h̄µν + ∂ν ∂ ρ ∂ σ h̄ρσ − ∂ µ ∂ ρ ∂ν h̄ρµ − ∂ ρ ∂ µ ∂µ h̄ρν = −16π∂ µ Tµν .
(2.14)
The left hand side of Eq. (2.14) is zero since partial derivatives commute in flat spacetime,
implying that ∂µ T µν = 0 (in the full theory, we have ∇µ T µν = 0 instead). For instance,
if T µν is the energy-momentum tensor of a point-like particle, the particle will follow
geodesics of flat spacetime [48]. Therefore, at the linear level, the particle does not “feel”
its own gravitational field, or any other one present in the spacetime. Physically, this
means that there is no energy exchange between the sources and the perturbations of the
gravitational field: the sources “live” in the background flat spacetime (determined by
ηµν ), not in the physical one (determined by g̃µν ) [47, 49]. We have to consider higher
order expansion in hµν to better model the equations of motion of the sources.
We can calculate the energy-momentum tensor generated by hµν and consider it as
source for new corrections. These corrections will also carry an energy-momentum contribution, so they will generate new corrections and so on. This procedure leads us to
the full Einstein’s equation, that is the corrections to the background metric will turn it
in an exact solution, the original unperturbed metric ηµν will not be relevant anymore.
Instead the spacetime will be governed by a metric that is solution of the full Einstein’s
equation. In summary, if we start with a perturbed theory around the Minkowski metric
and then allow the perturbations to couple with everything, even with itself, the resulting
theory will be general relativity. For more information about the subject, we direct the
interested reader to Refs. [39, 50]. Alternatively, we can consider only up to second order
2.1. GRAVITATIONAL PERTURBATIONS IN FLAT SPACETIMES
28
effects, to define an effective energy-momentum tensor for gravitational perturbations,
which sources the background spacetime. This is called short-wave expansion and can be
found in Refs. [51, 52].
2.1.2
Gauge invariance
Besides the usual Poincaré invariance, there is another type of invariance in the theory
of linearized gravity. This occurs because Eq. (2.2) is not unique. In other words, there
are several coordinate systems in which Eq. (2.2) is valid. In fact, if we consider local
infinitesimal transformations, that is
xµ → x0µ = xµ + ξ µ (x),
(2.15)
gµν = ηµν + h0µν ,
(2.16)
we obtain
with the following transformation for hµν
hµν (x) → h0µν (x0 ) = hµν (x) − (∂µ ξν + ∂ν ξµ ).
(2.17)
The derivatives |∂µ ξν | have to be, at most, the same order of magnitude of |hµν |. In these
conditions, we have that |h0µν | 1 is still satisfied. So, this diffeomorphism constitutes
a symmetry of the linear theory, which we call gauge invariance. The quantities ξ µ are
called generators of gauge transformations. There is a formal discussion about gauge
invariance, using the concepts of diffeomorphism mapping and Lie derivatives. To the
interested readers, we direct them to Refs. [37, 38].
Using gauge invariance we can simplify the equation (2.13) of gravitational perturbations. We can choose a suitable gauge, namely
∂ α h̄αµ = 0.
(2.18)
This is called the de Donder-Lorenz gauge, harmonic gauge or Hilbert gauge. We can
always choose a hµν that satisfies this gauge. Indeed, the gauge transformation for h̄µν is
h̄µν −→ h̄0 µν = h̄µν − (∂µ ξν + ∂ν ξµ − ηµν ∂ρ ξ ρ ),
(2.19)
(∂ ν h̄µν )0 = ∂ ν h̄µν − η λρ ∂λ ∂ρ ξµ .
(2.20)
then
We need to choose the functions ξµ (x) so that the righthand side of the equality (2.20)
vanishes. Then we have
η λρ ∂λ ∂ρ ξµ = ∂ ν h̄µν .
(2.21)
2.2. GRAVITATIONAL PERTURBATIONS IN CURVED SPACETIMES29
Equation (2.21) always admits solution, since the flat d’Alembertian η λρ ∂λ ∂ρ is an invertible operator [47]. It is interesting to note that we can still make subsequent gauge
transformations, without breaking the de Donder-Lorenz gauge. In fact, if is sufficient
that the generators satisfy
η λρ ∂λ ∂ρ ξµ = 0.
(2.22)
In the de Donder-Lorenz gauge, the perturbed Einstein’s equation (2.13) admits a much
simpler form, namely
η λρ ∂λ ∂ρ h̄µν = −16πTµν .
(2.23)
In vacuum (T µν = 0), Eqs. (2.23) and (2.18) are exactly the equations satisfied by a
non-massive field of spin 2, propagating in flat spacetime [38]. If this field suffers nonlinear self-interaction, we recover the complete theory of gravitation, as we mentioned
earlier. However, in the full theory, the flat spacetime is not relevant anymore and the
notions of mass and spin do not make sense, because Poincaré invariance plays no special
role in general relativity. That way, the parallel with the spin-2 field has precise meaning
only in the linear theory [38].
2.2
Gravitational perturbations in curved spacetimes
Since de Sitter spacetime is a curved spacetime we will need to compute quantities
related to gravitational perturbations around a general exact solution of Einstein’s equation. The procedure is basically the same as in flat spacetime, but some concepts have to
be slightly modified, such as gauge invariance. In this Section we aim to generalize the
results given in Sec. 2.1 to curved spacetimes.
2.2.1
Einstein’s equation
We start with Eq. (2.1), using the background metric gµν to raise and lower indices in
quantities of first order in hµν . Then, we have
g̃ µν = g µν − hµν ,
(2.24)
hµν = g αµ g σν hασ .
(2.25)
with
The perturbed Christoffel symbol is given by
where
Γ̃κµν = Γκµν + δΓκµν + O(h2 ),
(2.26)
1
δΓκµν = g κα (∂µ hαν + ∂ν hαµ − ∂α hµν ) − g κα Γβµν hαβ .
2
(2.27)
2.2. GRAVITATIONAL PERTURBATIONS IN CURVED SPACETIMES30
This can be rewritten as
1
δΓκµν = g κα (∇µ hαν + ∇ν hαµ − ∇α hµν ),
2
(2.28)
in which ∇µ is the covariant derivative with respect to the background metric gµν . It is
interesting to note that the perturbed connection is a tensor, while the connection is not.
We then calculate the perturbed Ricci tensor, namely
δRµν = ∂α δΓαµν − ∂ν δΓαµα − δΓβµα Γανβ − Γβµα δΓανβ + δΓβµν Γααβ + Γβµν δΓααβ
= ∇α δΓαµν − ∇ν δΓαµα .
(2.29)
The equation (2.29) is known as Palatini’s identity [39]. In terms of hµν we can rewrite
Eq. (2.29) as
1
δRµν = g λρ (∇λ ∇ν hρµ + ∇λ ∇µ hρν − ∇λ ∇ρ hµν − ∇µ ∇ν hλρ ).
2
(2.30)
We recall that we are retaining terms only up to the first order in hµν .
Now, we demand that the metric g̃µν satisfies Einstein’s equation, with T̃µν = Tµν +δTµν
as the the energy-momentum tensor. Tµν is just the source of the background metric, while
δTµν is a generic perturbation that sources the gravitational perturbations. Since gµν is
an exact solution of Einstein’s equation, with energy-momentum tensor Tµν , we obtain
the perturbed Einstein’s equation, namely
1
1
1
δRµν = 8π(δTµν − gµν δT ρ ρ + gµν hλη T λη − hµν T λ λ ).
2
2
2
(2.31)
The other terms besides the 8πδTµν arise because we have to evaluate the trace of the
energy-momentum tensor using the full metric g̃µν since, in principle, Tµν is of zeroth
order in hµν . We observe that the source term obeys the following law of conservation:
∇µ δT νµ + T νλ δΓµµλ + T λµ δΓνµλ = 0.
2.2.2
(2.32)
Gauge invariance
As in the case of perturbations around flat spacetime, the decomposition (2.1) is not
unique. Then, which gauge transformations retain the form of Eq. (2.1)? We consider
the gauge transformations given by
xµ → x0µ = xµ + ξ µ (x).
(2.33)
2.2. GRAVITATIONAL PERTURBATIONS IN CURVED SPACETIMES31
Therefore, we have
0
g̃µν
(x0 ) = g̃µν (x) − (∇ν ξµ + ∇µ ξν ) + ξ α ∂α g̃µν
= gµν (x0 ) + hµν − (∇ν ξµ + ∇µ ξν ).
(2.34)
We note that there is now an additional requirement on the gauge transformations - the
gauge generator ξ µ must be of the same order as hµν . Otherwise, we would not be able to
consider the term ξ α ∂α hµν in ξ α ∂α g̃µν negligible in comparison to the other terms in Eq.
(2.34). In that way, we obtain the gauge transformations for gravitational perturbations
around a curved background spacetime:
hµν → h0µν = hµν − (∇ν ξµ + ∇µ ξν ).
(2.35)
We can define a gauge transformation for any quantity in the linear theory. In general
the gauge transformation will have the form
S 0λ1 ...λn σ1 ...σm = S λ1 ...λn σ1 ...σm − Lξ S λ1 ...λn σ1 ...σm ,
in which Lξ is the Lie derivative in the direction of ξ µ [39].
(2.36)
Chapter 3
Gauge-invariant formalism for
perturbations
Since we want to analyze de Sitter spacetime in the static patch, which presents explicitly its spherical symmetry, it is convenient to work with perturbations in a formalism
appropriate to this symmetry. As we noted before, this can be achieved by writing the
perturbations in terms of special tensor functions, which possess some kind of relation
with spherical symmetry. We will see later on that these functions obey harmonic equations. Due to the special properties of these harmonic functions, the perturbations can
be treated separately and classified according to the type of tensor function used in the
expansion. These functions are called scalar, vector and tensor (rank-2) spherical harmonics, generating scalar, vector and tensor gravitational perturbations, respectively. This
type of expansion allows us to write a master equation for each type of perturbation,
consisting in second order self-adjoint differential equations. As we noted earlier, this
formalism generalizes the Regge-Wheeler gauge formalism.
In this Chapter, we review a gauge-invariant formalism developed in Refs. [30, 35] for
spherically symmetric background spacetimes. In Sec. 3.1, we present the properties of the
background spacetime, that is de Sitter spacetime in the static patch with n+2 dimensions.
In Sec. 3.2, we present explicitly the expansion of gravitational perturbations in terms
of harmonic tensors and proceed to write down the master equations that each type of
perturbation satisfy. We also present solutions for perturbations in de Sitter spacetime
in the static patch. From now on, we sometimes refer to gravitational perturbations as
gravitons, even in the classical level.
3.1
Properties of the background spacetime
The background spacetime is a manifold M with dimension 2 + n. This manifold
possesses a spatial isometry group called Gn , isomorphic to the isometry group of a ndimensional space Kn , with sectional constant curvature K = 0, ±1. The case K = 1
3.1. PROPERTIES OF THE BACKGROUND SPACETIME
33
corresponds to spherical symmetry. Then, we can locally write M as the product:
M2+n = N 2 × Kn 3 (y a , xi ) = (z µ ).
(3.1)
N 2 is called the orbit spacetime and is described by a Lorentzian metric, with two dimensions. Eq. (3.1) also establishes the notation used. We use the first letters of the
latin alphabet (a, b, c, ...) to refer to indices in quantities in the orbit spacetime N 2 , while
quantities in Kn are written with indices starting from i (i, j, k, ...). We are going to use
greek letters for indices in the background spacetime.
If the background spacetime is a solution of vacuum Einstein’s equation, then by
Birkhoff’s theorem, its line element may be written as [30]
ds2 = −f (r)dt2 +
with
f (r) = K −
dr2
+ r2 dσn2 ,
f (r)
2M
r2
−
rn−1
l2
(3.2)
(3.3)
and dσn2 is the line element of the space Kn . Both solutions of de Sitter and Schwarzschild
spacetimes are contemplated in Eqs. (3.2) and (3.3) (with K = 1) (and Schwarzschild-de
Sitter spacetime as well), where M is the mass of the spherically symmetric object and l
is de Sitter radius. The line element of the orbit spacetime will be:
ds2orb = gab (y)dy a dy b = −f (r)dt2 +
dr2
.
f (r)
(3.4)
We denote the covariant derivatives and Christoffel symbols as follows:
ds2 ⇒ ∇µ , Γαµν ,
ds2orb ⇒ Da , Γabc (t, r),
dσn2 = γij (x)dxi dxj ⇒ D̂i , Γ̂ijk (x).
(3.5)
(3.6)
(3.7)
We can write expressions for the Christoffel symbols of the background spacetime in
terms of their correspondent quantities in the other spaces N 2 and Kn , namely
Γabc (z) = Γabc (t, r),
Da r i
Γiaj (z) =
δ j,
r
Γaij (z) = −rDa rγij (x),
Γijk (z) = Γ̂ijk (x).
(3.8)
In the static patch of (n + 2)-de Sitter spacetime, we have K = 1 and M = 0. Hence
3.2. GRAVITATIONAL PERTURBATIONS - THREE DIFFERENT
TYPES OF PERTURBATIONS
34
the line element will be
ds2 = −(1 − r2 )dt2 +
dr2
+ r2 dσn2 ,
1 − r2
(3.9)
where we adopted l = 1 for simplicity. From now on we will restrict ourselves to this
spacetime, although most of the discussions here can be easily extended to other spacetimes. Besides, we will be interested in the propagation of perturbations in vacuum, that
is, with δTµν = 0.
3.2
Gravitational perturbations - three different types
of perturbations
We can show that the metric perturbation’s components transform differently with
respect to a spatial rotation. A spatial rotation, in spherical coordinates, can be given by
t → t0
r → r0
θ1 → θ10
θ2 → θ20
θn → θn0
=
=
=
=
..
.
=
t,
r,
θ10 (θ1 , θ2 , ..., θn ),
θ20 (θ1 , θ2 , ...θn ),
θn0 (θ1 , θ2 , ..., θn ),
(3.10)
(3.11)
where θ1 , θ2 , ... and θn are the angular variables. The transformations between the angular
variables will have a form that depends on the number of dimensions. For instance, in
four dimensions, a rotation about the z-axis by an angle α will have the form
t → t0
r → r0
θ → θ0
φ → φ0
=
=
=
=
t,
r,
θ,
φ + α.
(3.12)
3.2. GRAVITATIONAL PERTURBATIONS - THREE DIFFERENT
TYPES OF PERTURBATIONS
35
Rotation about the y-axis is a lot more complicated:
t → t0 = t,
r → r0 = r,
θ → θ0 = arccos (cos α cos θ − sin α sin θ cos φ) ,
−1
cot θ
0
φ → φ = arctan cot φ cos α + sin α
.
sin φ
(3.13)
And rotation about the x-axis is given by
t → t0 = t,
r → r0 = r,
θ → θ0 = arccos (sin α sin θ sin φ + cos α cos θ) ,
cot θ
0
.
φ → φ = arctan cos α tan φ − sin α
cos φ
(3.14)
Note that, in spherical coordinates, the rotation transformations are not linear. Therefore,
a general rotation has a much more involved form than in cartesian coordinates.
The metric perturbations components will transform as
h0µν =
∂xα ∂xβ
hαβ .
∂x0µ ∂x0ν
(3.15)
The r and t coordinates remain unchanged under a spatial rotation and the components
hab transform as scalars under rotations. In fact, we see that
∂xα
= δaα ,
0a
∂x
(3.16)
where, as we already noted, x0a = xa is the time or radial coordinate of the new coordinate
system. The hai ’s change as
∂xj
h0ai =
haj ,
(3.17)
∂x0i
implying that they transform as vectors under spatial rotations. The hij components
transforms as 2-tensors under spatial rotations, namely
h0ij =
∂xk ∂xl
hkl .
∂x0i ∂x0j
(3.18)
Therefore, if we restrict ourselves to the Kn , which is a compact submanifold (for
K = 1), the components hµν change differently to coordinate transformations and we
can consider expansions for the components of hµν in terms of functions that have these
3.2. GRAVITATIONAL PERTURBATIONS - THREE DIFFERENT
TYPES OF PERTURBATIONS
36
transformation properties.
As scalars under rotations, it is natural to expand the hab components in scalar spherical harmonics, so that they carry the symmetry properties of the background spacetime.
We will show later the equation that scalar spherical harmonics satisfy. That way we have
the following expansion:
X (lσ)
hab =
fab (t, r)S(lσ) ,
(3.19)
l,σ
with S(lσ) being the scalar spherical harmonic with labels l and σ. The coefficients
(lσ)
fab (t, r) will in general be functions of t and r, since we were initially doing the expansion in Kn , that is, for fixed t and r. Actually, σ is a set of eigenvalues depending on
the number of dimensions of the spacetime. For example, in 4 dimensions, the eigenfunctions will be just the usual spherical harmonics and we have the expansion:
hab =
X
(l,m)
fab
(t, r)Y (l,m) (θ, φ).
(3.20)
l,m
Any vector in a compact manifold can be decomposed according to the Helmholtz
decomposition [53], that is, as the gradient of a function plus a divergenceless vector.
Then, for a vector Vi , we have
D̂i W i = 0.
Vi = D̂i v + Wi ,
(3.21)
We can again expand the function v in terms of scalar spherical harmonics. The vector
Wi can be expanded in terms of the vector spherical harmonics, which have vanishing
divergence. We then have the following expansion for the hai ’s:
hai =
X
ga(lσ) (t, r)D̂i S(lσ) +
l,σ
X
(lσ)
u(lσ)
a (t, r)Vi
(3.22)
l,σ
Now, for 2-tensors Tij , we can decompose them in the form [53]:
Tij = D̂i Vj + D̂i Vj + Wij ,
(3.23)
where Wij is a transverse tensor (D̂i Wij = 0). In a space with constant curvature we can
uniquely decompose the transverse tensor in the form [53]:
Wij = Pij + γij ψ,
D̂i Pij = P i j = 0,
(3.24)
where ψ is an arbitrary function and Pij is a transverse-traceless 2-tensor. If we do the
3.2. GRAVITATIONAL PERTURBATIONS - THREE DIFFERENT
TYPES OF PERTURBATIONS
37
Helmholtz decomposition for the vector Vi , we obtain
Tij = D̂i D̂j ϕ + γij ψ + D̂i Lj + D̂j Li + Pij .
(3.25)
Now we have two arbitrary functions ϕ and ψ, a divergenceless vector Li and a transversetraceless 2-tensor Pij . We already know how to expand the functions and the vector in
terms of harmonic objects, so we have to expand the 2-tensor Pij in terms of the harmonic
2-tensors, which are transverse and traceless. Then we have the following expansion for
the hij components:
i
Xh
(lσ)
(lσ)
(lσ)
(lσ)
s (t, r)D̂i D̂j S + t (t, r)γij S
← scalar part
hij =
l,σ
+
X
+
X
(lσ)
(lσ)
← vector part
p(lσ) (t, r) D̂i Vj + D̂j Vi
l,σ
(lσ)
q (lσ) (t, r)Tij ← tensor part.
(3.26)
l,σ
Now we can consider these expansions separately. In other words, we can study the
perturbations that depend just on a particular type of harmonic object. We call them
scalar, vector or tensor perturbations, which depend on scalar, vector or tensor harmonics,
respectively.
3.2.1
Scalar perturbations
We now present the scalar spherical harmonics Slσ , which satisfy the following equation:
ˆ n + k 2 )S(lσ) = 0,
(∆
S
(3.27)
ˆ n = D̂i D̂i is the Laplace-Beltrami operator on Kn . For K = 1, which is our case,
where ∆
the set of eigenvalues kS2 is discrete and has the form
kS2 = l(l + n − 1),
(3.28)
The label l is a nonnegative integer and σ represents all labels other than l. They can be
shown to satisfy orthonormality according to the following inner product
Z
0
0
dΩn S(lσ) S(l0 σ0 ) = δ ll δ σσ ,
0 0
(3.29)
where S(l0 σ0 ) is the complex conjugate of S(l σ ) and the integration is over the unit hypersphere S n (with K = 1, the space Kn is isomorphic to S n ). In the 4-dimensional case,
3.2. GRAVITATIONAL PERTURBATIONS - THREE DIFFERENT
TYPES OF PERTURBATIONS
38
the Laplace-Beltrami operator is just the angular part of the usual Laplacian in spherical
coordinates, that is
∂
1 ∂2
1 ∂
ˆ
sin θ
+
∆2 ≡
.
(3.30)
sin θ ∂θ
∂θ
sin2 θ ∂φ2
Hence, in the 4-dimensional case, the S(lσ) functions are just the usual scalar spherical
harmonics Y (l,m) (θ, φ) with labels l and m and eigenvalues given by:
ˆ 2 Ylm (θ, φ) = −l(l + 1)Ylm (θ, φ),
∆
∂
Lz ≡ −i Ylm (θ, φ) = mYlm (θ, φ).
∂φ
(3.31)
(3.32)
Following the expansion used in (3.2), we can decompose the metric perturbation hµν
in terms of its scalar part, that is, the part that depends only on the scalar functions S(lσ)
(and its gradients). We can write each mode of the scalar perturbation as
(S;lσ)
hab
(S;lσ)
hai
(S;lσ)
hij
(l)
= fab S(lσ) ,
=
=
(lσ)
rfa(l) Si ,
(l)
2r2 (γij HL S(lσ)
(3.33)
(3.34)
+
(l) (lσ)
HT Sij ),
(3.35)
where
(lσ)
= −
(lσ)
=
Si
Sij
(l)
(l)
(l)
1
D̂i S(lσ) ,
kS
1
1
D̂i D̂j S(lσ) + γij S(lσ) ,
2
kS
n
(l)
(3.36)
(3.37)
and the coefficients fa , fab , HL and HT are all functions of t and r and are gaugedependent quantities. We note that we are using a slightly different expansion than the
used at (3.26), but this is just a redefinition of the coefficients to make a clear distinction
between a longitudinal part (containing HL ) and a transverse part (containing HT ). Notice
(lσ)
that the tensors Sij are chosen to be traceless.
The modes with l = 0, 1 are special cases (for which some of the coefficients above
(lσ)
(lσ)
are not defined). For l = 0, S(lσ) is a constant, and Si and Sij are not defined. The
perturbed spacetime will be spherically symmetric, but the only solution to the Einstein’s
equation in this case is Schwarzschild-de Sitter spacetime by Birkoff’s theorem [30]. So this
perturbation consists of a time-independent change of the background metric in the form
of a shift in the mass parameter. Obviously, this is singular at the origin, since it changes
the original de Sitter metric into the Schwarzschild-de Sitter one, thereby introducing a
(l)
small black hole. Hence, we exclude this case. For l = 1, the variable HT is not defined.
(1)
However we can set HT = 0 and regard this equation as a gauge condition. We find that
there is no corresponding nonzero gauge-invariant perturbation as shown in the Appendix
B of Ref. [30]. Hence we can impose the condition l ≥ 2.
3.2. GRAVITATIONAL PERTURBATIONS - THREE DIFFERENT
TYPES OF PERTURBATIONS
39
We can define gauge-invariant quantities by suitable combinations of the gauge variables. Since the infinitesimal gauge transformations are generated by a vector, we can do
the Helmoltz decomposition and define a gauge transformation of the scalar type:
(lσ)
ξa(lσ) = Ta(l) (t, r)S(lσ) ,
ξi
(lσ)
= rL(l) (t, r)Si .
(3.38)
This type of gauge transformation is defined so that it does not mix the different types of
perturbations. With this gauge transformation, the expansion coefficients of the metric
perturbation transforms as
(l)
δfab
δfa(l)
(l)
= −Da Tb − Db Ta(l) ,
(l) kS
L
= −rDa
+ Ta(l) ,
r
r
δXa(l) = Ta(l) ,
kS
Da r (l)
(l)
δHL = − L(l) −
T ,
nr
r a
kS (l)
(l)
δHT =
L ,
r
(3.39)
(3.40)
(3.41)
(3.42)
(3.43)
(l)
where Xa is a gauge-dependent quantity defined by
Xa(l)
r
=
kS
r
(l)
(l)
fa + Da HT .
kS
(3.44)
With that we can form two gauge-invariant quantities given by
1 (l) 1 a
H + D (rXa(l) ),
n T
r
(l)
(l)
= fab + Da Xb + Db Xa(l) .
(l)
F (l) = HL +
(l)
Fab
(3.45)
(3.46)
Now, using the perturbed Einstein’s equation in vacuum, that is, Eq. (2.31) with
δTµν = 0, we can decompose Eq. (2.31) in terms of components δRab , δRai , δRij . Substi-
3.2. GRAVITATIONAL PERTURBATIONS - THREE DIFFERENT
TYPES OF PERTURBATIONS
40
tuting Eqs. (3.45) and (3.46) in Eq. (2.31), we obtain four equations, namely
Dc r
(l)
(l)
(l)
(l)
(−Dc Fab + Da Fcb + Db Fca
)
− Fab + Da Dc Fbc (l) + Db Dc Fac (l) + n
r
2
kS
1
1
(l)
(l)
(l)
(2)
c (l)
+
Da rDb F + Db rDa F
+ 2R − 2(n + 1) Fab − Da Db Fc − 2n
r2
r
r
2n
2n c d
n(n − 1) c d
(l)
(l)
− Dc Dd F cd(l) + Dc rDd Fcd +
D D r+
D rD r Fcd
r
r
r2
k2 − n
2n(n + 1)
− 2nF (l) −
Dr · DF (l) + 2(n − 1) S 2 F (l) − 2nDa Db F (l)
r
2
r
1
k
n
S
+ R(2) Fcc (l) gab = 0,
− Fcc (l) − Dr · DFcc (l) +
(3.47)
r
r2
2
1
rn−2
Db (r
n−2
(l)
Fab )
+ rDa
1 b (l)
F
+ 2(n − 1)Da F (l) = 0,
r b
(3.48)
(l)
2(n − 1) a b (l) n − 1
a
b
a b
D rD Fab −
(n
−
2)D
rD
r
+
2rD
D
r
Fab
r
r2
1 (2) (n − 1)kS2
n−1
Dr · DFcc (l) +
R −
Fcc (l)
+ Fcc (l) +
r
2
nr2
2(n − 1)(n − 2)(kS2 − n) (l)
2n(n − 1)
Dr · DF (l) −
F = 0, (3.49)
+ 2(n − 1)F (l) +
r
nr2
− Da Db F ab(l) −
Faa (l) + 2(n − 2)F (l) = 0,
(3.50)
where R(2) is the Ricci scalar in N 2 and the is the d’Alembertian operator in the
two-dimensional orbit spacetime with line element ds2orb , namely
=−
1 ∂2
∂
∂
+ (1 − r2 ) .
2
2
1 − r ∂t
∂r
∂r
(3.51)
Due to the Bianchi identity, Eqs. (3.47)-(3.50) are not independent. It can be shown that
a set of independent equations of motion for the scalar perturbations are given by Eqs.
(3.47), (3.50) and the following equation [30, 54]:
Db (F̃ab(l) − 2F̃ (l) δab ) = 0,
(3.52)
3.2. GRAVITATIONAL PERTURBATIONS - THREE DIFFERENT
TYPES OF PERTURBATIONS
41
where
(l)
(l)
F̃ab = rn−2 Fab , F̃ (l) = rn−2 F (l) .
(3.53)
We can write the gauge-invariant quantities in terms of a master variable, namely
F̃ (l) =
(l)
F̃ab
1
(l)
( + 2)ΩS ,
2n
= Da Db ΩS −
n−1
n−2
+
n
n
(3.54)
(l)
ΩS gab .
(3.55)
(l)
With Eqs. (3.54) and (3.55), one can show that Fab and F (l) solve the perturbed Einstein’s
equation given by Eqs. (3.47), (3.50) and (3.52), if the master variable satisfies the
following wave equation [35, 54]:
(l)
ΩS
n
(l)
− (Dα r)(Dα ΩS ) −
r
(l)
kS2 − n
(l)
+ n − 2 ΩS = 0.
2
r
(3.56)
(l)
We can make the change ΩS = rn/2 ΦS to obtain what we call the scalar master
equation:
VS
(l)
(l)
ΦS −
ΦS = 0,
(3.57)
2
1−r
where the effective potential is given by
VS =
1 − r2
[4l(l + n − 1) + n(n − 2)
4r2
− (n − 2)(n − 4)r2 .
(3.58)
One can find solutions with Fourier components proportional to e−iωt and regular at the
origin, which are given by:
(ωl)
(ωl)
ΦS (t, r) = AS e−iωt rl+n/2 (1 − r2 )iω/2
1
n+1 2
1
×F
(iω + l + n − 1), (iω + l + 2); l +
;r ,
2
2
2
(3.59)
where the function F (α, β; γ; z) is Gauss’ hypergeometric function [55]. The normalization
(ωl)
constants AS will be determined later. This solution is the same as the one obtained in
Ref. [56], with d = n + 2 e j = 5.
3.2. GRAVITATIONAL PERTURBATIONS - THREE DIFFERENT
TYPES OF PERTURBATIONS
3.2.2
42
Vector perturbations
(lσ)
The vector-type perturbations are expanded in terms of harmonic vectors Vi , which
satisfy
ˆ n + k 2 )V(lσ) = 0,
(∆
(3.60)
V
i
D̂j Vj(lσ) = 0.
(3.61)
kV2 = l(l + n − 1) − 1,
(3.62)
Here,
(lσ)
where l = 1, 2, ... and σ again represents all labels other than l. The Vi
thonormality properties, that is
Z
(lσ)
(l0 σ 0 )
dΩn γ ij Vi Vj
Z
0
0
= δ ll δ σσ ,
(lσ) (l0 σ 0 )
dΩn γ ij Vi Si
satisfy or-
(3.63)
= 0.
(3.64)
In four dimensions, the solutions to Eqs. (3.60) and (3.61) for the vector harmonics
on the S 2 can be written as [46, 57]:
(l,m)
Yi
ij
(θ, φ) = p
∂ j Y (l,m) (θ, φ),
l(l + 1)
(3.65)
where ij is the totally antisymmetric tensor defined by:
θθ = φφ = 0,
θφ = −φθ = sin θ.
(3.66)
(3.67)
The metric perturbations of the vector type read
(V ;lσ)
= 0,
(V ;lσ)
= rfa(l) Vi ,
(V ;lσ)
= 2r2 HT Vij ,
hab
hai
hij
with
(lσ)
Vij = −
(lσ)
For l = 1, the tensors Vij
(3.68)
(lσ)
(l)
(3.69)
(lσ)
(3.70)
1
(lσ)
(lσ)
(D̂i Vj + D̂j Vi ).
2kV
(3.71)
(l)
vanish, rendering the coefficient Fa undefined. In this case
3.2. GRAVITATIONAL PERTURBATIONS - THREE DIFFERENT
TYPES OF PERTURBATIONS
43
one defines a gauge-invariant quantity given by
(1)
(1)
Fab = rDa (fb /r) − rDb (fa(1) /r).
(3.72)
For this specific mode, substituting the gauge-invariant quantity given by Eq. (3.72) in
the perturbed Einstein’s equation (2.31), we have the following equation that it has to
satisfy
1
(1)
Db (rn+1 Fab ) = 0.
(3.73)
n+1
r
The solution to the equation above is
C
(1)
Fab = ab
rn+1
,
(3.74)
in which C is an arbitrary constant and ab is the Levi-Civita tensor in the N 2 space.
Note that this tensor ab is different from the one defined by Eqs. (3.66) and (3.67).
The solution (3.74) gives rise to a rotational perturbation, similar to the Myers-Perry
solution [30, 58] if the black hole mass is nonzero. This means that in our case with no
black hole, there is no nonzero gauge-invariant vector-type perturbation with l = 1. So,
we again consider only modes with l ≥ 2.
A gauge transformation of the vector type is given by
(lσ)
ξa(lσ) = 0,
ξi
(lσ)
= rL(l) (t, r)Vi
(3.75)
and the expansion coefficients in Eqs. (3.68)-(3.70) transform as
δfa(l)
(l)
δHT
= −rDa
=
L(l)
r
,
kV (l)
L .
r
(3.76)
(3.77)
As in the scalar case, we define a gauge-invariant quantity for l ≥ 2 as follows:
Fa(l) = fa(l) +
r
(l)
Da HT .
kV
(3.78)
(l)
From the perturbed Einstein’s equation, we obtain, for the gauge-invariant quantity Fa :
kV
Da (rn−1 F a(l) ) = 0,
n
r
(3.79)
3.2. GRAVITATIONAL PERTURBATIONS - THREE DIFFERENT
TYPES OF PERTURBATIONS
1
r
Db
n+1
(
"
(l)
rn+2 Db
Fa
r
!
(l)
− Da
Fb
r
!#)
−
kV2 − (n − 1) (l)
Fa = 0.
r2
44
(3.80)
(l)
This quantity is related to a master variable ΦV by
(l)
rn−1 F (l)a = ab Db (rn/2 ΦV ).
(3.81)
which is shown to satisfy [35]:
(l)
ΦV −
with
VV
(l)
ΦV = 0,
2
1−r
1 − r2
n(n − 2)
2
(1 − r ) .
VV =
l(l + n − 1) +
r2
4
(3.82)
(3.83)
The solutions of Eq. (3.82) regular at the origin are
(ωl)
(ωl)
ΦV (t, r) = AV e−iωt rl+n/2 (1 − r2 )iω/2
1
1
n+1 2
(iω + l + 1), (iω + l + n); l +
;r .
×F
2
2
2
(3.84)
(ωl)
The normalization constants AV will be determined later. The solution given by (3.84)
is the same as the one obtained in Ref. [56], with d = n + 2 and j = 3.
3.2.3
Tensor (rank-2) perturbations
The tensor-type perturbations of the metric can be expanded in terms of symmetric
(lσ)
harmonic tensors of second rank Tij . They obey the following equations:
(lσ)
ˆ n + k 2 )T
(∆
T
ij
= 0,
(3.85)
Ti i(lσ) = 0,
D̂j Ti j(lσ) = 0.
(3.86)
(3.87)
kT2 = l(l + n − 1) − 2.
(3.88)
The set of eigenvalues is given by
3.2. GRAVITATIONAL PERTURBATIONS - THREE DIFFERENT
TYPES OF PERTURBATIONS
45
(lσ)
The label l is an integer larger than or equal to 2 (l ≥ 2). The Tij satisfy orthonormality
properties, namely
Z
(l0 σ 0 )
(lσ)
dΩn γ ik γ jl Tij Tkl
Z
Z
(lσ) (l0 σ 0 )
dΩn γ ik γ jl Tij Skl
(lσ)
0
0
= δ ll δ σσ ,
(l0 σ 0 )
dΩn γ ik γ jl Tij Vkl
(3.89)
= 0,
(3.90)
= 0.
(3.91)
It is a well-known fact that solutions to Eqs. (3.85), (3.86) and (3.87) do not exist on
S 2 [46, 59]. A concise proof of this fact can be found in Ref. [57]. Thus, we do not have
tensor-type modes for gravitational perturbations in 3 + 1 dimensions.
The harmonic modes of the metric perturbation are written as
hab
(T ;lσ)
= 0,
(T ;lσ)
hai
(T ;lσ)
hij
= 0,
= 2r
(3.92)
(3.93)
2
(l) (lσ)
HT Tij .
(3.94)
(l)
The quantity HT is already gauge-invariant since the generator of a gauge transformation
is a vector, which can only be expanded in terms of scalar and vector harmonics according
to the Helmholtz decomposition. Therefore, there is no gauge transformation of the tensor
(rank-2) type.
(l)
It is convenient to introduce a new variable ΦT by
(l)
(l)
ΦT = rn/2 HT .
(3.95)
(l)
Then the perturbed Einstein’s equation for ΦT reads
(l)
ΦT −
VT
(l)
ΦT = 0,
2
1−r
(3.96)
where the effective potential is
VT
1 − r2
n(n − 2)
=
l(l + n − 1) +
2
r
4
n(n + 2) 2
−
r .
4
(3.97)
3.2. GRAVITATIONAL PERTURBATIONS - THREE DIFFERENT
TYPES OF PERTURBATIONS
46
The solutions of Eq. (3.96) regular at the origin are given by
(ωl)
(ωl)
ΦT (t, r) = AT e−iωt rl+n/2 (1 − r2 )iω/2
1
n+1 2
1
(iω + l + n + 1), (iω + l); l +
;r ,
×F
2
2
2
(3.98)
(ωl)
where the normalization constants AT will be determined later. The solution given by
(3.98) is the same as the one obtained in Ref. [56] with d = n + 2 and j = 1.
Chapter 4
Graviton two-point function in de
Sitter spacetime
We wish to construct the physical1 graviton two-point function in a free field theory
with gauge invariance such as linearized gravity (see, e.g. Refs. [22, 23])2 . Due to gauge
invariance, linearized gravity cannot be quantized in a straightforward manner. One way
to overcome this difficulty is to insert a gauge fixing term. Another way is to fix the gauge
completely. We adopt the latter approach here.
We firstly present a way of quantizing the gravitational perturbations studied in Chapter 3. Then, in Sec. 4.1, we compute the normalization constants according to a particular
inner product. Finally, in Sec. 4.2, we compute the graviton two-point function inside
the cosmological horizon of de Sitter spacetime, i.e. in the static patch, and show that
the two-point function is finite in the infrared limit.
Let us assume the theory is described by a Lagrangian density L, where L is a local
function of hµν and ∇λ hµν . Though we use a symmetric tensor field theory in our explanation for obvious reasons, the construction presented here works for any other linear
field theories.
The Lagrangian density of linearized gravity (in vacuum) can be written as
√
−g λµνλ0 µ0 ν 0
∇λ hµν ∇λ0 hµ0 ν 0 + S µνλρ hµν hλρ ],
[K
L=
2
0 0 0
0 0 0
(4.1)
0 0 0
where K λµνλ µ ν = K λ µ ν λµν = K λνµλ µ ν and S µνλρ = S λρµν = S νµλρ = S µνρλ . We define
the conjugate momentum current pλµν by
1
∂L
0 0 0
= K λµνλ µ ν ∇λ0 hµ0 ν 0 .
pλµν := √
−g ∂(∇λ hµν )
1
(4.2)
The word “physical” is used here in the sense that all gauge degrees of freedom are fixed.
We are assuming the reader to be familiar with quantum field theory in curved spacetimes. The
interested reader can consult, for example, Refs. [60, 61].
2
48
The Euler-Lagrange equation (i.e., the perturbed Einstein’s equation in vacuum) will be
∇λ pλµν − S µνλρ hλρ = 0.
(4.3)
0 0 0
By comparison with Eq. (2.31), we can readily find the expressions for both K λµνλ µ ν
and S µνλρ , however they are not especially useful. For the interested reader, they can be
found in Ref. [22].
To quantize the gravitational perturbations, we will make an expansion of the field
ĥµν (y), where y represents all spacetime coordinates, in the following way:
ĥµν (y) =
X
(n)
(n)
[an hµν
(y) + a†n hµν (y)],
(4.4)
n
(n)
where hµν (y) and its complex conjugate form a complete set of classical solutions and n
represents all the possible labels for the solutions, which can be continuous or discrete.
The operators an and a†n are possible candidates to be annihilation and creation operators,
respectively. However, we have to define an invariant inner product, through which we can
define the usual equal-time canonical commutation relations for these operators. What
will be shown is that the gauge invariance of the theory is an obstacle to this method.
For any two solutions hµν and h0µν and their conjugate momentum currents pλµν and
p0λµν computed from (4.2), we define the current
J λ := hµν p0λµν − pλµν h0µν .
(4.5)
Then this current is conserved, namely:
∇λ J λ = ∇λ hµν p0λµν + hµν ∇λ p0λµν − ∇λ pλµν h0µν − pλµν ∇λ h0µν
= 0,
(4.6)
as can be shown through the equations of motion (4.3). We define the symplectic product
[62] by
Z
Ω(h, h0 ) = −
dΣnα (hµν p0αµν − pαµν h0µν ),
(4.7)
Σ
where Σ is a Cauchy surface and nα is the future-directed unit normal vector to Σ. It can
readily be shown that Ω(h, h0 ) is independent of the choice of Σ, since the integrand is a
conserved current [63].
Now, suppose that the symplectic product Ω is non-degenerate, i.e. that there are
(null)
no solutions hµν satisfying Ω(h, h(null) ) = 0, for all solutions hµν . Suppose further
(n)
(n)
that hµν and their complex conjugates hµν form a complete set of solutions such that
(n)
(m)
Ω(h(n) , h(m) ) = 0, for all n and m — i.e., Ω is nonzero only between hµν and hµν — and
49
define the inner product of two solutions by
hh(m) , h(n) i = iΩ(h(m) , h(n) ).
(4.8)
Then, the equal-time canonical commutation relations for the operators ĥµν (y) are
equivalent to
[am , a†n ] = (M −1 )mn ,
(4.9)
[am , an ] = [a†m , a†n ] = 0,
(4.10)
where M −1 is the inverse of the matrix M mn = hh(m) , h(n) i.
Unfortunately, linearized gravity cannot be quantized in this manner because the
matrix M defined by Eq. (4.8) is degenerate due to the gauge invariance: a pure-gauge
(g)
solution of the form hµν = ∇µ ξν + ∇ν ξµ has vanishing symplectic product with any
solution. So, if we include gauge solutions in the expansion (4.4), the matrix M will not be
invertible. However, if we fix the gauge completely so that the matrix M is non-degenerate,
then we can expand the field operator ĥµν (y) using only the solutions satisfying the gauge
conditions in Eq. (4.4) and quantize this field by requiring the commutation relations
given by Eq. (4.9). This procedure is the gauge-fixed version of the gauge-invariant
quantization formulated in Ref. [22].
Note that, if we normalize the solutions in a given gauge by requiring M mn = δ mn in
Eq. (4.8), then we have
[am , a†n ] = δmn .
(4.11)
With that, an and a†n will be the usual annihilation and creation operators, respectively.
Then, on the vacuum state |0i annihilated by the operators an , the two-point function is
h0|ĥµν (y)ĥµ0 ν 0 (y 0 )|0i =
X
(n)
0
h(n)
µν (y)hµ0 ν 0 (y ).
(4.12)
n
In the next sections we normalize the gravitational perturbations found in Chapter 3 so
that we have M mn = δ mn . This will make the construction of the two-point function
straightforward.
4.1. NORMALIZATION OF GRAVITATIONAL PERTURBATIONS
4.1
50
Normalization of gravitational perturbations
With a suitable normalization of the gravitational perturbation hµν , the part of the
Lagrangian density involving derivatives of hµν reads (after some integration by parts)
"
√
1
L =
−g ∇µ hµλ ∇ν hνλ − ∇λ hµν ∇λ hµν
2
1
+ (∇µ h − 2∇ν hµ ν )∇µ h
2
#
+ terms involving just hµν .
(4.13)
Hence, the conjugate momentum current is
1
∂L
pλµν := √
−g ∂(∇λ hµν )
λµ
= g ∇κ hκν + g λν ∇κ hκµ − ∇λ hµν
+g µν (∇λ h − ∇κ hλ κ )
1
− (g λν ∇µ h + g λµ ∇ν h).
2
(m)
(4.14)
(n)
Then, the inner product (4.8) between two solutions hµν and hµν can be written as
(m)
hh
(n)
,h
Z
i := −i
(m)
(m)λµν ,
dΣnλ hµν p(n)λµν − h(n)
p
µν
(4.15)
Σ
where the integration is to be carried out on a t = constant Cauchy surface of the static
patch of de Sitter space. Next, we find the normalization constants such that the inner
product (4.15) is simply δ mn (which also involves Dirac’s delta function because ω is a
continuous label). The calculation will closely follow Ref. [43].
4.1.1
Normalization of the tensor-type modes
(T ;ωlσ)
We are going to denote the tensor-type perturbations by hµν
since we are considering
them as being made from the solutions given by (3.98). The correspondent conjugate
momentum is
p(T ;ωlσ)λµν = −∇λ h(T ;ωlσ)µν
(4.16)
since the tensor-type perturbations given by Eqs. (3.92)-(3.94) are transverse (∇µ hµν = 0)
and traceless (hµ µ = 0). Note that there is no need to fix the gauge for the tensor
perturbations since they are already gauge-invariant, then they only have physical degrees
of freedom. Noting that r = 1 is the position of the cosmological horizon, we find the
4.1. NORMALIZATION OF GRAVITATIONAL PERTURBATIONS
51
inner product defined by Eq. (4.15) to be
(T ;ωlσ)
hh
(T ;ω 0 l0 σ 0 )
,h
ρ
Z
0
i = (ω + ω ) lim
ρ→1
0
rn
dr
1 − r2
Z
(T ;ωlσ) (T ;ω 0 l0 σ 0 )ij
dΩn hij
h
,
(4.17)
where dΩn integration is over the unit hypersphere S n . Noting that
Z
0 0
(lσ)
dΩn Tij T(l σ )ij =
1 ll0 σσ0
δ δ ,
r4
(4.18)
since the harmonic tensors are now being considered as quantities in the whole spacetime,
(lσ)
then we raise indices on the Tij with g ij = γ ij /r2 .
We then have
(T ;ωlσ)
hh
(T ;ω 0 l0 σ 0 )
,h
Z
ll0 σσ 0
0
i = 4(ω + ω )δ δ
ρ
lim
ρ→1
0
dr
(ωl) (ω 0 l)
ΦT ΦT .
2
1−r
(4.19)
We have to evaluate the following integral:
Z
ρ
Iρ = lim
ρ→1
(ωl)
Using Eq. (3.96), satisfied by ΦT
0
dr
(ωl) (ω 0 l)
ΦT ΦT .
2
1−r
(4.20)
(ωl)
and ΦT , we find
d
ω 02 − ω 2 (ωl) (ω0 l)
(ωl)
(ωl)
(ω 0 l)
(ω 0 l)
2 d
2 d
Φ ΦT
=
. (4.21)
ΦT (1 − r ) ΦT − ΦT (1 − r ) ΦT
1 − r2 T
dr
dr
dr
Integrating the above equation from 0 to ρ and then taking the limit ρ → 1, we find
Z
lim
ρ→1
0
ρ
"
0
1
dr
(ωl) (ω l)
(ω 0 l) d
(ωl)
(ωl) d
(ω 0 l)
2
Φ ΦT
=
lim (1 − r ) ΦT
Φ − ΦT
Φ
,
1 − r2 T
ω 02 − ω 2 ρ→1
dr T
dr T
r=ρ
(4.22)
(ω 0 l)
(ωl)
where we have used that ΦT (0) = ΦT
We can write, for r ≈ 1 [64],
(ωl)
Φ(ωl) = AT
where
Bωl =
h
(0) = 0.
i
Bωl (1 − r2 )−iω/2 + Bωl (1 − r2 )iω/2 ,
Γ(l + n+1
)Γ(iω)
2
.
1
1
Γ( 2 (l + iω))Γ( 2 (l + iω + n + 1))
(4.23)
(4.24)
4.1. NORMALIZATION OF GRAVITATIONAL PERTURBATIONS
52
Then we have
Z
ρ
dr
0
Φ(ωl) Φ(ω l)
2
0 1−r
(ωl) i|AT |2
i 0
l
l
2
=
B−ω
B−ω
0 exp[ (ω + ω) ln(1 − ρ )]
0
ω +ω
2
i 0
l
l
2
− Bω Bω0 exp[− (ω + ω) ln(1 − ρ )]
2
(ωl) 2 i|A |
i 0
l
2
+ 0T
Bωl B−ω
0 exp[ (ω − ω) ln(1 − ρ )]
ω −ω
2
i 0
2
l
l
− B−ω Bω0 exp[− (ω − ω) ln(1 − ρ )] ,
2
Iρ =
(4.25)
l
noting that Bωl = B−ω
. Dropping the terms rapidly oscillating as functions of ω and ω 0
in the ρ → 1 limit, we find
0
(ωl)
2|AT |2 |Bωl |2
ω −ω
1
Iρ =
sin
ln
.
ω0 − ω
2
1 − ρ2
(4.26)
sin Lx
= πδ(x),
L→∞
x
(4.27)
Using that
lim
we have
(ωl)
I1 = lim Iρ = 2π|AT |2 |Bωl |2 δ(ω 0 − ω).
ρ→1
(4.28)
Next, we choose
(ωl)
1
,
16πω|Bωl |2
2
sinh πω Γ( 21 (l + iω))Γ( 21 (l + iω + n + 1))
=
,
2
16π 2 Γ(l + n+1
)
2
|AT |2 =
(4.29)
where we have used
|Γ(iω)|2 =
π
.
ω sinh πω
(4.30)
Then, the inner product between two modes of the tensor type is just
0 0 0
0
0
hh(T ;ωlσ) , h(T ;ω l σ ) i = δ ll δ σσ δ(ω − ω 0 ).
(4.31)
4.1. NORMALIZATION OF GRAVITATIONAL PERTURBATIONS
4.1.2
53
Normalization of the vector-type modes
We need firstly to fix the gauge, that is, we have to choose a convenient gauge. In our
case, we choose a gauge such that the components hij vanish. For the vector-type modes,
with a gauge transformation of the vector type given by hµν → hµν + ∇µ ξν + ∇ν ξµ , where
ξa = 0, ξi = r2 φVi ,
(4.32)
we find
(l)
(l)
HT → HT − kV φ,
fa(l) → fa(l) + rDa φ.
(4.33)
(4.34)
Note that we are using a slightly different gauge transformation than the one used in Eq.
(l)
(l)
(l)
(l)
(3.75). By letting φ = HT /kV , we have HT = 0 and Fa = fa . This choice of gauge
leads to
(V ;lσ)
(lσ)
(4.35)
hab
= rFa(l) Vi
i
h
1
(lσ)
b
n/2 (l)
= n−2 ab D r ΦV Vi ,
r
= 0,
(V ;lσ)
hij
= 0.
(4.37)
hai
(V ;lσ)
(4.36)
Then, we find the (gauge-invariant) inner product (4.15) for the vector-type modes as
(V ;ωlσ)
hh
(V ;ω 0 l0 σ 0 )
,h
Z
i = 2i
0 0 0
0 0 0
dΩn drrn h(V ;ωlσ)bi p(V ;ω l σ )t bi − h(V ;ω l σ )bi p(V ;ωlσ)t bi , (4.38)
where p(V ;ωlσ) is expressed in terms of h(V ;ωlσ) as
p(V )λµν = g λµ ∇κ h(V )κν + g λν ∇κ h(V )κµ − g µν ∇κ h(V )λκ − ∇λ h(V )µν ,
(4.39)
since the vector-type perturbations are traceless (hµ µ = 0). We then have
(V ;ωlσ)
(V ;ωlσ)
p(V ;ωlσ)a bi = δ a b g ρν ∇ρ hνi
− g ac ∇c hbi
nDc r (V ;ωlσ)
(V ;ωlσ)
a
cd
= δ b g Dc hdi
+
hci
r
Dc r (V ;ωlσ)
(V ;ωlσ)
ac
−g
Dc hbi
−
h
r bi
(4.40)
4.1. NORMALIZATION OF GRAVITATIONAL PERTURBATIONS
54
and
(l0 σ 0 )
V(lσ)i Vi
(ωl)
=
−
De ΩV be
bi
n−2
r!
( "
#
0 0
0 0
f (ω l )
c
f (ω l )
D
Ω
D
rD
Ω
V
V
× δ a b df g cd Dc
+ ncf
rn−2
rn−1
"
!
#)
0 0
0 0
f (ω l )
f (ω l )
D
Ω
D
rD
Ω
c
V
V
−g ac bf Dc
− bf
,
rn−2
rn−1
0 0 0
h(V ;ωlσ)bi p(V ;ω l σ )a
(ωl)
where ΩV
(4.41)
(ωl)
= rn/2 ΦV . This can be simplified as
(l0 σ 0 )
0 0 0
h(V ;ωlσ)bi p(V ;ω l σ )a
bi
V(lσ)i Vi
rn−2
=
(ω 0 l0 )
D
e
(ωl)
ΩV
(ω 0 l0 )
Da rDe ΩV
−n
rn−1
De Da ΩV
rn−2
(ω 0 l0 )
Da rDe ΩV
−
rn−1
(ω 0 l0 )
De rDa ΩV
+2
rn−1
!
.
(4.42)
Now we calculate the integral
I(ΩV , Ω0V )
Z
0 0 0
dΩn drrn h(V ;ωlσ)bi p(V ;ω l σ )t bi
Σ
Z 1
(ωl)
ll0 σσ 0
= 2iδ δ
dr∂ t ΩV
0
Z 1
t
r
∂t ∂
(ωl)
(ω 0 l)
t ∂
dr∂ r ΩV
×
+ Γtr n−2 ΩV +
n−2
r
r
0
t
t
t
∂r ∂
∂
∂
(ω 0 l)
t
× n−2 + Γrt n−2 + 2 n−1 ΩV
,
r
r
r
= 2i
(4.43)
(4.44)
where we used the fact that
Z
(l0 σ 0 )
dΩn V(lσ)i Vi
=
1 ll0 σσ0
δ δ .
r2
(ω 0 l)
We use the following equation to eliminate the term ∂t ∂ t ΩV
(ω 0 l)
∂t ∂ t ΩV
rn−2
(ω 0 l)
= −∂r
∂ r ΩV
rn−2
!
(ω 0 l)
∂ r ΩV
+ 2 n−1
r
(4.45)
in Eq. (4.44):
(ω 0 l)
[l(l + n − 1) − n]ΩV
+
rn
.
(4.46)
(ω 0 l)
This equation is just the master equation given by Eq. (3.82), written in terms of ΩV
.
4.1. NORMALIZATION OF GRAVITATIONAL PERTURBATIONS
55
(ωl)
We multiply this equation by ∂ t ΩV and integrate with respect to r. We use integration
by parts for the second term, dropping the boundary term because it oscillates rapidly as
a function of ω and ω 0 , unless ω = ω 0 , and hence can be neglected as a distribution of ω
and ω 0 . We substitute the resulting expression into Eq. (4.44) and find the inner product
to be
0 0 0
hh(V ;ωlσ) , h(V ;ω l σ ) i = I(ΩV , Ω0V ) − I(Ω0V , ΩV )
Z
ll0 σσ 0
= 2iδ δ (l − 1)(l + n)
(4.47)
(ω 0 l)
1
dr
ΩV
(ωl)
∂ t ΩV
0
(ωl)
(ω 0 l)
− ΩV ∂ t ΩV
rn
,
(4.48)
i.e.
(V ;ωlσ)
hh
(V ;ω 0 l0 σ 0 )
,h
ll0 σσ 0
i = 2iδ δ
Z
(l − 1)(l + n)
0
1
dr
(ωl)
(ω 0 l)
(ω 0 l)
(ωl)
(ΦV ∂t ΦV − ΦV ∂t ΦV ).
2
1−r
(4.49)
(ωl)
From this equation we find the normalization constants AV in Eq. (4.50) in the same
way as in the tensor case. With the same reasoning as in the tensor case, we find
(ωl)
|AV |2
2
sinh πω Γ( 12 (iω + l + 1))Γ( 21 (iω + l + n))
.
=
2
8π 2 (l − 1)(l + n) Γ(l + n+1 )
(4.50)
2
4.1.3
Normalization of the scalar-type modes
(ωl)
Now, we shall find the normalization factors AS for the scalar-type modes. We first
choose a convenient gauge. Under the gauge transformation with the gauge generator
vector ξµ given by
ξa = ψa (t, r)S,
ξi = φ(t, r)Si ,
(4.51)
(4.52)
we find that the gauge-dependent functions transform as
(l)
fab
fa(l)
(l)
HT
(l)
HL
(l)
→ fab + Da ψb + Db ψa ,
φ
kS
(l)
→ fa + rDa
− ψa ,
2
r
r
kS
(l)
→ HT − 2 φ,
r
kS φ Da r
(l)
→ HL + 2 +
ψa .
nr
r
(4.53)
(4.54)
(4.55)
(4.56)
4.1. NORMALIZATION OF GRAVITATIONAL PERTURBATIONS
56
Hence by choosing
r2 (l)
H ,
kS T
r
1 (l)
(l)
= r
f + 2 Da HT ,
kS a
kS
(4.57)
φ =
ψa
(l)
(4.58)
(l)
we can set the functions fa and HT to zero. Then, the perturbations will be
(S;lσ)
hai
(S;lσ)
hab
(S;lσ)
hij
= 0,
(4.59)
(l)
Fab S(lσ) ,
2r2 γij F (l) S(lσ) ,
=
=
(l)
(4.60)
(4.61)
(l)
where F (l) and Fab are given in terms of the master variable ΦS by Eqs. (3.54) and (3.55),
respectively.
The conserved inner product (4.15), with the conjugate momentum current defined
by Eq. (4.14), can be found as
(S;lσ)
hh
0(S;lσ)
,h
Z
i = −2i
dΣna J a ,
(4.62)
Σ
where the conserved current J a is given by
J
a
=
2 c (l)ab (l0 )
0
(l)
Dr F
Fbc − F (l )ab Fbc
r
0
1
a (l )
(l0 )bc a l
(l)bc
− F
D Fbc − F
D Fbc
2
i
0
0
+2(2 − n) F (l) Da F (l ) − F (l ) Da F (l) .
0 0
S(l σ ) S(lσ)
(4.63)
(l)
Though it would be possible to express the inner product (4.62) in terms of ΦS
directly in the static coordinate system, it is much easier to do so if we use the EddingtonFinkelstein coordinates and evaluate it on the future horizon. Thus, we define the new
coordinate
1
1+r
u = t − log
.
(4.64)
2
1−r
This coordinate ranges over all real values. The line element of the orbit spacetime
becomes
ds2orb = −(1 − r2 )du2 − 2dudr.
(4.65)
On the future cosmological horizon we have ds2orb = −2dudr with −∞ < u < ∞.
Hence, if Σ is the constant-r hypersurface, a future-pointing vector orthogonal to this
4.1. NORMALIZATION OF GRAVITATIONAL PERTURBATIONS
57
hypersurface, which is spacelike if r > 1, is −∇λ r. Then, the unit future pointing normal
vector is
nλ = (r2 − 1)−1/2 ∇λ r
λ
λ
1
∂
∂
2
−1/2
2
= (r − 1)
+ (r − 1) 2
.
∂u
∂r
(4.66)
Now, the surface element of this hypersurface is
dΣ = dΩn du(r2 − 1)1/2 .
(4.67)
Hence,
dΣnλ = dΩn du
"
∂
∂u
λ
+ (r2 − 1)
∂
∂r
λ #
.
(4.68)
Then, in the limit r → 1, i.e. as it approaches the future cosmological horizon, we have
λ
lim dΣn = dΩn du
r→1
∂
∂u
λ
.
(4.69)
Thus, the inner product (4.62) can be evaluated on the future cosmological horizon as
(S;lσ)
hh
0(S;lσ)
,h
Z
i = −2i
dΩn duJu .
(4.70)
One can readily see that the first term in the conserved current (4.63) does not contribute because, on the horizon, we have Da r = −(∂/∂u)a and
0
(l0 )
(l)
(∂/∂u)a Dc r F (l)ab Fbc − F (l )ab Fbc = 0.
(4.71)
(l)
This equality follows from the fact that Fab is a symmetric tensor on the two-dimensional
orbit spacetime. Then, after dropping terms that are total derivatives with respect to u,
which do not contribute in the integral (4.70), we find that the current Ju on the horizon
can be written as
Ju =
h
(l)
(l)
(l) (l0 )
(l0 )
(l0 )
(l0 ) (l)
2Frr ∂u Fuu
− 2Frr
∂u Fuu − 4Frr Fuu
+ 4Frr
Fuu
i
0
0
0 0
+2n(n − 2) F (l) ∂u F (l ) − F (l ) ∂u F (l) S(lσ) S(l σ ) ,
(4.72)
4.1. NORMALIZATION OF GRAVITATIONAL PERTURBATIONS
58
where the relation F (l)a a = −2(n − 2)F (l) has been used. On the horizon we find, from
Eqs. (3.54) and (3.55), that
(l)
(l)
(l)
Frr
= Dr Dr (rn/2 ΦS ) = ∂r2 (rn/2 ΦS ),
(l)
(4.73)
(l)
(l)
Fuu
= Du Du (rn/2 ΦS ) = (∂u2 − ∂u )ΦS ,
r2−n
(l)
F (l) =
( + 2)(rn/2 ΦS )
2n 2
2
1
(l)
ΦS ,
= − (∂u + 1) 1 + ∂r −
2
n
n
(4.74)
(4.75)
where we have used = 2(∂u + 1)∂r on the horizon. We substitute these formulae into
(l)
Eq. (4.72) and use Eq. (3.57) satisfied by ΦS on the horizon. We then find
Ju =
n−1
l(l − 1)(l + n − 1)(l + n)
n
(l)
(l0 )
(l0 )
0 0
(l)
×(ΦS ∂u ΦS − ΦS ∂u ΦS )S(lσ) S(l σ ) .
(4.76)
Details of this calculation can be found in Appendix A.
The inner product is obtained by substituting Eq. (4.76) into Eq. (4.70). This inner
product can be rewritten as
(S;ωlσ)
hh
(S;ω 0 l0 σ 0 )
,h
i=
(n − 1)l(l − 1)(l + n − 1)(l + n)
lim
i
r→1
n
(ωl)
(ω 0 l)
×(ΦS ∂λ ΦS
(ω 0 l)
− ΦS
Z
0 0
dΣnλ S(lσ) S(l σ )
(ωl)
∂λ ΦS ).
(4.77)
Now, evaluating Eq. (4.77) on a t = constant Cauchy surface in the original tr coordinates,
we have
0 0 0
hh(S;ωlσ) , h(S;ω l σ ) i =
l(l − 1)(l + n − 1)(l + n)(n − 1)
n
Z 1
dr
0
0
(ωl)
(ω 0 l)
(ω 0 l)
(ωl)
×δ ll δ σσ
(ΦS ∂t ΦS − ΦS ∂t ΦS ).
2
0 1−r
i
(4.78)
We then require the same normalization condition as in the tensor case, i.e. Eq. (4.31).
(ωl)
Then, the normalization constants AS defined by Eq. (3.59) can be found to be
(ωl)
|AS |2
2
n sinh πω Γ( 21 (iω + l + 2))Γ( 21 (iω + l + n − 1))
=
2 .
2π 2 (n − 1)l(l − 1)(l + n − 1)(l + n) Γ(l + n+1 )
2
(4.79)
4.2. INFRARED FINITE TWO-POINT FUNCTION
4.2
59
Infrared finite two-point function
We want to compute the Wightman two-point function [65] for the gravitational perturbations studied in this dissertation. Since we have fixed the gauge, as we have seen
in Subsections 4.1.1, 4.1.2 and 4.1.3, we can make a proper expansion of the graviton
quantum field as
ĥµν (y) =
∞ XZ
X X
P =S,V,T l=2
(P )
σ
∞
i
h
(P ;ωlσ)
(P )
(P )†
(P ;ωlσ)
(y) .
dω alσ (ω)hµν
(y) + alσ (ω)hµν
(4.80)
0
(P )†
The operators alσ (ω) and alσ (ω) have the properties of annihilation and creation operators, respectively. They now have the simple interpretation of creating and annihilating
a graviton of a particular type, with frequency ω and quantum numbers given by l and
σ. In four dimensions, i.e. n = 2, we have just two types of gravitons, the scalar and the
vector ones, since there is no tensor contribution of second rank.
(T ;ωlσ)
(V ;ωlσ)
We can analyze the behavior of the normalized mode functions hµν
, hµν
and
(S;ωlσ)
hµν
in the infrared limit, that is, their low-ω behavior. We can easily see that this
(ωl)
(ωl)
(ωl)
behavior coincides with the behavior of the master variables ΦT , ΦV and ΦS . We
readily find that they all behave like ω 1/2 in the limit ω → 0, once that l ≥ 2. In fact,
the behavior of the master variables in the infrared limit is the same as the behavior of
the normalization constants. The normalization constants will all be
(ωl)
AP
∝
√
sinh πω,
(4.81)
when ω → 0. Therefore, they go to zero like ω 1/2 . In the Poincaré patch of de Sitter
spacetime, the graviton field can be decomposed into two minimally coupled massless
scalar fields [12], by choosing the transverse-traceless-synchronous gauge, that is,
∇µ hµν = hµµ = h0ν = 0.
(4.82)
Therefore, the low-ω behavior of the graviton mode functions is supposed to be similar
to the behavior of the minimally coupled massless scalar field. The minimally coupled
massless scalar field is shown to satisfy the same equation that the scalar master equation
satisfy in the static patch, that is Eq. (3.57) [43]. Then, the behavior of the graviton
mode functions should be, in fact, very similar to the behavior of the scalar field. However,
one must be careful because we have seen that the graviton modes have l ≥ 2, while the
scalar one starts with l = 0. In fact, the low-ω behavior of the scalar field is identical to
the behavior of graviton mode functions, except for the l = 0 mode, which behaves like
ω −1/2 [43].
Finally, to compute the two-point function, we have to choose an appropriate vacuum
state. Now, it is well known that the vacuum state in which the two-point function
has the form given by Eq. (4.12) is unphysical because it will have singularities in the
stress-energy tensor on the horizon. This state is analogous to the Rindler vacuum [66]
4.2. INFRARED FINITE TWO-POINT FUNCTION
60
in Minkowski spacetime and to the Boulware vacuum [67] in Schwarzschild spacetime. A
physically acceptable state is the de Sitter-invariant Bunch-Davies state [26], which is the
thermal state with temperature H/2π [31], where H is the Hubble constant. This state
is analogous to the Hartle-Hawking state [68] in Schwarzschild spacetime.
In this thermal state of temperature 1/2π (we are now setting H = 1), we have
(P )†
1
0
0
0
δ P P δ ll δ σσ δ(ω − ω 0 ),
−1
(P 0 )
halσ (ω)al0 σ0 (ω 0 )i =
e2πω
(4.83)
(P )
(P 0 )†
halσ (ω)al0 σ0
1
0
0
0
(ω 0 )i =
δ P P δ ll δ σσ δ(ω − ω 0 ),
−2πω
1−e
(4.84)
(P )
(P 0 )
(P )†
(P 0 )†
with halσ (ω)al0 σ0 (ω 0 )i = halσ (ω)al0 σ0 (ω 0 )i = 0, since this thermal state is not a quantum superposition of states with different particle numbers. In other words, it is a mixed
state. Thus, we find the graviton two-point function to be
D
E
ĥµν (y)ĥµ0 ν 0 (y ) =
0
∞ XZ
X X
P =S,V,T l=2
σ
∞
dω
0
1
(P ;ωlσ)
(P ;ωlσ)
hµν
(y)hµ0 ν 0 (y 0 )
−1
1
(P ;ωlσ) 0
(P ;ωlσ)
h
(y)hµ0 ν 0 (y ) .
+
1 − e−2πω µν
×
e2πω
(P ;ωlσ)
(4.85)
As we have seen, all mode functions hµν
(y) tend to zero as ω → 0 like ω 1/2 .
Hence, the two-point function (4.85) computed in the Bunch-Davies-like state is finite in
the infrared. Note that the two-point function for the minimally-coupled massless scalar
field, which takes a similar form, is IR-divergent (even if there were no thermal factors)
because the l = 0 mode functions behaves like ω −1/2 in the limit ω → 0.
By construction, the two-point function given by (4.85) is time-translation invariant.
This is easily seen since the two-point function only depends on the difference between
the two time coordinates t and t0 , and then, as any time-translation only shifts the value
of the time coordinates, their difference will remain invariant. In contrast to this, the
two-point function in the Poincaré patch is infrared divergent and grows as a function of
time [17]. The two-point function computed in the global patch also grows as a function
of time [28].
Many authors state that infrared gravitons have an impact in the background de Sitter
spacetime, therefore breaking de Sitter symmetry [69–74]. Some of these effects would
cause a change in time of the cosmological constant [75–78]. The two-point function given
by Eq. (4.85) does not have infrared divergences, so it seems to be in contrast with these
claims of de Sitter breaking effects.
Conclusion and perspectives
Our Universe, according to recent observations, is presently expanding in an accelerated
rate. Moreover, in the past, it presented an exponential growth rate, an epoch known
in cosmology as inflationary expansion. Due to the present expansion in an accelerated
rate, the Universe can become more and more similar to the one in the inflationary
epoch. Then, de Sitter spacetime is the best spacetime to model our Universe in these
periods. Besides, quantized gravitational perturbations, which can be realized as lowenergy manifestations of quantum gravity, may have an impact in the evolution of the
Universe in these phases. In particular, infrared divergences of the graviton two-point
function could be a mechanism that changes significantly the evolution of the Universe.
However, based on gauge invariance, it is more likely that infrared divergences are gauge
artifacts.
In this dissertation, we have shown that, inside the cosmological horizon of de Sitter
spacetime, we can construct an infrared finite graviton two-point function. This region is
the one causally accessible to an inertial observer and models our own point of view in
the Universe. The two-point function is finite in the infrared because all modes behave
as ω 1/2 in the limit ω → 0. Moreover, the two-point function found in this dissertation
is, by construction, time-translation invariant.
The presentation of this dissertation was structured such that we started studying
the general properties of d-dimensional de Sitter spacetime. We aimed to analyze its
most important properties from the point of view of general relativity. We then analyzed
the theory of linearized gravity, which can be used to study many problems in general
relativity and quantum field theory. We focused in obtaining the equations of motion for
the classical gravitational perturbations as well as studying the gauge invariance of the
theory. Reviewing a very useful gauge-invariant formalism for gravitational perturbations,
we specified it for our case, that is, we studied gravitons in the static patch of de Sitter
spacetime. After presenting a gauge-fixed method of quantization for the gravitational
perturbations, the calculation of the two-point function has been performed, using the
solutions we have found for the case of static de Sitter patch.
The results presented in this dissertation are promising, in the sense that they constitute a significant addition to the discussion of whether or not infrared divergences really
appear in the evolution of an inflationary universe such as ours. The two-point function
we have found appears to be in conflict with the claims that infrared graviton breaks
de Sitter symmetry. We have used a gauge-fixed method of quantization. It would be
useful to have a perturbation theory for gravitons in the covariant formulation. This
Conclusion and perspectives
62
could decide once and for all if there are physical de Sitter breaking effects due to infrared
gravitons. One particular perspective as a continuation of our work is to study interaction field theory of gravitons in the static patch. This would be relevant to the discussion
of de Sitter symmetry breaking effects, since most of the works on the subject rely on
interacting infrared gravitons [69–74].
Appendix A
Calculation of the inner product for
the scalar-type modes
(l)
To express the conserved current in terms of the master variable ΦS , we first simplify
(l)
Eq. (4.75), which expresses F (l) in terms of ΦS , using the field equation (3.57), which
reads, on the horizon,
(l)
(l)
(l)
ΦS = 2(∂u + 1)∂r ΦS = An,l ΦS ,
(A.1)
with
An,l = (l + 1)(l + n − 2),
and
F
(l)
1
=−
2
2
An,l
(l)
−
∂u + 1 −
ΦS .
n
n
(A.2)
(A.3)
Then, we find
F (l) ∂u F
(l0 )
2 An,l
(l)
ΦS
∂u + 1 − −
n
n
2 An,l
(l0 )
ΦS
×∂u ∂u + 1 − −
n
n
"
2 #
(l0 )
∂u ΦS
2 An,l
(l)
2
≈ −
∂u − 1 − −
ΦS .
4
n
n
1
=
4
(A.4)
Here we indicated the equivalence up to a total derivative with respect to u by the symbol
“≈” because we will integrate this quantity over u to obtain the symplectic product
between two scalar-type modes that tend to zero as u → ±∞.
64
Similarly, we find
0
(l)
(l0 )
0
(l)
(l )
(l )
2Frr ∂u Fuu
− 4Frr Fuu
≈ 2∂u ΦS (∂u2 + 3∂u + 2)
n(n − 2)
(l)
2
ΦS ,
× ∂r + n∂r +
4
(A.5)
so that
0
(l)
(l)
0
0
(l )
(l )
− 4Frr Fuu
+ 2n(n − 2)F (l) ∂u F (l )
2Frr ∂u Fuu
(l0 )
(l)
≈ 2∂u ΦS (∂u + 1)(∂u + 2)(∂r2 + n∂r )ΦS
3n(n − 2)
(l0 )
(l)
(l)
(l0 )
∂u ΦS ∂u ΦS + n(n − 2)ΦS ∂u ΦS
+
2
2
n(n − 2)
2 An,l
(l0 )
(l)
+
ΦS ∂u ΦS .
1− −
2
n
n
(A.6)
We can rewrite the first term of Eq. (A.6), using Eq. (A.1), as
(l0 )
(l)
2∂u ΦS (∂u + 1)(∂u + 2)(∂r2 + n∂r )ΦS
(l0 )
(l)
= 2∂u ΦS (∂u + 1)(∂u + 2)(∂r2 + 2∂r )ΦS
(l0 )
(l)
−(n − 2)An,l ∂u ΦS (∂u + 2)ΦS . (A.7)
Substituting this equation into Eq. (A.6), we find
0
(l)
(l)
0
0
(l )
(l )
2Frr ∂u Fuu
− 4Frr Fuu
+ 2n(n − 2)F (l) ∂u F (l )
(l0 )
(l)
≈ 2∂u ΦS (∂u + 1)(∂u + 2)(∂r2 + 2∂r )ΦS
3n(n − 2)
(l0 )
(l)
+
− (n − 2)An,l ∂u ΦS ∂u ΦS
2
"
2 #
4An,l
2 An,l
n(n − 2) (l)
0
+ 2−
+ 1− −
ΦS ∂u Φ(l ) .
n
n
n
2
(A.8)
Next, we note that
i
1h
(l)
(l)
(l)
(r2 ΦS ) − 2ΦS = 2(∂u + 1)(∂u + 2)(∂r2 + 2∂r )ΦS .
2
(A.9)
(l)
To calculate (r2 ΦS ), we write
(l)
ΦS =
Bn,l + Cn,l r2 (l)
ΦS ,
r2
(A.10)
65
with
Bn,l =
1
[4l(l + n − 1) + n(n − 2)]
4
(A.11)
and
Cn,l = −
(n − 2)(n − 4)
.
4
(A.12)
It is important not to let r = 1 in Eq. (A.10), because we are going to differentiate this
expression with respect to r. Then, we have, noting that An,l = Bn,l + Cn,l for r = 1,
(l)
(l)
(l)
(r2 ΦS ) = (A2n,l − 4Cn,l )ΦS − 4Cn,l ∂u ΦS .
(A.13)
Substituting Eq. (A.13) into Eq. (A.9) and using the resulting expression in Eq. (A.8),
we obtain
(l)
0
(l)
0
0
(l )
(l )
2Frr ∂u Fuu
− 4Frr Fuu
+ 2n(n − 2)F (l) ∂u F (l )
2
An,l
n(n − 2)
− An,l − 2Cn,l +
≈
2
2
"
2 #)
4An,l
2 An,l
(l)
(l0 )
× 2−
+ 1− −
ΦS ∂u ΦS
n
n
n
3n(n − 2)
(l)
(l0 )
+
− (n − 2)An,l − 2Cn,l ∂u ΦS ∂u ΦS .
2
(A.14)
Substituting this equation into Eq. (4.62), we find for the inner product between two
scalar-type modes
0 0 0
hh(S;ωlσ) , h(S;ω l σ ) i = i
(n − 1)l(l − 1)(l + n − 1)(l + n)
n
Z
×
0 0
(ωl)
(ω 0 l)
dΩn duS(lσ) S(l σ ) (ΦS ∂u ΦS
(ω 0 l)
− ΦS
(ωl)
∂u ΦS ). (A.15)
In tr coordinates and on the t = constant Cauchy surface, the inner product (A.15) can
be rewritten as
0 0 0
(n − 1)l(l − 1)(l + n − 1)(l + n)
n
Z 1
dr
0
0
(ωl)
(ω 0 l)
(ω 0 l)
(ωl)
×δ ll δ σσ
(Φ
∂
Φ
−
Φ
∂t ΦS ). (A.16)
t
S
S
S
2
1
−
r
0
hh(S;ωlσ) , h(S;ω l σ ) i = i
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