Cosmic Inflation : An Overview of the Current Themes

Imperial College London
Masters Thesis
MSc Quantum Fields and Fundamental Forces
Cosmic Inflation :
An Overview of the Current Themes
Author:
Amy Tee
Supervisor:
Prof. Carlo R. Contaldi
Submitted in partial fulfilment of the requirements
for the degree of Master of Science of Imperial College London
Theoretical Physics Group
Department of Physics
November 22, 2016
Abstract
In this disseration the current themes in the field of cosmological inflation in relation
to the recent results of the Planck collaboration are discussed. Starting with motivations
for inflation it is shown how a brief but accelerated period of inflation can resolve certain
issues within the hot Big Bang theory and the minimum number of e-foldings required
for resolution. Having chosen to primarily focus on scalar field inflation, the slow roll
parameters are derived and their role in the inflationary epoch. This epoch is investigated in terms of cosmological perturbations (scalar and tensor) which arose, and their
associated equations. As the focus is on the results of the Planck data releases (2013 and
2015) an overview of a select number of inflationary models is given noting their viability. Due to the favourability of R2 (Starobinsky) inflation in the results, f(R) gravity is
reviewed in both the Einstein and Jordan frames. Finally a number of numerical simulations based on themes presented throughout this disseration are given based on the
scalar field potential and the R2 potential.
i
For Diana,
with all my love,
Acknowledgements
First and foremost, I’d like to express my gratitude and thanks to the project supervisor Professor Carlo Contaldi, who ignited my interest in the field of cosmology, and
whose guidance and invaluable support allowed me to produce this piece of work.
I wish to also express my gratitude to my friends for their help and moral support
during the 2014/15 academic year, in particular John Ronayne, Doogesh Kodi Ramanah,
and Daniel Matos.
In addition I thank my friends during the 2015/16 academic year, in particular
Miguel Garcia Cutillas, Amilios Pagouropoulos and Mir Mehedi Faruk. Futhermore,
a grateful mention to Nellie Marangou for her support during the project.
iii
List of Figures and Tables
List of Figures
1
Fate of the Universe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2
Timeline of the Universe. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
3
The density parameter evolution for an open or closed universe. . . . . . . .
16
4
The effect of inflation on Ω(t). . . . . . . . . . . . . . . . . . . . . . . . . .
17
5
Evolution of observable wavelengths as a function of time during inflation
and other cosmological epochs. . . . . . . . . . . . . . . . . . . . . . . . .
27
6
Graceful exit and inflation. . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
7
Scalar field rolling down a potential. . . . . . . . . . . . . . . . . . . . . . .
38
8
The evolution of scales. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
9
Old inflationary model. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
10
New inflation model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
11
Planck 2015 contour confidence results for the spectral index and the running
of the spectral index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
12
Planck 2015 confidence contours for ns and r at 68% and 95% confidence level. 66
13
Planck 2015 results for ns and r0.001 . . . . . . . . . . . . . . . . . . . . . . .
78
14
Plot of the potential against the field for Starobinsky inflation. . . . . . . . .
86
15
Plot of the scalar potential as a function of time. . . . . . . . . . . . . . . . 103
16
Plot of the scale factor as a function of time. . . . . . . . . . . . . . . . . . . 104
17
Plot of the number of e-folds as a function of time. . . . . . . . . . . . . . . 105
18
Plot of the number of e-folds as a function of the scalar field. . . . . . . . . . 106
19
Plot of the real part of the Mukhanov equation. . . . . . . . . . . . . . . . . 111
20
Plot of the imaginary part of the Mukhanov equation. . . . . . . . . . . . . 112
21
Plot of the modulus of the Mukhanov equation. . . . . . . . . . . . . . . . 113
22
Primordial Spectrum of Tensor Perturbations. . . . . . . . . . . . . . . . . 115
23
Scalar potential power spectrum . . . . . . . . . . . . . . . . . . . . . . . . 116
24
Starobinsky potential power spectrum . . . . . . . . . . . . . . . . . . . . . 117
25
Plot of the slow roll parameters. . . . . . . . . . . . . . . . . . . . . . . . . 118
26
Spectral index for scalar potential inflation. . . . . . . . . . . . . . . . . . . 119
iv
27
Spectral index for Starobinsky inflation. . . . . . . . . . . . . . . . . . . . . 120
List of Tables
1
Fate of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
8
Symbols
a(t)
scale factor
k
Curvature of the Universe, scaled to +1, 0, −1
r
Proper distance
Gµν
Einstein tensor
Rµν
Ricci tensor
R
Ricci Scalar
Tµν
Stress-energy tensor
gµν
Metric
G
Newton’s gravitational constant
L
Langragian
ηµν
Minksowski metric, diag(-1,+1,+1,+1)
σ
Rµνρ
Riemannian curvature tensor
ρ
enery density
P
Pressure
ω
Pressure to energy density ratio
ρ0
cosmological energy density today
a0
Scale factor today
ρc
Critical energy density
Ω(t)
Ratio of energy density to critical energy density
H(t)
Hubble parameter
H0 (t) Hubble parameter today
q0
Deceleration parameter
Γσαµ
Christoffel symbol
Ω0 (t)
Pressure to energy density ratio today
T
Temperature
t
Time
vi
MP l
Planck mass
h
Planck constant
c
Speed of light
MP c
Mega parsec
l
Length scale
D
Proper distance
v
Velocity
E
Energy
Λ
cosmological constant
N
number of e-foldings
λ
wavelength
R
Curvature perturbation
P
Power Spectrum
g
Determinant of the metric
κ
8πG
ns
Scalar tilt
nt
Tensor tilt
Slow roll parameter - measures slope of potential
η
Slow roll parameter - measures curvature of potential
σ
Standard deviation
As
Scalar spectral amplitude
V
Potential
Please note this is not an exhaustive list, a number of terms which feature subscripts
or superscripts (such as Ek are not listed) but their meaning can be inferred from here and
within the dissertation. Some symbol’s are repeated with different meanings e.g k is reserved
for both the Universe’s curvature and to describe a mode.
vii
Contents
1
2
3
4
5
Introductions
1
1.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
The Cosmological Standard Model
4
2.1
Recalls of the Standard Model of Cosmology . . . . . . . . . . . . . . . . .
4
2.2
ΛCDM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
Setting the Scene for Inflation
11
3.1
Motivations for Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
3.1.1
The Problem of Initial Conditions . . . . . . . . . . . . . . . . . .
12
3.1.2
The Flatness Problem . . . . . . . . . . . . . . . . . . . . . . . . .
15
3.1.3
The Horizon Problem . . . . . . . . . . . . . . . . . . . . . . . . .
20
3.1.4
The Monopole Problem and Relic Particles . . . . . . . . . . . . . .
24
3.1.5
The Origin of perturbations . . . . . . . . . . . . . . . . . . . . . .
25
3.2
What Is inflation? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
3.3
Repulsive Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
3.4
General Conditions For a Successful Inflationary Model . . . . . . . . . . .
28
Inflationary Dynamics
31
4.1
Scalar Field Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
4.2
Slow Roll Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
4.2.1
Slow Roll Parameters . . . . . . . . . . . . . . . . . . . . . . . . . .
35
4.2.2
Background Evolution . . . . . . . . . . . . . . . . . . . . . . . . .
37
4.3
End of Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
4.4
Evolution of Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
4.5
Initial Conditions for Inflation . . . . . . . . . . . . . . . . . . . . . . . . .
42
4.5.1
Old Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
4.5.2
New Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
Cosmological Perturbations
46
5.1
46
Scalar and Tensor Perturbations . . . . . . . . . . . . . . . . . . . . . . . .
viii
5.2
5.3
5.4
5.5
5.6
6
Perturbations and Gauge Freedom . . . . . . . . . . . . . . . . . . . . . . .
47
5.2.1
Basic Recalls on Quantization of a Free Scalar Field In Flat Space-Time 49
5.2.2
Definition of The Mode Function . . . . . . . . . . . . . . . . . . .
50
5.2.3
Free Field in Curved Space-Time . . . . . . . . . . . . . . . . . . . .
51
5.2.4
Quantum to Semi-Classical Transition . . . . . . . . . . . . . . . .
51
Tensor Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
5.3.1
General Equations . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
5.3.2
Solutions During De-Sitter Stage
. . . . . . . . . . . . . . . . . . .
55
5.3.3
Long Wavelength Solution During and After Inflation . . . . . . . .
55
5.3.4
Primordial Spectrum of Tensor Perturbations . . . . . . . . . . . .
56
Scalar Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
5.4.1
General Equations . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
5.4.2
Solution During Quasi de-Sitter Stage . . . . . . . . . . . . . . . . .
58
5.4.3
Long Wavelength Solution During and After Inflation . . . . . . . .
59
5.4.4
Primordial Spectrum of Scalar Perturbations . . . . . . . . . . . . .
60
Computing Smooth Spectra Using Slow Roll Expansion . . . . . . . . . . .
61
5.5.1
Simple Method for Computing The Tilts . . . . . . . . . . . . . . .
63
5.5.2
Slow Roll Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
Planck Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
Inflationary Models
67
6.1
Inflationary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
6.2
Chaotic Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
6.2.1
The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
6.2.2
Large ϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
6.2.3
Planck Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
Hybrid Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
6.3.1
Behaviour of the Fields φ and σ . . . . . . . . . . . . . . . . . . . .
72
6.3.2
Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
6.3.3
Planck Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
Hill-Top Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
6.4.1
Slow Roll Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
6.4.2
The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
6.3
6.4
ix
Planck Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
Natural Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
6.5.1
The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
6.5.2
Spectrum of Density Perturbations . . . . . . . . . . . . . . . . . .
77
R2 inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
6.6.1
The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
Planck Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
6.4.3
6.5
6.6
6.7
7
f(R) Gravity
81
7.1
f(R) Gravity Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
7.1.1
f(R) Model: Equations of Motion . . . . . . . . . . . . . . . . . . .
82
7.1.2
f(R) Models : Inflationary Dynamics . . . . . . . . . . . . . . . . .
84
7.1.3
Dynamics In The Einstein Frame . . . . . . . . . . . . . . . . . . .
86
7.1.4
Perturbations Generated During Inflation . . . . . . . . . . . . . . .
88
7.1.5
Gauge Invariant Quantities . . . . . . . . . . . . . . . . . . . . . .
90
Perturbations Generated During Inflation . . . . . . . . . . . . . . . . . . .
92
7.2.1
Curvature Perturbations . . . . . . . . . . . . . . . . . . . . . . . .
92
7.2.2
Tensor Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . .
95
7.2.3
The Spectra of Perturbations in inflation based on f(R) Gravity . . .
96
7.2.4
Power Spectra in the Einstein Frame . . . . . . . . . . . . . . . . .
98
7.2
8
Numerical Simulations
8.1
8.2
100
The Cosmological equations . . . . . . . . . . . . . . . . . . . . . . . . . . 100
8.1.1
Dimensionless Cosmological Equations . . . . . . . . . . . . . . . . 101
8.1.2
Evolution of the Potential as a function of time . . . . . . . . . . . . 103
8.1.3
Evolution of the scale factor as a function of time . . . . . . . . . . 104
8.1.4
Number of e-folds for the scalar potential . . . . . . . . . . . . . . . 105
8.1.5
Scalar field versus the number of e-folds . . . . . . . . . . . . . . . . 106
Mukhanov - Sasaki Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 107
8.2.1
Bunch - Davis Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . 108
8.2.2
Mukhanov-Sasaki Equation in Physical Time . . . . . . . . . . . . . 109
8.2.3
Real Mode Function . . . . . . . . . . . . . . . . . . . . . . . . . . 111
8.2.4
Imaginary Mode Function . . . . . . . . . . . . . . . . . . . . . . . 112
x
8.2.5
8.3
Power Spectrum PQ (k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
8.3.1
9
Modulus of the Mode Function . . . . . . . . . . . . . . . . . . . . 113
Primordial Spectrum of Tensor Perturbations . . . . . . . . . . . . 114
8.4
Slow Roll Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
8.5
Spectral Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Conclusions
121
9.1
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
9.2
An Appraisal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
10 An Abbendum
123
A Appendix
124
A.1 Energy-Momentum Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
A.2 Alternative Method for Computing the Spectra Using Slow Roll Expansion
125
A.3 Newtonian Gauge Line Element - Equation 106 . . . . . . . . . . . . . . . . 126
B Recalls of the Cosmological Standard Model - Derivations
130
B.1 The FLRW Metric - Equation 1 . . . . . . . . . . . . . . . . . . . . . . . . 130
B.2 The Einstein Tensor - Equation 2 . . . . . . . . . . . . . . . . . . . . . . . 131
B.3 The Riemannian Curvature Tensor - Equation 3 . . . . . . . . . . . . . . . 132
B.4 The Friedmann Equation - Equation 6 . . . . . . . . . . . . . . . . . . . . . 133
B.5 The Conservation Equation - Equation 7 . . . . . . . . . . . . . . . . . . . 134
B.6 The Acceleration Equation - Equation 8 . . . . . . . . . . . . . . . . . . . . 134
B.7 ρ As a Function of the Scale Factor - Equation 9 . . . . . . . . . . . . . . . 135
C Setting the Scene For Inflation - Derivations
136
C.1 Ratio of Total Initial Energy to Initial Kinetic Energy - Equation 21 . . . . . 136
C.2 Ratio of Total Initial Energy to Initial Kinetic Energy Upper Limit - Equation 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
C.3 Density Parameter Ω initially close to unity - Equation 24 . . . . . . . . . . 136
C.4 The Amount of Inflation At the Time of Nucleosynthesis - Equation 32 . . 137
C.5 Horizon Distance dh (t1 , t2 ) - Equation 39 . . . . . . . . . . . . . . . . . . . 137
C.6 How the Horizon grows - Equation 47 . . . . . . . . . . . . . . . . . . . . 138
xi
C.7 Equation 51 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
C.8 H 2 + Ḣ - Equation 57 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
D Inflationary Dynamics - Derivations
140
D.1 Density Equation - Equation 64 . . . . . . . . . . . . . . . . . . . . . . . . 140
D.2 Pressure Equation - Equation 65 . . . . . . . . . . . . . . . . . . . . . . . . 140
D.3 Klein Gordon Equation - Equation 71 . . . . . . . . . . . . . . . . . . . . . 141
D.4 Relation between Ḣ and ϕ̇ - Equation 73 . . . . . . . . . . . . . . . . . . . 141
D.5 N In Terms of the Potential and Scalar Traversed - Equation 91 . . . . . . . 142
D.6 N - Equation 92 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
D.7 Quantities Relative to the Present Co-moving Scale - Equation 93 . . . . . . 143
E Cosmological Perturbations - Derivations
144
E.1 Hamiltonian for L = 12 ∂µ χ∂ µ χ - Equation 110 . . . . . . . . . . . . . . . . 144
E.2 Fourier Transform of the Hamiltonian - Equation 111 . . . . . . . . . . . . 144
E.3 Fourier Transformed Hamiltonian - Equation 112 . . . . . . . . . . . . . . 145
E.4 Lagrangian for Tensor Perturbations - Equation 131 . . . . . . . . . . . . . 145
E.5 The Hamiltonian of the Tensor Lagrangian - Equation 132 . . . . . . . . . . 146
E.6 Fourier Equation of Motion - Equation 133 . . . . . . . . . . . . . . . . . . 146
E.7 Large Wavelength Limit Solution - Equation 140 . . . . . . . . . . . . . . . 147
E.8 Variable hλ - Equation 141 . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
E.9 Long Wavelength Solution in Exact De-Sitter Limit - Equation 163 . . . . . 147
E.10 First order derivation ns − 1 - Equation 187 . . . . . . . . . . . . . . . . . . 149
E.11 Tensor Tilt - Equation 189 . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
F Inflationary Models - Derivations
150
F.1
Equation of Motion: Chaotic Inflation - Equation 195 . . . . . . . . . . . . 150
F.2
Chaotic Inflation Einstein Equation - Equation 197 . . . . . . . . . . . . . . 150
F.3
H=
F.4
ϕ̇ derivation - Equation 199 . . . . . . . . . . . . . . . . . . . . . . . . . . 151
F.5
Starobinsky Action - Equation 226 . . . . . . . . . . . . . . . . . . . . . . . 152
ȧ
a
- Equation 198 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
G f(R) Gravity - Derivations
155
G.1 f(R) Equation of Motion - Equation 235 . . . . . . . . . . . . . . . . . . . . 155
xii
G.2 Ricci Scalar F(R) theory - Equation 242 . . . . . . . . . . . . . . . . . . . . 156
G.3 Field Equation In a Flat FLRW background - Equation 243 . . . . . . . . . 158
G.4 Second Modified Field Equation in flat FLRW background - Equation 244 . 159
G.5 The first modified field equation in the absence of matter - Equation 246 . . 160
G.6 H 2 - Equation 247 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
G.7 Need to think up a name - Equation 248 . . . . . . . . . . . . . . . . . . . . 161
G.8 The solution for H - Equation 251 . . . . . . . . . . . . . . . . . . . . . . . 162
G.9 Inflationary Dynamics in terms of H - Equation 252 . . . . . . . . . . . . . 163
G.10 Inflationary Dynamics H - Equation 254 . . . . . . . . . . . . . . . . . . . 163
G.11 Evolution of the Scale Factor During Inflation - Equation 255 . . . . . . . . 164
G.12 Evolution of R During Inflation - Equation 256 . . . . . . . . . . . . . . . . 164
G.13 Calculating tf - Equation 259 . . . . . . . . . . . . . . . . . . . . . . . . . . 164
G.14 Number of E-foldings Between ti and tf - Equation 260 . . . . . . . . . . . 165
G.15 N - Equation 261 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
G.16 V,ϕϕ - Equation 265 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
G.17 t̃ - Equation 266 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
G.18 Evolution of ã - Equation 267 . . . . . . . . . . . . . . . . . . . . . . . . . 166
G.19 ˜1 - Equation 273(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
G.20 ˜2 - Equation 273(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
G.21 Ñ - Equation 275 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
G.22 ˜1 as a function of Ñ - Equation 276 (a) . . . . . . . . . . . . . . . . . . . . 169
G.23 ˜2 as a function of Ñ - Equation 276 (b) . . . . . . . . . . . . . . . . . . . . 170
G.24 α - Equation 314 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
G.25 A - Equation 315 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
H Python Code - Scalar Potential
172
I
178
Python Code - Starobinsky Potential
xiii
1
Introductions
’At the moment we can’t afford to go to
other planets. We don’t have the ships
to take us there. There may be other
people out there (I don’t have any
opinions about Life Out There, I just
don’t know) but it’s nice to think that
one could, even here and now, be
whisked away just by hitchhiking.’
Don’t Panic: The Official Hitchhikers
Guide to the Galaxy Companion Douglas Adams
This introductory chapter gives a brief outline of the development of modern cosmology
and gives an outline of the direction taken in this dissertation.
1.1
Introduction
We have all looked up at the sky at night as a child and wondered what was out there, what
is this strange dark void before our eyes, and what are these twinkling things which seem
so ever far away. The desire to know what is beyond our earthly limits can arguably be
seen as the need to comprehend our existence - our origins. History has shown us that this
need stretches back to at least when records began, indeed 16th century BCE Mesopotamia
believed Earth was a flat circular disc in a cosmic ocean, and for most of human history the
study of the Universe1 (it’s origin, structure and evolution) was a branch of metaphysics and
religion.
The passage to cosmology as a science could arguably originate with Copernicus and
his heliocentric model of the solar system and Newton’s laws of physics. Cosmology as we
understand it now, accelerated under the advent of Einstein’s Theory of General Relativity
and Theory of Special Relativity, combined with numerous theoretical and experimental
observations, such as Hubble’s discovery that our Universe contains galaxies and Slipher
1
This dissertation makes a clear distinction between our Universe (spelt with capital U) and a generic (not
necessarily refering to the Universe we exist in) universe spelt with a lower case u.
1
showing our Universe is expanding. The start of the revolution into understanding our
origins led to the Big Bang theory by Lemaìtre to explain the Universe from its earliest
stages through to large scale structural evolution, and important for this dissertation, the
idea of an expanding universe. Observational evidence in the 1990s which included the
Cosmic Microwave Background (CMB) and galaxy red-shift surveys formed part of what is
the (now) standard model of cosmology.
Numerous problems were identified with the Big Bang theory concerning primarily dark
energy, dark matter, the horizon problem, magnetic monopoles and the flatness problem.
To the rescue came inflation - exponential expansion of the Universe - first proposed by
Guth (1980)[40] which seemed to resolve some of these issues. However Guth’s proposed
mechanism for inflation lead to a much too granular universe, i.e. large density variations
resulting from bubble wall collisions. This was solved by Linde (1982)[70] with his proposal
of slow roll inflation.
Our knowledge of the Universe has grown tremendously due to high precision satellites
in the last couple of decades. The COBE (COsmic Background Explorer) satellite (1989)
took the first measurements of the CMB, showing that measurements in its anisotropies
were approximately 1 part in 1000, with a background radiation of 2.7K. In addition it was
shown at the CMB is an almost perfect black body.
The launch of WMAP (Wilkinson Microwave Anisotropy Probe) satellite (2001) provided a more accurate mapping of the CMB, encoding information about the early Universe.
May 2009 saw the launch of the Planck satellite and its purpose included to serve as
a rigorous test of early Universe predictions, detect primordial gravitational waves and its
spectrum, and study scalar spectrum perturbations.
Inflation sees the realization of the cosmological principle and numerous observational
experiments are in agreement with the concept of inflation. This dissertation is written in
celebration of the results of the Planck experiment and reports on a number of its observations related to inflation.
1.2
Outline
This dissertation explores inflation from the resolution of problems inherent in the hot Big
Bang theory, to the numerous models that have evolved from work by Guth in the 1980s,
and look at perturbations in the Universe, viewed as the seeds of large scale cosmological
2
structure such as galaxies.
It is natural for theorists to want to incorporate inflation within established ideas and so
start off with a brief review of Friedmann-Lemaìtre-Robertson-Walker (FLRW) cosmology
based on the principles of homogeneity and isotropy before looking at the motivations for
inflation. It is shown how a brief, but necessary, period of accelerated expansion is required
to solve the various motivations.
Inflation is explored in the context of slow roll, and thus review the slow roll parameters
required for a successful scalar field inflation model. In light of this, an overview of a number
of inflationary models - primarily slow roll - investigated by the Planck satellite and a very
promising theory introduced by Starobinsky known as R2 inflation are presented.
FLRW cosmology is based on two principles, however these principles only hold on large
scales, but a completely homogeneous and isotropic model cannot account for the large scale
structures present in our Universe. To describe spatial inhomogeneity and anisotropy a
perturbative approach is used to derive the scalar and tensor perturbation equations whose
values have been measured by Planck.
A review R2 inflation as part of the theory of f(R) gravity is given, with an in depth look
at the equations of motion, inflationary dynamics and perturbations.
Finally, a number of numerical simulations have been produced based on the cosmological equations, exploring the relationship between the potential, e-foldings and scale factor.
The Mukhanov equation is tranformed from conformal time to physical time and presented
numerically. The power spectrum and spectral index for both the scalar potential and the
Starobinsky potential are also plotted.
The dissertations begins by recalling basic concepts associated with FLRW cosmology.
Please note that the majority of the equations in this dissertation have their derivations in
the appendix, along with the Python code for the numerical simulations.
3
2
The Cosmological Standard Model
’From now on, we live in a world where
man has walked on the moon. And it’s
not a miracle, we just decided to go. On
Apollo 8, we were so close. Just 60
nautical miles down, and it was as if I
could just step out, and walk on the face
of it.’
Film - Apollo 13
This chapter recalls some of the basic key mathematical foundations of cosmology used
throughout this dissertation. A summary of the Friedmann-Lemaìtre-Robertson-Walker
Metric, the Einstein, Friedmann, acceleration, and conservation equations, fluids and finally
the density parameter are explored.
2.1
Recalls of the Standard Model of Cosmology
The Friedmann-Lemaìtre-Robertson-Walker Metric
At the level of background quantities (i.e averaging over spatial fluctuations) the Universe
is described by the much celebrated Friedmann -Lemaìtre-Robertson-Walker (FLRW) metric
[28] [116],

!
dr2
ds2 = dt2 − a(t)2 
+ r2 dθ2 + sin2 θφ2 .
1 − kr2
(1)
The line element (1) describes the path of an object through space-time where θ and φ
are the standard cylindrical co-ordinates. a(t) is the scale factor representing the relative expansion of the universe. t is the proper time measured by a free falling observer which is
titled cosmological time2 . It relates the proper distance3 r between a pair of objects moving
in an FLRW universe describing the distance of an object from an origin point in space at a
2
Cosmic time is commonly used in models of the Big Bang theory. Existing in a homogeneous expanding
universe the time co-ordinate is chosen for a universe with has the same density everywhere at each moment in
time. Cosmic time is used as the standard basis for solutions to the Einstein equations in an FLRW universe.
3
This is different to the co-moving distance which factors out the expansion of the Universe.
4
specific moment of cosmological time. Here k relates to the curvature of the universe, which
is usually scaled to: k=1 for a positively curved closed universe, k=0 for a flat universe and
k=-1 for a negatively curved open universe.
The Einstein Equations
With no formal discussion of metrics and geodesics presented here, Donaldson [30]
writes that the concept of a metric coupled with the realization that non-trivial metrics affect
geodesics exist independently of general relativity and that gravitation can be described by
a metric. Another idea is that the metric can be related to matter and energy in the Universe, inherent within the Einstein equations, and these equations relate components of the
Einstein tensor describing the geometry of the Universe to the energy-momentum tensor
describing energy [30].
The Einstein tensor4 Gµν is given by
1
Gµν ≡ Rµν − gµν R = 8π G Tµν .
2
(2)
Tµν is the stress-energy tensor (defined as Tµν = ∂µ ϕ∂ν ϕ − gµν L), Rµν is the Ricci tensor
(dependent on the metric and its derivatives), R the Ricci scalar (also known as the curvature
scalar and is the trace of the Ricci tensor), G is Newton’s constant and gµν the metric. The
left hand side of the Einstein tensor (2) is a function of the metric and the right hand side a
function of energy so the Einstein tensor relates curvature to matter.
The Ricci Tensor and Ricci Scalar
The Riemann curvature tensor is the only tensor which can be constructed from the
metric and is most conveniently expressed in terms of Christoffel symbols,
σ
Rαβµ
= ∂β Γσαµ − ∂α Γσβµ + Γναµ Γσβν − Γνβµ Γσαν .
(3)
The Ricci tensor5 is found by contracting the Riemann tensor
Rµν = Rµανα ,
4
(4)
In fact the Einstein tensor is usually written without the cosmological constant term so Gµν = Rµν −
1
2 gµν R
+ gµν Λ. Λ is the cosmological constant defined as Λ ≡ 8πGρ and is an inverse length squared, where
length gives the radius of curvature associated to the cosmological constant.
5
There are only two non-vanishing components, one with µ = ν = 0 and the other µ = ν = i.
5
and by taking the trace gives the Ricci scalar (5),
R = g µν Rµν = Rµ µ .
(5)
The Riemann tensor (3) is derived from derivatives of the metric tensor, giving the best
picture of the curvature of space. In flat space this tensor is zero. The Ricci tensor (4) arises
from the need for a tensor with two indices and is essentially the averaging of certain portions of the Riemann tensor (3). By averaging the Ricci tensor (4) leads to the the Ricci
scalar (5) and is the simplest measure of curvature in general relativity.
The Friedmann, Conservation and Acceleration Equations
Cosmology is based on the principles of homogeneity and isotropy6 which are inherent
within the FLRW metric. One can derive the Friedmann Equation (6) from the Gtt component of the Einstein tensor, and the acceleration equation (8) from the Einstein tensor’s
Gij component7 . The conservation equation (7) is derived by utilising stress energy conservation, ∇µ Tµν = 0, where ∇ is the divergence vector operator. The Friedmann equation
directly determines the Hubble function, H(t) ≡ ȧa .
ȧ
a
!2
+
k
8πG
=
ρ,
2
a
3
ȧ
ρ̇ + 3 (ρ + P ) = 0,
a
(6)
(7)
!
4πG
ä
=−
ρ + 3P .
a
3
(8)
ρ is the total cosmological density, P is total cosmological pressure, G is the gravitational
constant, an over-dot is a derivative with respect to time and two over-dots is the second
derivative with respect to time. a(t) is the scale factor where ȧ(t) is it’s time derivative.
6
Isotropy: No preferred direction in the Universe, so when the Universe is viewed from a particular point
it looks the same everywhere regardless of direction. Homogeneity: At a given instant the Universe looks the
same everywhere.
7
An alternative derivation is by taking the derivative of (6) with respect to time, and substituting in the
conservation equation, cancelling common factors and then substituting in the Friedmann equation we also
arrive at the acceleration equation (8).
6
Fluids
By considering perfect fluids with cosmological symmetry - doesn’t interact with matter
- the equation of state is P = ωρ for constant ω:
• Dust/cold matter: ω = 0, so P ' 0,
• radiation/hot matter: ω = 13 , so P ' 13 ρ,
• and vacuum energy: ω = −1.
The equation of state coupled with the FLRW equations of an isotropic universe filled
with a perfect fluid can be used to describe its evolution. From the conservation equation
(7) by substituting in the equation of state and integrating then,
3(1+ω)

a0
ρ = ρ0  
a
(9)
where a0 (t) is the scale factor today and ρ0 (t) is the density today. After substituting in
the above aforementioned values:
• Dust/cold matter: ω = 0, ρ '
1
a3
so matter dilutes,
• Radiation/hot matter: ω = 13 , so ρ '
1
,
a4
the radiation also dilutes,
• Vacuum energy: ω = −1, ρ =constant8 .
The Density Parameter
Many of the key characteristics of the Universe today are determined by three parameters (i) the Hubble parameter H0 (t) which determines the expansion rate of the universe, (ii)
the deceleration parameter q0 which determines the rate of expansion9 and, (iii) the dimensionless Ω(t) parameter, and Ω0 (t) which determines the density of matter in the universe
today.
The Ω(t) parameter implies the fate of the universe i.e. will it expand forever or will it
collapse. From the Friedmann equation (6) for k = 0 there is a critical density ρc , such that
ρc (t) ≡
8
3H 2
.
8πG
(10)
The vacuum energy can never dilute, and is equivalent to the cosmological constant Λ. This is valid for
the energy of a vacuum scalar field in its fundamental state and indeed a classical field in a state of equilibrium.
9
Topic not discussed here but a brief outline can be read [90].
7
Ω(t) can therefore be defined as
Ω(t) ≡
8πGρ
ρ(t)
=
.
ρc (t)
3H 2
(11)
By taking the Friedmann equation (6) dividing through by H 2 and re-writting it as
ρ
k
− 2 2 = 1,
ρc a H
(12)
it can then be seen after a simple substitution of the definition of the Ω(t) parameter (11)
that
Ω−1=
k
a2 H 2
(13)
.
The sign of of the right hand side of equation (13) is fixed by the sign of k (k is either +1,
0, −1). Ω(t) can never pass from being greater than unity, to less than unity and vice versa10 .
Table 1 contains a concise description of how the total density, the density parameter and
the geometry of the Universe ultimately determines it’s fate. The scenarios described in this
table have been shown pictorially in figure 1, highlighting how the curvature of the Universe
affects the scale factor as time progresses. For a closed universe the scale factor at ti = 0 was
a(t) = 0 and returns to this value at tf . The subscript i stands for an initial time (the Big
Bang) and f is the time at the death of the universe.
Table 1: Fate of the Universe
ρ > ρc
Ω>1
k = +1(closed)
Re-collapses
ρ = ρc
Ω=1
k = 0 (Flat)
Just expands forever
ρ < ρc
Ω<1
k = −1 (open)
Expands forever
Table 1: This table highlights the relationship between the density parameter, the curvature of
the Universe and its ultimate fate.
2.2
ΛCDM Model
The ΛCDM (Lambda Cold Dark Matter) model is a parametrization of the Big Bang cosmological model of our Universe with cosmological constant (Λ), dark energy and cold
10
A great discussion of the Ω(t) parameter is discussed in numerous standard cosmology textbooks, but
Professor James Lindsay’s cosmology notes chapter 4 provides a useful and clear explanation.
8
Figure 1: The fate of our universe, each scenario dependant on the data in Table 1. One can
see that for an open universe, it continues to expand forever at a much slower rate as time
progresses. Image courtesy of [78].
dark matter (CDM). This is the widely accepted model for our Universe, based on six independent parameters: physical baryon density, physical dark matter density, the age of the
Universe, scalar spectral index, curvature fluctuations amplitude and re-ionization optical
depth [4, 106].
The ΛCDM models assumes general relativity in that flat geometry holds on cosmological scales (k = 0) and incorporates the Standard Model of Particle Physics [26]. Despite being
a simple model it has held up against current observations concerning the CMB [4], large
scale structures [10], Hubble constant measurements and the Universe’s expansion [100, 36].
Despite many successes of the ΛCDM model a number of (currently) unexplained anomalies feature such as the nature of dark matter and dark energy, with the proposition that dark
energy is the driving the current accelerated expansion of the Universe [93, 101], which
throws up a number of theoretical issues [26, 117, 92].
9
Figure 2: A brief timeline of our Universe, outlining the main events. Compatible with a
ΛCDM model. Image courtesy of [20].
10
3
Setting the Scene for Inflation
’[On how to use a reentry module to
get back to Earth] You point the
damned thing at Earth. It’s not rocket
science.’
Film - Gravity
The study of our Universe’s origins gained pace in the 1980s, so begin here with an
overview of the main themes in the field of inflationary cosmology which evolved from
a paper by Guth (1981) [40], proposing a solution to the horizon and flatness problems.
Guth’s paper considered to be one of the original defining papers on the subject, gave a
mechanism for inflation no longer viewed as accurate but arguably set the groundwork for
this field [42].
This chapter reviews the motivations for inflation including the horizon and flatness
problems - mainly to appease several issues within the hot Big Bang theory - and a brief
overview of the nature of inflation.
3.1
Motivations for Inflation
In order to understand the structure of the Universe in it’s present form, it is necessary to
look towards the history of our vast expanse in a quest to understand the initial conditions of
the Universe that led to the theory of inflation. There are certain initial conditions required
to appease and accommodate the hot Big Bang theory (the Big Bang referring to a time when
the scale factor saw a(t) = 0), which describes the Universe’s earliest stages. It is known that
inflation cannot proceed forever, due to nucleosynthesis and the creation of the CMB - which
requires a universe that transcends from radiation domination to matter domination and the
observed baryon-anti-baryon asymmetry of the Universe, all imply that inflation must have
ceased by a certain time for these events to occur [64]. Figure 2 provides a timeline for the
major events in the history of the Universe, indicating that Inflation ended around 10−32 s
after the Big Bang.
According to Guth [40], the initial conditions of the Universe encompass the following:
the standard model of cosmology has a singularity taken to be at time t = 0, as t → 0
then the temperature T → ∞. It is not possible to define physics at t = 0 as T becomes
11
of order of the Planck mass or greater and quantum gravitational effects are expected trump
all standard model physics. By defining/assuming the Big Bang temperature to be below
the Planck mass.11 MP l , where the subscript ’Pl’ stands for Planck, one can then form a
description of the initial conditions and we can then use equations of motion to describe the
evolution of the Universe, which so far fit with current observations [1].
The initial conditions of the hot Big Bang theory present their own problems which
can be resolved with a sufficiently long period of inflation. Simultaneously, inflation also
explains why from our observations the Universe appears isotropic and homogeneous on
scales larger than 50 MPc [78]. It was with these principles and problems in mind, that Guth
introduced the idea of inflation through accelerated expansion [40].
3.1.1
The Problem of Initial Conditions
Mukhanov [86] looks at the problem of initial conditions whose work is followed in this
chapter. Briefly c, the speed of light, is not set equal to one in line with Mukhanov’s derivations.
On scales larger than a few hundred parsecs matter is distributed re-markedly evenly,
CMB data has shown that at the time of recombination12 the Universe was extremely homogeneous and isotropic on all scales up to the present horizon (accurate to 10−4 ) [86] [17]. A
common question is what were the initial conditions which led to homogeneity and isotropy.
To examine the initial conditions a number of assumptions need to be made:
• Inhomogeneity cannot be dissolved by expansion, an idea supported by general relativity [86].
• Non-perturbative quantum gravity does not play a role at sub-Planckian curvatures
[86].
• As non-perturbative quantum gravity effects dominate when the curvature reaches
Planckian values and classical space-time breaks down, initial conditions are looked at
around the Planckian time ti = tP l ' 10−43 s [86].
q
~c
There are two definitions of the Planck mass (1): MP l =
G or (2) the reduced Planck mass used by
q
~c
particle physicists and cosmologists MP l = 8πG
12
Recombination refers to the epoch when charged electrons and protons were first bound to form neutral
11
hydrogen atoms around 378, 000 years after The Big Bang.
12
These problems can be resolved if the Universe undergoes a brief period of accelerated
expansion.
Two independent sets of initial conditions characterise matter [86]:
• Spatial distributions described by its energy density ρ(x) and,
• the initial field of velocities.
The Homogeneity and Isotropy Problem - Horizon Problem
The extent of homogeneity and isotropy is at least as large as the present horizon scale
ct0 ' 1028 cm [86], which at one time was smaller by the ratio of corresponding scale factors
ai
.
a0
Assuming inhomogeneity cannot be resolved by expansion, the size of the homogeneous
isotropic region from which our Universe originated li at t = ti is
li ' ct0
ai
,
a0
(14)
and comparing this to the size of the causal region lc ' cti :
li
t0 ai
'
.
lc
ti a0
As a(t) ∝
1
T
(15)
and assuming primordial radiation dominates around ti ' tP l , then TP l '
1032 K and taking T0 = O(1) then,
ai
T0
'
' 10−32
a0
Tpl
(16)
as a ∝ T1 . It can then be concluded
li
1017
' −43 10−32 ' 1028 .
lc
10
(17)
Around the Planckian time the Universe exceeded causality by 28 orders of magnitude,
which implies 1084 causally disconnected regions where the energy density was smoothly
distributed with variation on the order of
δρ
ρ
' 10−4 [86]. Due to the maximal propagation
of light no causal physical processes can account for the smooth matter distribution.
Re-writing equation (15) as
li
ȧi
'
lc
ȧ0
and having used the approximation
a
t
(18)
' ȧ, then we can see that the size of our Universe
was initially larger than the size of a causal patch. Assuming gravity would slow down any
13
expansion then equations (17) and (18) highlight how the homogeneity scale is larger than
the causal scale. The horizon marks the boundary of causally connected regions (regions
light rays could have reached since the Big Bang). The regions grow over time as light travels, coming inside the horizon and becoming causally disconnected. Hence the homogeneity
problem is sometimes called the horizon problem.
Initial velocities (Flatness) Problem
Initial velocities must obey the Hubble Law, v = H0 D where v is the velocity of an object,
D is the proper distance and H0 13 is the scale factor today to account for the matter distribution in the Universe. This idea can be explored by considering a large spherically symmetric
cloud of matter and compare its total energy E t , with the kinetic energy due to Hubble
expansion E k . The total energy composed of positive kinetic energy and negative potential
energy of the gravitational self interaction E p is conserved
E t = Eik + Eip = E0k + E0P ,
(19)
where E0 is the energy measured today and Ei is some initial energy value. Kinetic
energy is proportional to the velocity squared,

2
ȧi
Eik = E0k  
ȧ0
(20)
therefore,

2
Eit
Eik + Eip
E0k + E0p  ȧi 
=
=
.
Eik
Eik
E0k
ȧ0
Now E0k ' |E0p | and
ȧi
ȧ0
(21)
≤ 10−56 [86] so
Eit
≤ 10−56 .
k
Ei
(22)
For a given energy density distribution, the initial Hubble velocities need to compensate
for the huge negative gravitational energy of matter against the huge positive kinetic energy
to an accuracy of 10−54 % [86]. Mukhanov [86] notes that the accuracy of this is that if there
was an error in the initial velocities exceeding 10−54 %, this would result in the Universe
either re-collapsing or becoming ’empty’ too early.
13
Currently taken to be 67.3 ± 1.2 (Km/s)/(MPc)[3]
14
It can be viewed in terms of the flatness problem by rewriting the Friedmann equation
(6) in terms of the density parameter Ω(t),
Ω−1=
k
.
(aH)2
(23)
Taking the ratio of Ω(t) − 1 at the time ti with t0 and using the result that Ω =
Ep
Ek
[86]
then

2
ȧi
(Ha)20
= (Ω0 − 1)  ≤ 10−56 ,
(Ωi − 1) = (Ω0 − 1)
2
(Ha)
ȧ0
(24)
which follows from equation (22). It can be seen from equation (24) that the density
parameter must initially have been extremely close to unity. Hence this is also known as the
flatness problem.
Initial Perturbation Problem
To understand large scale structures14 is the need to understand the origin of primordial
inhomogeneities, which must be initially on the order
δρ
ρ
' 10−5 [86]. Inhomogeneities
aggravate the problem of isotropy and homogeneity in the Universe but in the following
chapters it is shown how a sufficiently long period of inflation solves these problems.
3.1.2
The Flatness Problem
The flatness problem, arguably celebrated less than the Horizon Problem, but stressed by
Dicke and Peebles [98], concerns the reasoning behind why the density of the Universe today
is so comparable to the critical density ten billion years after the Big Bang. The hot Big Bang
model however predicts that ȧ decreases with time due to gravitational attraction of cosmic
structures causing the Ω(t) parameter to move away from unity and away from the critical
density. The nature of the flatness problem has been introduced and will now be examined
further. The beauty contained within the flatness problem is presented here, along with a
resolution.
Recall the Friedmann equation (13), assuming k 6= 0,
Ω − 1
14
=
1
ȧ2
(25)
Not covered in this literature review. Introductory reading on this field includes Liddle and Lyth’s text
’Cosmological Inflation and Large-Scale Structure’
15
˙2
where a2 H 2 = a2 aa2 = ȧ2 . In a universe dominated by matter or radiation (or both)
then the acceleration equation (8) has ä < 0 always, as density and pressure are non-negative
quantities. The expansion of the Universe slows down as ȧ is decreasing over time. As a
consequence the left hand side of equation (25) is increasing further away from unity as time
passes and the Universe expands (see figure 3).
Figure 3: The density parameter moves away from unity in an open or closed universe. For
k = 0 then Ω = 1 for all time. Image courtesy of [51].
Consider the evolution of the Ω(t) parameter back in time, it gets closer to unity the
further back you go. To see this, recall that the density of radiation falls as ρr ∝ a(t)−4 , and
for pressure-less matter the relation is ρm ∝ a(t)−3 . Both of these decrease more rapidly than
the curvature term in the Friedmann equation (6).
Observations today give Ω0 = O(1) [4] [3] , implying the Universe is spatially flat
(k = 0). However it has been shown that if |Ω − 1| is small at the present epoch, it
must have been much smaller at earlier times. This raises the question of how close to unity
was Ω(t) when the Universe was very young.
The Universe underwent the period of primordial nucleosynthesis, denoted n, around
1
tn ' 1s after the Big Bang, and assuming the Universe was radiation dominated, a ∝ t 2 , with
the correction due to the Universe becoming matter dominated and the epoch of radiationmatter equality considered negligible, hence
|Ω − 1|0
ȧ2
t0
= n2 = .
|Ω − 1|n
ȧ0
tn
16
(26)
Taking the value for the age of the Universe to be t ' 1017 s and |Ω − 1|0 to be O(1),
|Ω − 1|n ' 10−17 .
(27)
To highlight how narrow a band this is - and the beauty of the flatness problem [51],
0.99999999999999999 ≤ Ωn ≤ 1.00000000000000001
(28)
i.e the Universe is equal to the critical density to 1 part in 1017 . Therefore the Ω(t)
parameter has always been close to unity from the time of nucleosyntheis to today. How
was this achieved?
One way to solve the flatness problem, and understand why the value of Ω(t) has always
been close to unity (at least until the period of primordial nucleosynthesis), is to suppose
that the Universe underwent a period of inflation where if the expansion were to accelerate
for a period of time then ȧ is pushed towards unity, and is pushed even closer to unity the
longer the accelerated expansion lasts for.
Figure 4: The effect of introducing a period of inflation to Ω. Between t = 0 and ti the
Ω parameter is pushed away from unity. However, if the Universe undergoes a period of
accelerated expansion between ti and tf then Ω is pushed back towards unity before slowly
moving away from unity until today t0 . Image courtesy of [77].
Figure 4 explores how inflation could solve the flatness problem. From t = 0 to ti the
Ω(t) parameter moves away from unity, however between ti to tf the Universe then undergoes a period of accelerated expansion, i.e inflation. This period of accelerated expansion,
17
pushed Ω(t) back towards unity. It once again starts moving away from unity after time tf ,
as although the Universe is still expanding, the rate it does this is decelerating. The value of
Ω0 (t) today is dependent on how close Ω(t) was pushed to unity at the end of inflation.
It should be noted that if Ω < 1 it remains so for the entirety of the Universe’s history
and similarly for Ω > 1 it remains greater than unity forever. Ω = 1 corresponds to a
universe that is spatially flat i.e k = 0, and Ω = 1 remains constant for all time which can be
seen in equation (13). To pass from greater than unity to less than unity would alter the sign
of k.
A question arises of how long does inflation have to last for to solve the flatness problem,
with |Ω − 1|n ' 10−17 (27). Taking the Friedmann equation (23) it is important to see that
|Ω − 1|a2 H 2 is a constant for any k, so it is concluded that
|Ωi − 1|a2i Hi2 = |Ωf − 1|a2f Hf2 ,
(29)
where the subscript i denotes quantities at the beginning of inflation, and f at the end
of inflation. By taking |Ωi − 1| = O(1), since inflation occurs in the early Universe and
curvature is negligible at sufficiently early times, then Hi = Hf is constant during exponential inflation (as the change between the two values would be negligible in comparison to
inflation). Equation (29) therefore simplifies to
af
1
'q
.
ai
|Ωf − 1|
(30)
From the Friedmann equation (23), and relating the values of Ω(t) at the end of inflation
to the time when primordial nucleosyntheis occurred,
|Ωf − 1|
a2 H 2
ȧ2
tf
= n2 n2 = n2 = .
|Ωn − 1|
af Hf
ȧf
tn
(31)
Substituting in the appropriate values and rearranging, the amount of inflation that must
occur to keep Ω(t) close to unity at the time of nucleosynthesis must exceed

− 1
af
tf 
' 109 
ai
sec
2
.
(32)
Taking tf ' 10−34 s,
af
> 1026 .
ai
18
(33)
If the Universe inflated more than 1026 times its original size then the flatness problem is
solved. This toy model gives an idea of the size of the expansion parameter required during
inflation.
An alternative way to view this is to assume once again inflation occurs between ti and tf ,
the scale factor grows from ai to af with exponential expansion, with constant H(t), so the
total density ρT remains constant between these two times. Working with the assumption
that at the end of inflation ρT is fully converted to radiation energy then ρr ∝ a−4 between
tf and today t0 .
By going back to the Friedmann equation (6) the effective curvature density can be def
3 k
fined15 , ρef
≡ − 8πG
[63], which scales as a−2 . Assuming de-Sitter expansion so that
k
a2
between ti and tf the scale factor grows from ai to af , with constant Hubble rate Hi then
the total density ρT is constant between ti and tf . Another assumption is that ρT is converted
into radiation energy ρr which scales as ρr ∝ a−4 between tf and t0 . The final assumption is
f
that ρef
is equal to ρT at ti and ρr at t0 , so we can then write
k

2

4
f
ρef
ρr (a0 )
ρr (a0 )
ρr (a0 )  af 
ai
k (a0 )
.
=
=
=
=  =
ef f
a0
ρT (ai )
ρT (af )
ρr (af )
a0
ρk (ai )
(34)
As expansion has slowed down since the end of the inflationary period, the scale factor from the beginning of inflation, ti , to the end of inflation, tf grew much quicker than
between the end of inflation and today, t0 , leading to the assumption
af
a0
≥ .
ai
af
(35)
Minimum Number of E-foldings to Solve the Flatness Problem
A useful measure of expansion is titled the e-fold number16 , defined as
(36)
N ≡ ln a,
which quantifies the amount of expansion between times t1 and t2 , where the number of
e-folds ∆N = N2 − N1 = ln aa21 = ln a. An e-fold is simply an expansion of the universe from
≈ 2.72 times its original size. Hence
15
The Einstein equation relates curvature to matter, and in the case of a homogeneous and isotropic universe
described by the Friedmann metric, the Einstein equation gives the Friedmann equation. The spatial curvature
1
radius is Rk ≡ ak − 2 and the Hubble radius is RH = H −1 =
reformulated as
3
2
RH
±
3
2
Rk
a
ȧ
so that the Friedmann equation can be
= 8πGρ. Setting the Hubble radius to zero we can derive the effective curvature
density.
16
A funky way of measuring how an exponential quantity grows/increases by a factor of e- the base e
analogue of doubling time.
19
Nf − Ni ≥ N0 − Nf ,
(37)
where the number of e-folds during inflation should be more than or equal to the number
of e-folds post inflation to the present day. Note that there is no upper bound to the left
hand side of equality (37) only a lower bound to solve the flatness problem, therefore ∆N =
N0 − Nf .
Computing ∆N as a function of the energy density at the end of inflation, ρr (af ) then
taking the values of ρr (a0 ) to have order O(10−4 eV )4 and the inflationary energy scale to be
of the order O(1016 GeV )4 [63] gives,
1

4
a0
ρr (af ) 
∆N = ln
= ln
≤ ln 1029 ' 67.
af
ρr (a0 )
(38)
67 is the minimum number of e-folds required if inflation takes place around the 1016 GeV
scale to solve the flatness problem. 67 e-folds corresponds to a universe having increased
≈ 1.14 × 1026 times its original size.
3.1.3
The Horizon Problem
The Horizon problem was identified in the late 1960s by Misner [83] which highlights that
due to one of the principles of the Theory of Special Relativity [32] it is not possible to
communicate with a distant observer faster than the speed of light, as such there is a finite
distance light could have travelled since the Big Bang. For each observer in the Universe
there is a maximal distance beyond which space cannot be observed, this is called the horizon
distance.
Defining the horizon distance dH (t1 , t2 ) to be the physical distance between two particles
recorded at time t2 emitted from the same point in space, travelling at the speed of light in
opposite directions at time t1 . Taking the origin to coincide with point of emission the
physical distance can be computed by integrating over small distance elements dl between
the origin and position r2 of one particle and multiplying the integral by two
dH (t1 , t2 ) = 2
Zr2
dl = 2
0
Zr2
0
a(t2 ) √
dr
,
1 − kr2
(39)
which is derived from the FLRW metric (1). The geodesic equation (1) for relativistic
dr
particles travelling in a radial path has ds = 0, so dt = a(t) √1−kr
2 which can be integrated
along particle path,
20
Zt2
t1
r2
Z
dt
dr
= √
.
2
a(t)
1
−
kr
0
(40)
Substituting this into equation (39),
Zt2
dH (t1 , t2 ) = 2a(t2 )
t1
dt
.
a(t)
(41)
To highlight the horizon problem, it is beneficial to remove the time element from equation (41) by making the substitution dt =
eter H =
2
ȧ
a
da
aH
which was obtained from the Hubble param-
,
dH (a1 , a2 ) = 2a2
Za2
a1
da
a2 H(a)
(42)
so the Hubble parameter is no longer a function of t. Taking t1 and t2 to be during the
radiation domination period (RD), and from the Friedmann equation H ∝ a−2 , so making
a new parametrization H(a) = H2
a2
a
2
,
dH (a1 , a2 ) = 2a2
Za2
a1
da
a22 H2
=
2 (a2 − a1 )
.
H2
a2
(43)
Taking t2 to be sufficiently long after t1 such that a2 >> a1 , the approximate expression
for the horizon is,
dH (a1 , a2 ) '
2
.
H2
(44)
It can be concluded that the horizon equals twice the distance of the Hubble radius,
R = H −1 . At time t2
dH (t1 , t2 ) = 2RH (t2 ).
(45)
The horizon is equivalent to a casual distance17 in the Universe, in that if an event occurs
at t1 , and information cannot travel faster than the speed of light, an event cannot affect
distances larger than 2RH (t2 ) at time t2 . Thus the horizon provides a coherence scale for an
event.
The Surface of Last Scattering
The Big Bang theory states that in the early Universe it was sufficiently hot for all matter to be fully ionised so electromagnetic radiation was scattered very efficiently by matter
17
A good illustration of this is to look at light cones.
21
and the Universe was in a state of thermal equilibrium. However as the Universe cooled
electrons were able to combine into atoms and scattering decreased in a period known as
recombination (see figure 2). When recombination was complete, photons could propagate
freely throughout the Universe - and this radiation is known as the CMB. This radiation
appears to come from a spherical surface around the observer - a radius which is the distance
each photon has travelled since it was last scattered - this surface is the last scattering surface.
This subchapter discusses the last scattering surface in the context of the hot Big Bang
model providing no explanation for the homogeneity of the temperature of the photons after
decoupling, and its resolution via inflation.
Before photon decoupling, it is expected that the Planck temperature of photons at a
given point depends on their local density [104], and that the Universe would have random
values of the local density. The coherence scale at which the density is nearly homogeneous
is given by the horizon distance (45). If t1 and t2 are two times during radiation domination,
this value cannot exceed 2RH (t2 ). Therefore the temperature of the photons at t2 should not
be homogeneous on scales larger than 2RH (t2 ).
However CMB experiments [17] that mapped the photon temperature at the time of
last scattering (which occurred around the time of matterradiation equality18 , tdec [63]),
mapped the CMB to be homogeneous on scales up to 2RH (tdec ). It is found that the distance
2RH (tdec ) subtends an angle of order a few degrees in the sky, rather than encompassing
the diameter of the surface of last scattering [61]. It seems the last scattering surface is
composed of ' 1083 causally disconnected pieces [40]. However (and here is the crux),
CMB anisotropies are only of the order 10−5 , highlighting how homogeneous the surface of
last scattering actually is. This appears to be a paradox within the hot Big Bang theory.
Looking at where the problem originates from, resolves the paradox. In computing the
integral (42) the integral converged with respect to the boundaries a1 and a2 , even if the initial
time is taken to be infinitely early. However, if the integral was divergent, then an infinitely
large horizon at time t2 is obtained, providing a1 be small enough. The convergence of
Za2
a1
Z a2
da
da
=
,
2
a H(a)
a1 aȧ
(46)
as a1 → 0 depends on if the expansion is accelerated or decelerated. To reach an arbitrarily large value for the horizon at the time of decoupling then it is required that the radiation
18
Which occurred approximately 75,000 years after The Big Bang.
22
dominated phase is to be preceded by an infinite stage of accelerated expansion. This would
explain the homogeneity of the last scattering surface, and it has been calculated that the
Universe needs to only undergo a small amount of accelerated expansion to boost the horizon by a factor of ' 103 with respect to the Hubble radius to account for the homogeneity
of the CMB.
To determine how large this period of expansion was, the number of e-folds needs to
be determined to see whether inflation is a viable option. Imagine two photons emitted at
the same time ti at the beginning of inflation, by assuming the expansion of the Universe
is exponential (with constant Hi ) until time tf and switches back to radiation domination,
during inflation the horizon grows as
dH (ti , tf ) = 2af
Za2
a1


da
2  af
=
− 1,
a2 Hi
Hi ai
(47)
so that
(48)
dH (ti , t) = dH (ti , tf ) + 2RH (t).
The problem of the horizon distance can be resolved if the co-moving scale19 corresponding to the horizon at the end of inflation is at least equal to the diameter of the surface of last
scattering. Therefore the surface of last scattering is entirely in causal contact.
To compute the diameter of the surface of last scattering (i.e what is the largest wavelength observed today), then imagine two photons emitted at tdec travelling towards us from
this surface, coming from opposite directions and reaching us on Earth today. By resolving
the coordinates such that origin of co-mobile coordinates places us at the origin it is possible
to compute the co-mobile radius coordinate rdec from the time of emission, which would
give a total diameter of 2rdec . The mathematics for this is similar to the steps followed in
equations (39-41),
dH (tdec , t0 ) = 2
rZdec
dl = −2
tdec
Z
t0
0
dr
a(t0 ) √
1 − kr2
(49)
where the minus sign accounts for photons travelling towards us. By neglecting the recent
2
stage of dark energy domination, integrating equation (49), by taking a ∝ t 3 during matter
domination, and it is found dH (tdec , t0 ) ' RH (t0 ), an order one factor.
19
A co-moving scale is a scale divided by a(t) the scale factor, i.e it factors out the expansion of the Universe.
23
If the co-moving horizon is at least equal to the co-moving diameter of the surface of last
scattering, we can write this as
dH (ti , tf )
RH (t0 )
≥
.
af
a0
(50)
Assuming the Hubble parameter is constant during inflation (Hi ' Hf ) and ignoring
numerical factors we obtain the equality,


1
1  af
− 1 ≥
.
af Hf ai
a0 H0
(51)
To evaluate the right hand side, we note the radiation domination epoch (aH ) scales like
a−1 and ignoring the matter and dark energy domination we can approximate the right hand
side by
a0
,
af
af
a0
≥ .
ai
af
(52)
Equation (52) highlights that the number of inflationary e-folds should at least equal the
number of post inflationary e-folds. If it is larger, then the size of the observable universe
is smaller than the horizon. It is important to highlight how equation (52) is the same condition met in the flatness problem (35) which implies that inflation solves both the horizon
and flatness problems.
To summarize, in the hot Big Bang model the distance over which co-moving causal
−1
interactions can occur before photon decoupling (180Ω0 2 h−1 M P c) leading to the CMB is
much less than the distance radiation travels after decoupling (5820h−1 M P c in a flat universe) [64]. Photons separated by more than the horizon distance at last scattering could not
have interacted before decoupling. The hot Big Bang model provides no explanation for the
homogeneity of the temperature of the photons after decoupling. To account for this observation, the Universe is assumed to have undergone a period of inflation before the radiation
phase in the history of the Universe.
3.1.4
The Monopole Problem and Relic Particles
Grand Unified theories predict the existence of massive, stable, relic particles, occurring
around 10−35 s after the Big Bang [51], the focus here is on the magnetic monopole, a stable particle, with a high mass of O(1016 ) GeV. Due to their high mass this dominates the
24
energy of the particle, even with a high temperature universe. The particle becomes nonrelativistic once the temperature of the Universe falls to T ' 1028 K, which is around the
same temperature where monopoles form, hence they are essentially non-relativistic soon
after formation.
The problem with magnetic monopoles is that the density of matter falls as a−3 whereas
radiation falls as a−4 . By allowing sufficient time to elapse then magnetic monopoles should
come to dominate relativistic particles in the Universe. This could potentially lead to a
sufficient number forming and dominating before the epoch of primordial nucleosyntheis
which would have disastrous consequences. However, observations indicate that primordial
nucleosynthesis did proceed. It must be assumed there was a constraint on the density of
monopoles when they formed, it must be sufficiently low enough for primordial nucleosynthesis to take place.
To solve this problem it is assumed that inflation took place after the creation of magnetic
monopoles, such that the exponential increase in volume of the Universe, lead to a negligible
density of them throughout the Universe. From the flatness problem (33) it was shown that
if the density of the universe decreased by a factor of 1078 then this solves the problem20 .
3.1.5
The Origin of perturbations
Guth [40] demonstrated that large scale homogeneity and isotropy of the Universe are part
of the initial conditions on which cosmological models are based however the Universe is
not perfectly homogeneous and these inhomogeneities led to large scale structures. Consequently it is important to find an explanation for these cosmological perturbations, or to
assume it is also part of the initial conditions of the Universe.
Inhomogeneities can be expanded in a co-moving Fourier space and have wavelength,
λ(t) =
2πa(t)
,
k
(53)
1
dependent on the scale factor. During radiation domination, a(t) ∝ t 2 and RH ∝ t. The
Hubble radius grows faster than the perturbation wavelengths, so observable perturbations
were originally super-Hubble fluctuations. Returning to the horizon problem the Hubble
radius RH (t2 ) gives an upper bound on the causal horizon for any t1 (45), so super-Hubble
20
we calculated in equation (33) that the Universe needed to expand more than 1026 times, which translates
to a volume of 1078
25
fluctuations are expected to be out of causal contact. However the mechanism for generating
coherent fluctuations on a-causal scales is not well understood. One way to resolve this is to
assume all observable cosmological perturbations were generated inside the causal horizon.
Mechanism to Generate Perturbations
A mechanism needs to be sought which allows for the largest wavelength observable today λmax (t0 ) ' RH (t0 ), to have been inside the causal horizon at an earlier time ti . Assuming
the Universe was in decelerated expansion before time ti then the causal horizon was of the
order RH (ti ). So this raises the problem of if λmax ≤ RH at ti how can λmax ' RH today.
As the Hubble radius grows faster than the physical wavelengths one cannot have a Universe
dominated by radiation or matter between ti and t0 . To resolve this
λ(t)
2πa(t) ȧ(t)
2π ȧ(t)
=
=
,
RH (t)
k a(t)
k
(54)
so that during inflation the physical wavelengths grew faster than the Hubble radius.
From ti to tf the Universe undergoes accelerated expansion so that λ < RH at ti , then the
scale λmax can exit the Hubble radius during inflation and re-enter today (see figure 5).
To find how much inflation is required, it is assumed λmax ' RH at ti and t0 today, from
which a lower bound on the number of e-folds during inflation can be sought.
kmax =
2πa(ti )
2πa(t0 ) ai
H0
=
,
=
,
RH (ti )
RH (t0 ) a0
Hi
(55)
where kmax is accosted with λmax . Assuming de-Sitter expansion between ti and tf with
constant Hubble parameter, Hi = Hf . Between tf and t0 the Friedmann equation has
3
H ∝ a−2 during radiation domination and H ∝ a− 2 during matter domination. By taking
radiation domination to last through to today then
H0
Hf
can be estimated by saying more
e-folds occur during radiation domination than matter domination. So
H0
Hf
'
af
a0
2
and
equation (55) becomes

2
ai
af
=  ,
a0
a0
af
a0
= .
ai
af
(56)
The number of inflationary e-folds during inflation should at least be equal to the post
inflationary e-folds.
A discussion of each part of figure 5:
• 1. Inflation starts at ti where the potential energy of the scalar field dominates the total
energy density of the Universe. The curvature fraction Ωk is of order one or larger.
The curvature gets diluted and after a few e-folds Ωk << 1.
26
Figure 5: Evolution of observable wavelengths as a function of time during inflation and
other cosmological epochs. The largest wavelength we see today is equal to the Hubble
radius. This scale exits the Hubble radius between 37 to 67 e-folds before the end of inflation
depending on energy scales. The numbers refer to different epochs described in the chapter
on generating perturbations. Image courtesy of [63].
• 2. Assuming the previous bullet point holds, then all observable cosmological wavelengths exist within the Hubble radius.
• 3. Starting from the largest wavelengths the perturbation wavelengths exit the Hubble
radius and the quantum fluctuations of the scalar field and metric undergo a semiclassical transition.
• 4. At the end of inflation the scalar field decays into particles. This stage is called
re-heating. During this stage the large wavelength perturbations vanish but metric
perturbations survive. The metric perturbations have λ >> RH and are frozen against
micro-physics taking place on smaller scales.
• 5. After the decay of the scalar field the Universe becomes dominated by the energy
of relativistic particles and so the Universe enters into the radiation dominated epoch.
The long wavelength perturbations of the metric couple gravitationally to the radiation and matter perturbations. These perturbations lead to the CMB anisotropies.
27
3.2
What Is inflation?
Liddle and Lyth say ’The definition of inflation is extraordinarily simple: it is any period of
the Universe’s evolution during which the scale factor, describing the size of the Universe is
accelerating. This leads to a very rapid expansion of the Universe [64].’ It has been calculated
that inflationary period started around 10−36 s after the Big Bang, and continued to between
10−33 − 10−32 s after the Big Bang [86]. Following this rapid expansion, the Universe has
continued to expand but at a less accelerated rate.
3.3
Repulsive Gravity
For inflation to be possible requires gravitational repulsion, and many modern particle theories suggest there is a state of matter which alters the effect of gravity, and it is assumed the
Universe contained a small patch of this material. General relativity predicts the possibility of such a material with a negative pressure to create repulsive gravity. Modern particle
theories say materials with negative pressure are easy to construct - hence a possible cause of
inflation.
Imagine a patch with negative pressure then it will rapidly exponentially expand due to
gravitation repulsion, however the density remains constant. This patch of universe was in
a false vacuum state which eventually decays. The equation of state is P = ωρ and for a
material with negative pressure P = −ρ as ω = −1. This arises because in a region with
negative pressure, pressure (65) and density (64) are dominated by their potential energy
and so the ratio of these gives ω = −1. Turning to the acceleration equation (8) then when
ρ + 3P > 0 is satisfied, ä < 0 and so gravity acts to decelerate the expansion. Gravitational
acceleration requires ä > 0 and so ρ + 3P < 0.
3.4
General Conditions For a Successful Inflationary Model
Any successful inflationary model needs to be sufficiently long to solve the problem of initial
conditions as has been shown, but also lead to a graceful exit. Taking the Friedmann equation
(6) and differentiating it with respect to time, multiplying by aȧ , noticing Ḣ =
ä
a
−
ȧ2
a2
and
rearranging,
ä
= H 2 + Ḣ.
a
28
(57)
ä, the second time derivative with respect to the scale factor, should be negative during
the graceful exit21 implying the derivative of the Hubble constant, Ḣ is also negative. The
ratio
Ḣ
H2
grows towards the end of inflation and the graceful exit occurs when Ḣ is on the
order of H 2 . Making the assumption H 2 changes much faster than Ḣ ( that is Ḧ < 2H Ḣ ),
then the estimate for the duration of inflation is
tf '
Hi
.
Ḣi
(58)
At t ' tf , the right hand side of equation (58) changes sign and the Universe begins to
decelerate.
Figure 6: After the graceful exit the Universe enters a non accelerated period of expansion.
Although the Universe is still expanding it does so at a much slower rate. Image courtesy of
[86].
Inflation needs to last sufficiently long enough for the Universe, to be at a minimum, the
size of the observable Universe.
ȧi
ȧ0
< 10−5 , which can be re-written as
ȧi ȧf
ai Hi ȧf
=
< 10−5 ,
ȧf ȧ0
af Hf ȧ0
21
(59)
Cosmic expansion cannot go on forever if we want to arrive at our slowly expanding FLRW Universe.
There needs to be a mechanism to halt the exponential expansion, called the graceful exit. Graceful exit depends
on the potential and temperature, V (φ, T ). I have chosen not to discuss this topic here but a very thorough
review can be found in Matts Roos’ book Introduction to Cosmology [102]. A brief discussion can also be
seen in Mukhanov’s book [86] in chapter 5.5.1.
29
where
ȧf
ȧi
> 1028 . Hence inflation is only successful provided,
Hi
af
> 1033
.
ai
Hf
(60)
An estimate from this of the minimum number of e-folds required to solve the initial
conditions by assuming Ḣi << Hi2 and neglecting the Hubble parameter can be calculated.
It is roughly estimated,
af
Hi2
' exp(Hi tf ) ' exp
ai
Ḣi
>> 1033 .
(61)
The initial conditions are solved provided tf > 75Hi−1 giving the minimum number of
e-foldings as 75.
30
4
Inflationary Dynamics
’As with most of life’s problems, this
one can be solved by a box of pure
radiation.’
Andy Weir - The Martian
This chapter details the history of the study of inflation, and introduces a number of
conditions that a successful inflationary model needs to follow. The chapter also looks at the
evolution of modes during inflation and the various quantities that are measured.
4.1
Scalar Field Inflation
There are many models within the field of inflationary cosmology, however, the focus of
this disserstation is to look at a select number of inflationary models which solve numerous
aforementioned problems connected with isotropy, homogeneity, CMB distribution, magnetic monopoles etc and feature in the relevant 2013 and 2015 Planck papers [6][5]. Inflation
also explains temperature anisotropies and is the seed of large scale structure [99].
This dissertation predominantly looks at scalar field inflation with slow roll conditions,
but also R2 inflation. It must be stressed that currently the mechanism responsible for inflation is not known, but the basic theory is in line with current observations [96][114]. The
field deemed responsible for inflation is called the inflaton [39]. The inflaton is a spin-zero
scalar field particle, similar to other quantum fields [114], and it has recently been suggested
that the Higgs Boson could act as an inflaton [82].
Inflation solves a number of the ’fine tuning’ problems and initial conditions seen in the
hot Big Big model and in Guth’s 1981 paper [40], it was noticed that with regards to the
monopole problem, their density should ’over-close’ the Universe. The monopole contribution to Ω(t) is greater than the observed upper bound on the density parameter, Ω(t) > 4
[97]. To rationalise this, Guth concluded that symmetry breaking associated with scalar
fields must cause the Universe to enter a rapid period of inflation, thus diluting the density
of monopoles in the Universe.
The first models of inflation included the existence of a phase where the Universe was
close to de-Sitter space-time, and a slow rolling expansion field which governs the inflation.
31
Inflationary cosmology started to emerge around the 1970s, when it became apparent
that the energy density of a scalar field is equivalent to the vacuum energy/cosmological
constant [68] but changing due to cosmological phase transitions [57]. It was noted that these
changes can occur discontinuously due to first order phase transitions from a supercooled
vacuum state (false vacuum) [58].
The first semi-realistic model of inflation was proposed by Starobinksy [108], however
his rather complicated model faced numerous challenges. Guth [40] proposed a much simpler model now called ’old inflation’ based around supercooling during cosmological phase
transitions [58]. It was found to be incompatible, but laid the foundations for subsequent
efforts, due to the paper explaining how inflation solved many of the aforementioned cosmological problems.
In Guth’s scenario, exponential inflation occurs from the Universe being in a supercooled
false metastable vacuum state without any fields or particles, but has a large energy density.
During expansion the energy density does not change, and the empty space remains empty.
This leads to inflation in a false vacuum, leading to a big and flat universe. The false vacuum
then decays, and bubbles22 of the new phase collide leading to a hot universe. However if
the probability of bubble formation is large then bubbles of the new phase form near each
other. If inflation is deemed to be too short to solve any problems, the bubbles would lead to
an extremely inhomogeneous universe. On the contrary, a long inflation period would lead
to bubbles forming away from each other and with a low probability of formation. Each
bubble would represent a small open universe with vanishingly small Ω(t). Both scenarios
do not tally with today’s observations [43].
A new inflation theory emerged between 1981-1982 [70][71], whereby inflation may begin with either a false vacuum or an unstable state at the top of a potential. The inflation field
ϕ then rolls down to the minimum of it’s effective potential. This is critical, as density perturbations produced during slow roll inflation are inversely proportional to ϕ [88][44][110].
This new scenario explained the homogeneity of the Universe, and does not occur in the false
vacuum state (ϕ = 0). This model encountered problems, namely that it requires the effective potential to have a very flat plateau near ϕ = 0. In extensions to this, the inflation field
22
Quantum uncertainty allows for the temporary creation of bubbles of energy, or pairs of particles (e.g
electron-positron pairs) out of nothing, provided they vanish after a short period of time. The less energy used
to create the particles the longer they last for.
32
has a small coupling constant meaning it could not be in thermal equilibrium with other
matter fields, and the theory of cosmological phase transitions does not work. In addition,
thermal equilibrium requires many particle interactions. The new inflation theory could
only explain why our Universe is so large if it contained many particles from the beginning
[74].
Both old and new inflation require changes to the Big Bang theory, which does not sit well
with the assumption that the Universe was in a state of equilibrium from the beginning, that
it was relatively homogeneous and large enough to survive until the beginning of inflation,
and that inflation was just a small stage in the history of the Universe [76]. Even in the
1980s these assumptions seemed natural and unavoidable, as observational evidence (CMB,
abundance of light elements etc.) suggested the Universe was created in a hot big bang.
In 1983, the problems associated with old and new inflation were resolved with the concept of chaotic inflation [72], which did not require thermal equilibrium in the early universe. Chaotic inflation occurs in any theory with a potential with a sufficiently flat region
allowing for the slow roll regime [72].
4.2
Slow Roll Conditions
Consider a real homogeneous inflation scalar field, ϕ, coupled with the action for a scalar
field evolving in a potential V (ϕ) (without a gradient the particle wouldn’t be forced to
move) and follow the derivations found in [95][63].
S=−
Z
√

#
1
d x −g  ∂µ ϕ∂ µ ϕ − V (ϕ)
2
4
(62)
Defining the energy-momentum tensor Tµν = ∂µ ϕ∂ν ϕ − gµν L,

Tµν

1
= ∂µ ϕ∂ν ϕ − gµν  ∂α ϕ∂ α ϕ + V (ϕ).
2
(63)
Here Tµν = diag(ρ, −P, −P, −P ). Cosmic time, energy density and pressure are given
respectively by
ρϕ =
ϕ̇2
+ V (ϕ)
2
(64)
Pϕ =
ϕ̇2
− V (ϕ).
2
(65)
and
33
From the acceleration equation (8), ρϕ + 3Pϕ = 2(ϕ̇2 − V ), and for accelerated positive
expansion ä > 0, as such the Universe enters a period of expansion as soon as ϕ̇2 < 0. Indeed
the expansion is quasi-exponential provided
ϕ̇2 << 0,
(66)
in the scalar field slow roll regime. The potential energy dominates over the kinetic
energy. In the limit of zero kinetic energy, the energy momentum tensor (63) equals the
cosmological constant and expansion would never halt, therefore
ϕ̇2 << 0
(67)
is the first slow roll condition required to necessitate a long finite stage of acceleration.
The evolution of a single scalar field is a second order equation, and the first slow roll
condition can apply instantaneously and not just for an extended period therefore the time
derivate must hold giving the second slow roll condition
ϕ̈
<<
∂V (ϕ) .
∂ϕ (68)
Taking the definition for ρϕ in (64) the Friedmann equation can be re-written as,


k
8πG  1 2
H2 =
ϕ̇ + V  − 2 .
3
2
a
(69)
Following a similar procedure for the acceleration equation (8)


ä
8πG 
=
V − ϕ̇2 .
a
3
(70)
To obtain the Klein Gordon equation in an expanding universe multiply the Lagrangian
in equation (62) by a3 (due to volume expansion) and derive the equation using the classical
Euler-Lagrange equation to study the temporal evolution of the inflation field,
ϕ̈ + 3H ϕ̇ + V 0 (ϕ) = 0.
(71)
The second term in the Klein Gordon equation (71) is referred to as Hubble damping acts as a friction term to ensure slow roll. By virtue of the Klein Gordon equation the second
condition (68) can be re-written as
|ϕ̈| << 3H|ϕ̇|
34
(72)
Using relation (57) takingthe Friedmann equation (69) and re-writting it using equation
(70) with k = 0 gives
Ḣ ' −4πGϕ̇2 .
(73)
To summarize, the two slow roll conditions (67),(68) have been given and from these it
is now possible to define the slow roll parameters.
4.2.1
Slow Roll Parameters
Many inflationary models are underpinned by the slow evolution of some scalar field23 ϕ in
a potential V (ϕ). For this slow roll approximation was introduced - independent of inflationary model - which neglects slowing changing terms in the equations of motion where
the potential is less than the kinetic energy. This approximation must eventually fail for
inflation to end. One of the advantages is that knowledge of the exact form of the potential
is not required to model the predictions of inflation.
k = 0 as the curvature term decreases as a−2 and is negligible in comparison to the
contribution from the scalar field which has almost constant energy density. If inflation lasts
sufficiently long Ḣ < 0 and |Ḣ| < H 2 if ä > 0. If the scalar field is rolling slowly, then ϕ '
constant and Ḣ ' 0 [86].
Before introducing the slow roll parameters it is useful to see how a flat potential in an
inflationary field arises. As the scalar field must be in the slow roll regime (and thus varies
little), this leads to two further equalities:
1 2
ϕ̇ << V
2
(74)
ϕ̈ << 3H ϕ̇.
(75)
With these two conditions (74,75) equations (69, 71) and (73) can be recast as
8πG
V,
3
(76)
Ḣ = −4πGϕ̇2 ,
(77)
∂V
.
∂ϕ
(78)
H2 '
3H ϕ̇ ' −
23
A vacuum like period which drives inflation must be dynamical and must be able to violate the strong
energy condition to get a system with P = −ρ, and this requires scalar fields. Scalar fields have the same
quantum numbers as the vacuum and can mimic a vacuum like state.
35
Implying,
|Ḣ|
3 ϕ̇2
<< 1
<<
H2
2V
(79)
d V,ϕ 
V,ϕϕ ϕ̇
V,ϕ V,ϕϕ
1 V,ϕ V,ϕϕ
ϕ̈ ' − 
'−
'
,
'
2
dt 3H
3H
9H
48πG V
(80)
as ϕ̇2 << V .
Peter and Uzan [91] note that


and so using (80) and conditions (74, 75) it is seen
2

V
 ,ϕ  << 48πG,
V
(81)
and
V,ϕϕ V (82)
<< 48πG.
For inflation to occur the potential needs to be very flat to get a friction dominated term.
Introducing the slow roll parameters first defined by Liddle and Lyth [67]:
2
1
∂V /∂ϕ 
=
16πG
V
η=
(83)
1 ∂ 2 V /∂ϕ2
8πG
V
(84)
where the slow roll conditions implies << 1 and η << 1.
Alternative Definitions
• However Peter and Uzan [91] use alternative definitions [86]:
−−
Ḣ
˙
, η =−
,
2
H
2H
(85)
Which only depend on the space time geometry and are therefore not restricted to the
single field case.
• The slow roll parameters can also be defined as [66]:


M2 V 0
= Pl  
16π V
36


M 2 V 00
η = P l  .
8π
V
(86)
• They can also be defined as [85]
2

1 V,ϕ
V,ϕϕ
=   , η=
2 V
V
(87)
• Baumann [15] uses
=−
4.2.2
Ḣ
dlnH
< 1.
=−
2
H
dN
(88)
Background Evolution
Although the exact form of the potential does not need to be known, it must allow for a
sufficient number of inflationary e-foldings to solve the aforementioned problems in the Big
Bang theory.
The number of e-foldings during inflation is
a(tf )
a(ti )
eN =
(89)
where N = log af − log ai .
Expressing this as an integral
Zaf
N=
ai
da
a
(90)
and using the slow roll conditions (67,68):
H2 '
8πG
V
3
3H ϕ̇ ' −
∂V
,
∂ϕ
an equation for N in terms of the potential and scalar traversed can be sought
N = −8πG
Zϕf
ϕi
V
,
V,ϕ
(91)
where the minus sign is because V 0 < 0 gives slow roll in the positive ϕ direction.
From (81) this implies,
N >>
Zϕf
8πG √
ϕi
>> √
37
1
dϕ
48πG
1 ∆ϕ
48πG
(92)
where ϕf is defined by (ϕf ) = 1 as inflation would end through violation of the slow
roll conditions.
A large number of e-folds requires the inflation field to move a long way in comparison
to the gravitational scale G, this can lead to problems in effective field theory.
4.3
End of Inflation
Figure 7: We can see the potential at V0 is not a stable point, and so the scalar field slow rolls
down to the true potential which has that value V (ϕf ), inflation ceases. Image courtesy of
[99].
The end of inflation is quite simply the end of the slow roll regime, where the value
of the field ϕf has max (, η) ' 1. The universe becomes very flat, k = 0, and curvature
terms can be neglected in the primordial phase. In addition entropy has become dilated.
Inflation requires a potential that is flat enough to allow slow roll, however as the scalar field
approaches the bottom of the potential (see figure 7) the slow roll conditions (67,68) will be
violated. In this regime, the scalar field / particle will roll back and forth around the bottom
of the potential, slowing down due to the Hubble damping, and the scalar field decays into
standard model particles.
During this phase the inflationary universe transitions into a hot universe (high entropy,
radiation dominated), titled the reheating phase, and this connects inflation with the hot Big
Bang theory.
38
4.4
Evolution of Scales
This chapter delves into the evolution of scales, leaving out a lot of the mathematics, but
rather focusing on the results presented in various papers on the topic.
For a given scale, there is a need to know whether it is bigger or smaller than the horizon,
which is also commonly called the Hubble length24
Density perturbations25 are identified by their co-moving wave-number k (not to be confused with the curvature constant), which arises from the Fourier decomposition of density
perturbations. A scale is equal to the horizon when k = aH.
Only during inflation, is the Hubble length (horizon) decreasing. At the beginning of
inflation a fixed co-moving length scale k −1 may begin its evolution much smaller than
H −1
,
a
and at the end of inflation be considerably larger. The Hubble radius, k = aH is achieved
when a scale crosses the horizon during inflation, which can be related to the number of
e-foldings that occur after that time [64]. An example of the evolution of scales can be seen
in figure 8, which looks at the evolution of a scale k presently equalling the Hubble radius
k = a0 H0 .
The evolution of the Universe can be used to describe scales. An example given by Liddle
and Lyth [64] breaks evolution up into the following:
• From the time the scale k −1 equals the Hubble radius up to the end of inflation.
• From the end of inflation until it reconnects with the hot Big Bang theory, and makes
the assumption that the Universe behaves as if its matter dominated.
24
The Hubble length is defined as LH =
c
H0
' 4000M P c. The Hubble length is a distance increasing at the
rate c. At any given time distances longer than the Hubble length increase faster than c. At each epoch in the
Universe’s history there has been a different Hubble length. During inflation the Hubble length would have
remained steady. The Hubble length or radius is the threshold for superluminal recession. Objects inside the
Hubble radius are receding slower than c. This can be thought of in terms of a photon of light heading in our
direction, if it is closer than the Hubble radius it will reach us. However if that photon of light was travelling
to us from outside the Hubble radius it will not reach us. Although the radius could expand, encompass the
photon and it would reach us.
25
Density perturbations form part of cosmological perturbation theory which describes the evolution of
structure. It uses general relativity to compute the gravitational forces required to create small perturbations to
generate the structures in the Universe. Cosmological perturbation theory is only valid during periods when
the Universe is predominantly homogeneous such as during inflation.
39
• The radiation era which lasts from the end of the reheating phase to the time of matterradiation equality teq .
• teq to the present.
Figure 8: Two pictures of a co-moving scale k relative to the Hubble length. During inflation
the Hubble length is decreasing. The first graph is in terms of physical coordinates, and the
second in terms of a co-moving length.The figures demonstrate how a scale starts within the
horizon, and leaves sometime before the end of inflation, re-entering the horizon long after
the end of inflation. Image courtesy of [64].
However other evolutions are possible, such as one incorporating thermal inflation,
where extra periods of inflation occur at lower energy scales [64].
Assuming instantaneous transitions between the regimes identified then measuring quantities relative to the present co-moving scale
H0−1
,
a0
k
ak Hk
af areh aeq Hk
=
= e−N (k)
.
a0 H0
a0 H0
areh aeq a0 H0
(93)
Where k indicates quantities evaluated at k = aH during inflation, the subscripts f and
40
reh indicates values at the end of inflation and after reheating26 . The first fraction on the
right hand side represents the number of e-foldings N(k) of inflation when the scale k equals
the Hubble radius.
Some useful factors are [64],
aeq Heq
= 219Ω0 h
a0 H0
Heq = 5.25 × 106 h3 Ω20 H0
H0 = 1.75 × 10−61 hMpl with h ' 0.7
where during the slow roll approximation during inflation [65],
Hk2 '
8πVk
,
3Mpl2
where Vk is the potential evaluated at k = aH. In evaluating equation (91), there are
various uncertainties in the energy scales used, leading to a different number of e-foldings,
but these factors are not deemed to be too large. Liddle and Leach compute equation (91)27
with the following values [65]:
v
u
u 8πVk 1
k
1 ρreh 1 ρeq
N (k) = −ln
+ ln
+ ln
+ lnt
+ ln 219 Ω0 h.
a0 H0 3 ρf
4 ρreh
3Mpl2 Heq
(95)
Following the assumptions given by Liddle and Leach [65], they impose a maximum on
the number of e-foldings at which the present scale k = a0 H0 equalled the Hubble scale
during inflation. It is assumed there is no significant drop in the energy density during the
26
During the inflationary epoch the temperature of the Universe dropped by a factor of ' 100000 [38]. At
the end of this exponential expansion the Universe returns to its pre-inflationary temperature in a process titled
reheating, whereby the large potential energy of the inflaton field decays into standard model particles beginning the radiation domination phase. Further reading includes Phase transitions in the universe by M.Gleiser
[8].
27
It is important to mention that the Planck 2015 results [6] compute the number of e-foldings at the end of
inflation with
k∗
N∗ ' 67 − ln
a0 H0
!
1
V∗2
+ ln
4
MP4 l ρend
!
1 − 3ωint
ρth
+
ln
12(1 + ωint )
ρend
!
−
1
ln(gt ).
12
(94)
ρend is the energy density at the end of inflation, a0 H0 is the present horizon scale, V∗ is the potential energy
when k∗ left the Hubble radius during inflation, and gth is the number of effective degrees of freedom at the
energy scale ρt h. The results use k∗ = 0.003M P c−1 and gth = 103 . The parameter ωint characterises the
effective equation of state between the end of the inflation and the energy scale ρth .
41
last stages in inflation, so Vk = ρf , where ρf is the energy density at the end of inflation. To
maximise the number of e-foldings, the transition from inflation to reheating is considered
instantaneous so that ρf = ρreh . Looking at the current horizon scale the maximum number
of e-foldings corresponding to the horizon scale of,
max
Nhor
v
u
u 8πVk 1
1 ρeq
= ln
+ lnt
+ ln 219 Ω0 h.
4 ρreh
3Mpl2 Heq
(96)
where the subscript "hor" refers to quantities evaluated at the horizon scale can be estimated. Substituting in the relevant known values gives [65],
1 Vhor
max
Nhor
= 68.5 + ln 4 .
4 Mpl
(97)
To evaluate this, it is noted that the potential energy is bounded due to the requirement
that perturbations have an observed amplitude. By assuming all perturbations are entirely
from density perturbations whose amplitude is given by the slow roll approximation [64]
P=
8V 1
3Mpl4 (98)
where observations restrict the slow roll parameter to ≤ 0.05, and the large scale observed perturbation amplitude is P ' 2.6 × 10−9 [62] gives
1
max
Nhor
= 63.3 + ln .
4
(99)
Liddle and Leach give an upper limit of around 62 e-foldings, with the actual value expected to be around 10 less [65]. However, it is not necessary to have a definite number of
e-foldings for which the present scale k = a0 H0 equalled the Hubble scale during inflation.
Indeed, in their paper they highlight how various different models such as λφ4 has fewer uncertainties than other models, leading to a fairly definite value of 64 e-foldings. A discussion
of how the values can be obtained for equation (95) can be read about in [65].
4.5
Initial Conditions for Inflation
To set the scene for inflation a brief foray into the initial conditions of the inflationary period
must be discussed.
The CMB results place an upper bound on the vacuum energy density during the inflation epoch [55],
42
V '
MP4 l
16
' 10 GeV
4
(100)
so it is imagined that the inflationary period begins around the Planck scale [40]. There
are two reasons behind this assumption. One is that this prevents the Universe from collapsing within a Planck time assuming Ω(t) is initially bigger than 1. The second, which also
applies to the case Ω(t) < 1, protects an initially homogeneous region from invasion by the
surrounding inhomogeneous regions [64].
Slow roll inflation erases the memory of what happened in the Universe before inflation
occurred, so the Universe contains no memory of the era before it left the horizon. However,
a complete model of inflation should specify the potential and how the inflation field can find
itself slow rolling down the appropriate part of the potential, when the Universe leaves the
horizon.
One approach is to explore what happened around the Planck scale, and looking at the
conditions which are termed chaotic, whereby the Universe took on wide range of values
in different regions of the Universe [64]. The Universe is anticipated to emerge from the
Planck era with a scalar field well displaced from any minimum, with an energy around the
Planck scale. Provided a region has a suitable value, then inflation occurs.
This early era of inflation has no observable consequences therefore predicted density
perturbations could be on the order of unity, which leads to eternal inflation28 in which
quantum fluctuations in the scalar field dominate over the classical behaviour so that the
field can diffuse up the potential as well as to roll and diffuse down [42][69][64]. A higher
energy density leads to more rapid expansion, implying the Universe may be dominated by
regions moving up the potential. Therefore some parts of the Universe continue to inflate
forever.
Inflation does not need to occur around the Planck scale, just that the Universe be an
inflation region. Regardless of the nature of the theory, it is assumed that the Universe
underwent a period of inflation which starts within the horizon and ends some number
of e-folds after leaving the horizon. To explain the large scale structures of the Universe,
the nature of inflation must be slow roll, where the field is either moving away from the
maximum of a potential or coming in from large field values.
28
A hypothetical inflationary model where the inflationary phase lasts forever in some parts of the universe.
These exponentially expanding regions imply most of the volume of the universe is inflating. All eternal
inflationary models produce a hypothetically infinite multiverse. Further reading see [69], [42] and [41].
43
A brief outline of old and new inflation is presented before moving onto cosmological
perturbations.
4.5.1
Old Inflation
Models of old inflation (similar to the one proposed by Guth (1981) [40]) require a first
order phase transition mechanism (See figure 9). This can be compared to a scalar field
trapped in the local minimum with a constant energy density V (ϕi ) which is equivalent
to the cosmological constant. The Universe is able to expand exponentially according to
de-Sitter space-time.
Via tunnelling mechanisms the field can end up at its global minimum V (ϕf ) = 0 creating bubbles29 of the true vacuum which correspond to a non-inflationary universe.
Figure 9: Old inflationary models based on a first order phase transition from a local minimum to the true minimum. Here the false vacuum can allow the tunnelling of a scalar field
to the true minimum. Image courtesy of [84].
As the reader can probably guess, these old inflationary models are disfavoured for many
reasons including that associated with the properties of de-Sitter space-time. Depending on
choice of co-ordinates the Universe can be considered to be expanding, contracting or static.
In the metastable phase there is no preferred hyper-surface or time-like direction to choose a
slicing30 , as such a phase transition can occur on any hyper-surface. The result is each bubble
of the true vacuum will have different physical properties leading to an inhomogeneous
universe.
29
30
An overview of bubbles can be seen in A.Guth’s 1981 paper, whilst a basic definition can be found at [105].
More on slicing can be read about in chapter 8.2.4.2 in Primordial Cosmology by Peter and Uzan [91].
44
4.5.2
New Inflation
New inflationary models have a scalar field ϕ which exits a false vacuum by slowly rolling
towards the true one. Slow-roll leads to the density perturbations required to generate largescale structures. New inflation models require the potential to have a very flat plateau at
ϕ = 0 which is artificial. The theory of cosmological phase transitions is not required as the
inflaton is not in thermal equilibrium with other matter fields.
Figure 10: In contrast to figure 9, the scalar field is able to roll slowly towards its true
minimum of the potential. Image courtesy of [84].
These models have also been abandoned as no realization of this model has come to
fruitation [91].
45
5
Cosmological Perturbations
’A great day this has turned out to be.
I’m suicidal, me mate tries to kill me,
me gun gets nicked and we’re still in
fucking Bruges.’
Film - In Bruges
This chapter introduces the scalar and tensor perturbations that were created during
inflation and reviews their evolution and how they are numerical quantified.
5.1
Scalar and Tensor Perturbations
So far this dissertation has focused on a homogeneous and isotropic universe, which is a
good approximation to take on sufficiently large scales, and indeed measurements of the
CMB confirm how isotropic and homogeneous the Universe is [17]. However the Universe
also contains non-linear structures such as galaxies, clusters, superclusters, voids etc, and one
of the merits of inflation is that these inhomogeneities can be described.
During inflation two types of perturbations (small fluctuations) were created, scalar (density) and tensor (metric) perturbations. Both have different origins, scalar perturbations arise
from quantum fluctuations in the inflaton field before and during its evolution. Tensor perturbations come from the space-time metric within quasi de-Sitter space-time. The Planck
2013 results [5] argue that ’The quantum fluctuations in the inflaton, and in the transverse
and traceless parts of the metric, are amplified by the nearly exponential expansion, yielding
the scalar and tensor primordial power spectra respectively’.
During inflation two types of perturbations arose31 :
• Scalar perturbations: These arose from quantum fluctuations in the inflaton field before and during its evolution. Scalar perturbations due to the metric are interesting
since they couple to the density of matter and radiation and therefore determine a lot
of the inhomogeneities and anisotropies that exist.
31
There are also vector perturbations, but no vector perturbations are produced during scalar field inflation
and they also decay in an expanding Universe so we choose to ignore them here.
46
• Tensor perturbations: These come from the space-time metric, and are sometimes
called gravity waves. These are not coupled to the density and are therefore not responsible for large scale structures in the Universe, but their signature can be seen in
the CMB [30].
To understand perturbations during inflation it is taken that the Universe consists of
primarily a uniform scalar field with a uniform background metric. The scalar fields fluctuate
quantum mechanically but at any time the average fluctuation is zero, however the average
of the square of the fluctuations is not zero.
The Planck 2013 results [5] comment that without quantum fluctuations inflationary
theory would fail. They argue that initial spatial curvature (or gradients in the scalar field,
and any inhomogeneities in other fields) would decay away rapidly during quasi-exponential
expansion [5]. As such quantum fluctuations must exist, otherwise the Universe would be
too homogeneous and isotropic going against observations [5]. In addition the paper further
comments that quantum fluctuations must exist to satisfy the uncertainty relations derived
from the canonical commutation relations seen in quantum field theory [5].
This chapter reviews both scalar and tensor perturbations which lead to observable values
that the Planck papers produced results for [6][5].
5.2
Perturbations and Gauge Freedom
Assume a homogeneous universe with flat metric given by


gµν = diag 1, −a(t)2 , −a(t)2 , −a(t)2 ,
(101)
decomposing this and the field into a homogeneous background plus spatial perturbations:
gµν (t, x) = g µν (t) + δgµν (t, x)
ϕ(t, x) = ϕ(t) + δϕ(t, x)
where g µν is the Friedmann metric.
The symmetric 4 tensor δgµν (t, x) has in principle ten degrees of freedom:
• Four scalar degrees of freedom describing generalized Newtonian gravity,
• four vector degrees of freedom describing gravito-magnetism,
47
(102)
• and two tensor degrees of freedom describing gravitational waves [63].
At first order these scalar, vector and tensor sectors are uncoupled in perturbation theory
and have independent equations of motion.
Scalar metric perturbations couple with the scalar field perturbations, δϕ(t, x), vector
modes only have decaying solutions so can be neglected but tensor perturbations although
uncoupled with the field have non-decaying solutions.
There are a number of non-physical degrees of freedom in the metric which produce
unobservable equations of motion which arise from the non-unique way of defining perturbations at a given point. Lets consider the perturbed but perfectly defined quantity ρ(t, x).
A change of co-ordinates will re-map the field ρ but not change it, however the perturbations are changed δρ(t, x) = ρ(t, x) − hρ(t, x)ix (where hρ(t, x)ix is the spatial average at
time t). The perturbation is non-local since the average is performed along the hyper-surface
of simultaneity32 at time t. Therefore a change of co-ordinates changes the hyper-surfaces
of simultaneity and the new quantity δρ(t0 , x0 ) is defined by comparing with the local value
ρ(t0 , x0 ) with different physical points on a different hyper-surface of simultaneity [63].
Assuming the Universe is slightly perturbed then there exists an infinite number of
gauges (ways to define the hyper-surfaces of simultaneity whilst keeping perturbations small),
so that metric perturbations can be changed by a group of gauge transformations. However,
gauge transformations only have four degrees of freedom therefore the number of physical
degrees of freedom is reduced to six comprised of two scalars, two vectors and two tensors.
As physical observables are gauge invariant one can fix a gauge (leading to a unique slicing of space-time into hyper-surfaces of simultaneity), so that although variables are not
gauge invariant their equations of motion give the correct number of independent solutions. This prescription will be used in the following derivations for looking at scalar perturbations which requires a perturbed diagonal metric, called the longitudinal or Newtonian
gauge33 [63]:
32
A hyper-surface is the generalisation of the concept of a hyperplane. If we have a manifold M with
n dimensions, then any sub-manifold of M with n − 1 dimensions is a hyper-surface. The FLRW class of
solutions of the Einstein equations describe space-times which can be sliced into a one parameter set of spacelike hyper-surfaces which are inhomogeneous and isotropic. It is convenient to label the hyper-surfaces to be
the proper time, so a hyper-surface is then ’space at time t. Each horizontal hyperplane in this space-time
diagram is a hyper-surface of simultaneity
33
A brief derivation of the metric in the Newtonian gauge is provided in the appendix, but a very through
48


2
2
2
gµν = diag (1 + 2φ), −a (1 − 2ϕ), −a (1 − 2ϕ), −a (1 − 2ϕ).
(103)
gµν has the line element ds2 = a2 (1 + 2φ), 1(1 − 2ϕ)δij dxi dxj and a discussion of this
can be seen in the appendix. Tensor perturbations are gauge invariant whereas gravitational
waves are usually described by 2 independent components (or polarizations) part of the
traceless 3 × 3 tensor hij ,
δgij = −a(t)2 hij
(104)
where hii = 0 and ∀j, ∂i hij = 0. Therefore the tensor can be decomposed into its two
independent polarizations,
hij = h1 e1ij + h2 e2ij
(105)
where h1 and h2 are functions of space and time, and e1ij , e1ij are two orthogonal transverse
vectors of norm
1
2
each,
X
eλij eλij =
ijλ
5.2.1
1 1
+ = 1.
2 2
(106)
Basic Recalls on Quantization of a Free Scalar Field In Flat Space-Time
For a quantum harmonic oscillator with equation of motion ẍ + ω 2 x = 0, the wave function
of the fundamental state is a Gaussian,
1
2
Ψo (x) = N e 2 ωx .
(107)
This has a probability of P(x) = |Ψ0 (x)|2 to find the system in a position x is a Gaussian
of variance
s
σ=
1
.
2ω
(108)
A massless scalar field in flat space-time (i.e Minkowski metric) can easily be quantised as
each Fourier mode is analogous to a harmonic oscillator, with Lagrangian
1
L = ∂µ χ∂ µ χ.
2
This has a Hamiltonian of
description of this concept can be seen in Cosmological Perturbation Theory by Daniel Baumann [16].
49
(109)


1Z 3  2
H=
d x χ̇ + ∂i χ∂ i χ.
2
(110)
The real field χ can be Fourier transformed to the complex field, χk = χ∗−k . The Fourier
equation of motion reads
χ̈k + k 2 χk = 0,
(111)
so that the Hamiltonian becomes

"
1Z 3
d x χ̇k χ̇∗k + k 2 χk χ∗k .
H=
2
(112)
The wave function of each fundamental mode is given by
1
Ψ0 (χk ) = N e− 2 k|χk |
2
(113)
so that the probability P(χk ) = |Ψ0 (χk )|2 where each Fourier mode of wave number k
has an amplitude χk is a Gaussian of variance
s
σ=
1
.
2k
(114)
There are two issues which arise when describing perturbations via quantization, namely
non-quadratic self coupling terms and curved space-time. Within inflation the first issue does
not matter as we are not quantizing the field itself but rather its small perturbations however
the curvature of space-time cannot be neglected.
5.2.2
Definition of The Mode Function
The Fourier equation of motion (111) has solutions of the form
χk = Ak e−ikt + Bk e−ikt .
(115)
In flat space-time there is no ambiguity in the definition of time as the metric is invariant
under time translations, so that
∂
∂t
is a killing vector. Eigenfunctions with positive frequency
solutions obeying
∂
fk = −ikfk
∂t
(116)
are the only physical solutions and in equation (115) positive frequency solutions are
those given when Bk = 0. Of these, one is normalized according to the commutation
relation [x, p] = [χ, χ̇] = i. Therefore the commutation relation becomes
50
ik|Ak |2 − (−ik|Ak |2 ) = i
(117)
and up to an arbitrary phase
s
Ak =
1 −ikt
e .
2k
(118)
Thus the mode function is defined as the positive frequency solution of the classical
equation of motion which has been normalised to the commutation relation.
5.2.3
Free Field in Curved Space-Time
In curved space-time it is impossible to construct positive frequency solutions as the time
derivative is no longer a killing vector. However it is possible to define an ’instantaneous
fundamental state’ by building up creation and annihilation operators which evolve according to Bogolioubov transformations34 but it is not invariant in time.
In the context of inflation, ambiguity can be avoided by treating quantum fields as if
they were in flat space-time. Despite living in an expanding Universe space-time curvature
effects can be neglected on small scales with distances and wavelengths much smaller than
the curvature radius Rk and Hubble radius RH .
During inflation observable Fourier modes start within the Hubble radius but wavelengths grow faster than RH and a point will be reached where λ ' RH which in terms of
wave-numbers is equivalent to k ' aH. The problem of curvature can be solved by defining
the initial state of the field much before this time when k aH and the system is effectively
in flat space-time, removing ambiguities. It is taken that the system stays in its fundamental
state whilst k aH will evolve and no longer be in the vacuum state and particle creation
will occur.
5.2.4
Quantum to Semi-Classical Transition
The Universe needs to undergo a quantum to semi-classical transition. Quantum fluctuations
become classical when their wavelengths cross the horizon and the wish is to review cosmological perturbations in terms of classical fields. The transition from quantum to classical
occurs around the horizon exit.
34
A unitary transformation from a unitary representation of some canonical commutation relation into
another unitary representation.
51
The shift is quantified by P(χk ) as the system undergoes a quantum to classical transition, and at late times quantum perturbations are indistinguishable from classical stochastic
perturbations which should not be confused with decoherance.
A classical stochastic system has an equation of motion with an initial distribution of
probability in phase space. The distribution of probability at late times is equal to that at
an initial time ’transported’ by an equation of motion, which serves as a mapping. This
mapping is a solution of the Hamilton-Jacobi equation [63]:
x(t) = α(t)x(t0 ) + β(t)p(t0 )
(119)
x(t) = γ(t)x(t0 ) + δ(t)p(t0 )
(120)
and the initial phase space distribution is a normalized function of P(x, p). A subclass of
these systems leads to
β(t)
δ(t)
= lim
= 0.
t→∞ α(t)
t→∞ γ(t)
(121)
lim
Arbitrary statistical momentum is the solution of the following integral [63],
m
hxn pm i =
Z
dxxn
γ(t) 
x P(x, t).
α(t)
(122)
In curved space-time there is a class of quantum fields which obey the Hamilton-Jacobi
equations (119,120) for each mode k but also precisely equation (121). The modes which
obey equation (121) also have a corresponding Wigner function which concentrates along
a line of equation pχk =
γ(t)
χ
α(t) k
and the mode is called a squeezed state with momenta
indistinguishable from equation (122). The quantum momenta can be computed as
hχnk pm
χk i
=
Z
∗
dχk Ψ
(χk , t)χnk
∂
i∂χk
!m
Ψ(χk , t)
(123)
and in the semi-classical limit this becomes,
hχnk pm
χk icl
=
Z
dχk χnk
γ(t)
χk
α(t)
!m
|Ψ(χk , t)|2 .
(124)
In the squeezed state in the Schroedinger picture this becomes [63]
n m
hχnk pm
χk i − hχk pχk icl
−→ 0
hχnk pm
χk i
provided
β(t)
α(t)
and
δ(t)
γ(t)
(125)
are vanishingly small. In this limit the system becomes classically
stochastic.
52
Quantum inflationary perturbations can be viewed in this form, and two independent
solutions for the Euler equation can be sort, a growing mode and a decaying mode which
can both be matched up with the positive frequency solution before horizon crossing. Near
the horizon crossing both modes have comparable amplitudes (matching provides the normalization constant for each mode), but after some time the decaying mode’s amplitude
becomes vanishingly small. This is equivalent to finding
β(t)
α(t)
and
δ(t)
γ(t)
both small.
Equation (125) can be written in the Heisenberg picture [63] as
n
ˆ o
ˆ
hχk | χˆk , p†χk |χk i >> h|χk |[χˆk , p†χk ]|χk i.
(126)
The semi-classical transition occurs when the absolute value of the mean value of the
anti-commutator is much larger than that of the commutator which equals 1.
Within inflation the instantaneous number of particles operator is defined as
N̂ =
Z
d3 k â†k âk
(127)
so the condition on when a semi-classical transition occurs is equivalent to hN̂ i 1.
Therefore the condition for the semi-classical approach is to get the number of particles
created in the vacuum to be much larger than 1 which occurs for inflationary perturbations
after Hubble crossing.
To summarize quantum fluctuations can be ignored except when specifying the normalization of the mode function which is related to the positive frequency condition and the
commutation relations. However, in the late times inflationary perturbations which are
stochastic and Gaussian are considered.
5.3
Tensor Perturbations
In this subchapter and the following one conformal time η is used, rather than cosmic time
t. The two are related by a change of variable dt = adη so that the metric is conformally
invariant, ds2 = a2 (η)[dη 2 − dx2 ]. As previously discussed, tensor perturbations arise from
the space-time metric within quasi de-Sitter space-time, and are sometimes called gravity
waves.
5.3.1
General Equations
Tensor perturbations are described by two independent polarizations h1 , h2 unaffected at
first order by the scalar sector. The Lagrangian for h1 , h2 is contained within
53
Lgrav =
q
|g|
R
,
16πG
(128)
where R is the Ricci scalar.
Writing this in conformal time then
q
|g| = a4 and the part of the Lagrangian describing
tensor perturbations becomes [63]

Ltensors

1
a4  1
∂µ h1 ∂ µ h1 + ∂µ h2 ∂ µ h2 + div ,
=
16πG 4
4
(129)
where div stands for irrelevant divergent terms.
Note from the Lagrangian (129) that both h1 and h2 share the same Lagrangian and
equation of motion so after quantization and Hubble crossing they will be two equivalent
stochastic Gaussian variables. One of the modes can be ignored, so look at one mode hλ . To
quantize this involves a change of variable and a rescaled field
ahλ
y≡√
,
32πG
(130)
which has Lagrangian

1  0 a0
y − y
Ly =
2
a
!2
− ∂i y

2 
(131)
,

where the superscript 0 corresponds to derivatives with respect to conformal time.
The Hamiltonian is

1 Z 3  0 a0
H=
dx y − y
2
a
!2

2
+ ∂i y ,
(132)
and the equation of motion in Fourier space is


a0
yk00 + k 2 − yk = 0.
a
(133)
The Hamiltonian can therefore be re-written as


2
1 Z 3  0
a0 2
H=
d k yk − yk + k 2 yk .
2
a
(134)
Notice that in the small wavelength limit k >> aH equation (134) reduces down to the
flat space-time ideal


1Z 3  0 2
d k |yk | + k 2 |yk |2 
H=
2
(135)
which is to be expected, as modes within the Hubble radius do not see space-time curvature and can be quantised. Therefore the mode function is
s
yk >> aH =
54
1 ikη
e .
2k
(136)
5.3.2
Solutions During De-Sitter Stage
Exact de-Sitter sees a(t) = eHt , and translating to conformal time can be found by integrating
over dη = e−Ht dt giving η =
−1
.
aH
During de-Sitter a(t) goes from very small to very large
values which corresponds to η going from −∞ to zero. The Hubble crossing occurs when
k ' aH i.e when kη ' −1. The equation of motion becomes


2
yk00 + k 2 −  = 0,
η
(137)
which has two solutions




i
i
yk = Ak 1 − e−ikη  + Bk 1 − eikη .
kη
kη
(138)
The mode function reads
s
yk =


1
i
1 − eikη ,
2k
kη
(139)
and in the large wavelength limit k aH which is equivalent to kη −→ 0 then
ykaH = − √
i
iaH
=√
.
2k 3 η
2k 3
(140)
Therefore
√
hλk =
32πG iaH
= iH
a
2k 3
s
16πG
.
k3
(141)
As discussed earlier the modulus of the mode function is equivalent to the Gaussian
variance of the classical stochastic mode after the Hubble crossing. In the long wavelength
limit, each degree of polarization has a squared variance,
16πGH 2
h|hλk | i =
.
k3
2
(142)
During the de-Sitter stage combining results the gravitational wave tensor hij is

h|
X
ij
5.3.3
|2 i = h|h1k |2 i
X
ij
e1ij e1ij + h|h2k |2 i
X
ij

1 1
16πGHi2
e2ij e2ij = h|hλk |2 i +  =
.
2 2
k3
(143)
Long Wavelength Solution During and After Inflation
Gravitational waves which are generated during inflation can be seen through their contribution to CMB anisotropies, and their evolution after they re-enter inside the Hubble radius
55
can be computed. An initial condition required is that the amplitude must be equivalent to
the Gaussian invariance of gravitational waves on super-Hubble scales.
The evolution of super-Hubble scales is trivial since k −→ 0 so
yk00
a00
− yk = 0.
a
(144)
One solution has yk ∝ a corresponding to constant hλk . The other solution is a decaying
mode so it can be concluded that hλk is constant on super-Hubble modes.
5.3.4
Primordial Spectrum of Tensor Perturbations
Its been demonstrated that hλk is constant on super-Hubble scales and the primordial spectrum in exact de-Sitter (143) required for computing observables can be done at the end of
inflation. The power spectrum can be defined as the squared variance multiplied by k 3 [63]:
Ph ≡ k
3
2 +
*
X
hijk = 16πGH 2 ,
(145)
ij
therefore Ph is scale invariant (independent of k).
Literature cites additional ways of expressing the power spectrum
• Lesgourges uses [63]
2
Ph = (8πG)2 V,
3
(146)
• whereas Liddle and Lyth [64] use
P̃h ≡
2k 3 X
1
16GH 2
2
h|
h
|
i
=
P
=
,
ijk
h
2π 2 ij
π2
π
(147)
• and Donaldson [30] uses both
Ph (k) = 16πG
|y(k, η)|2
8πGH 2
16πG 1
,
P
(k)
=
=
.
h
2
3
a
k
a2 2k 3 η 2
(148)
• The Planck paper [6] use the following equation which will be used when quoting
observational results:



dnt
ln
2 dlnk
nt + 1
k3
k
Pt (k) = 2 |h1k |2 + |h2k |2  = At  
2π
k∗
where At is the tensor amplitude, nt is the tensor spectral index,
k
k∗
dnt
dlnk
+....
(149)
is the running of
the tensor spectral index. k∗ is the pivot value of k A number of these terms will be
met later.
56
5.4
Scalar Perturbations
Determining scalar perturbations is more complicated than tensor perturbations due to the
presence of perturbations in the scalar field driving inflation, which are coupled to the field.
Scalar perturbations are responsible for large scale structure, arising from deviations in a
homogeneous and isotropic universe.
5.4.1
General Equations
Scalar perturbations in the longitudinal (Newtonian) gauge have three degrees of freedom:
• Diagonal metric perturbations φ,
• diagonal metric perturbations ϕ
• and inflaton field perturbations δϕ.
The equations of motion dictate whether the three fields need to be treated as independent [63]
δGji = ∂i ∂ j (φ − ϕ) = 8 π G ∂i δϕ ∂ j ∂ϕ
(150)
for i 6= j. By noticing that the right hand side of equation (150) vanishes at first order
in perturbations then either φ − ϕ is either null or a quadratic function of the spatial coordinate. However, the latter solution is impossible as far from the origin φ − ϕ is arbitrarily
large and perturbations are no longer small. At first order in perturbation theory ϕ = φ,
leaving two variables φ and δϕ which evolve according to the Klein Gordon and Einstein
equations,


∂V
k2 ∂ 2V
δ ϕ̈k + 3Hδϕk +  2 +
(ϕ̄)δϕk = 4ϕ̄˙ φ̄˙ k − 2
(ϕ̇)φk
2
a
∂ϕ
∂ϕ
(151)
ϕ̇k + Hφk = 4 π G ϕ̄˙ δϕk
(152)
Taking the first equation (151) to describe propagation with the second (152) as a constraint then scalar metric perturbations are not additional independent fields, they just follow matter perturbations. Without matter perturbations, scalar metric perturbations would
vanish so only need to quantize one degree of freedom and can do this by combining Einstein equations to find a second order differential equation involving a single master variable.
57
Lesgourgues [63] highlights how the master variable is not unique, and gives the following
second order differential equation for the metric perturbation only as




ϕ̈¯
k2
ϕ̈¯
φ̈k + H − 2 ¯ φ̇k − 2H ¯ + 8πGϕ̇¯2 2 φk = 0.
a
ϕ̇
ϕ̇
(153)
This is rather complicated but work by Mukhanov [86] showed that metric and scalar
field perturbations can be combined into a gauge invariant quantity called the MukhanovSusaki variable [91] written in the Newtonian gauge as [63]
ξ = δϕ +
ϕ̄0
φ.
H
(154)
The action for a general scalar field in curved time is given by
S=−
Z
q
d4 x |g| Lg + Lϕ ,
(155)
where
Lg =
R
16πG
1
Lϕ = ∂µ ϕ∂ν ϕ − V (ϕ).
2
and
(156)
Mukhanov [86] found that in any gauge the part of the action (155) describing first order
perturbations is entirely contained within the Lagrangian [63],


z 00
1
Lξ = ξ 02 − (∂i ξ)2 + ξ 2 + div 
2
z
(157)
where div is irrelevant total divergent terms and
z≡
ϕ̄0
.
H
(158)
The Hamiltonian is


1 Z 3  02
z 00
H=
d x ξ + (∂i ξ)2 − ξ 2 + div .
2
z
5.4.2
(159)
Solution During Quasi de-Sitter Stage
By making the assumption that over a relevant range of time (a few e-folds before and after
the horizon crossing), H and ϕ̄˙ are approximately constant. z is proportional to a so that
η=
−1
aH
remains true so
z 00
z
=
a00
a
=
−2
η2
= −2(aH)2 , then first order calculations of the scalar
power spectrum can be performed.
58
In the sub-Hubble limit k aH then the Hamiltonian reduces to the flat space-time
counterpart, and the equation of motion for ξ becomes


a00
ξk + k 2 − ξk = 0
a
00
(160)
and so the mode function is
s
ξk =
5.4.3


i
1
1 − e−ikη .
2k
kη
(161)
Long Wavelength Solution During and After Inflation
At any stage of the Universe, during and post inflation, the metric perturbation φ has an
exact analytical solution in the long wavelength limit k −→ 0 [63][86]:
t
HZ
4πG H

φk = C1 (k) 1 −
a(t)dt − C2 (k) 2
a
k a


(162)
where C1 and C2 are the coefficients for the non-decaying and decaying modes. The
integration can be performed explicitly during each stage (inflation, radiation domination,
matter domination, dark energy domination), here will naturally focus on the integral during
inflation but will review other cases too.
• In the exact de-Sitter limit during inflation the non-decaying mode vanishes as φk decays like
H
.
a
As such, we need to work at second order and approximate H(t) near
some time ti as Hi + Ḣi (t − ti ). Noting H =
a(t) =
d ln a
dt
 t
Z

a(ti ) exp

H(t)dt
ti
'
 t
Z

a(ti ) exp
(Hi

+ Ḣt)dt
ti


(t2 − t2i ) 
= a(ti ) exp Hi (t − ti ) + Ḣi
2
(163)
Plugging equation (163) into equation (162) and computing the result at first order in
the small parameter
Ḣi
,
Hi2
φk = −C1 (k)
59
Ḣi
.
Hi2
(164)
During slow roll inflation the non-decaying mode is slowly varying with a dominant
contribution given by
φk = −C1 (k)
Ḣ(t)
.
H 2 (t)
(165)
1
• During radiation domination (RD): a(t) ∝ t 2 and the non-decaying mode is


1 2
2
φk = −C1 (k)1 − ×  = C1 (k)
2 3
3
(166)
so φk is constant on super-Hubble scales.
2
• During matter domination: a(t) ∝ t 3 and the non-decaying mode is

φk = −C1 (k) 1 −
2 3 3
= C1 (k)
×
3 5
5
(167)
so again φk is constant.
5.4.4
Primordial Spectrum of Scalar Perturbations
The power spectrum during radiation domination times the factor
2

PφRD ≡ k 3 h|φk |2 iM D

9
10
is
2
9Hi4
3 Hi2 
3 Hi2 4πG ¯ 
√
= k3
h|φk |2 ii = k 3 
˙ϕi =
5 Ḣi
5 Ḣi 2k 3
50 ˙¯ϕ2i
(168)
and does not depend on k, it is scale invariant. This can be re-expressed in terms of the
scalar potential during inflation,
PφRD
4
9
8πG
=
50
Vi4
.
(∂V /∂ϕ)2i
(169)
Using the definition of the slow roll parameter (83) and call this i to show equation
(169) is computed with constant values Hi and ϕ̄˙ i ,
PφRD
3
3
=
8πG
100
Vi
.
i
(170)
The ratio of tensor spectrum to the scalar spectrum is
Ph
200
i .
=
RD
Pφ
9
Spectrum of Curvature Perturbations
60
(171)
Some of the literature [63] prefers to parametrize scalar perturbations not with the scalar
metric perturbations, φ (called the Bardeen potential or generalised gravitational potential),
but with the curvature perturbation R which relates the perturbation of the spatial curvature
radius R(3) on co-moving hyper-surfaces. During the radiation era φ is constant and the
literature states that R = 23 φRD [63]. However during the matter dominated era, R = 52 φM D
[63]. As such
2
5
PR =
3
PφRD
2
1
=
8πG
12
Vi
2πGHi2
=
,
i
i
(172)
which gives a tensor to curvature ratio of
Ph
= 8i .
PR
(173)
However Liddle and Lyth [64] use alternative definitions for the power spectra:
P̃R ≡
1
PR
2π 2
P̃h ≡
2
Ph .
2π 2
(174)
and taking the ratio, r,
r≡
P̃R
= 16i .
P̃h
(175)
r is a very important ratio as it is commonly used when defining the tensor-to-scalar ratio
and is used when comparing inflationary models with observations.
The Planck data [6] use the following equation which will be used when quoting observational results:
PR (k) =
k3
|Rk |2
2
2π


= As 
dns
ln
2 dlnk
ns −1+ 1
k
k∗


2
k
)+ 61 d ns2
k∗
dlnk
ln
k
k∗

+...
(176)
,
where As is the scalar amplitude, ns is the scalar spectral index, k∗ is a pivot value
the running of the scalar spectral index and
d2 n
s
dlnk2
dns
dlnk
is
is the running of the running of the scalar
spectral index. A number of these terms will be met later.
5.5
Computing Smooth Spectra Using Slow Roll Expansion
Looking over this chapter, tensor perturbations have been examined in the exact de-Sitter
stage with a constant H and scalar perturbations assuming a quasi de-Sitter stage with con˙ These assumptions are centered around scalar and tensor spectra only being
stant H and ϕ̄.
61
sensitive to a small number of e-folds and the slow roll conditions (which see Ḣ H 2 )
which imply variations in background quantities being very small.
For k aH, the mode function ξk (scalars) and yk (tensors) are oscillatory with am1
plitude (2k) 2 which is independent of background dynamics, as variation in H and Ḣ does
not affect the mode amplitude. To contrast in the k << aH regime, the mode function can
be matched to the analytical long wavelength approximation which takes into account background variation until the end of inflation and beyond. This matching can be seen for φk
using the solution of equation (162). For curvature perturbations, then Rk is time invariant
once k aH and for tensor perturbations the time invariance of hλk for k aH was used.
To demonstrate the number of e-folds that occur during slow-roll expansion note that the
primordial spectra depends on a small time range when the modulus of (ξk , yk ) does not have
1
the amplitude (2k) 2 and departs from equation (162), with Ṙk = 0 and ḣλk = 0. However
these approximations have yet to be justified. These assumptions occur when k and aH have
the same order of magnitude between k = 10aH and k = 0.1aH [63]. During this interval
the scale factor increases by a factor of 100,
∆N = ∆ln a = ln 100 ' 4.6
(177)
therefore the results are so far valid provided negligible variation of H over about 5 efolds.
To take this further but avoid resolution of the exact mode function equations (133,160)
using the slow roll expansion scheme it is possible to Taylor expand one of the background
functions (e.g H(t), H(a), H(N ), ϕ̄, V (ϕ̄)) around a pivot value (denote t∗ , a∗ , N∗ , ϕ̄∗ ). Assuming higher order derivatives vanish, then the mode function can be evaluated at a given
order. The approximations work best if the pivot value is chosen to occur at the same time, t∗
a typical observable wavelength k∗ crosses the Hubble radius during inflation, so k∗ = a∗ H∗
The Wentzel-Kramers-Brillouin (WKB) approximation is a useful method in calculating
inflationary power spectra at a given order in slow roll. For example the first order WKB
2
− 21
solution of ẍ + ω(t) x is x ∝ ω(t)
exp ± iω(t)t and this approximation is useful when
the time variation of ω(t) over one period of oscillation is negligible. However higher order
solutions include higher orders in the time derivatives. When moving to higher orders in the
slow roll approximation a much simpler method for obtaining results is preferred.
Noting first order solutions of Ph and Pφ the scalar and tensor tilts respectively which
are dependent on second order solutions can be introduced:
62
d ln Ph nt ≡
d ln k k=k∗
(178)
d ln Pφ .
d ln k k=k∗
ns − 1 ≡
(179)
The convention is that the scale invariant scalar spectrum corresponds to ns = 1 whilst
the scale invariant tensor spectrum corresponds to nt = 0. At second order the tilts are also
scale invariant and give way to exact power laws [63],

Ph (k) = Ph (k∗ )

nt
k
k∗
ns −1
k
Pφ (k) = Pφ (k∗ ) 
k∗
5.5.1
(180)
.
(181)
Simple Method for Computing The Tilts
The scale dependence of the spectra follows from the time dependence of the Hubble parameter and is quantified by the spectral indices (178) and (179).
Taking the scalar tilt this can be split into two factors:
dlnPh
dlnPh
dN
=
×
dlnk
dN
dlnk
(182)
By noting that the derivative with respect to the e-folds [15] is given by
dlnH dln
dlnPh
=2
−
.
dN
dN
dN
(183)
where for just this small chapter an alternative definition of given by Baumann [15] is
used, = − dlnH
. The first term is then just −2 and the second term has been evaluated
dN
using the result found in Appendix D of Baumann’s notes [15]:
d ln = 2( − η), where
dN
η=−
d ln H,φ
.
dN
(184)
The second term in (182) can be evaluated by recalling the horizon condition k = a∗ H∗
which can be written as
(185)
ln k = N + ln H.
From which it can be seen
dN
d ln k −1
d ln H
=
= 1+
' 1 + .
d ln k
dN
dN
63
(186)
To first order in the Hubble slow roll parameters then
ns − 1 = 2n − 4.
(187)
To calculate the tensor tilt note that [45]
dlnH
dlnPφ
= nt = 2
= −2
dlnk
dk
(188)
nt = −2.
(189)
so
The alternative method to deriving the spectra can be seen in the appendix.
5.5.2
Slow Roll Results
In the slow roll approximation
' i ,
η ' ηi − i
(190)
hence the scalar spectral index is
ns − 1 = 2ηi − 6i .
(191)
nt = −2ηi .
(192)
The tensor spectral index is
5.6
Planck Results
The Planck 2013 data [5] give the following results:
• The primordial power spectrum was modelled using the power law PR (k) = As ( kk∗ )ns −1
sees As = 2.20 × 10−9 and ns = 0.9603 for a pivot scale k∗ = 0.05M P c−1 . (ns =
0.9655 ± 0.0062 in the latest Planck 2015 results [6]). Please see figure 11.
This value automatically rules out the Harrison-Zeldovich (HZ) ns = 1 model (not
discussed here) at 5σ [5], and the Planck 2013 results [5] further comment that extensions to
this model are also ruled out.
Although this dissertation does not discuss the running of the spectral index, it is worthwhile to note that the Planck data does not prefer ’a generalization of a simple power law
s
spectrum to include a running of the spectral index ( ddn
= −0.0134 ± 0.0090)’ [5].
ln k
64
Figure 11: Planck 2015 contour confidence results for the spectral index and the running of
the spectral index at 68% and 95% confidence level with Planck 2013 results. The thin black
strip represents the predictions for single field inflation models with 50 < N∗ < 60. Image
courtesy of [6].
• Inflation models that exhibit tensor fluctuations have a a 95 % confidence level bound
on the tensor-to-scalar ratio of r0.002 < 0.12 using Planck+WP data. From this a bound
on r emits an upper limit on the inflation energy scale, 1.9 × 1016 GeV and the Hubble
parameter H∗ < 3.7 × 10−5 MP l at a 95% confidence level [5]. Planck 2015 data results
[6] give a slightly different value of H∗ < 3.6 × 10−5 MP l in conjunction with the
Planck TT + low P data sets. See figure 12.
65
Figure 12: Planck 2015 confidence contours for ns and r at 68% and 95% confidence level.
Image courtesy of [6].
66
6
Inflationary Models
’In the face of overwhelming odds, I’m
left with only one option, I’m gonna
have to science the sh*t out of this’
Novel - The Martian
This chapter looks at the implications of the Planck data [6, 4] on cosmological inflation
by reviewing a number of simple models that featured in the Planck inflationary papers [5, 6]
which are increasingly favoured due to the absence of the more dangerous non-Gaussian
perturbations [69]. In particular emphasis is place on the Starobinksy model which was first
proposed in 1980 [109] before inflation became a topic and this literature review chooses
to look at this model in depth in light of the results 2013 and 2015 Planck results [6, 5]
and in terms of f(R) gravity, a modified gravity theory which generalises Einstein’s General
Relativity in the following chapter.
6.1
Inflationary Conditions
In building a simple slow roll inflation model a number of features common to inflationary
models are generally met, and the following have been confirmed by the Planck results [4, 6]:
• The Universe must be flat, and most models have Ωtotal = 1 ± 10−4 . Planck results
have calculated that
Ω = 1.0005 ± 0.0066
(193)
at 95% confidence [69, 4].
• Perturbations of the metric produced during inflation are adiabatic.
• Perturbations are generally Gaussian. The non-Gaussian parameter was found to be
zero [4].
• In the slow roll regime inflationary perturbations have (η, ) 1 and they have a flat
spectrum with ns ' 1, Planck results have so far confirmed this with
ns = 0.9629 ± 0.0057
at 68% confidence [69, 4].
67
(194)
• The spectrum of inflationary perturbation is slightly non flat and deviations from
ns ' 1 is one of the distinguishing features of inflation.
• Although the metric perturbations could be scalar, tensor or vector inflationary theories tend to involve scalar perturbations. There can be tensor perturbations with
nearly flat spectrum but inflation does not produce vector perturbations.
• Inflationary perturbations produce specific peaks in the CMB spectrum. See [31, 87]
for a thorough explanation.
• A number of models featured in both the 2013 and 2015 Planck papers [5][6] will be
discussed in this chapter in the neighbourhood of a pivot scale k∗ .
The 2013 Planck paper [5] suggests that the strongest constraint on inflation comes
from the amplitude of the primordial power spectrum. The paper comments how it is
a free parameter in many models and that for successful structure formation R ' 10−5
or As ' 10−10 .
Although many of these features can be violated it tends to lead to complicated models
[69]. Planck results [5] have shown that cosmological perturbations in the CMB are nearly
Gaussian, and adiabatic. In addition assuming perturbations can be ascribed to single field
models, then the Planck findings put restrictions on associated inflationary parameters [80].
Briefly a number of models discussed in the Planck results are looked at here [6, 5], but
a deep foray into R2 inflation is presented due to favourable Planck results, in terms of f (R)
theories in the next chapter [5, 6].
6.2
Chaotic Inflation
Chaotic inflation results in many distinct universes which formed from different regions of a
mother universe, parts of which inflate and move away from the mother universe. Our Universe would then have grown out of quantum fluctuations which grew out of a pre-existing
region of a much larger space-time. Other universe’s could be borne from the mother universe or regions within our own Universe. A universe formed by this process would have its
own laws of physics could be completely different from ours.
The appeal of chaotic inflation is the notion of there being a multitude of universes
catering to all possible values of the fundamental parameters, removing the need to ’fine
68
tune’ the parameters to understand what makes our Universe habitable. Indeed it would
then imply our Universe is one of a few (or many) biological universes amongst the infinite
number of universes.
6.2.1
The Model
The simple model presented here focuses on the efforts of Linde and his avocation of chaotic
inflation [69].
Consider a scalar field ϕ having a mass m, and potential energy density V (ϕ) =
m2 2
ϕ,
2
with a minimum at ϕ = 0, hence L = 21 ∂µ ϕ∂ µ ϕ − 12 m2 ϕ2 . It is expected that the scalar field
should oscillate around this minimum, and is the case where the universe doesn’t expand.
The equation of motion is
ϕ̈ = m2 ϕ.
However, due to expansion with Hubble constant
(195)
ȧ
a
an additional term appears and the
equation of motion, equivalent to equation (71), becomes
ϕ̈ + 3H ϕ̇ = −m2 ϕ,
(196)
where the second term has already been identified as a friction term.
The Einstein equation is

H2 +

k
1
= ϕ̇2 + m2 ϕ2 
2
a
6
(197)
where k corresponds to the curvature of the universe, k = 1, 0, −1, and MP−2l = 8πG = 1.
The way chaotic inflation evolves depends on the value of ϕ.
6.2.2
Large ϕ
According to equation (197) a large value of ϕ implies a large value of the Hubble parameter,
H. Therefore the friction term 3H ϕ̇ in equation (196), is also very large, and can picture this
as a ball rolling slowly through a very viscous liquid down a hill, and the energy density of
the scalar field remains almost constant. As inflation is thought to have been rapid, coupling
this with the slow motion of ϕ, the beginning of this regime sees ϕ̈ 3H ϕ̇, H 2 ϕ̇2 m2 ϕ2 .
The equations can therefore be re-written as
69
k
a2
and
ȧ
a
H=
(198)
s
ϕ̇ = −m
2
3
(199)
where the first equation (198) can be integrated to show that the Universe grows as
eHt , and ends when ϕ is much smaller than Mpl = 1. Solving these equations, it has been
demonstrated that after a long stage of inflation the Universe initially at ϕ 1 grows as [74]
ϕ2
(200)
a = a0 e 4 .
This report looks at one particular chaotic inflation potential energy density, but sufficiently flat potential V (ϕ), of any form of polynomial can be applied [72]. Linde goes
further to highlight how chaotic inflation can proceed with or without spontaneous inflation, how all possible initial conditions can be explored without requiring the Universe to
be in a state of thermal equilibrium, but with ϕ at the minimum of its effective potential.
To explore the initial conditions consider a closed universe of size l ' 1 (Planck units)
with a Planck density of ρ ' 1. In this classical universe the sum of kinetic energy density,
gradient energy density and potential energy density is of the order of unity:
2
1 2 1
ϕ̇ +
∂i ϕ
2
2
(201)
+ V (ϕ) ' 1.
There are no a priori constraints on the initial value of ϕ, bar the condition described in
equation (201) and the theory must be invariant under a shift of symmetry.
One condition is placed on the amplitude of the field if the potential is not constant and
grows to become greater than the Planck density at ϕ > ϕP l where V (ϕP l ) = 1. This regime
gives ϕ ≤ ϕP l , but ϕ does not need to be much smaller so in a theory can expect ϕ ' ϕP l .
As long as aforementioned initial conditions are met then when
2
1 2 1
ϕ̇ +
∂i ϕ
2
2
(202)
≤ V (ϕ)
inflation begins and continues within the Planck time provided
1 2
ϕ̇
2
and
1
2
2
∂i ϕ
be-
comes much smaller than V (ϕ). Therefore chaotic inflation prefers initial conditions with
V (ϕ) ' 1 [74][73].
The advantage of this simple model is that inflation may begin immediately after the
Universe’s creation with the largest possible energy density of O(1), the smallest possible
size, the Planck length, smallest possible mass m = O(1) and smallest possible entropy of
70
O(1). With these initial conditions then problems with the hot Big Bang model such as the
horizon and flatness problems amongst others are solved [74].
Various other scenarios such as models with several non-interacting scalars, models with
several interacting scalars and low scale inflation (V (ϕ) 1) are also examined in [69] and
concludes that inflation can occur in certain regimes.
6.2.3 Planck Results
According to Kehagias et al [56] Chaotic models with ϕn with n ≥ 2 are disfavoured, and
the most simple chaotic models with m2 ϕ2 are ruled out with a 95% confidence level.
Planck results [69][6] disfavour the chaotic model of inflation, however as ever in the
scientific community, theories are increasingly adapted rather than dropped to try and fit
with later observations. As such a brief outline of chaotic inflation was included here as it is
still promoted by cosmologists such as Linde [69].
6.3
Hybrid Inflation
Hybrid inflation could be seen as an extension of chaotic inflation. It has been mentioned
that inflation usually ends by a slow rolling of the inflation field which becomes faster and
faster or a first order phase transition. However hybrid inflation ends with the rapid rolling
of a scalar field σ, sometimes called the ’waterfall’ effect, triggered by another scalar field
φ. As such, the scalar field starts off in slow roll, along a relatively flat potential, but due
to another field the scalar field rapidly rolls down a sharp inclined potential. This model
features the same potential seen in chaotic inflation, V (φ) =
symmetry breaking potential of the form V (σ) =
1
4λ
m 2 φ2
and
2 2
M 2 − λσ
an added spontaneous
. In this theory the latter
stages of inflation are supported by the non-inflationary potential V (σ). In this particular
flavour of Hybrid inflation the following potential is used [75]
1
M 2 − λσ 2
4λ
V (σ, φ) =
2
+
m2 2 g 2 2 2
φ + φσ ,
2
2
(203)
where the field σ is considered to be the Higgs field, which remains as a physical degree
of freedom in a theory with spontaneous symmetry breaking, and only takes on positive
values. λ, g are the effective coupling constants.
The effective mass squared for the field σ is −M 2 + g 2 φ2 leading to two scenarios:
71
• For φ > φc =
M
g
the minimum of the effective potential V (σ, φ) is at σ = 0. Linde
notes that the curvature of the effective potential in the σ-direction is much greater
than in the φ-direction, implying that at the beginning of inflation the field σ rolled
down to σ = 0, in contrast to φ which could remain larger for a longer period of time
[75]. This model uses σ = 0 with large φ.
M
g
• For φ < φc =
2
then a phase transition with symmetry breaking occurs. If m2 φ2c =
4
m2 Mg m Mλ the Hubble constant at the point of the phase transition is
H2 =
By assuming M 2 λm2
g2
2πM 4
.
3λMP2 l
(204)
and m2 H 2 then
s
M 2 mMP l
3λ
.
2π
(205)
For φ > φc inflation occurs, even for m2 > H 2 . Towards the end of the inflationary
epoch inflation is driven not by the inflation field energy density φ but rather by the
vacuum energy density V (0, 0) =
6.3.1
M2
.
4λ
Behaviour of the Fields φ and σ
This chapter reviews the behaviour of the fields φ and σ at time ∆t.
s
∆t = H −1 =
3λ MP l
,
2π M 2
(206)
which is after time tc when φ = φc . As the equation of motion for the field φ is given by
3H φ̇ = m2 φ,
(207)
then during the time interval ∆t = H −1 the field φ decreases from φc by the amount
∆φ =
m2 φc
λm2 MP2 l
=
.
3H 2
2πgM 3
(208)
The mass squared value for the field σ is given by
M 2 (φ) =
λm2 MP2 l
.
πM 3
(209)
Here M 2 (φ) is much greater than H 2 for M 3 λmMP2 l . In a time ∆t = H −1 then
the field σ rolls to its minimum at σ(φ) =
M√(φ)
,
λ
72
and rapidly oscillates around this value
dissipating energy to the expanding universe. However the field is unable to dampen due to
the effective potential V (φ, σ) at σ(φ) has a non vanishing partial derivative
∂V
gφM 2 (φ)
= m2 φ +
,
∂φ
λ
(210)
which forces the field φ to rapidly roll to the bottom of its effective potential within a
√
time much smaller than H −1 provided M 3 λgmMP2 l . Consequently inflation ends in
this theory once the field φ reaches φc =
6.3.2
M
.
g
Generalizations
In taking the potential seen in equation (203) it can be modified so that
with
φ4
λφ
4
m 2 φ2
2
is replaced
. This would lead to two disconnected stages of inflation. The first stage would
MP l
3
occur for large φ (akin to chaotic inflation) and end at φ <
field rolls down its potential and oscillates until φ '
oscillations '
q
M2
.
λφ M P l
for M 2 << λφ MP2 l . The
At this point the frequency of
λφ φ becomes smaller than the Hubble constant and switches to the second
stage of inflation. This stage of inflation ends when φc =
M
.
g
These two stages of inflation
produce a density perturbation spectrum with a non trivial structure.
6.3.3 Planck Results
This model was reviewed in light of the 2013 Planck results [5] which disfavour hybrid
models which feature a potential of the form V (φ) =
m2 φ2
2
due to the high tensor-to-scalar
ratio. Models which feature V (φ) ' λ4 are also disfavoured due to predicting ns > 1 [5].
However not all models that encompass hybrid inflation have been ruled out. Models
which predict ns < 1 include potentials of the form
 
φ
V (φ) = αh λln 
µ
(211)
where µ >> 13MP l . Here αh > 0 is a dimensionless parameter. This potential predicts
h)
ns − 1 ' − (1+3α
N∗ and r '
2
8αh
.
N∗
However this model is disfavoured when it has the
following variables: αh << 1, and N∗ ' 50, giving ns ' 0.98 are disfavoured by the
Planck+WP+BAO data results [5]. The results from Planck 2013 mention that ’permissive
reheating prior allowing N∗ < 50 or a non-negligible αh give models that are consistent with
the Planck data’ [5].
73
6.4
Hill-Top Models
Boubekeur and Lyth [23] look at inflation near the maximum of the potential. Their avocation of hill-top inflation originates from the following beliefs:
• The flatness problem can be solved provided inflation occurs near the (local) maximum
of the potential, hence the term ’hill-top’, and forms part of the theories termed ’new
inflation’.
• Slow roll conditions are more easily satisfied. While the first slow roll condition highlights how inflations starts at a local maximum, the second slow roll condition can
be relaxed to |η| < 6 [23]. This leads to fast roll inflation [59], and alleviates the
η-problem.
• From an effective field theory standpoint, hill-top model’s potentials are favoured due
to the variation in the inflation field during inflation ≤ MP l . Similar potentials are
seen in particle physics models which see broken symmetries.
• Hill-top inflation incorporates eternal inflation, hence the discussion on initial conditions becomes irrelevant.
6.4.1
Slow Roll Inflation
Boubekeur and Lyth [23] use the following slow roll parameters (flatness conditions):
2

V 00 (φ) M 2 V 0 (φ) 
<< 1.
<< 1, |η| ≡ MP2 l ≡ Pl 
2
V (φ)
V (φ) (212)
Using the slow roll approximation 3H φ̇ ' −V 0 (φ) and the critical density condition
ρ = 3H 2 MP2 l with the slow roll parameters it follows that 3H 2 MP2 l ' V (φ).
Slow roll inflation ends when either one of the flatness conditions is violated or the
potential is destabilised by some waterfall field (c.f hybrid inflation).
The number of e-foldings from horizon exit (when a scale leaves leaves the horizon) to
the end of inflation φend is given by
N (k) =
Zφ
MP−1l
dφ
q
φend
74
2(φ)
,
(213)
where k is the wave number associated to φ, and d N ' −d ln k. As cosmological scales
leave the horizon during about 10 e-folds, beginning with the exit of the whole universe
which corresponds to k = H0 , then N = N (H0 ). For post-inflationary cosmology


1016 GeV 
N ' 60 − ln
1
V4
(214)
The differential form of equation (213) is
dN = MP−1l =
dφ
,
2
(215)
where is exponentially small.
At the time of the Horizon exit, the vacuum fluctuations of the field is converted to
a classical perturbation, a Gaussian with spectrum Pφ =
H
2π
2
. However after the hori-
zon exit this corresponds to a position dependent shift back and forth along the inflation
trajectory which corresponds to a time independent curvature perturbation
PR (k) =
1
24π 2 MP4 l
V
.
(216)
With a spectral index of the perturbation
ns − 1 =
d ln PR
= 2η − 6.
d ln k
(217)
If there is tensor perturbations their spectrum is some fraction r of PR , given by
(218)
r = 16.
6.4.2
The Model
As hill-top inflation occurs need the top of the potential then the potential will have the
form [59],
1
λφp
V = V0 ± m2 φ2 − p−4 + ...
2
Mpl


1 φ2
φp
= V0 1 + η0 2  − λ p−4 .
2 MP l
MP l
with V ' V0 , η0 =
±m2 MP2 l
V0
(219)
and the dots indicating higher order terms in the power series
expansion and considered negligible during inflation35 [59] and p is just the power to which
35
We only need to keep the first two terms as if p is an integer then we would like φ << MP l after the
largest cosmological scale leaves the horizon (around N = 60).
75
the field is raised. φ = 0 as an origin is chosen purely for convenience. The tacyonic mass m
is characterised by η0 < 0, which is the value of η at the maximum.
In certain hill-top models there is a bound on the r term where the slope parameter (φ)
increases monotonically. Therefore equation (213) tells us
2

1 φend 
,
2 < 2 
N MP l
(220)
for when a cosmological scale leaves the horizon.
Using then equation (218)

2 
2
60
φend 
r < 000.2  
.
N
MP l
(221)
Using this bound for r then the height of the hill is

V
1
4
15
< 7.0 × 10
GeV 
1 
2
1
2
60   φend 
.
N
MP l
(222)
6.4.3 Planck Results
Reviewing the Planck 2013 results [5], p = 2 for large field inflationary models and this
predicts ns − 1 ' −
4MP2 l
µ2
+
3r
8
and r '
32φ2∗ MP2 l
.
µ4
These values are agreed upon by the
Planck+WP+BAO joint work with a 95% confidence level for super-Planckian values of µ,
µ > 9MP l .
(p−1)
Hill-top models with P ≥ 3 predict ns −1 ' − N2 (p−2)
when r ' 0. However the hill-top
potential for p = 3 lies outside the 95 % confidence level by the Planck+WP+BAO joint
work.
P = 4 sees tension with Planck+WP+BAO but is fine within the 95 % confidence level
for N∗ ≥ 50.
6.5
Natural Inflation
The penultimate theory looked at is natural inflation, which employs a pseudo-NambuGoldstone Boson (PNGB) to serve as the field for inflation, the inflaton. Freese [37] recommends this model due to its three main attractors:
• No ’fine-tuning’ of the inflation potential. In natural inflation, a PNGB gives a flat
potential without any fine-tuning of parameters.
76
• Density fluctuation spectrum can have extra power on large scales. Primordial density
fluctuations generated by quantum fluctuations in the inflation field is a power law
P(k) ∝ k n .
• Particle theory indicates a PNGB type particle is required for inflation.
6.5.1
The Model
In ’rolling’ models of inflation, as the inflaton ’rolls down’ the potential that dominates
the energy density of the universe, the universe is simultaneously expanding. These two
constraints coupled with sufficient inflation required to solve initial conditions and the amplitude of density fluctuations in agreement with observations indicate that the potential
required must be very flat [2].
The potential for the inflaton is

V (φ) = Λ4 1 ± cos

φ 
.
f
(223)
The height and width of the potential are given by 2Λ4 and πf respectively. Λ here is the
approximately MGU T ' 1016 GeV , and f ' MP l .
The ratio of the height of the potential to the width to the fourth power can be quantified
as
χ=
6.5.2
∆V
height
=
≤ 10−6 .
4
(∆φ)
width
(224)
Spectrum of Density Perturbations
Density perturbations due to quantum fluctuations in the inflation field is given by P(k) =
|δk |2 ' k n where n ' 1 −
MP2 l
.
8πf 2
Although many models give n = 1, the scale invariant
spectrum for natural inflation predicts 0.6 ≤ n ≤ 1. This lower limit arises because successful
reheating in the model predicts r ≥ 0.3MP l . Hence a tilted spectrum with n < 1.
It is worth noting that natural inflation provides negligible gravitational modes. The
same mechanism for density perturbations also provides gravity wave modes.
6.6
R2 inflation
R2 inflation, named for the Ricci curvature term which appears in the model, is also known
as the Starobinksly model of cosmic inflation. This model takes into account higher curva77
ture corrections to the Einstein-Hilbert action of gravity.
Figure 13: Data from the Planck 2015 results [6] showing joint 68% and 95% confidence
levels for ns and r0.002 compared with theoretical predictions. Image courtesy of [6].
R2 inflation is one of the models discussed in the Planck results and is in perfect agreement
with the data gathered [5, 6], which can be seen in figure 13. As a result, R2 inflation is
seeing renewed interest. The model’s agreement with Planck data arises due to featuring a
1
N
suppression (N is the number of e-folds from the beginning to end of inflation) of r with
respect to the prediction for the scalar spectral index ns .
There are a number of models which are also in agreement with the Planck data such as
Higgs inflation [18] and universal attractor models [54][53] which have the same predictions
to leading order as Starobinksy theory. In a paper by Kehagias, Dizgah and Riotto [56] they
argue that the similarity arises because they they are at the heart, all Starobinksy inflationary
models. Some people are in agreement with this idea [19] whilst others disagree [52], and
indeed the Planck results treat the Starobinksy and Higgs models are separate. Further discussion on this debate can be seen in [56], but this report focuses on the starobinksy model
without reference to other similar models.
Takeiko, Satoshi et al [11] argue that an advantage of Starobinksy inflation over other
models is that it does not require the introduction of an inflation field, the inflaton degree of
freedom arises from a higher order gravitation term.
78
6.6.1
The Model
The starobinksy model [109] is described by the Lagrangian (ignoring the additional matter
lagrangian term),


1
1Z 4 √ 
S=
d x −g R +
R2 
2
6M 2
(225)
which sees the propagation of a spin 2 particle (graviton) and a scalar degree of freedom
and m 1. In the infra-red limit, R M 2 , the action reduces to general relativity, however
when R ' M 2 the second term in equation (227) dominates [11]. Takeiko, Satoshi et al [11]
note that if one interprets M 2 as the expansion parameter in equation (225), then higher
order terms such as R3 and R4 can appear (not including terms arising from Ricci tensors,
Riemann tensors and derivatives which are neglected due to the introduction of ghosts from
negative powers of M 2 ). Equation (225) can therefore be re-written with this in mind in a
manner seen in [11].
Takeiko, Satoshi et al [11] go on to say that the introduction of higher orders in O(R3 )
spoils the successes of this successful inflationary model (according to Planck [5, 6]), by
modifying the potential which can be seen in equation (228).
Alexandre, Houston and Mavromatos note [7] that the scalar curvature-squared term
seen in equation (225) can be viewed on the microscopic level as one loop level quantum
fluctuations of conformal matter fields of of various spins which have been integrated out in
a curved background space-time [11, 21]. They comment that Starobinsky inflation is not
driven by rolling scalar fields and is conformally equivalent to an Einstein-gravity coupled
scalar field with a potential that drives inflation which can be reviewed in their paper [11].
By making a conformal transformation to transform to the Einstein frame then a scalar
field version of the Starobinksky model emerges which manifests an extra degree of scalar
freedom,
2 
√
R̃
1 µν
3 2
− 2κ
ϕ 

3
S = d x −g̃
− g̃ ∂µ ϕ∂ν ϕ − M 1 − e
.
2
2κ
2
4
4
√


(226)
√ 2κ
˜ +
where MP l = 1. Here g̃ = Ω2 gµν = F gµν 
= e− 3 gµν , 
κ2 = 8πG and R̃ = ΩR2 − 6ω
˜ = √1 ∂µ √−g̃g µν ∂ν ω . A derivation for the action in
6g̃ µν ∂µ ω∂ν ω where ω = lnΩ and −g̃
the Einstein frame can be seen in the appendix. During inflation for large values of ϕ then
the dynamics are dominated by the vacuum energy
79
3
V = M 2.
4
(227)
Using this theory (226) then for the scalar tilt and tensor-to-scalar ratio [5]:
ns − 1 '
−8(4N∗ + 9)
4(N∗ + 3)
192
.
(4N∗ + 3)2
r'
It is worthwhile to note that the extra
1
N
(228)
(229)
in the tensor-to-scalar ratio could be indicative
of gravitational waves. The normalization of CMB anisotropies gives m ' 10−5 [56]. These
results fit with the Planck data [5]. Planck data uses ns = 0.963 for N∗ = 55 [56].
In Kehagias, Dizgah and Riotto’s paper [56] they write these values as
ns − 1 ' −
r'
where r has an additional
1
N
2
N
12
.
N2
(230)
(231)
suppression indicative of gravitational waves. The Planck
results also constrain m to m ' 10−5 .
6.7
Planck Results
The Planck 2013 results [5] constraint on r places an upper bound on the energy scale of
inflation
V∗ =
3π 2 As
r∗
rMP4 l = (1.94 × 1016 GeV )4
,
2
0.12
at 95% confidence level where As is the scalar amplitude and approximately As u
where ν is given by equation (87)
80
(232)
V
,
24π 2 MP4 l ν
7
f(R) Gravity
’How hard can it be?’
Jeremy Clarkson
This chapter introdues the theory of f(R) gravity and reviews it’s dynamics in relation to
inflation. The perturbations which arise from this model are covered and also reviewed in
the Einstein frame.
7.1
f(R) Gravity Action
The Einstein-Hilbert action yields the Einstein equations, and f (R) gravity generalises Einstein’s general relativity, by relaxing the hypothesis that the Einstein-Hilbert action for a
gravitational field is linear in the Ricci curvature scalar R. Initially the motivation to extend
the Einstein-Hilbert action was borne from what would happen if the fundamental action
was different, to compare and contrast alternate theories of Einstein’s gravity. However
it seems higher order corrections are needed to understand what happens when quantum
corrections become important as they appear to at scales close to the Planck length. Stelle
[112] was the first to show that in constructing a re-normalizable gravity action by including
terms quadratic in the Riemann curvature tensor, then such terms appear as counter terms
in re-normalisable theories involving scalar fields coupled to the curvature tensor suggesting the Einstein-Hilbert action is the only effective action produced by the vacuum [115]
[103]. f (R) gravity is a family of theories defined by a different function of the Ricci scalar
- this dissertation makes no comment on the form of the function - however a wide range of
phenomena can be produced from various functions.
One takes the Einstein-Hilbert action and generalises it to
S=
Z
d4 x
Z
1 √
1 √
−gR
−→
S
=
d4 x 2 −gf (R).
2
2κ
2κ
(233)
This approach is designed to appease problems within the standard cosmological model
and the lack of complete quantum gravity theory, considered part of the Extended Theories
of Gravity [24]. These theories incorporate higher order curvature invariants and minimally
or non-minimally coupled scalar fields into dynamics related to effective theories of some
fundamental theory. This new approach to gravitational interactions arises from the belief
Einstein’s general relativity needs to be extended to problems arising at the ultra-violet and
81
infra-red scales. This is due to an incomplete quantum gravity theory. From a cosmological
point of view there is a desire to encompass ideas such as dark energy and dark matter
under one geometric standard under the related idea that gravitational interaction depends
on the scales. This geometric view preserves the results of general relativity. Despite the
promising Planck results related to f (R) inflation [5, 6], no f (R) model addresses all the
phenomenological issues on quantum to cosmological scales [35].
7.1.1
f(R) Model: Equations of Motion
The dissertation focuses on f (R) = R + αR2 , (α > 0) - none other than the Starobinksy
model [109] - however a number of the following derivations are kept general without specifying the form of f (R).
The four-dimensional action in f (R) gravity is given by


Z
1 Z 4 √
S= 2
d x −gf (R) + d4 x Lm gµν , Ψm ,
2κ
(234)
where κ2 = 8πG, g is the determinant of the metric gµν , Lm is the matter Lagrangian
and R is the Ricci scalar. Varying the action with respect to gµν gives the following equation
of motion [35]
1
(m)
Σµν ≡ F (R)Rµν − f (R)gµν − ∇µ ∇ν F (R) + gµν 2F (R) = κ2 Tµν
2
where F (R) =
∂f
∂R
(235)
(m)
and Tµν
is the energy momentum tensor of the matter fields
2 δLm
(m)
Tµν
= −√
−g δg µν
(236)
which satisfies the continuity equation36 . Taking the trace37
36
It is worthwhile to note that
0 = ∇µ T0µ
= ∂µ T0µ + Γµµλ T0λ − Γλµ0 Tλµ
ȧ
= −∂0 ρ − 3 (ρ + P )
a
is little more than the conservation equation!!!
37
It is worthwhile to note that the equation of motion (235) also satisfies its own continuity equation, that
is ∇µ Σµν = 0. The trace of the field equation (235) gives 3F (R) + F (R)R − 2f (R) = κ2 T . This would be
an interesting path to explore at a later date as Einstein gravity without a cosmological constant has f (R) = R
and F (R) = 1 so then 3F (R) in the trace vanishes. Then interestingly R = −κ2 T and so the Ricci Scalar R
is determined by matter.
82
3F (R) + F (R)R − 2f (R) = κ2 T.
(237)
Hence the equation of motion (235) can be re-written as [13, 107],


(m)
(D) 
Gµν = κ2 Tµν
+ Tµν
(238)
where38 Gµν = Rµν − 21 gµν R and
(D)
κ2 Tµν
≡ gµν
f −R
+ ∇µ ∇ν F − gµν 2F + (1 − F )Rµν .
2
(239)
There is an exact de-Sitter point which corresponds to a vacuum solution (T = 0) where
the Ricci scalar is constant. 2F (R) = 0 at this point so from the trace,
(240)
F (R)R − 2f (R) = 0,
is satisfied for the model f (R) = αR2 , giving rise to the exact de-Sitter solution [109]. In
the Starobinsky model, f (R) = R + αR2 , inflationary expansion ends when αR2 becomes
much smaller than R.
Consider the spatially flat FLRW metric in cosmic time t given by
ds2 = gµν dxµ dxν = −dt2 + a2 (t)dx2
(241)
which has Ricci scalar R
2
(242)
R = 6 2H + Ḣ .
The energy momentum tensor Tν(M )µ = diag − ρM , PM , PM , PM
where ρM is the
energy density and PM the pressure density. From the field equation (235) in the flat FLRW
background then,
3F H 2 =
(F R − f )
− 3H Ḟ + κ2 ρM
2
− 2F Ḣ = F̈ − H Ḟ + κ2 (ρM + PM ).
38
(M )
Briefly, ∇µ Gµν = 0 and ∇µ Tµν
(D)
(243)
(244)
= 0 so then ∇µ Tµν = 0, which is useful for when investigating the
dark energy equation of state [46][107] and the equilibrium dynamics for the horizon entropy [12].
83
7.1.2
f(R) Models : Inflationary Dynamics
To explore the Starobinsky model (n = 2)
f (R) = R + αRn , (α > 0, n > 0),
where α =
1
.
6M 2
(245)
In the absence of matter (ρM = 0) then equation (243) reads
1
3 1 + 2αR H 2 = αR2 − 6αH Ṙ.
2
(246)
Inflationary acceleration occurs when F = 1 + 2αR 1. Under the approximation
F ' 2αR and dividing equation (246) by 6αR to get


1
Ṙ
H 2 ' R − 12H .
12
R
(247)
During inflation it is assumed the Hubble parameter evolves slowly such that
Ḧ
H Ḣ
Ḣ
H
1 and
1 , hence equation (246) can be recast as
Ḣ
' −1
H2
(248)
2−n
.
(n − 1)(2n − 1)
(249)
for n = 2. Where 1 is
1 =
In the starobinksy model H is constant for F 1.
For starobinsky’s model equation (245) is
f (R) = R +
R2
6M 2
(250)
where the constant M has dimensions of mass.
Integrating this for 1 > 1 gives the following solution39 :
H'
1
,
1 t
1
(251)
a ' t 1 .
As mentioned earlier the linear term R causes inflation to end so neglecting that,
39
Ḣ 2 1 2
Ḧ −
+ M H = −3H Ḣ
2H 2
(252)
√
(1+ 3)
.
2
For n = 2 then 1 = 0 which means
We have cosmic acceleration fro 1 < 1 which requires n >
H is constant for F >> 1. We aren’t looking at models with n > 2 but this leads to super inflation which has
Ḣ > 0 so standard inflation occurs for
√
(1+ 3)
2
< n < 2.
84
R̈ + 3H Ṙ + M 2 R = 0.
(253)
However during inflation the first two terms in equation (252) can be ignored so Ḣ '
2
− M6 which has the solutions:
H ' Hi −
M2
(t − ti ),
6

(254)

M2
a ' ai exp Hi (t − ti ) −
(t − ti )2 ,
12
(255)
R ' 12H 2 − M 2 ,
(256)
where Hi and ai are the Hubble parameters and scale factor at the beginning of inflation
(t = ti ).
Expansion continues as long as the slow roll parameter
1 = −
Ḣ
M2
'
≤ 1,
H2
6H 2
(257)
is smaller than unity i.e H 2 ≥ M 2 . Inflation ends (at t = tf ) when f ' 1, so that
Hf '
M
√
6
corresponding to R ' M 2 . WMAP results constrain M ' 1013 GeV [35].
At time t = tf , the slow roll parameter is on the order of unity i.e 1 ' 1 and inflation
stops. In this regime
Hf
M
' 1 −→ Hf ' √ .
2
H
6
(258)
Inserting this into equation (254)
tf '
6Hi
+ ti .
M2
(259)
The number of e-foldings from t = ti to t = tf is given by
N≡
Ztf
Hdt ' Hi (tf − ti ) −
ti
Inflation ends at tf ' ti +
6Hi
M2
M2
(tf − ti )2 .
12
(260)
so,
N'
3Hi2
1
'
.
2
M
21 (ti )
(261)
To solve the horizon and flatness problems it is required that N ≥ 70 [64] so then
1 (ti ) ≤ 7 × 10−3 .
85
7.1.3
Dynamics In The Einstein Frame
Consider dynamics in the Einstein frame with
R2
6M 2
f (R) = R +
(262)
and the absence of matter (Lm = 0). Taking the action in the Einstein frame (234) with
a field ϕ defined by
s
ϕ=
31
lnF =
2κ
s
R
31
ln 1 +
2κ
3M 2
(263)
with the potential in equation (234),
√ 2
3M 2
− 23 κϕ
.
V (ϕ) =
1−e
4κ2
(264)
Slow roll inflation occurs for κϕ 1 which gives a nearly constant potential, V (ϕ) '
1
M 2 ϕ2
2
and the field oscillates around ϕ = 0 with Hubble damping. The second derivative
of V with respect to ϕ is
2
V,ϕϕ = −M e
√2
3
κϕ
√2 1 − 2e 3 κϕ .
(265)
Figure 14: The potential given in equation (264), normalised by its asymptotic value at
ϕ −→ ∞. Image courtesy of [27].
As F '
4H 2
M2
during inflation [35] there is a relation between cosmic time t̃ in the Einstein
frame and the Jordan frame given by
86
t̃ =
Zt
ti


√
2
M2
(t − ti )2 ,
F dt '
Hi (t − ti ) −
M
12
(266)
where t = ti corresponds to t̃ = 0. The end of inflation at time tf given by equation
√
(259) corresponds to t̃f = M2 N , so the scale factor ã = F a in the Einstein frame evolves as

ã(t̃) ' 1 −
where ãi =
2Hi ai
.
M

2
M t̃
M
ãi e 2 ,
M
t̃
12Hi2
(267)
The evolution of the Hubble parameter is given by H̃ =
H
√
F
1+
F
2HF
,
and substituting in appropriate values,

M2
M2
M
1−
1
−
M t̃
H̃(t̃) '
2
6Hi2
12Hi2
−2

(268)
.
In the Einstein frame the Universe evolves quasi-exponentially.
The field equations in the Einstein frame are simply the standard ones seen in general
relativity as the Lagrangian density is
1
Lϕ = − g̃ µν ∂µ ϕ∂ν ϕ − V (ϕ).
2
(269)
By varying the action, equation (226), but with no matter fields the following field equations arise

1 dϕ
3H̃ 2 = κ2 
2 dt̃

2
(270)
+ V (ϕ),
d2 ϕ
dϕ
+ 3H̃
+ V,ϕ = 0.
2
dt̃
dt̃
Defining the slow roll approximations as
dϕ
dt̃
2
(271)
V (ϕ) and
d2 ϕ
dt̃2
H̃ dϕ
then 3H̃ 2 '
dt̃
κ2 V (ϕ) and 3H̃ dϕ
+ V,ϕ = 0. Defining the slow roll parameters as
dt̃
2
˜ t̃
1  Vϕ 
dH/d
˜1 ≡ −
' 2
,
2κ V
H̃ 2


˜2 ≡
d2 ϕ/dt̃2
Vϕϕ
' ˜1 −
.
dϕ
3H̃ 2
H̃( dt̃ )
(272)
which for the potential (246) give
−2
4 √2
˜1 ' e 3 κϕ − 1 ,
3

√2
M 2 √ 2 κϕ 
κϕ 
3
3
˜2 ' ˜1 +
e
1
−
2e
2
3H̃


(273)
which during inflation are much smaller than 1. The end of inflation sees (1 , 2 ) = O(1).
Solving for ˜1 = 1 gives ϕf ' 0.19MP l .
87
The number of e-foldings in the Einstein frame is defined as
Ñ =
Zt̃f
2
H̃dt̃ ' κ
Zϕi
ϕf
t̃i
V
dϕ,
V,ϕ
(274)


where ϕi is at the beginning of inflation. Now H̃dt̃ = Hdt1 +
slow roll limit:
Ḟ
2HF
'
Ḣ
H2
Ḟ 
2HF
so Ñ = N in the
1, so combining this with κϕi 1 then
Ñ ' e
√2
3
κϕi
(275)
.
Taking Ñ ' 70, ϕi ' 0.11MP l . Using the approximation H̃ '
˜1 '
3
,
4Ñ 2
˜2 '
M
2
leads to
1
,
Ñ
(276)
where for ˜2 terms of order O(M 2 ) are dropped. These results can be used to estimate
the spectra of density perturbations.
7.1.4
Perturbations Generated During Inflation
f (R) theories have one extra degree of freedom compared to the ΛCDM model, resulting
in more freedom for the background. Starting with a general perturbed metric about the
FLRW background [35, 33, 111, 34]
2
i
ds = −(1+2α)dt −2a(t) ∂i β −Si dtdx +a (t) δij +2ψδij +2∂i ∂j γ +2∂j Fi +hij dxi dxj
2
2
(277)
where
• α,β,ψ and γ are scalar perturbations,
• Si and Fi are vector perturbations [35],
• and hij are the is the tensor perturbations.
Considering the following general action [35]

S=
Z

√
1
1
d4 x −g  2 f (R, φ) − ω(φ)g µν ∂µ φ∂ν φ − V (φ) + SM (gµν , ΨM )
2κ
2
(278)
where f (R, φ) is a function of the Ricci scalar R and scalar field φ, ω(φ) and V (φ) are
functions of φ and SM is the matter action.
88
Noting the results of [35] from varying the action with respect to gµν and φ gives the
following equations of motion,
1
F Rµν − f gµν − ∇µ ∇ν F + gµν 2F
2
 


1
(M ) 
= κ2 ω ∇µ φ∇ν φ − gµν ∇λ φ∇λ φ − V gµν + Tµν
,
2
(279)
and


f,φ
1 
ω,φ ∇λ ∇λ − 2V,φ + 2  = 0
2φ +
2ω
κ
(280)
For this action, the background equations sans perturbations are [35]:


1
1
3F H = (RF − f ) − 3H Ḟ = κ2  ω φ̇2 + V (φ) + ρM 
2
2
(281)
− 2F Ḣ = F̈ − H Ḟ + κ2 ω φ̇2 + κ2 (ρM + PM )
(282)
2
1
f,φ
φ̈ + 3H φ̇ +
ω,φ φ̇2 + 2V,φ − 2
2ω
κ
(283)
=0
(284)
ρM
˙ + 3H(ρM + PM ) = 0.
Defining
χ ≡ a(β + aγ̇),
A ≡ 3(Hα − ψ̇) −
∆
χ
a2
(285)
then perturbing the Einstein equations at linear order leads to the following equations
[48, 49, 47, 9, 25, 50, 89, 94]:

!
∆
1
∆
ψ + HA = − 3H 2 + 3Ḣ + 2 δF
2
a
2F
a

!
1
−3Hδ Ḟ + κ2 ω,φ φ̇2 + 2κ2 V,φ − f,φ δφ+
2

!
κ2 ω φ̇δ φ̇ + 3H Ḟ − κ2 ω φ̇2 α + Ḟ A + κ2 δρM ,

(286)

1  2
Hα − ψ̇ =
κ ω φ̇δφ + δ Ḟ − HδF − Ḟ α + κ2 (ρM + PM )v ,
2F


1
χ̇ + Hχ − α − ψ = δF − Ḟ χ,
F
89
(287)
(288)


!
∆
δ
1
Ȧ + 2HA + 3H + 2 α = − 3δ F̈ + 3Hδ Ḟ − 6H 2 + 2 δF + 4κ2 ω φ̇δ φ̇+
a
2F
a
2κ2 ω,φ φ̇2 − 2κ2 V,φ + f,φ δφ − 3Ḟ α̇ − Ḟ A−

4κ ω φ̇ + 3H F +˙ 6F̈ α + κ2 δρM + δPM ,(289)

2
2

∆ R
2
1 2
δ F̈ + 3Hδ Ḟ −  2 + δF + κ2 φ̇δ φ̇ +
κ ω,φ φ̇2 − 4κ2 V,φ + 2f,φ δφ =
a
3
3
3
1 2
2
1
κ (δρM − 3δPM ) + Ḟ (A + α̇) + 2F̈ + 3H Ḟ + κ2 ω φ̇2 α − F δR,
3
3
3

∆
ω,φ
ω,φ
φ̇ δ φ̇ +  − 2 +
δ φ̈ + 3H +
ω
a
ω

2V,φ−f,φ φ̇
δφ =
+
2ω
,φ 2
,φ
ω,φ 2
1
φ̇ α + φ̇A +
F,φ δR,
ω
2ω
φ̇α̇ + 2φ̈ + 3H φ̇ +
∆
+ PM ) A − 3Hα + 2 v ,
a
δρM + 3H(δρM + δPM ) = (ρM
(290)
(291)
(292)


δPM
1
d 3
a (ρM + PM )v  = α +
,
3
a (ρM + PM ) dt
ρ M + PM
(293)
where δR is,


∆
∆
δR = −2Ȧ + 4HA + 2 + 3Ḣ α + 2 2 ψ 
a
a
7.1.5
(294)
Gauge Invariant Quantities
A number of gauge invariant quantities need to be constructed to avoid the appearance of unphysical modes which could arise in cosmological perturbations. Consider the gauge transformations:
t̂ = t + δt,
x̂i = xi + δ ij ∂j δx
(295)
where δt and δx represent the spatial time slicing and threading respectively. The scalar
metric perturbations (α, β, ψ and γ) transform according to [13][14]:
α̂ = α − δ ṫ,
(296)
β̂ = β − a−1 δt + aδ ẋ,
(297)
ψ̂ = ψ − Hδt,
(298)
90
(299)
γ̂ = γ − δx.
Decomposing φ and F into homogeneous and perturbed parts so that φ = φ + δφ and
F = F + δF , then the energy-momentum tensor for a perfect fluid with perturbations is
T00 = −(ρM + δρm ),
Ti0 = −(ρM + PM )∂i v,
Tji = (PM + δPM )δji
(300)
The matter perturbations such as δφ and δρ transform according to:
δ φ̂ = δφ − φ̇δt,
(301)
δ ρ̂ = δρ − ρ̇δt.
(302)
with δF following a similar transformation rule. The scalar part of the three momentum
can be written as
δTi0 = ∂i δq.
(303)
δq = −φ̇δφ
(304)
δq = −(ρM + PM )v
(305)
For the scalar field and perfect fluid:
where v is the velocity potential of the fluid so that δq transforms as
(306)
δ q̂ = δq + (ρ + P )δt.
A number of gauge invariant quantities can be constructed from (295) [35]:
β
d 2
a γ−
,
Φ=α−
dt
a
β
,
Ψ = −ψ + a2 H γ̇ +
a
H
R=ψ+
δq,
ρ+P
(307)
(308)
(309)
H
,
φ̇
(310)
H
δF,
Ḟ
(311)
δρq = δρ − 3Hδq.
(312)
Rδφ = ψ −
RδF = ψ −
Note that:
91
• R is identical to Rδφ since δq = −φ̇δφ for a potential V (φ) in scalar field inflation,
• From equation (312) can also construct another gauge invariant quantity for the matter
perturbation of perfect fluids:
δM =
where ωM =
δρM
+ 3H(1 + ωM )v
ρM
(313)
PM
.
ρM
• The tensor perturbation hij is invariant under the transformations described by Malik
and Wands [81].
7.2
Perturbations Generated During Inflation
Lets review the perturbations generated during inflation for the theory seen in equation (278)
but ignoring the matter equation (such that SM = 0). In f (R) gravity the contribution of the
field φ or rather δφ is absent and so not present in the perturbation equations (286)-(291). As
such the gauge condition sees δF = 0, so RδF = ϕ. It is usually taken in tensor-scalar theory
that F is a function of φ alone, and so the gauge δφ = 0 imply Rδφ = ϕ. As δF = F,φ δφ = 0
then RδF = Rδφ = ϕ.
This report explores single field theory f (R) gravity and scalar-tensor theory with coupling F (R)φ having chosen the gauge conditions δφ = 0 and δF = 0. The curvature
perturbations RδF and Rδφ are denoted R.
7.2.1
Curvature Perturbations
Setting δφ = 0 and δF = 0 in (287) then
α=
Ṙ
H+
Ḟ
2F
(314)
.
So then plugging equation (314) into (286)


∆
3H Ḟ − κ2 ω φ˙2 
 R+
A=−
Ṙ .
Ḟ
Ḟ
a2
H + 2F
2F H + 2F
1
(315)
Equation (289), gives


Ḟ
3Ḟ
3F̈ + 6H Ḟ + κ2 ω φ̇2 ∆ 
Ȧ + 2H +
A+
α̇ + 
+ 2 α = 0,
2F
2F
2F
a
92
(316)
having utilised equation (282).
Plugging equations (314) and (315) into (316) and re-arranging it can be seen that the
curvature perturbations in Fourier space obey
R̈ +
(a3˙Qs )
k2
Ṙ
+
R
a3 Q s
a2
(317)
where k is the co-moving wave number and
ω φ̇2 +
Qs = H+
3Ḟ 2
2κ2 F
2
Ḟ
2F
(318)
.
√
Defining Zs = a Qs and u = zs R equation (317) can be expressed as
z 00
u + k + s u=0
zs
00
2
(319)
where the primes are derivates with respect to conformal time.
To derive the spectrum of curvature perturbations during inflation requires defining the
following variables:
Ḣ
,
H2
φ̈
2 ≡
,
H φ̇
Ḟ
3 ≡
,
2HF
Ė
4 ≡
,
2HE
1 ≡ −
(320)
where
3Ḟ 2
E =F ω+ 2 2
2κ φ̇ F
(321)
So that Qs can be re-expressed as
Qs = φ̇2
E
.
+ 3 )2
F H 2 (1
(322)
Taking 1 to be constant then [113]
η=
−1
.
(1 − 1 )aH
(323)
Felice and Tsujikawa note in their paper [35] that if ˙ = 0 then
vR −
zs00
=
zs
η2
93
1
4
(324)
with
2
=
vR
1 (1 + 1 + 2 − 3 + 4 )(2 + 2 − 3 + 4 )
+
4
(1 + 1 )2
(325)
Then (319) can be expressed in terms of Hankel functions
√
u=


πη i(1+2vR ) π 
(1)vR
4 c H
e
(kη) + c2 H (2)vR (kη)
1
2
(326)
where c1 and c2 are integration constants.
• If during inflation i 1 then
zs00
zs
' (aH)2 .
• Deep inside the Hubble radius (k aH, kη 1) then the perturbation u is none
other than the equation of canonical Minkowski space-time u00 + k 2 u ' 0.
• After the Hubble crossing (k = aH ) but during inflation, then the gravitational term
zs00
zs
becomes important.
• In the super Hubble limit (k aH, kη 1) the last term in (317) can be neglected
giving
R = c1 + c2
dt
Z
a3 Qs
(327)
.
The second term is the decaying mode which decays during inflation, so that after the
Hubble crossing the curvature perturbation approaches the constant value c1 .
To fix the integration constants it is best to look towards the asymptotic past (η −→
−∞), where the solution to (319) is determined by the vacuum state [22] where u −→
so that the constants become c1 = 1 and c2 = 0 so then
√
πη i(1+2vR ) π (1)
4H
e
u=
vR (kη).
2
eikη
√
,
2k
(328)
The power spectrum of curvature perturbations is defined as
PR ≡
4πk 3 2
R
(2π)3
(329)
So that the power spectrum using (328) becomes [49]
2 

3−2vR
1 
Γ(vR ) H   kη 
PR =
(1 − 1 ) 3
Qs
2
Γ( 2 ) 2π
94
,
(330)
having used the relation Hv(1) (kη) −→ −( πi )Γ(v)( kη
)−v in the asymptotic limit kη −→ 0.
2
As the curvature perturbations freeze at the Hubble radius (k = aH ) this is where the
spectrum should be evaluated at. The spectral index of R is defined by
dlnPR nR − 1 =
dlnk (331)
k=aH
hence
nR − 1 = 3 − 2vR .
(332)
Provided i 1 during inflation then the spectral index becomes
nR − 1 ' −41 − 22 + 23 − 24 .
(333)
The spectrum is now close to scale invariant (nR ' 1) so that the curvature power
spectrum of perturbations reduces to
1 H
PR '
Qs 2π
7.2.2
2
.
(334)
Tensor Perturbations
In chapter 5.3 an introduction to tensor perturbations was given for a general model, here
that knowledge is applied
It was noted in section 5.3.2 that there are two independent tensor polarizations hij , h1
and h2 . In keeping with the notation used in [35] they are henceforth labelled λ = +, ×
[64]. Equation (105) becomes
×
hij = h+ e+
ij + h× eij .
(335)
For the action (278) then the Fourier components hλ obey the following [47],
ḧλ +
√
Defining zt = a F and uλ =
(a3˙F )
k2
ḣ
+
hλ = 0.
λ
a3 F
a2
√zt hλ
16πG
(336)
then
u00λ + k 2 −
zt00
uλ = 0.
zt
(337)
Following the same procedure for curvature perturbations in the previous chapter taking into account the polarization states and quoting the results in [35] which gives for the
spectrum of tensor perturbations
95
2 

3−2vT
Γ(vT )
16πG 4πk 3 2
16 H 2 1 
kη
)
(1 − 1 ) 3   
PT = 4 × 2
uλ ' (
3
a F (2π)
π MP l F
2
Γ( 2 )
(338)
which like the curvature perturbations should be evaluated at the Hubble crossing, so
that the spectral index is
(339)
nT = 3 − 2vT
and in the limit i 1 this reduces to
nT ' 21 − 23 .
(340)
The amplitude of tensor perturbations is given by
16 H
PT '
π MP l
2
1
.
F
(341)
64π Qs
PT
≈ 2
.
PR
MP l F
(342)
The tensor-to-scalar ratio is given by
r≡
7.2.3
The Spectra of Perturbations in inflation based on f(R) Gravity
Here a brief foray into into the spectra for scalar and tensor perturbations in metric f (R)
gravity is made. Defining two new variables,
E=
3Ḟ 2
,
2κ2
4 =
F̈
H Ḟ
(343)
so that
Qs =
5F 23
E
=
.
κ2 (1 + s )2
F H 2 (1 + 3 )2
(344)
In this regime the kinetic term ( φ̇) is absent so 2 is absent in equations (325) and (333).
Again taking i 1 equation (333) becomes
nR ' −41 + 23 − 24 .
(345)
So then in the absence of the matter fluid equation (244) this becomes
1 = −3 (1 − 4 )
and taking 4 1 then 1 = −3 so that
96
(346)
(347)
nR − 1 ' −61 − 24 .
The amplitude can be estimated to be
PR '
2
H
1
3πF MP l
1
.
23
(348)
Once again using 1 = −3 then the spectral index of tensor perturbations is
(349)
nT ' 0
which vanishes at first order in slow roll approximations. The tensor-to-scalar ratio is
therefore
r ' 4823 ' 4821 .
(350)
Storobinksy Model:
This model is not completely scale invariant and because inflation occurs in the regime
R M 2 and Ḣ H 2 then the following approximation can be made F '
R
3M 2
'
4H 2
,
M2
so
that the power spectra read
PR '
1
M
12π MP l
4 M
PT '
π MP l
2
1
21
(351)
.
(352)
2
The evolution of the Hubble parameter is given by (254) so at the Hubble crossing at
time tk requires
M2
(tk
6
− ti ) Hi allowing the approximation H(tk ) ' Hi . The number of
e-foldings can be approximated by
Nk '
1
.
21 (tk )
(353)
The curvature perturbation is then given by
PR '
Nk2 M
3π MP l
2
(354)
Data from WMAP PR = (2.445 ± 0.096) × 10−9 at the scale k = 0.002M P c−1 [60].
Taking Nk = 55 then the mass M is constrained to
M ' 3 × 10−6 MP l .
The spectral index (345) can be reduced further by taking F '
giving
97
(355)
4H 2
M2
so that 4 ' −1
Nk
2
= −3.6 × 10−2
nR ' −41 ' −
Nk
55
−1
(356)
.
For Nk = 55 then nR = 0.964 which is in agreement with WMAP results [60].
Finally an estimate the tensor-to-scalar ratio is
12
Nk
r ' 2 ' 4.0 × 10−3
Nk
55
−2
(357)
in agreement with the current bound of r < 0.22 [60].
7.2.4
Power Spectra in the Einstein Frame
The perturbed metric (277) under a conformal transformation g̃µν = F gµν becomes:
ds̃2 = F ds2
= −(1 + 2α̃)dt̃2 − 2α̃(t̃) ∂i β̃ − S̃i dt̃ dx̃i
+ ã2 (t̃) δij + 2ψ̃δij + 2∂i ∂j γ̃ + 2∂j F̃i + h̃ij dx̃i dx̃j .
In addition the conformal factor can also be split into the background and perturbed
parts,
δ(t, x)
F (t, x) = F̄ (t) = 1 +
F̄ (t)
(358)
and the bar notation will now be dropped. In this regime the transformation of the scalar
metric perturbations is
α̃ = α +
δF
,
2F
β̃ = β,
ψ̃ = ψ +
δF
,
2F
(359)
γ̃ = γ,
where vector and tensor perturbations remain invariant:
S̃i = Si ,
F̃i = Fi ,
(360)
h̃ij = hij .
Taking equation (311) and using the above transformation laws R = R̃ under a conformal transformation. Hence, because the tensor perturbation is invariant it is expected
r̃ = r.
Taking the Starobinsky model (262) with the action in the Einstein frame (see appendix
F.5 for derivation):
S=
Z
√


Z
1
1 µν


d x −g̃
R̃ − g̃ ∂µ ϕ∂ν ϕ − V (ϕ) + d4 xLm F −1 (φ)g̃µν , ΨM
2
2κ
2
4
98
(361)
the slow roll parameters ˜3 and ˜4 vanish. Hence using the spectral index (333) and
equation (276) the spectral index of curvature perturbations in the Einstein frame is given
by
64π dφ
r̃ = 2
MP l dt̃
2
1
12
' 16˜1 ' 2 ,
2
H̃
Ñk
which is consistent with the tensor-to-scalar ratio (357).
99
(362)
8
Numerical Simulations
"From out there on the Moon,
international politics look so petty. You
want to grab a politician by the scruff of
the neck and drag him a quarter of a
million miles out and say, ’Look at that,
you son of a b***h.’"
Edgar Mitchel - Apollo 14 Astronaut
In this chapter numerical simulations are performed based on a number of equations
and themes presented throughout this dissertation in relation to inflation using the scalar
field potential. The associated Python code can be found in the appendix. Please note that
for plots created for the scalar field potential please see appendix H and for information
associated with the plots concerning the Starobinsky potential please see appendix I.
8.1
The Cosmological equations
The cosmological equations in flat space are given by:
2
ȧ
a
=
8πG
ρφ
3
(363)
(364)
φ̈ + 3H φ̇ + V,φ = 0
ä
4πG
=−
ρ φ + Pφ
a
3
(365)
where ρφ = 21 φ̇2 + V (φ) and Pφ = 12 φ̇2 − V (φ).
During inflation the potential dominates hence ρφ ' V (φ) and the equation of state
P = ωρ sees ω = −1, therefore Pφ ' −ρφ . Taking the scalar potential V (φ) = 12 m2 φ2 then
intitially H02 =
8πG
V
3
(φ0 ), hence
H02 =
8πG 1 2 2
m φ0 .
3 2
(366)
Introducing the slow roll formalism introduced in chapter 4 and using equation (68) i.e
φ̈ ' 0 with φ̈+3H0 φ˙0 +V,φ (t0 ) = 0 and the scalar potential V (φ) = 21 m2 φ20 then rearranging,
Vφ (t0 )
m2 φ20 3
1
1 1
=
−
φ˙0 = −
=−
.
2
3H0
3 4πG m2 φ0
4πG φ0
100
(367)
8.1.1
Dimensionless Cosmological Equations
To plot the evolution of the cosmological equations in a manner that python can interpret
the first task is to re-write the equations in a dimensionless form by first employing the use
of the reduced planck mass mpl =
q
1
:
8πG
v
u
1 √
ȧ u
= t 2 ρφ
a
3mpl
(368)
1
ä
= − 2 ρφ + 3Pφ .
a
6mpl
(369)
1
x
mpl
Introducing the following changes of variable: φ = mpl y, t =
leads to
d
dt
and a =
1
.
mpl ã
This
d
= mpl dx
, φ̇ = m2pl y 0 and
d
da
mpl dã
a = mpl
=
= ã0
dt
dx
mpl dx
(370)
d
dy
d
φ = mpl φ = m2pl
= m2pl y 0 .
dt
dx
dx
(371)
d
d
dy
d2 y
φ̇ = (m2pl ) = m2pl 2 = m3pl y 00 .
dt
dt
dx
dx
(372)
Taking the equation of motion (364) this can be re-expressed dimension free as
ã0
V 0 (mpl y)
y 00 + 3 y 0 +
= 0.
ã
m3pl
Thus w0 (y) =
V 0 (mpl y)
m3pl
so w(y) =
(373)
V (mpl y)
m4pl
The Friedmann equation (368) can be written as:
v
u
ã0 u 1 √
mpl = t 2 ρφ
ã
3mpl
1
=
q
=
q
=q
1 2
φ̇ + V (φ)
2
1 2 0 2
(mpl y ) + w(y)m4pl
2
3m2pl
1
3m2pl
1
3m2pl
1
m2pl
2
1
2
1
1 02
y + w(y)
2
2
Which gives
˙
ã0
1 1 02
=√
y + w(y)
ã
3 2
101
(374)
Finally taking the acceleration equation (369):
˜00
1
= − 2 ρφ + 3Pφ
ã
6mpl
1
3
1 2
=− 2
φ̇ + V (φ) + φ̇2 − 3V (φ)
6mpl 2
2
1
= − 2 2φ̇2 − 2V (φ)
6mpl
1 1
2
φ̇
−
V
(φ)
=−
3 m2pl
1 1
2 0 2
4
=−
(m
y
)
−
w(y)m
pl
pl
3 m2pl
1 2 02
= − mpl y − w(y) .
3
a
m2pl
So our final equation is
a˜00
1 02
= − y − w(y) .
ã
3
(375)
The set of equations (373, 374, 375) can now be solved numerically using initial conditions and setting mpl = 1 and m = 1. The initial s
conditions were taken to have a(0) = 1,
√
t(0) = 1, φ(0) = 18, φ̇(0) = − 36 10118 , and H(0) = 13 12 φ̇2 + 12 φ2 [29].
To plot the various evolutions, the python package ’scipy.integrate.odeint’ was made use
of which solves a system of ordinary differential equations using various methods. Please
note that the time scales in the code were altered for each graph for relevancy, details of
which can be found in the figure captions.
The python code can be seen in Appendix H.
102
8.1.2
Evolution of the Potential as a function of time
By using ’ODEint’ the evolution of the scalar potential as a function of time with slow roll
formalism (figure 15) was investigated.
Figure 15: Plot of the scalar potential as a function of time. To obtain this graph a range of
400 evenly spaced data points were taken over a time interval of 0 < t < 40. Noting that all
√
quantities are dimensionless so the initial values are the following: φ0 = 18, φ̇ = −
a0 = 1 and H0 =
q
1 1 2
( φ̇
3 2
6 1
,
3 1018
+ 21 φ2 ).
Inflaton starts off at the top of the potential and rolls slowly down until it reaches zero,
at this point the inflaton is gently oscillating at the bottom of the potential where φ is small
and can be seen in figure 15. Assuming homogeneous inflation then the equation of motion
(364) still holds. However the expansion scale becomes much longer than the oscillation
period and so the damping term becomes trivial and the field undergoes oscillations with
frequency m. The inflaton then decays into standard model particles.
103
8.1.3
Evolution of the scale factor as a function of time
The relative expansion of the Universie is parameterized by the dimensionless scale factor, a,
and so as the Universe expands the value of the scale factor should increase.
Figure 16: Plot of the scale factor as a function of time. To obtain this graph a range of 40000
evenly spaced data points were taken over a time interval of 0 < t < 4000. Noting that all
√
quantities are dimensionless so the initial values are the following: φ0 = 18, φ̇ = −
a0 = 1 and H0 =
q
1 1 2
( φ̇
3 2
+ 21 φ2 ).
104
6 1
,
3 1018
8.1.4
Number of e-folds for the scalar potential
Figure 17: Plot of the number of e-folds as a function of time. To obtain this graph a range
of 400 evenly spaced data points were taken over a time interval of 0 < t < 40. Noting that
√
all quantities are dimensionless so the initial values are the following: φ0 = 18, φ̇ = −
a0 = 1 and H0 =
q
1 1 2
( φ̇
3 2
6 1
,
3 1018
+ 21 φ2 ).
Scalar field inflation usually sees 60-70 e-folds (although there is no upper limit), and
here have approximately 80 e-folds (see figure 17). This is fine as long as get the minimum
60 in which to solve the horizon and flatness problems. Indeed the largest scales observed
today were created about 60 e-folds before the end of inflation which the graph comfortably
accomodates.
105
8.1.5
Scalar field versus the number of e-folds
An interesting graph to plot is that of the number of e-folds against the scalar field φ (see
figure 18). This can be imagined as the Universe rapidly expanding as the inflaton rolls down
the potential. The expansion rate slows down the closer the inflaton is to the bottom of
the potential and as the field oscillates around the bottom leads to the final few e-folds of
expansion.
Figure 18: Plot of the number of e-folds as a function of the scalar field. To obtain this
graph a range of 40000 evenly spaced data points were taken over a time interval of 0 < t <
400000. Noting that all quantities are dimensionless so the initial values are the following:
√
φ0 = 18, φ̇ = −
6 1
,
3 1018
a0 = 1 and H0 =
q
1 1 2
( φ̇
3 2
+ 12 φ2 ).
Figure 18 seems to be in agreement with figure 17 where the increase in the number of
e-folds towards the end of inflation is a lot fewer compared with the start of inflation.
106
8.2
Mukhanov - Sasaki Equation
The Mukhanov (or Mukhanov-Sasaki equation) equation was alluded to in chapter 5, with a
more formal introduction here. The Mukhanov equation is
νk00
z 00
+ k −
νk = 0
z }
|
{z
2
(376)
≡ωk2 (τ )
where z =
φ0
H
where derivatives (represented by dashes) are with respect to conformal
time,τ and the Fourier modes for each mode k are given by
νk (τ ) ≡
Z
d3 x e−ik.x ν(τ, x).
(377)
The Mukhanov variable is defined as
φ0
νk ≡ a σφk + Φk ,
h
(378)
where σφk represents perturbations in the inflaton field, Φ is the metric perturbation and
h≡
a0
.
a
The evolution of the Mukhanov variable is given by the Mukhanov equation (376).
The equation of motion was obtained by varying the following action and expanding in
Fourier modes [79] 40 :
z 00
1Z
dτ d3 x (ν 0 )2 − (∇ν)2 + ν 2
S=
2
z
(381)
There are two special limits:
• For modes with wavelengths much smaller than the horizon,
νk00 + k 2 νk = 0 (Subhorizon).
(382)
Which produces the oscillating solution: νk ∝ e±ikτ .
40
The action is equivalent to the action for a harmonic oscillator with a time dependent mass
Z
1
1
S = dτ d3 x − η µν ∂µ ν∂ ν ν − m2ef f (τ )ν 2
2
2
(379)
where
m2ef f (τ ) ≡ −
z 00
H ∂ 2 aφ̇ =−
.
z
aφ̇ ∂τ 2 H
107
(380)
• For modes larger than the horizon
z 00
2
νk00
=
≈ 2 . Superhorizon
νk
z
τ
(383)
This represents the growing solution νk ∝ z ∝ τ −1 .
As the Universe expands the absolute value of |τ | decreases, so for a given mode k, the
value of k|τ | eventually becomes smaller than unity. When k becomes of order of the curvature scale, i.e when k|τ | ' 1, then this is the moment of the event horizon crossing and
refer to modes with k|τ | 1 and k|τ | << 1 as the subhorizon and superhorizon modes
respectively [120].
Quantum theory allows the mode function ν to be quantized in the normal way giving
a general solution of
1 Z d3 k
∗
ik.x
ν(t, x) = √
âk + νk e−ik.x âk+ ,
3 νk (τ )e
2 (2π) 2
(384)
where the creation and annihilation operators obey the standard commutation relations
provided the mode function obeys the normalization condition
νk’ νk∗ − νk νk’∗ = 2i.
(385)
Thus νk (τ ) is a complex solution of the second order differential equation (376). There
are two solutions, real and imaginary, with the initial conditions given by
√
νk’ (τ0 ) = i ωk eiαk’ (τ0 ) .
1
νk (τ0 ) = √ eiαk (τ0 )
ωk
8.2.1
(386)
Bunch - Davis Vacuum
At sufficently early times all modes are deep inside the horizon,
k
∼ |kτ | 1
aH
(387)
which implies all observable modes had time independent frequencies i.e,
ωk2 = k 2 −
2
−→ k 2 .
τ2
(388)
These modes are not affected by gravity and behave as in Minkowski space,
νk00 + k 2 νk = 0.
(389)
Thus now solving the Mukhanov equation with the Minkowski initial condition:
1
lim νk (τ ) = √ e−ikτ ,
τ →−∞
2k
which defines the unique physical vacuum, the Bunch-Davis vacuum.
108
(390)
8.2.2
Mukhanov-Sasaki Equation in Physical Time
The Mukhanov equation (376) is currently presented in conformal time, τ , and to plot the
evolution of a mode k requires the equation to be rewritten in physical time, t, which is
related to conformal time τ , by the following relations:
ds2 = −dt2 + a2 (t)dx
ds2 = a2 (−dτ 2 + dx2 )
a2 dτ 2 = dt2
dτ
1
=
dt
a
dν dτ
dν
=
dt
dτ dt
dν
= ν̇a.
dτ
Taking equation (376) it is worth noting that z =
to write ν 00 and
z 00
z
in terms of ν̇ and z̈z . Initially
Noting that adτ = dt and
dφ
dτ
=
dφ dt
dt dτ
z 00
z
aφ0
h
=
φ0
,
H
a00
'
a
i.e h = aH and it is required
which simplies the derivation.
then φ0 = a1 φ̇. It can be inferred that
φ0
1
z=
−→ z =
φ̇.
H
aH
(391)
Taking the derivative with respect to conformal time
0
0φ
φ0
z =a +a
− 0 .
h
h
h
0
00
φ
(392)
To transform the scale factor, a, note that a0 = a1 ȧ and so
ȧ0
ȧ
ä
1 ȧ2
1 ä ȧ2
a = − a0 2 = 2 −
=
−
.
a
a
a
a a2
a a a2
(393)
z 00
a00
1 ä
'
= 2
− H2 .
z
a
a a
(394)
00
Hence,
The focus is to now write ν 00 in terms of cosmic time by noting
d2 ν
d dν
=
2
dτ
dτ dτ
1 d dν
a dt dτ
=
Thus putting everything together,
109
d
dτ
=
1 d 1 dν
a dt a dt
1 d2 ν
1 dν
= 2 2 − 3 .
a dt
a dt
1 d
.
a dt
=
(395)
1
1 ä
1
2
− H2
ν̈
−
ν̇
+
k
−
a2
a3
a2 a
ν = 0,
(396)
ν = 0.
(397)
which can be written as
ν̈ − H ν̇ + a2 k 2 −
1 ä
− H2
a2 a
Equation (394) is the Mukhanov equation (376) written in cosmic time and is the form
of the equation used in the python code. As the Mukhanov equation is complex it will have
two solutions, real, r, and imaginary, i, with different initial conditions based on the BunchDavis vacuum. The initial conditions for the mode functions were taken to be νr = √12k ,
√
ν˙r = 0, νi = 0 and ν̇ = 2k. The associated phase seen in equation (390) can be set to 1.
The same initial conditions for φ(0), φ̇(0), a(0) and H(0) are the same as before.
Using the same method to plot graphs of the cosmological equations ’ODEint’ within
Python was used to solve the ordinary differnential equations for both the complex and
imaginary parts.
The python code can be found in Appendix H.
110
8.2.3
Real Mode Function
A plot of the real part of the mode function (397) is given in figure 19 for a single mode k.
Figure 19: A plot of the real part of the mode function. This graph plots the real comoving
part of the mode function (397) against a comoving scale. The initial values are given by
q
√
ν̇r = 1/ k, ν̈r = 0 and k = 100 a0 H0 where H0 = 13 ( 21 φ̇2 + 12 φ2 ) and a0 = 1.
It can be seen that the mode function is highly oscillatory at first and then ’calms down’
with the mode stopping around 3.2 aH
where it exits the horizon.
k
111
8.2.4
Imaginary Mode Function
A plot of the imaginary part of the mode function (397) is given in figure 20.
Figure 20: A plot of the imaginary part of the mode function. This graph plots the imaginary
comoving part of the mode function (397) against a comoving scale. The initial values are
q
√
given by ν̇i = 0, ν̈i = k and k = 100 a0 H0 where H0 = 13 ( 12 φ̇2 + 12 φ2 ) and a0 = 1.
Comparting figures (19) and (20) where the only difference in their set up was the initial
conditions, there is a difference in the points at which the mode function becomes less oscillatory and smoothes out and for the imaginary part although it exits the horizon at the same
point, it does so at a negative value of the mode function.
112
8.2.5
Modulus of the Mode Function
Figure 21 shows a plot of the modulus of both the imaginary and real parts of the mode
function over the scale factor against a comoving scale.
It is expected that the scale should cut out at 1 which represents a mode crossing the
horizon, however on all three graphs (figures 18,19,20) the line on the graph stops instead
around 3.2 which isn’t inconceivable and likely due to the set up involved.
Figure 21: A plot of both the imaginary and real parts of the mode function. The initial
√
√
values are given by ν̇i = 0, ν̈i = k, ν̇r = 1/ k, ν̈r = 0 and k = 100 a0 H0 where
H0 =
q
1 1 2
( φ̇
3 2
+ 21 φ2 ) and a0 = 1.
113
8.3
Power Spectrum PQ (k)
When investigating inflationary models, the comoving curvature (R) and tensor (h) flucations are amplified due to the nearly exponential expansion from quantum fluctations to
become high squeezed (classical looking) states.
This is described by the conformal evolution, τ , of the mode functions of the gauge
invariant inflation fluction δφ and for the tensor fluctions, h, descibed by the Mukhanov
equation (376) where the curvature fluctuation, R, and inflaton fluctation, δφ, are related by
R = H δφ
. This subchapter looks at a theoretical power spectrum (devoid of any potential)
φ̇
and the scalar and Starobinsky potential power spectrums.
8.3.1
Primordial Spectrum of Tensor Perturbations
A theoretical graph (figure 22) of the power spectrum based on equation (145) has been
created to highlight how the power spectrum should look for a single mode k, not incorportating any information on the form of the potential. Normally a power spectrum is created
based on observational data which can then be used to determine cosmological parameters.
Ph = 16πGH 2
114
(398)
Figure 22: A plot of the primoridal spectrum of tensor perturbations according to equation
(145) with G = 1.
115
In contrast a power spectrum for the scalar potential has been created (figure 23), which
is very similar to the power spectrum for the Starobinksy potential (figure 24).
Figure 23: A plot of the power spectrum for the scalar potential over the entirety of inflation.
Created using the code that can be found on [122].
116
Figure 24: A plot of the power spectrum for the starobinsky potential over the entirety of
inflation. Created using the code that can be found on [122].
117
8.4
Slow Roll Parameters
The slow roll parameters given equations (83,84) and shown below:

2
1  V,φ 
=
16πG V
(399)
1 V,φ,φ
,
8πG V
(400)
η=
which will be used in the numerical calculations. In the numerical setup G = 1 and
using a simple integration within the ODEint package in Python the following graph (figure
25) was produced which shows both slow roll parameters leveling out towards the end of
inflation which is to be expected as the potential becomes more and more flat. measures
the slow of the potential whilst η measures the curvature.
Figure 25: A plot of the slow roll parameters and η as a function of the efolding number n,
using the scalar potential.
118
8.5
Spectral Index
The scalar spectral index, ns , and met in equation (179) describes how density fluctuations
vary with scale. ns = 1 implies variations are the same on all scales where ns is an input
parameter to the ΛCDM model which influencs the size of structure formation. ns having
been measured close to unity is compatible with current models of inflation.
The size of density fluctuations depends on the inflationary motion when the quantum
fluctuations become super horizon sized. All inflationary models have their own spectral
indices dependent on the slow roll parameters and the gradient and curvature of the potential. ns 6= 1 supports inflation as inflation can produce a not quite scale invariant gaussian
spectrum of density fluctuations.
Figure 26: A plot of the spectral index for scalar potential inflation. Created using the code
that can be found on [122].
119
Figure 27: A plot of the spectral index for Starobinsky inflation. Created using the code that
can be found on [122].
120
9
Conclusions
’They say it all started out with a big
bang. But, what I wonder is, was it a big
bang or did it just seem big because
there wasn’t anything else drown it out
at the time?’
Karl Pilkington
This chapter summarises the work of this dissertation and culminates in an apparasial of
the progress in the field of cosmology.
9.1
Conclusion
The aim of this dissertation was to give a broad overview of a number of the current themes
within the topic of inflation whilst giving a deeper understanding of the mathematics involved to develop an encompassing view on the field of inflation whilst utilising the Planck
data.
The need for inflation to occur in the early Universe has been explored in great depth
with a focus on the horizon and flatness problems and their resolution via rapid expansion of
the Universe. Description was given to what a successful inflationary model must incorprate
such as not violating the slow roll parameters and the need for a minimum number of efoldings. A number of simple inflationary models were presented to highlight how there
is no definative resolution on how the Universe inflated but that a number of theoretical
models can be ruled out, thanks to the success of the Planck mission.
The seeds for the evolution of large scale structures which are a consequence of inflation
due to cosmological perturbations has been explored. Both the tensor and scalar modes
fluctuations have been explored and a much more in depth look at the dynamics behind R2
inflation has been presented via f(R) gravity. Finally a number of numerical simulations have
been made focusing primarily on scalar field inflation.
The Planck mission expects to release an improved data set in 2016 based upon the 2015
data sets [121].
121
9.2
An Appraisal
’These moments when we dare to aim
higher, to break barriers, to reach for
the stars, to make the unknown known.
We count these moments as our
proudest achievements.’
Film- Interstellar
There is something rather comforting in the idea that regardless of where we are on Earth
and regardless of background we are all united in looking uptowards the same sun, moon and
stars. We are a mere pinprick full of colour, beauty and life in the vast emptiness of space,
and the last few decades have witnessed immense progress in our understanding of the dark
void out there.
The field of cosmology and theoretical physics has seen remarkable breakthroughs in
piecing together the story of our early Universe, due to advancements in technology having
enabled humanity (or rather the lucky few) to take our first steps beyond our Earthly confines, and put people and satellites out there. The vast collection of observational data has
givnen us a glimpse into what is out there and how we came to be, allowing us to fine tune
(or in some cases throw away) theories and put constraints on cosmological parameters. Despite this wealth of knowledge, it is easy to feel sadness that complete knowledge will come
long after you or I have passed on, for now we watch Star Trek and only wonder.
We look forward to future work in the field of cosmology, in particular in the field of
cosmic neutrinos and gravitational waves.
122
10
An Abbendum
The majority of this dissertation (up to the numerical simulations) was written during the
summer of 2015. In October of that year rumours started circulating that the LIGO experiment had detected gravitational waves, and was offically confirmed in February 2016
[119].
The form of the gravitational waves detected, originated from a binary black hole merger
and it is known that the production of gravitational waves is a fundamental prediction of
any cosmological inflationary model. Naturally, in light of this recent milestone it is worth
briefly noting how the possible future detection of primordial gravitational waves is of great
importance to cosmology. Indeed primordial gravitational waves are not expected in early
Universe models not featuring inflation, thus "making them a smoking gun probe of inflation." [118].
123
A
Appendix
A.1
Energy-Momentum Tensor
The energy-momentum tensor is periodically referred to throughout this disseration, so presented here is a derivation based on Neother’s theorem. Consider the Lagrangian for a real
scalar field
1
1
L = η µν ∂µ ϕ∂ ν ϕ − m2 ϕ2
2
2
1 2 1
1
2
= − ϕ̇ + (∇ϕ) − m2 ϕ2 .
2
2
2
Looking to Neother’s theorem and defining the transformation
δϕ(x) = X(ϕ),
to be a symmetry of the Lagrangian provided it changes by a total derivative
δL = ∂µ F µ
for some function F µ . Begin by making an arbitrary transformation of the fields δϕ:
δL =
∂L
∂L
δϕ +
∂µ ϕ
∂ϕ
∂(∂µ ϕ)

=




∂L
∂L 
∂L 
− ∂µ 
δϕ + ∂µ 
.
∂ϕ
∂µ ϕ
∂(∂µ ϕ)
Assuming the equations of motion are satisfied (the first term in the square brackets = 0)
then


∂L
δϕ.
δL = ∂µ 
∂(∂µ ϕ)
However for the symmetry transformation δϕ = X(ϕ), δL = ∂µ F µ giving
∂µ J µ = 0,
Jµ =
∂L
X(ϕ) − F µ (ϕ).
∂(∂µ ϕ)
Now consider an infinitesimal transformation for infinitesimally small parameter :
xν −→ xν − ν
=⇒ δJ µ
=⇒
ϕ(x) −→ ϕ(x) + ν ∂ν ϕ(x).

ν

δL  ν
=
∂ν ϕ(x) + Lν
∂(∂µ ϕ)


δL
=
(∂ ν ϕ) − Lg µν ν .
∂(∂µ ϕ)
Define the term inside the square brackets as the energy momentum tensor
T µν =
δL
(∂ ν ϕ) − Lg µν .
∂(∂µ ϕ)
124
A.2
Alternative Method for Computing the Spectra Using Slow Roll
Expansion
The derivations here are an addition for the content that appears in chapter 5.5.1.
Starting from the lowest order solution
Ph (k) = 16 π G H 2
Pφ (k) =
9 H4
5ϕ̄˙ 2
where H and ϕ̄˙ are assumed constant during inflation. This assumption is based on
defining the slow roll expansion using ϕ̄(t) or V (ϕ̄) which is equivalent to assuming constant
˙ and
H, ϕ̄(t), V (ϕ̄), Ḣ, V (ϕ̄)
∂V
.
∂ϕ
These formulas apply when H and ϕ̄(t) are slightly varying
precisely at the time k = aH, use this assumption so that the tilts at first order in slow roll
are,
Ph (k) = 16 π
2
GH k=aH
9 H4 Pφ (k) = ¯2 5ϕ̇ .
k=aH
¯ when k = k∗ and the
These equations are evaluated using the difference between (H, ϕ̇)
time when it equals


H2
k∗ + dk = (a∗ + da)(H∗ + dH) = a∗ H∗ + a∗ 1 + ∗ dH.
Ḣ∗
Using the slow roll approximation
dk ≈
a∗ H∗2
dH.
Ḣ∗
Tensor tilt
nt u
ln Ph (k∗ + dk) − ln Ph (k∗ )
ln (H∗ + dH)2 − ln H∗2
2k∗ dH
u
u
ln Ph (k∗ + dk) − ln (k∗ )
(dk/k∗ )
H∗ dk
therefore
nt ≈
2k∗ Ḣ∗
a∗ H∗3
so
125
(401)
nt ≈
2Ḣ∗
.
H∗2
This can be re-expressed as a function of the potential derivatives giving
nt ≈ −
(∂V /∂ϕ)2∗
= −2∗ .
8πGV∗2
Scalar tilt
Following the same method:
d lnPφ ns − 1 =
dlnk k∗
dln(H 4 /ϕ̄˙ 2 ) =
dlnk
k∗
dlnH dlnϕ̇ =4
.
−2
dlnk dlnk k∗
k∗
It is already known that
dlnH dlnk =
k∗
Ḣ∗
,
H∗2
and rewriting the second term as
dlnϕ̄˙ dlnk k∗
dlnH dlnϕ̇ =
dlnk dlnH k∗

H∗



H∗ ϕ̄¨∗ 
Ḣ∗
ϕ̄¨∗
=  2 × 
=
˙∗
H∗
H∗ ϕ̄˙ ∗
Ḣ∗ ¯ϕ
so that the final result is
ns − 1 = 4
Ḣ∗
ϕ̈∗
−2
.
2
H∗
H∗ ϕ̄˙ ∗
Re-writting this in terms of potential derivatives and the slow roll approximations,
ϕ̈∗ = −
(∂ 2 V /∂ϕ2 )ϕ̇ (∂V /∂ϕ)Ḣ
+
3H
3H 2
so that
ns − 1 = 6
A.3
Ḣ∗ 2(∂ 2 V /∂ϕ2 )∗
Ḣ∗ 2(∂ 2 V /∂ϕ2 )∗
+
=
6
+
= −6∗ + 2η∗ .
H∗2
3H∗2
H∗2
8πGV∗
Newtonian Gauge Line Element - Equation 106
Starting off with the metric,


gµν = diag (1 + 2φ), −a2 (1 − 2ϕ), −a2 (1 − 2ϕ), −a2 (1 − 2ϕ)
this can be split into
gµν = ḡµν + δgµν .
126
Through the Einstein equations, the metric perturbations will be coupled to perturbations in the matter distribution. The following derivations are with respect to flat FLRW
background space-time with line element


ds2 = a2 (t)dt2 − δij dxi dxj .
The metric perturbations can be categorized into three distinct types: scalar, vector and
tensor. Classification is based on symmetry properties of homogeneous and isotropic backgrounds which is invariant with respect to the group of spatial rotations and translations.
• δg00 behaves like a scalar −→ δg00 = 2a2 φi ,
• δg0i = a2 B,i + Si . Here B is a scalar with B,i =
∂B
,
∂xi
and S is a vector with zero
divergence. S,ii = 0 and has two independent components.


• δgij = a2 2ϕδij + 2E,ij + Fi,j + Fj,i + hij . E, ϕ are scalar functions, and the vector
Fi has zero divergence (F,ii = 0). hij is a three tensor with hii = 0 and hij,i = 0.
The number of independent functions forms the group δgαβ : 4 functions for scalar perturbations 4 functions for vector perturbations and 2 functions for tensor perturbations.
Consider the coordinate transformation:
xα −→ x̄α = xα + εα
where εα is a small function of space and time.
At a given point of the space-time manifold the metric tensor in the coordinate system x̄
can be calculated using the usual transformation law
g̃αβ (xρ ) ≈
∂xγ ∂xδ
gγδ (xρ ) ≈(0) gαβ (xρ ) + δgαβ −(0)gαδ εδ,β −(0) gγβ εγ,α
α
β
∂ x̃ ∂ x̃
where only linear terms have been kept. The (o) denotes with respect to the Friedmann
metric which depends on x̃.
Now
(0)
gαβ (xρ ) ≈(0) gαβ (x̃ρ ) −(0) gαβ,γ εγ ,
giving rise to the following transformation law
127
δgαβ −→ δg̃αβ = δgαβ −(0) gαβ,γ εγ −(0) gγβ εγ,α −(0) gαδ εδ,β .
Writing the spatial components of the infinitesimal vector εα ≡ (ε0 , εi ) as εi = εi⊥ + ζ ,i
then
• εi⊥ : 3 vector with zero divergence,
• ζ: scalar field.
It is concluded that
• δg̃0i = δg0i + a2 εi⊥i + (ζ i − ε0 ),i
• δg̃ij = δgij + a
2
0
2 aa δij ε0
+ 2ζ,ij + (ε⊥i,j + ε⊥j,i )
• δg̃00 = δg00 − 2a(aε0 )0 .
Where the primes are differentiation with respect to conformal time.
For scalar perturbations the metric takes the form:


ds2 = a2 (1 + 2φ)dτ 2 + 2B,i dxi dτ − (1 − 2ϕ)δij − 2E,ij dxi dxj .
Under a change of coordinates
1
φ −→ φ̃ − (aε0 )0
a
ϕ −→ ϕ̃ = ϕ +
a0 0
ε
a
B ←− B̃ = B + ζ 0 − ε0
E ←− Ẽ = E + ζ.
Only ε0 and ζ contribute and by choosing them appropriately any of the 4 functions
φ, ϕ, B, E vanish.
The simplest gauge group invariant linear combinations of these functions which span
the 2D space of physical perturbations are,
Φ=φ−
1h
a(B − E 0 )]0
a
Ψ=ϕ+
a0
(B − E 0 ).
a
128
In the longitudinal (conformal Newtonian) gauge B = E = 0. These conditions fix the
coordinate system uniquely. Therefore


ds2 = a2 (1 + 2φ), 1(1 − 2ϕ)δij dxi dxj 
From which the metric (103) can be read off.
129
B
Recalls of the Cosmological Standard Model - Derivations
B.1
The FLRW Metric - Equation 1
Homogeneity implies that at any given time, physical parameters such as density and temperature are the same everywhere, hence the following interval,
ds2 = dt2 − dl2 ,
where dl2 = gij dxi dxj . Here ds is the line element, t is cosmic time, and dl is the
small length element. Due to isotropy g0i is zero. gij is the metric with diag (-1,+1,+1,+1)
signature.
For dl2 look for a three dimensional space of constant curvature, analogous to the surface
of a sphere. Imagine embedding this in four dimensions, so taking the Cartesian co-ordinates
(x, y, z, ω) and replacing (x, y, z) by the spherical polars (ρ, φ, ϕ) giving
dl2 = dρ2 + ρ2 dΩ2 + dω 2
where dΩ2 is the angular terms, and
x2 + y 2 + z 2 + ω 2 = ρ 2 + ω 2 = R 2 .
Noticing
ρdρ + ωdω = 0,
this can be re-arranged to find
dω 2 =
ρ2 dρ2
ρ2 dρ2
=
,
ω2
R 2 − ρ2
meaning
dl2 = dρ2 +
ρ2 dρ2
+ ρ2 dΩ2 ,
2
2
R −ρ
therefore
dl2 =
dρ2
+ ρ2 dΩ2 .
1 − ( Rρ )2
130
Setting ρ = Rr the homogeneous isotropic space of all three possible curvatures k =
+1, 0, −1 can be expressed:


dr2
dl2 = R2 
+ r2 dΩ2 .
1 − kr2
For R to be an arbitrary function of time R(t) requires


dr2
+ r2 dφ2 + r2 sin2 φϕ.
ds2 = −dt2 − R2 (t)
1 − kr2
Replacing R2 (t) with a2 (t) to arrive at the final result:

2
2
2
ds = dt − a(t)
B.2
!
dr2
+ r2 dφ2 + sin2 φϕ2 .
2
1 − kr
The Einstein Tensor - Equation 2
Deforming Minkowski space into the Newtonian space-time gives the following:
gµν = ηµν − 2Φ(x)δµν + O(2 )
where the Newton gravity potential Φ(x) obeys the usual
δ ij ∂i ∂j = 4πGρ
for mass density ρ and is an infinitesimally small expansion parameter.
The stress-energy tensor Tµν for a perfect fluid is given by
Tµν = ρ + P Uµ Uν + P gµν
where ρ and P are mass density and pressure and Uµ gives the local 4 velocity of the
fluid. Ttt = ρ and Tti = Tij = 0. Any equation governing the geometry of space-time must
recover δ ij ∂i ∂j Φ = 4πGρ in the Newtonian case, hence stress- energy conservation must be
consistent with the Einstein equation. The Einstein tensor Gµν ≡ Rµν − 21 gµν R is conserved,
∆µ Gµν = 0, so the Einstein tensor must take the form
Gµν = kTµν .
Newtonian space-time tells us that
Gtt = 2δ ab ∂a ∂b Φ
131
Gti = Git = Gij = 0
Ttt = ρ
Tti = Tit = Tij = 0
to leading order in . The ti and ij components of the Einstein equation are consistent
and the tt component gives
2δ ab ∂a ∂b Φ = kρ
(402)
for some constant k. In Newtonian theory it is expected that δ ab ∂a ∂b = 4πGρ, leading
to the conclusion, k = 8πG therefore
Gµν = 8πGTµν .
Hence the Einstein tensor:
1
Gµν ≡ Rµν − R = 8πGTµν .
2
B.3
The Riemannian Curvature Tensor - Equation 3
Geometry is locally Euclidean/Minkowski, however deviations away from flat space leads to
curvature described by the Riemann curvature tensor.
The commutator of two covariant derivatives acting on a function f vanishes:


∇α , ∇β f
= ∇α ∂β f − ∇β ∂α f




= ∂α ∂β f − Γµαβ ∂µ f  − ∂β ∂α f − Γµβµ ∂µ f .
However a co-vector field Bµ generally does not vanish,


∇α , ∇β Bµ
ν
= Rαβµ
Bν
ν
where Rαβµ
is the Riemann tensor.
Evaluating this in terms of Christoffel symbols,
132



∇α ∇β Bν = ∂α ∇β Bν  −


= ∂α ∂β Bµ −

=
Γδβµ Bσ 

−

−

Γναµ ∇β Bν 

Γναβ ∇µ Bµ 

−
Γναµ ∂β Bν


−∂α Γσβµ Bσ


Γναβ ∇ν Bµ 
+
Γναµ Γσβν Bσ
+ ∂α ∂β Bν −

−
Γσβν Bσ 

Γναβ ∇ν Bµ 
− Γνβµ ∂α Bν − Γναµ ∂β Bν

= −∂α Γσβµ Bσ + Γναµ Γσβν Bσ + ...
where the ... terms are symmetric in α ↔ β. Hence


∇α , ∇β Bµ
σ
= Rαβµ
Bσ
σ
where Rαβµ
= ∂β Γσαµ − ∂α Γσβµ − Γναµ Γσβν − Γνβµ Γσαν
B.4
The Friedmann Equation - Equation 6
The results for Rtt and Rij are derived in appendix F.7. The results will be used with the
following line element.
ds2 = −dt2 + a2 (t)gµν dxµ dxν ,
Rtt = −3
ä
a
and


Rij = 2k + 2ȧ2 + aägij .
Using equation (2) with components µ = ν = 0
1
Gtt = Rtt − gtt R = 8πGTtt .
2
Hence the following Ricci scalar
R = Rµν g µν = Rtt g tt + Rij g ij ,
ä
1
= 3 + Rij 2 gij ,
a a
ä
1
= 3 + 2k + 2ȧ2 + äa 2 gij g ij ,
a
a 

2
ȧ
ä
ä
2k
= 3 + 3 2 + 2
+ ,
a
a
a
a
2
ä
k
ȧ
=6 +6 2 +6
a
a
a
133
.
Substituting this into the Einstein Equation (2)
2
ä
k
ä
ȧ
Gtt = −3 + 3 + 3 2 + 3
a
a
a
a
= 8πGρ(t),
rearranging gives the Friedmann equation (6):
2
ȧ
a
B.5
+
k
8πG
ρ
=
2
a
3
The Conservation Equation - Equation 7
This derivation makes use of the following identity
∇µ = g µν ∇ν .
The conservation equation is derived from stress energy conservation, that is ∇µ Tµν = 0,


∇µ Tµν = g µα ∇α Tµν = g µα ∂α Tµν − Γβαν Tµβ − Γβαν Tβν .
Taking ν = t:


µ µα 
∇ g
∂α Tµt −
Γβαt Tµβ
−
Γβαµ Tβt 
therefore,
g tt ∂t Ttt − g ij Γkjt Tik − g ij Γtij Ttt = 0
− ρ̇(t) −
1 ij ȧ
1
g δjk a2 P (t)gik − 2 g ij aȧgij ρ = 0
2
a
a
a
ȧ
ȧ
− ρ̇ − 3 P − 3 ρ = 0
a
a
Rearranging leads to the conservation equation (7):
ȧ
ρ̇ + 3 ρ + P
a
B.6
= 0.
The Acceleration Equation - Equation 8
This is calculated from the i, j component of the Einstein tensor,
1
Gij = Rij − gij R = 8πGTij .
2
Using a number of results from the derivation of the Ricci scalar (appendix B.4),
134

2
ä
k
ȧ
2k + 2ȧ + aä gij − 3a gij  + 2 +
a a
a
2
2


2
= k + ȧ + 2äa gij = −8πGa2 P gij .
From which it can be seen
2aä = −8πGa2 P − ȧ2 − k.
Eearranging gives the conservation equation,

2
ȧ
ä
1
= −4πGP − 
a
2
a
= −4πGP −

k
+ 2 ,
a
4πG
ρ,
3 4πG
ρ + 3P .
=−
3
B.7 ρ As a Function of the Scale Factor - Equation 9
This equation is derived using the conservation equation (7) and the equation of state P =
ωρ. Taking the conservation equation (7) it can be re-written as
dρ
1 da
= −3
ρ(1 + ω)
dt
a dt
dρ
da
= −3 (1 + ω),
ρ
a
for some constant a0 . After integrating,
ρ
ln
ρ0
a
= −3(1 + ω)ln
a0
=


3(1+ω)
a
0
.
ln
a
Taking natural logs leads to the equation (9).
ρ(t) = ρ0
a0
a
135
3(1+ω)
.
C
Setting the Scene For Inflation - Derivations
C.1
Ratio of Total Initial Energy to Initial Kinetic Energy - Equation
21
E t = Eik + Eip = E0k + E0p
2

ȧi
Eik = E0k  
ȧ0
Eit
Eik + Eip
=
Eik
Eik
Ek + Ep
= 0 k 0
Ei
=
C.2
E0k
+
E0k
 2
p
E0  ȧ0 
ȧi
Ratio of Total Initial Energy to Initial Kinetic Energy Upper Limit
- Equation 22
E0k ≈ |E0p |
ȧ0
≤ 10−28
ȧi
Eit
Eik


= 1 +
2
E0p  ȧ0 
E0k
ȧi
2

ȧ0
= 2 
ȧi
= 2 × 10−28
2
' 10−56
C.3
Density Parameter Ω initially close to unity - Equation 24

2
Ωi − 1
k (a0 H0 )2
a20 ȧ20 a2i
ȧ
 0
=
=
=
2
2
2
Ω0 − 1
(ai Hi )2
k
ai a0 ȧi
ȧi
Therefore

2
ȧ0
Ωi − 1 = Ω0 − 1   ≤ 10−56 .
ȧi
136
C.4
The Amount of Inflation At the Time of Nucleosynthesis - Equation 32
Ωf
Ωn
− 1
− 1
ȧ2n
tf
=
2
ȧf
tn
=
Taking tn ' 1 and Ωn − 1 ' 1017 ,
Ωf
− 1
10−17
'
tf
.
sec
Therefore
tf 1
a2i
'
2
af
sec 1017
1

ai
tf −17 
'
10
af
sec
2
so
− 1

2
af
tf 
'
ai
sec
C.5
109 .
Horizon Distance dh (t1 , t2 ) - Equation 39
Defining the line element as

!
dr2
ds2 = dt2 − a(t)2 
+ r2 dθ2 + sin2 θφ2 ,
1 − kr2
where for radial particles θ=φ = 0
ds2 = dt2 − a2 (t)
dr2
.
1 − kr2
Now ds2 = 0 hence
dr2
,
1 − kr2
dr
dt = a(t) √
.
1 − kr2
dt2 = a2 (t) −
Integrating where dl = ds is the length element
dH (t1 , t2 ) =
Zr2
r1
Zr2
dl =
a(t) √
r1
dr
1 − kr2
therefore
dH (t1 , t2 ) = 2
Zr2
dl = 2
0
Zr2
0
137
a(t2 ) √
dr
.
1 − kr2
C.6
How the Horizon grows - Equation 47
dH (ti , tf ) = 2af
Za2
a1
da
,
a2i Hi
a2
2af Z
=
da a−2 ,
Hi a2 a
1
2af a−1
,
=
Hi −1
af
2af ,
=−
Hi a ai
2af
2af
=−
+
,
Haf
Hai


2af  1 
,
=
H ai − af


2  af
=
− 1.
Hi ai
C.7
Equation 51
2 af
dH (ti , tf ) =
−1
Hi ai
hence
dH (ti , tf )
2
af
2
af
=
−1 =
−1
af
Hi af ai
Hf af ai
as Hi = Hf .
As 2RH (t0 ) = dH (tdec , t0 ) '
2
,
H0
substituting it into the equality gives
2
af
2
−1 ≥
,
Hf af ai
a0 H0
giving the final equality
1
af
1
−1 ≥
.
Hf af ai
a0 H0
C.8
H 2 + Ḣ - Equation 57
This derivation makes use of the Friedmann Equation (6): H 2 =
Ḣ = −4πG(ρ + P )
Differentiating with respect to t
138
ȧ2
a2
=
8πG
ρ
3
−
k
a2
where
ȧä ȧ3
ȧ
ȧ
− 3 − k 3 = −4πG ρ + P ,
2
a
a
a
a
multiplying through by
a
ȧ
ä ȧ2
k
− 2 − 2 = −4πG ρ + P ,
a a
a
where the first two terms
ä
a
−
ȧ2
a2
= Ḣ. Finally rearranging,
ȧ2
ä
k
= −4πG(ρ + P ) + 2 + 2 ,
a
a
a
= H 2 + Ḣ.
where k is set to zero.
139
D
Inflationary Dynamics - Derivations
D.1
Density Equation - Equation 64
Tµν = diag(ρ, −P, −P, −P )
Tµν = ∂µ ϕ ∂ν ϕ − gµν
1
∂α ϕ∂ α ϕ + V (ϕ)
2
Taking the T00 component:
T00 = ∂0 ϕ ∂0 ϕ − gtt
1
∂α ϕ∂ α ϕ + V (ϕ)
2
In the sum over α, only the α = 0 term is different from zero,
2
T00 = ∂0 ϕ
+
1
∂0 ϕ)2 + V (ϕ)
2
−
2
1
∂0 ϕ + V (ϕ)
=
2
2
1
ρ=
∂0 ϕ + V (ϕ)
2
ϕ̇2
=
− V (ϕ).
2
D.2
Pressure Equation - Equation 65
Following a similar procedure which was used to calculate the energy density (64), but this
time looking for the Tij component.
Tµν = ∂µ ϕ∂ν ϕ − gµν
Tij = ∂i ϕ∂j ϕ − gij
2
= ∂i ϕ
−
1
∂α ϕ∂ α ϕ + V (ϕ)
2
1 tt
g ∂α ∂ α ϕ + V (ϕ) ,
2
1
∂α ϕ∂α ϕ + V (ϕ)
2
In the sum over α only the α = 0 terms are different from zero
2
1
∂0 ϕ − V (ϕ)
2
ϕ̇2
=
+ V (ϕ).
2
P =
140
D.3
Klein Gordon Equation - Equation 71


1
L = a3  ∂µ ϕ∂ µ ϕ − V (ϕ).
2
L is the lagrangian seen in equation (62) where the Lagrangian has been multiplied by
the scale factor to account for the Universe’s expansion.
Using the classical Euler- Lagrange Equation of motion:


d  ∂L  ∂L
−
= 0,
dt ϕ̇
∂ ϕ̇
the following is calculated
∂V
∂L
= −a3
,
∂ϕ
∂ϕ
∂L
= a3 ϕ̇,
∂ ϕ̇
and


∂V
d  ∂L 
= 3a2 ȧϕ̇ + a3 ϕ̈ = −a3
.
dt ∂ ϕ̇
∂ϕ
Substituting in these values
ȧ
∂V
ϕ̈ + 3 ϕ̇ = −
,
a
∂ϕ
where
ȧ
a
= H(t). Therefore the final result is:
ϕ̈ + 3H ϕ̇ + V 0 (ϕ) = 0.
D.4
Relation between Ḣ and ϕ̇ - Equation 73
From Equation (57)
Ḣ =
ä
− H2
a
Substituting in equation (69)
8πG
8πG 1 2
Ḣ =
V − ϕ̇2 −
ϕ̇ + V
3
3 2
Setting k = 0
Ḣ ' −4πGϕ̇2 .
141
+
k
a2
D.5 N In Terms of the Potential and Scalar Traversed - Equation 91
8πG
V
3
H2 =
3H ϕ̇ ' −
∂V
,
∂ϕ
hence
H'−
∂V /∂ϕ
.
3ϕ̇
Dividing by H 2 /H
8πG
V
H2
3
= ∂V
/∂ϕ
H
− 3ϕ̇
=−
8πGV
,
∂V /∂ϕ
where ϕ̇ << V .
As N = ln
af
ai
=
Rtf
Hdt, then substituting in the solution it is seen
ti
af
N = ln
ai
=
Ztf
Hdt = −8πG
ϕi
ti
D.6 N - Equation 92
N = −8πG
Zϕf
ϕi
V
,
V,ϕ
and from equation (81)
2

V
 ,ϕ 
V
<< 48πG,
which can be inverted to give


V 
1

<< √
.
V,ϕ
48πG
Inserting this into the equation for N leads to
N >>
>>
Zϕf
ϕf
Z
1
√
dϕ
48πG
ϕi
1 √
∆ϕ.
48πG
142
V
.
V,ϕ
D.7
Quantities Relative to the Present Co-moving Scale - Equation 93
ak Hk
k
=
a0 H0
a0 H0
ak ae areh aeq Hk
=
ae areh aeq a0 H0
af areh aeq Hk
.
= e−N (k)
areh aeq a0 H0
143
E
Cosmological Perturbations - Derivations
E.1
Hamiltonian for L = 12 ∂µ χ∂ µ χ - Equation 110
Using the Lagrangian L = 12 ∂µ χ∂ µ χ with the Euler-Lagrange equation of motion:


∂L  ∂L
∂ρ 
−
= 0.
∂(∂ρ χ)
∂χ
Then,
1
∂µ η µν ∂ν χ = 0,
2


1 µν 
η
∂µ χδνρ + ∂ µ χδµρ  = 0,
2


1 ν ρ
∂ χδν + ∂ µ χδµρ  = 0,
2


1 ρ
∂ + ∂ ρ  = ∂ ρ χ = 0.
2
The Euler Lagrange equation is
∂ρ ∂ ρ χ = 0 −→ χ̈ − ∂i ∂ i χ = 0
The momentum conjugate is defined as P = π =
L
∂ χ̇
= χ̈ and the Hamiltonian density is
defined as H = π(x)φ̇(x) − L(x). It is easy to compute the Hamiltonian, H:

H=
Z

1
d3 xH = ẋ2 − ∂0 χ∂ 0 χ − ∂i χ∂ i χ
2


1Z 3  2
=
d x ẋ + ∂i χ∂ i χ.
2
E.2
Fourier Transform of the Hamiltonian - Equation 111
χ(x, t) =
Z
d3 k ik.x
e χ(k, t)
(2π)3
Therefore,


Z
d3 k ik.x
∂2


+
∇
e χ(x, t) = 0.
∂t2
(2π)3
Acting on this

χ̈k

+ (ik)(−ik)χk  = 0
therefore
χ̈k + k 2 χk = 0.
144
E.3
Fourier Transformed Hamiltonian - Equation 112


1Z 3  2
H=
d x ẋ + ∂i χ∂ i χ
2
(403)
Applying the Fourier transform

Hχ(x,t)

1 Z 3  ∂2
=
dx
χ + ∂i χ∂ i χ
2
∂t2


1Z 3 
d x χ̇k χ̇∗k + (ik ∗ )(ik)χk χ∗k 
=
2


1Z 3 
d x χ̇k χ̇∗k + k 2 χk χ∗k 
=
2
having used k ∗ k = χ2 .
E.4
Lagrangian for Tensor Perturbations - Equation 131
ahλ
y≡√
−→ hλ =
32πG

√
32πGy
a

a4  1
Lt =
∂µ hλ ∂ µ hλ + div ,
16πG 4


√
√
a4  1
32πG µ 32πG
=
∂µ
y∂
y + div ,
16πG 4
a
a


1
y y
= a4 ∂µ ∂ µ ,
2
a a




1
= a4  − ya−2 a0 + a−1 y 0  − ya−2 a0 + a−1 y 0  − ∂i (ya−1 )2 ,
2



2 

y
1
y0
a0
y0
a0
= a4  − y 2  − y 2  − ∂i  ,
2
a
a
a
a
a

0
02
02

2 

2 
1
y
a
a
y
= a4  2 − 2y 0 y 3 + y 2 4 − ∂i  ,
2
a
a
a
a

1
= y 02 a2 − 2y 0 ya0 a + y 2 a02 − ∂i ya .
2

1
a0
a0
=⇒ Ly = y 02 − 2yy 0 + y 2
2
a
a

1
a0
=  y0 −
2
a
2
2
− ∂i y

2
− ∂i y .
as Lt −→ Ly .
145
2

,
E.5
The Hamiltonian of the Tensor Lagrangian - Equation 132

a0
a0
1
Ly = y 02 − 2yy 0 + y 2
2
a
a
2
− ∂i y
2


The momentum conjugate is given by

∂L
a0
∂
a0
πy = Py = 0 = 0 y 02 − 2 yy 0 +
y
∂y
∂y
a
a
2


a0
= 2y 0 − 2 y
a
So it is seen
πy = y 0 −

H=
Z
d3 xH =
Z

d3 xπy ẏ − L
2

=
Z
a0
y.
a

a0
1
a0
d3 xy 0 − y  −  y 0 − y
a
2
a
2

1
a0
H = y 0 − y  + ∂i y
2
a
E.6
2
2
− ∂i y
2




Fourier Equation of Motion - Equation 133

1
a0
a0
Ly = y 02 − 2yy 0 + y 2
2
a
a
∂L
Using the Euler-Lagrange equation ∂µ ∂(y
0) −
y 00 −
∂L
∂y
2
− ∂i y
2


=0
a0
y + ∂i2 y = 0
a
which can be written as


∂2
a0

−
+ ∂i2 y(x, t) = 0
∂t2
a
Using the Fourier transform:
y(x, t) =
Z
d4 xeik.x y(k, t),
The equation of motion is Fourier transformed to
yk00 −
a0
yk + (ik)(−ik)yk = 0,
a
which simplifies to
yk00 + k 2 −
a0
yk = 0.
a
146
(404)
E.7
Large Wavelength Limit Solution - Equation 140
It is known
s
yk =


i
1
1 − e−ikη .
2k
kη
In the limit k << aH then kη −→ 0 and using kη = −1 and k = aH then
s
1 i
2k kη
i
= −√
2k 3 η
ikη
=√
2k 3 η
iaH
=√
2k 3 η
yk<<aH = −
E.8
Variable hλ - Equation 141
ahλ
−→ hλ =
y≡√
32πG
√
32πG
y
a
Therefore using previous results
√
hλk =
32πG iaH
√
a
2k 3
s
= iH
E.9
16πG
k3
Long Wavelength Solution in Exact De-Sitter Limit - Equation 163
H=
ȧ
−→ adt = H −1 da
a
This can be integrated by parts to see
Z
adt =
Z
H −1 da = H −1 a −
Z
ad(H −1 ).
Writing
d(H −1 ) =
d(H −1 )
dt
dt
then
Z
ad(H −1 ) =
Z
a
Z
d(H −1 )
d(H −1 ) 1
dt =
H da.
dt
dt
147
(405)
Therefore
Z
d(H −1 ) −1
H da,
dt


Z
−1
d(H −1 ) −1
d(H
)
=
H a − a d
H −1 ,
dt
dt
ad(H −1 ) =
Z


Z
d d(H −1 ) −1 
d(H −1 ) −1
H a− a 
H
dt,
=
dt
dt
dt


Z
d(H −1 ) −1
d  d(H −1 ) −1  −1
=
H a−
H
H da,
dt
dt
dt






Z
d(H −1 ) −1
d d(H −1 ) −1  −1
d d d(H −1 ) −1  −1 
=
H a− 
H
H a− a 
H
H
,
dt
dt
dt
dt dt
dt




Z
d d(H −1 ) −1  −1
d2  d
d(H −1 ) −1
=
H a− 
H
H a+
(H −1 )(H −1 )2 H −1 da,
dt
dt
dt
dt2 dt





d2 d
d(H −1 ) −1
d d
=
H a −  (H −1 )(H −1 )H −1 a +  2  (H −1 )(H −1 )2 H −1 da
dt
dt dt
dt dt

−


d3 d
a 3  (H −1 )(H −1 )3 dt.
dt dt
Z
Taking the first three terms:

Z



d
d d
d2 d
adt = (H −1 )H −1 a −  (H −1 )H −1 H −1 a + 2  (H −1 )(H −1 )3 H −1 a − .....
dt
dt dt
dt dt
n

= aH
−1
d  −1
H .
dt
Therefore,


HZ

φ1 (k) = C1 (k) 1 −
a(t)dt
a


H
d
= C1 (k)1 − a H −1
a
dt
n

H −1 .
At first order,


H
φ1 (k) = C1 (k)1 − aH −1 (−1)H −2 Ḣ 
a


Ḣ
= −C1 (k) 2 .
H
148
E.10
First order derivation ns − 1 - Equation 187
ns − 1 =
dlnPh
dlnk
Noting that
dlnPh
dlnPh
dN
=
×
dlnk
dN
dlnk
and
dlnH dln
dlnPh
=2
−
dN
dN
dN
= −2 − 2( − η).
Substituting this in:
dlnPh
= (−2 − 2( − η)) × (1 + )
dlnk
= −2 − 2 + 2η
= 2η − 4η
E.11
Tensor Tilt - Equation 189
nt =
dlnφ
dlnk
dlnPφ
dlnPφ
dN
=
×
dlnk
dN
dlnk
where
dlnPφ
dlnH
=2
= −2.
dN
dN
Substituting it all in
nt = −2(1 + )
nt ' −2.
149
F
Inflationary Models - Derivations
F.1
Equation of Motion: Chaotic Inflation - Equation 195
1
1
L = ∂µ ϕ∂ µ ϕ − m2 ϕ2 − m2 ϕ2 .
2
2
Using the Euler-Lagrange equation of motion
∂L
∂(∂µ ϕ)
−
∂L
∂ϕ
= 0,


∂L
∂ 1
1
=
∂σ ϕη µσ ∂σ ϕ + m2 ϕ − M 2 ϕ2 
∂(∂µ ϕ)
∂(∂ρ ϕ) 2
2
1
1
= η µσ ∂σ ϕδµρ + η µσ ∂σ ϕσµρ
2
2
1 ρ
1 ρ
= ∂ ϕ+ ∂ ϕ
2
2
= ∂ ρ ϕ.
∂L
= −2m2 ϕ − m2 ϕ = −m2 ϕ
∂ϕ
So the Euler-Lagrange equation is
∂ρ ∂ ρ + m2 ϕ = 0 −→ ϕ̈ = −m2 ϕ.
F.2
Chaotic Inflation Einstein Equation - Equation 197
The Lagrangian for this model is given by L = 21 ∂µ ϕ∂ µ ϕ − m2 ϕ2 − 12 m2 ϕ2 . The Einstein
tensor is given by Gµν = 8πGTµν . The Einstein equation gives us
H2 =
8πG
k
ρ − 2.
3
a
Which is equivalent to
H2 +
k
8πG
=
Ttt .
2
a
3
Where Tµν is given by

Tµν

1
= ∂µ ϕ∂ν ϕ − gµν  g ρσ ∂ρ ϕ∂σ ϕ − V (ϕ)
2
As such Ttt is



1
1
Ttt = ∂t ϕ∂ t ϕ − gtt  g tt ∂t ϕ∂t ϕ −  − m2 ϕ2 − m2 ϕ2 
2
2
1
m2 2
= ∂t ϕ∂ t ϕ +
ϕ.
2
2
150
Substituting this in


k
8πG  2
H2 + 2 =
ϕ̇ + m2 ϕ2 
a
6


1
= ϕ̇2 + m2 ϕ2 
6
Where 8πG = 1.
F.3 H =
ȧ
a
- Equation 198
To derive equation (198) requires the following results:


k
1
H 2 + 2 = ϕ̇2 + m2 ϕ2 ,
a
6
ϕ̈ + 3H ϕ̇ = −m2 ϕ,
and
ϕ̈ << 3H ϕ̇, H 2 >>
k
, ϕ̇2 << m2 ϕ2 .
a2
Therefore


k
1
H 2 + 2 = ϕ̇2 + m2 ϕ2 ,
a
6
where H 2 >>
k
.
a2
Then using ϕ̇2 << m2 ϕ2


1
H 2 = m2 ϕ2 ,
6
mϕ
ȧ
H= √ = .
a
6
F.4 ϕ̇ derivation - Equation 199
This derivation makes use of the equation of motion ϕ̈ + 3H ϕ̇ = −m2 ϕ and using ϕ̈ <<
3H ϕ̇
3H ϕ̇ = −m2 ϕ.
Using H =
mϕ
√
6
and substituting in
3ϕ̇
√ = −m,
6
so after a simple rearrangement:
s
ϕ̇ = −m
151
2
.
3
F.5
Starobinsky Action - Equation 226
This will be derived very generally, consider an f (R) theory where the action is given by,
S=
√
1
d4 x −g 2 f (R)
2κ
Z
where κ = 8πG. This can be defined in the Jordan frame where the energy-momentum
tensor of the matter fields is covariantly conserved, ∇µ Tµν = 0 and particles follow geodesics
of the metric.
Consider the conformal transformation:
g̃µν = Ω2 gµν .
As an immediate consequence
√
√
−g = Ω−4 −g̃.
Under a conformal transformation the Ricci scalar transforms according to


˜ − 6g̃ µν ∂µ ω∂ν ω ,
R = Ω2 R̃ + 6ω

where ω = lnΩ, ∂µ ω =
∂ω
∂ x̃µ
˜ =
and √
√1 ∂µ 
−g̃

−g̃g̃ µν ∂ν ω .
Writing the action in the form [35]:
S=
where U =
F R−f
.
2κ2
S=
=
Z
Z
Z

√

1
d4 x −g  2 F R − U 
2κ
The new action can be written as:
√

√


1
d4 x −g  2 F R − U 
2κ



1
˜ − 6g̃ µν ∂µ ω∂ν ω  − Ω−4 U .
d4 x −g̃  2 F Ω−2 2R̃ + 6ω
2κ
The Einstein frame is obtained for the choice Ω2 = F , F > 0 [35].
A new scalar field ϕ is introduced by
√
3
κϕ =
lnF.
2
Therefore ω = lnΩ as ω =
κϕ
√
.
6
√ ˜
R
Using Gauss’s theorem then d4 x −g̃ ω
vanishes and so the action becomes
152

S==
=
=
=
=
V =
U
F2
=



Z
√
1
˜ − 6g̃ µν ∂µ ω∂ν ω  − Ω−4 U ,
d x −g̃  2 F Ω−2 R̃ + 6ω
2κ
Z
√
1
d x −g̃  2 R̃ − 6g̃ µν ∂µ ω∂ν ω  − Ω−4 U ,
2κ
Z
√
1
kϕ kϕ
d x −g̃  2 R̃ − 6g̃ µν ∂µ √ ∂ν √  − Ω−4 U ,
2κ
6
6
Z
√
1
1
1
d4 x −g̃  2 R̃ − g̃ µν ∂µ ϕ∂ν ϕ − 2 U ,
2κ
2
F
Z
√
1
1
d4 x −g̃  2 R̃ − g̃ µν ∂µ ϕ∂ν ϕ − V (ϕ).
2κ
2
4






4


4




1 RF −f
.
2κ2 F 2
To evaluate V (ϕ) requires the following pieces of information:
√2
√2
R
∂f
2
kϕ
3
F = e 3 kϕ , f (R) =
=1+
,
R
=
3M
e
− 1
∂R
3M 2

In the following calculations x =
q
2
kϕ.
3

V (ϕ) =


1 F R
f
− 2 ,
2
2
2κ
F
F

2

R
1  R (R + 6M
2)
,
= 2
−
2κ F
F2


R
R2 
1 R
− 2−
,
= 2
2κ F
F
6M 2 F 2



1 R
1
R 
= 2  1 − −
,
2κ F
F
6M 2 F









1 3M 2 (ex − 1) 
(3M 2 (ex − 1)) 
1
= 2
−
1
−
,
2κ
ex
ex
6M 2 ex

1
(ex − 1) 
1 3M 2 (ex − 1) 
= 2
1
−
−
,
2κ
ex
ex
2ex

1 3M 2 (ex − 1)  x
ex − 1 
= 2
,
e
−
1
−
2κ
e2x
2

1 3M 2 (ex − 1)  1 x 1 
e −
,
= 2
2κ
e2x
2
2


1 3M 2
= 2  2x (ex − 1)(ex − 1),
4κ e
=
3M 2 (ex − 1)2
.
4κ2
e2x
153
Taking
(ex −1)2
e2x
this can be written as
(ex − 1)2
(ex − 1)(ex − 1)
=
,
e2x
e2x
e2x − ex − ex + 1
=
,
e2x
= 1 − 2e−x − e−2x ,
= (1 − ex )2 .
Therefore substituting everything back in leads to
2 
√ 2κ
3
R̃
1
S = d4 x −g̃  2 − g̃ µν ∂µ ϕ∂ν ϕ − M 2 1 − e− 3 ϕ  .
2κ
2
4
√


154
G
f(R) Gravity - Derivations
G.1
f(R) Equation of Motion - Equation 235
In metric f(R) gravity one arrives at the field equations by varying with respect to the metric
and treating the connection independently.
The variation of the determinant is given by
√
1√
δ −g = −
−ggµν δg µν .
2
The Ricci scalar is R + g µν Rµν and its variation with respect to the inverse metric is
δR = Rµν δg µν + g µν δRµν


ρ
ρ 
= Rµν δg µν + g µν ∇ρ δTνµ
− ∇ν δTρµ


Where δT λµν = 12 g λa ∇µ δgaν + ∇ν δgaµ − ∇a δgµν 
Therefore,
δR = Rµν δg µν + gµν δg µν − ∇µ ∇ν δg µν .
The variation of S as


Z
√
√
δS
1
= d4 x 2 δf (R) −g + f (R)δ −g ,
µν
δg
2κ

=

Z
√
1
1√
d4 x 2 F (R)δR −g −
−ggµν δg µν f (R),
2κ
2
Z
1 √
1
d4 x 2 −g F (R)Rµν δg µν + gµν δg µν − ∇µ ∇ν δg µν − gµν δg µν f (R).
2κ
2

=


So

δS =
Z



1 √
1
d4 x 2 −gδg µν F (R)Rµν − gµν f (R) + gµν − ∇µ ∇ν F (R).
2κ
2
For an equation of motion δS = 0 and so the equation of motion becomes

F (R)Rµν

1
− f (R)gµν + gµν − ∇µ ∇ν F (R) = k 2 Tµν ,
2
where
Tµν
√
2 δ( −gLm )
= −√
.
−g
δg µν
155
G.2
Ricci Scalar F(R) theory - Equation 242
ds2 = gµν dxµ dxν = −dt2 + a2 (t)dx2
with g00 = −1, g11 = g22 = g33 = gii = a2 (t).
Using the definition of the Christoffel symbol:


1
Γabc = g aγ ∂b gγc + ∂c ggγ − ∂γ gbc 
2
For this metric there are 64 Christoffel symbols of which only 9 are non-zero (note
0 = t, 1 = 2 = 3 = i):
•

Γ011

1
= g 00 ∂1 g01 + ∂1 g10 − ∂0 g11 ,
2


1
= g 00  − ∂0 g11 ,
2


1
= −  − ∂t a2 (t),
2
= a(t)ȧ(t).
Here Γ011 = Γ022 = Γ033 = a(t)ȧ(t)
•

Γ101 = Γ110

1
= g 11 ∂0 g11 + ∂1 g01 − ∂1 g01 ,
2


1
= g 11 ∂0 g11 ,
2


1 1  2 
=
∂t a (t) ,
2 a2 (t)


1 1 
=
2a(t)ȧ(t),
2 a2 (t)
=
ȧ(t)
.
a(t)
For this metric Γ101 = Γ110 = Γ202 = Γ220 = Γ303 = Γ330 =
ȧ(t)
.
a(t)
The Ricci tensor can be calculated from the Christoffel symbols however there are just
two non-zero components: R00 and Rii .
156
The Riemann curvature tensor (from which to calculate the Ricci tensor) is given by
δ
Rαβµ
= ∂β Γδαµ − ∂α Γδβµ + Γναµ Γδβν − Γνβµ Γδαν .
Therefore the two components are
•
Rtt = ∂β Γβtt − ∂t Γββt + Γνtt Γββν − Γνβt Γβtν ,
= 0 − ∂t Γββt + Γνtt Γββν − Γνβt Γβtν ,
ȧ
= 0 − 3∂t + 0 − Γiit Γiti ,
 a

2
= −3ȧ(−1)a−2 ȧ + a−1 ä − 3


ȧ
a
2

ä ȧ2
ȧ2
= −3 − 2  − 3  ,
a a
a
ä
= −3 .
a
•
Rii = ∂α Γαii − ∂i Γαiα + Γαα Γii − Γαi Γiα ,
= ∂α Γαii + Γα0α Γ0ii − Γ0ii Γii0 − Γii0 Γ0ii ,
= ∂α Γαii + 3Γi0i Γ0ii − 2Γ0ii Γii0 ,
= ∂α Γαii + Γi0i Γ0ii ,
= ∂0 Γ0ii + Γi0i Γ0ii ,
= ȧ2 + aä + ȧ2 ,
= 2ȧ2 + aä.
Using these results the Ricci scalar is
R = Rµν g µν = Rtt + Rii


ä
ȧ2
ä
= 3 + 3  + 2 2 ,
a
a
a

2
ä
ȧ
= 6 +
a
a


,

= 6Ḣ + H 2 .
Where Ḣ + H 2 = aä .
157
,
G.3
Field Equation In a Flat FLRW background - Equation 243
This derivation requires the following results:
1
(m)
,
Σµν ≡ F (R)Rµν − f (R)gµν − ∇µ ∇ν F (R) + gµν 2F (R) = κ2 Tµν
2
Tνµ(M ) = diag(−ρM , PM , PM , PM ),


R00 = −3Ḣ + H 2 ,
Rii = 2ȧ2 + aä,


R = 6Ḣ + 2H
2
.
Writing R00 in terms of the Ricci scalar and Hubble factor:


R00 = −3Ḣ + H 2 ,


1
= − 6Ḣ + 2H 2 − H 2 ,
2
1
= − R + 3H 2 .
2
An expression is sought for the energy-momentum tensor for the modified Friedmann
equation,
Tνµ
= diag − ρM , PM ,
Tµν = gαµ Tνµ .
Replacing the new definition of R00 into the equation of motion
1
Σµν = F (R)R00 − f (R)g00 − ∇0 ∇0 F (R) + g00 F (R) = κ2 g00 T00 .
2
Working out explicitly the fourth term on the left hand side:


√
1
g00 F (R) = (−1) √ ∂0  −gg 00 ∂0 F ,
−g


1 ∂
∂
= − 3  − a3 F  ,
a ∂t
∂t
=
1 2
a3
3a
ȧ
Ḟ
+
F̈ ,
a3
a3
= 3H Ḟ + F̈ .
158
Working out the third term with the covariant derivatives it will become ∂0 ∂0 F since
F (R) is a scalar quantity, and will cancel F̈ finally gives the modified field equation:


1
1
F 3H 2 − R + f (R + 3H Ḟ ) = κ2 ρM ,
2
2
so
3F H 2 =
G.4
(F R − f )
− 3H Ḟ + κ2 ρM .
2
Second Modified Field Equation in flat FLRW background - Equation 244
1
(m)
Σµν ≡ F (R)Rµν − f (R)gµν − ∇µ ∇ν F (R) + gµν 2F (R) = κ2 Tµν
2
and for µ = ν = i
1
F (R)Rii − f (R)gii − ∇i ∇i F (R) + gii F (R) = κ2 gii Tii .
2
Evaluating the last term on the left hand side:




√
1
gii F (R) = a √ ∂µ  −gg µν ∂ν F ,
−g
2


√
√
1
1
= a √ ∂0  −gg 00 ∂0 F  + a2 √ ∂i  −gg ii ∂i F ,
−g
−g
2


= a2


1
1 
1
∂0 − a3 Ḟ  + a2 3 ∂i a3 2 ∂i F ,
3
a
a
a
a2
a3
1
a
= − 3 3a2 ȧḞ − a2 3 F̈ +
∂i a ∂i F + ∂i ∂i F,
a
a
a
a
1
dt
∂
dt
∂
a
F + ∂i ∂i F,
= −3a2 H Ḟ − a2 F̈ +
a dx ∂t
dx ∂t
ȧ ∂t 2
2
2
= −3a H Ḟ − a F̈ +
Ḟ + ∇i ∇i F,
a ∂x
= −3a2 H Ḟ − a2 F̈ + a2 H Ḟ + ∇i ∇i F,
= −2a2 H Ḟ − a2 F̈ + ∇i ∇i F,
where for a flat FLRW universe
∂t
∂x
2
= a2 . Substituting the above into the equation of
motion gives


2
2ȧ
+ aäF −
a2
f − 2a2 H Ḟ − a2 F̈ = κ2 a2 PM ,
2
159




2ȧ2 + aä 
1

F − f − 2H Ḟ − F̈ = κ2 PM ,
2
a
2
ȧ2 
1
aä

+
3
F − f − 2H Ḟ − F̈ = κ2 PM .
2
2
a
a
2
Replacing the third term from the first modified field equation (243) with
−
f
1
= 3H 2 F − F R + 3H Ḟ − κ2 ρM
2
2
then the second modified field equation becomes


1
ḢF + 3H 2 F + 3H 2 F − F R + 3H Ḟ − 2H Ḟ − F̈ = κ2 ρM + P + M ,
2
1
ḢF + 6H F − F 6 2H 2 + Ḣ + H Ḟ − F̈ = κ2 ρM + PM ,
2
2
and so the field equation reads
2
− 2ḢF = F̈ − H Ḟ + κ
G.5
ρM + PM .
The first modified field equation in the absence of matter - Equation 246
Taking equation (243) and setting ρM = 0 this can be re-written as
3F H 2 =
For the specific model f (R) = R +
FR − R
− 3H Ḟ .
2
R2
6M 2
but to keep things general this is written as
f (R) = R + αRn .
∂
f (R),
∂R

∂
=
R + αRn ,
∂R
F =
= 1 + nαRn−1 .
To see what form the field equations take, requires the following product, F R = R +
nαRn and,
160
∂
∂
1 + nαRn−1
Ḟ = F =
∂t
∂t
= n(n − 1)αRn−2 Ṙ
Substituting this into the field equation
1
n−1
2
n
n
n−2
3 1 + nαR
H =
R + nαR − R + αR
− 3H n n − 1 αR Ṙ ,
2
1
2
2
2
R + 2αR − R − αR − 3H 2αṘ ,
3 1 + 2αR H =
2
1
3 1 + 2αR H 2 = αR2 − 6HαṘ.
2
G.6 H 2 - Equation 247
Cosmic acceleration can be realised in the regime F = 1 + nαRn−1 >> 1, so dividing
equation (246) by 3nαRn−1


n(n − 1)αRn−2 Ṙ
1 (n − 1)αRn 
−
3H
,
H2 ' 
6
nαRn−1
3nαRn−1 1
1n−1
Ṙ
R−H ,
6 n
R 
n − 1
Ṙ
'
R − 6nH .
6n
R
'
For n = 2


1
Ṙ
H 2 ' R − 12H 
12
R
G.7
Need to think up a name - Equation 248


∂
Ṙ = 6 2H 2 + Ḣ 
∂t
= 6 4H Ḣ + Ḧ .
6 4H Ḣ + Ḧ
Ṙ
= R
6 2H 2 + Ḣ
H Ḣ 4 +
=
H2 2 +
4H Ḣ
2H 2
Ḣ
=2 ,
H
=
161
Ḣ
H Ḣ

H 
Ḣ 2
where the approximations
Ḣ
2
H
<<
1
Ḣ
H Ḣ
and
<< 1 have been used. Again taking
equation (243) with ρM = 0,
3F H 2 =

and H =
n−1 
R
6n
FR − f
− 3H Ḟ ,
2

−
Ṙ 
6nH R


n − 1
Ḣ
H2 =
6 2H 2 + Ḣ − 6nH2 ,
6n
H


n − 1
=
6 2H 2 + Ḣ 2nḢ ,
6n


n − 1
Ḣ
=
2 − (2n − 1) 2 .
n
H
Now,
Ḣ
n
= 2 − (2n − 1) 2 ,
n−1
H
2−
Ḣ
n
= (2n − 1) 2 .
n−1
H
Therefore,
Ḣ
2(n − 1) − n
=
,
H2
(n − 1)(2n − 1)
n−2
=
,
(n − 1)(2n − 1)
≈ −1 .
G.8
The solution for H - Equation 251
Integrating for positive 1 :
dH
dt
H2
= −1 −→
dH
= −1 dt,
H2
Therefore
1
1
= 1 t −→ H =
.
H
1 t
The scale factor evolves as
ȧ
1
da
1
=
−→
=
dt.
a
1 t
a
1 t
Integrating this up,
ln a =
1
1
1
ln t = ln t 1 −→ a ∝ t 1 .
1
162
G.9
Inflationary Dynamics in terms of H - Equation 252
f (R) = R +
R2
6M 2
The presence of the linear term here in R causes inflation to end, so it cannot be neglected.
From before,

2
2
2
R = 6 2H + Ḣ




4

2
2
= 364H + Ḣ + 4H Ḣ 



∂
6 2H 2 + Ḣ  = 64H Ḣ + Ḧ 
∂t
Taking equation (246) 3 1 + 2αR H 2 = 12 αR2 − 6HαṘ, and substituting for α =
1
6M 2
and the other terms:
1 1
1
1
R H2 =
R2 − 6 2 H Ṙ,
2
2
M 
2 6M  M




1
1
1
2
2
4
2
31 + 2
6 2H + Ḣ H =
364H + Ḣ + 4H Ḣ  − 2 H 4H Ḣ + Ḧ ,
3M 2
12M 2
M
3 1+2
3M 2 H 2 + 12H 4 + 6ḢH 2 = 12H 4 + 3Ḣ 2 + 12ḢH 2 − 24ḢH 2 − 24ḢH 2 − 6H Ḧ.
And so
3M 2 H 2 + 6H Ḧ + 18H 2 Ḣ − 3Ḣ 2 = 0,
1
1 Ḣ 2
= 0,
Ḧ + M 2 H + 3H Ḣ −
2
2H
1 Ḣ 2 1 2
Ḧ −
+ M H = −3H Ḣ.
2H
2
G.10
Inflationary Dynamics H - Equation 254
Ḣ ≈ −
As Ḣ =
dH
dt
M2
6
this can be integrated:
ZH
Hi
dH ≈ −
Zt
ti
M2
dt.
6
Therefore,
M2
H ≈ Hi −
t − ti .
6
163
G.11
As
Evolution of the Scale Factor During Inflation - Equation 255
d
a
ȧ
−→ H = dt ,
a
a
this can be integrated to find an expression for a:
H=
Za
ai
t−t
t
Z i 2
da Z
M
≈ Hi dt −
(t − ti )d(t − ti ).
a
6
ti
0
Therefore,


M2
(t − ti )2 .
a ≈ ai exp Hi (t − ti ) −
12
G.12
Evolution of R During Inflation - Equation 256


R = 62H 2 + Ḣ ,
Ḣ = −
M2
6
So it is simple to see
R ≈ 12H 2 − M 2 .
G.13
Calculating tf - Equation 259
Ḣ
M
√
≈
1
−→
H
≈
f
H2
6
M2
Hf ≈ Hi −
t − ti
6
.
t=tf
Therefore after a simple re-arrangement
M2
M2
√ ≈ Hi −
(tf − ti ).
6
6
Taking
M
M2
M
√
≈ Hi +
ti − √ ,
6
6tf
6
the last term can be neglected as WMAPS [35] shows this value to be very small M ∝
1013 GeV .
So then
tf ≈
6Hi
+ ti .
M2
164
G.14
Number of E-foldings Between ti and tf - Equation 260
N≡
Ztf
dt
ti
As before, H =
d
a
dt
a
N=
Zaf
ai
tf
t−t
Z i 2
da Z
M
= Hi dt −
(t − ti )d(t − ti )
a
6
ti
0
M2
(tf − ti )2
' Hi (tf − ti ) −
12
G.15
N - Equation 261
tf ≈
6Hi
+ ti
M2
N ' Hi (tf − ti ) −
M2
(tf − ti )2
12
Substituting the first equation into the second:


2

M 2  6Hi
6H 2
+ ti − ti  ,
N ' Hi  2i + ti − ti  −
M
12 M 2
6Hi2 M 2 36Hi2
−
3,
M2
12 M 4
3Hi2
1
'
≈
,
2
M
21
'
where 1 =
G.16
M2
.
6H 2
V,ϕϕ - Equation 265
√
3M 2 
− 23 κϕ 
V (ϕ) =
1
−
e
,
4κ2




3M 2 
=
1 − 2e−xϕ + e−2xϕ ,
4κ2
where x =
q
2
κ.
3


dV
3M 2  −xϕ
=
2xe
− 2xe−2xϕ 
dϕ
4κ2
165


d2 V
3M 2  2 −xϕ
=
−
2x e
− 4x2 e−2xϕ ,
2
2
dϕ
4κ


3M 2
= − 2 2x2 e−xϕ 1 − 2e−xϕ ,
4κ
√
√2
κϕ 
− 23 κϕ 
3
1 − 2e
.
= −M e

V,ϕϕ
G.17

2
t̃ - Equation 266
Consider the flat FLRW space-time with the metric,
ds2 = gµν dxµ dxν = −dt2 + a2 (t)dx2
in the Jordan Frame. The metric in the Einstein frame is given by

2
2
2

2
2
2
ds̃ = Ω ds = F  − dt + a (t)dx
,
= −dt̃2 + ã2 (t̃)dx2 .
This leads to the following relations (for F > 0)
dt̃ =
√
F dt,
ã =
√
Fa
F,ϕ
Where F = e−2Qkϕ and Q = − 2κF
.
For our model F ≈
4H 2
M2
−→
t̃ =
2H
M
so it can be inferred
Ztf √
F dt.
ti
t
2 Z
=
Hdt.
M
ti

'
G.18
2

M
2
Hi (t − ti ) −
(t − ti )2 .
M
12
Evolution of ã - Equation 267
t̃ = −dt̃2 + ã2 (t̃)dx2


M2
a ' ai exp Hi (t − ti ) −
(t − ti )2 
12
Can write t̃ as
166


M2
2
t̃ ' 1 −
(t − ti )Hi ,
12Hi
M


M2
2
' 1 −
H (t − ti )Hi ,
2 i
12Hi
M



M2 1 M
2
M2
' 1 −
(t − ti )2 Hi ,
t̃
+
2
12 Hi 2
12
M


2
' 1 −


M 1 M
M
2
t̃ +  − t̃ + Hi (t − ti )2 Hi ,
2
12 Hi 2
2
M
2
' 1 −

M 1
t̃M +
12 Hi
2 M Hi

(t − ti ) Hi 12Hi2
M

2
.
t+ti
Therefore


M2 1
2Hi √
1 −

t̃M
= F.
12 Hi2
M
Now


t̃
M2
ai exp Hi (t − ti ) −
(t − ti )2  = ai eM 2 .
12
Therefore




2Hi M t̃
M2 1
ai e 2 .
t̃M 
ã = 1 −
2
12 Hi
M
t̃
M2 1
= 1 −
t̃M ãi eM 2 .
2
12 Hi
G.19
˜1 - Equation 273(a)

2
V,ϕ
˜1 ≈  
V
Previously,
2
V =

3M 
1 − e−
2
4κ
√2
3

κϕ 
,
dV
3M
=
dϕ
2κ
167
2
s
√2
2 −
e
3
3

κϕ1 − e−
√2
3

κϕ 
3M 2

˜1 '
'
2κ
1 
2κ2
2
q
2 −
e
3
√2

κϕ1 − e−
3

3M 2 
1
4κ2

2 4κ 
3 2κ2
e
−2
√2
3
−
−e
1−e
3

κϕ  
2


,
κϕ 
3

κϕ
√2
−2
√2
√2
κϕ
3
2 ,
√ −2
4  −2√ 2 κϕ
− 23 κϕ
,
3
e
1−e
'
3


√2
√2
4 √2
' e−2 3 κϕ 1 − 2e2 3 κϕ + e2 3 κϕ ,
3


√2
√2
4
' e−2 3 κϕ − 2e− 3 κϕ + 1,
3


2
4 √2
' e 3 κϕ − 1 .
3

G.20
˜2 - Equation 273(b)
˜2 ≈ ˜1 −
V,ϕϕ
2H̃ 2
√2
√
κϕ 
− 23 κϕ 
3
= −M e
1 − 2e

V,ϕϕ

2
√2
√
1 
2
κϕ 
− 23 κϕ 
3
−
M
e
1
−
2e
,
˜2 ≈ ˜1 −
3H̃ 2
√2 

√2
M 2 e 3 κϕ 
≈ ˜1 +
1 − 2e− 3 κϕ .
3H̃ 2

G.21


Ñ - Equation 275
Ñ =
Zt̃f
H̃dt̃ ≈ κ2
t̃i
V
dϕ
V,ϕ
2
√
3M 2 
− 23 κϕ 
V =
1
−
e
4κ2

dV
3M 2
=
dϕ
2κ
s
√2
2 −√ 2 κϕ 
1 − e− 3 κϕ 
e 3
3

168

√2

3M 2
V
=
V,ϕ
1 − e−
4κ2
q
3M 2
2κ
2 −
e
3
√2
3
− e−
1 
1
2κ
=
q
√2
3
=
=
1
1
2κ
2κ

q
q
2
2
3
3
s
3
,
κϕ 
κϕ 
,
κϕ
− e−

√2

√2

1
2κ 
q
1
2
3
1
=
2
1 − e−
κϕ 
3
2 −
e
3
κϕ 

√2

3
2
3

κϕ 
√2
e 3 κϕ ,

− 1,


1
3  2κ
κ q − 1.
2
2
3
Inserting this result into the definition of Ñ
1
2
s
1
≈ κ2
2
s
Ñ ≈ κ2
Zϕi
3
κ
2 ϕ
√2
e 3 κϕ − 1dϕ,


f
√2
ϕi
3  e 3 κϕ
− ϕ ,
κ q
2
2
κ

3
ϕf
1 3 √ 2 κϕ
e 3 ,
2 2
3 √2
≈ κ2 e 3 κϕ ,
2
√2
≈ e 3 κϕi .
s
≈ κ2
G.22
˜1 as a function of Ñ - Equation 276 (a)
−2
4  √ 2 κϕ
˜1 ≈
e 3 − 1 ,
3


3 √ 2 κϕi
Ñ = e 3 ,
4
Therefore
e
√2
3
κϕi
4
≈ Ñ .
3
Hence,
169
−2

4 4
˜1 '  Ñ − 1 ,
3 3
−1

4 16
4
'  Ñ 2 − 2 Ñ + 1 ,
3 9
3
4
3
'
 16 Ñ 2
9
,
− 2 43 Ñ + 1
3 1
11
4
−
+
,
4 Ñ 2 2 Ñ
3
3 1
'
.
4 Ñ 2
'
G.23
˜2 as a function of Ñ - Equation 276 (b)
√2 

√2
M 2 e 3 κϕ 
1 − 2e− 3 κϕ ,
˜2 ' ˜1 +
3H̃ 2
√2
3 √2
4
Ñ ' e 3 κϕ −→ e 3 κϕi = Ñ ,
4
3
H̃ ≈
M
.
2
Substituting in where appropriate,
√2 

√2
M 2 e 3 κϕ 
˜2 ' ˜1 +
1 − 2e− 3 κϕ ,
2
3H̃
√2
4 √2
' ˜1 + e 3 κϕ 1 − 2e− 3 κϕ ,
3


√2
√2
4
' ˜1 + e− 3 κϕi − 2e−2 3 κϕi ,
3




4 1
2
' ˜1 +  4 − 4 2 ,
3 3 Ñ
( 3 Ñ )
8
3 1
1
6
'
+
− 4 2,
4 Ñ 2 Ñ
( 3 Ñ )
3 1
1
3 1
'
+
,
−
2
4 Ñ
4 Ñ 2
Ñ
1
' .
Ñ
G.24
α - Equation 314
Taking equation (287) with δφ = 0 and δF = 0 and taking into account SM = 0
170


1 
Hα − ϕ̇ =
δ Ḟ − Ḟ α.
2F
Setting δ Ḟ = 0,


Ḟ 
= ψ̇
α H +
2F
and so
α=
ψ̇
Ḟ
2F
H+
.
Taking the gauge invariant quantities (309, 310, 311) then coupling with the aforementioned conditions Ṙ = ϕ̇ and so
α=
G.25
Ṙ
H+
Ḟ
2F
.
A - Equation 315
Taking equation (286)

!
1  2
∆
∆
ψ
+
HA
=
−
3H
+
3
Ḣ
+
δF
a2
2F
a2

!
1
−3Hδ Ḟ + κ2 ω,φ φ̇2 + 2κ2 V,φ − f,φ δφ+
2

!
κ2 ω φ̇δ φ̇ + 3H Ḟ − κ2 ω φ̇2 α + Ḟ A + κ2 δρM ,
and setting the relevant terms to zero (i.e δφ = δF δ φ̇ = SM = 0) and after substituting
in α (314),

!
!

∆
1 
ψ
+
HA
=
−
3H Ḟ − κ2 ω φ̇2 α + Ḟ A,
a2
2F

1
=− 
2F
3H Ḟ − κ2 ω φ̇2 R
H+

+ Ḟ A.
Ḟ
2F
Rearranging,



1
1
∆

Ḟ + H A = −  − 2 ψ2F +
2F
2F
a
3H Ḟ − κ2 ω φ̇2 R
H+
Ḟ
2F
Noticing that the gauge conditions (309, 310, 311) implies R = ψ then
A=−
1
Ḟ
2F

∆
 ψ+
+ H a2
171
3H Ḟ − κ2 ω φ̇2 R
H+
Ḟ
2F

.

.
H
1
Python Code - Scalar Potential
i m p o r t numpy a s np
i m p o r t math
3
import m a t p l o t l i b . pyplot a s p l t
from s c i p y . i n t e g r a t e i m p o r t o d e i n t
5
from s c i p y i m p o r t c o n s t a n t s
from s c i p y . i n t e r p o l a t e i m p o r t i n t e r p 1 d
7
# P l o t t i n g t h e e v o l u t i o n o f t h e c o s m o l o g i c a l e q u a t i o n s . We want t o p l o t t h e
e v o l u t i o n of the s c a l e f a c t o r a s a f u n c t i o n of time .
9
# To u s e ODEint a l l e q u a t i o n s must be w r i t t e n a s a s y s t e m o f 1 s t o r d e r ODES
s o we need t o re −w r i t e our e q u a t i o n s .
11
# y ’ ’+3 ( a ’ / a ) y ’ +w ’ ( y ) >>>> dy ’ / dx=−3 ( a ’ / a ) y ’ −w ’ ( y ) and s o dydx=y ,
n o t e t h a t y=p h i
# a ’= a ( 1 / ( s q r t 3 ) ) ( ( 1 / 2 ) y ’ ^ 2 +w( y ) ) ^2
13
# da ’ / dx= −a ( 1 / 3 ) ( y ’^2 −w( y ) ) , we w i l l however i g n o r e t h i s e q u a t i o n a s i t
i s not i n d e p e n d e n t .
15
# F i r s t want t o p l o t t h e e v o l u t i o n o f p a r a m e t e r s a s s o c i a t e d t o t h e
c o s m o l o g i c a l e q u a t i o n s s u c h a s t h e f i e l d phi , t h e s c a l e f a c t o r a e t c .
B a s e d on s c a l a r p o t e n t i a l
17
# P l e a s e n o t e t h a t a s mentioned i n t h e d i s s e r a t i o n t h e t i m e s p a c i n g was
d i f f e r e n t f o r e a c h g r a p h p r o d u c e d and c a n be found i n t h e g r a p h ’ s
accompanying c a p t i o n .
19
d e f cosmology ( X, t ) :
21
phi , dphi , a = X
23
# Vector
H = np . s q r t ( 1 / 3 . * ( 1 / 2 . * d p h i * * 2 + 1 / 2 . * p h i * * 2 ) )
da = H* a
# da / d t
25
d d p h i = −3. *H* d p h i − p h i
27
r e t u r n [ dphi , ddphi , da ]
# ddphi / dt
29
172
# hubble parameter
phi0 = 1 8 .
31
d p h i 0 = −np . s q r t ( 6 . ) / 3 . / 1 0 * * 18
a0 = 1 .
33
X0 = [ phi0 , dphi0 , a0 ]
# i n i t i a l values
H0 = np . s q r t ( 1 / 3 . * ( 1 / 2 . * d p h i 0 * * 2 . + 1 / 2 . * p h i 0 * * 2 . ) )
# i n i t i a l hubble
parameter
35
t=np . l i n s p a c e ( 0 . , 4 0 0 0 . , 4 0 0 0 0 . )
# t i m e s p a c i n g , i . e m e a s u r e i n g t i m e between
t =0 and t =4000 with 4 0 0 0 0 e v e n l y s p a c e d d a t a p o i n t s
37
from s c i p y . i n t e g r a t e i m p o r t o d e i n t
39
s o l = o d e i n t ( cosmology , X0 , t )
41
phi = s o l [ : , 0 ]
43
dphi = s o l [ : , 1 ]
a = sol [ : ,2 ]
45
n = np . l o g ( a )
H = np . s q r t ( 1 / 3 . * ( 1 / 2 . * d p h i * * 2 + 1 / 2 . * p h i * * 2 ) )
47
49
# Graph p l o t t i n g
51
53
plt . figure ( )
p l t . p l o t ( t , s o l [ : , 2 ] , l a b e l= ’ S c a l e f a c t o r ’ )
55
p l t . x l a b e l ( ’ time ’ )
plt . ylabel ( ’ Scale factor ’ )
57
p l t . t i t l e ( ’ S c a l e f a c t o r a s a f u n c t i o n of time ’ )
p l t . l e g e n d ( l o c= ’ b e s t ’ )
59
plt . s a v e f i g ( ’ s c a l e f a c t o r s c a l a r . jpg ’ )
61
plt . figure ( )
63
p l t . p l o t ( n , s o l [ : , 1 ] / s o l [ : , 2 ] , l a b e l= ’ S c a l e f a c t o r ’ )
p l t . x l a b e l ( ’ time ’ )
65
plt . ylabel ( ’ Scale factor ’ )
p l t . t i t l e ( ’ S c a l e f a c t o r a s a f u n c t i o n of time ’ )
173
67
p l t . l e g e n d ( l o c= ’ b e s t ’ )
p l t . s a v e f i g ( ’ eurgh . jpg ’ )
69
71
plt . figure ( )
p l t . p l o t ( t , s o l [ : , 0 ] , l a b e l= ’ P o t e n t i a l ’ )
73
p l t . x l a b e l ( ’ Time ’ )
p l t . y l a b e l ( ’ P o t e n t i a l , $V( \ p h i ) $ ’ )
75
p l t . t i t l e ( r ’ P o t e n t i a l a s a f u n c t i o n of time ’ )
p l t . l e g e n d ( l o c= ’ b e s t ’ )
77
plt . s a v e f i g ( ’ potential . jpg ’ )
79
plt . figure ( )
p l t . p l o t ( t , n , l a b e l= ’ e−f o l d s ’ )
81
p l t . x l a b e l ( ’ time ’ )
p l t . y l a b e l ( ’ number o f e−f o l d s ’ )
83
p l t . t i t l e ( ’ Number o f e−f o l d s a s a f u n c t i o n o f t i m e ’ )
p l t . l e g e n d ( l o c= ’ b e s t ’ )
85
plt . s a v e f i g ( ’ efolds . jpg ’ )
87
89
plt . figure ( )
p l t . p l o t ( n , s o l [ : , 0 ] , l a b e l= ’ e−f o l d s ’ )
91
p l t . x l a b e l ( ’ number o f e−f o l d s ’ )
p l t . y l a b e l ( ’ s c a l a r f i e l d $\ phi$ ’ )
93
p l t . t i t l e ( ’ B e h a v i o r o f t h e s c a l a r f i e l d and e−f o l d s ’ )
p l t . l e g e n d ( l o c= ’ b e s t ’ )
95
p l t . s a v e f i g ( ’ nphi . j p g ’ )
97
99
#Want t o p l o t t h e e v o l u t i o n o f t h e Mukhanov e q u a t i o n f o r a mode k .
101
d e f mukhanov ( F , t ) :
# S e t t i n g up our v a r i a b l e s t o c a l c u a l a t e t h e mode
f u n c t i o n i n t h e Mukhanov e q u a t i o n s
103
174
phi , dphi , a , U_r , dU_r , U_i , dU_i = F
# Our v e c t o r , U i s u s e d a s t h e
mode f u n c t i o n , e q u i v a l e n t t o \nu i n e q u a t i o n ( 3 7 6 )
105
H = np . s q r t ( 1 / 3 . * ( 1 / 2 . * d p h i * * 2 + 1 / 2 . * p h i * * 2 ) )
107
da = H* a
109
d d p h i = −3. *H* d p h i − p h i
111
dda = ( −1/3. * ( d p h i * * 2 −1/2. * p h i * * 2 ) ) * a
113
z = ( a * d p h i ) /H
115
ddU_r = (H* dU_r ) −( ( k * * 2/ a * * 2 ) −( ( dda / a )−H* * 2 ) ) * U_r
# Real part of the
mode f u n c t i o n
117
ddU_i = (H* dU_i ) −( ( k * * 2/ a * * 2 ) −( ( dda / a )−H* * 2 ) ) * U_i
o f t h e mode f u n c t i o n
119
121
r e t u r n [ dphi , ddphi , da , dU_r , ddU_r , dU_i , ddU_i ]
123
k= 1 0 0 . * a0 *H0
# Fixed k value
125
F0=[ phi0 , dphi0 , 1 . , 1 . / np . s q r t ( 2 . * k ) , 0 . , 0 . , np . s q r t ( 2 . * k ) ]
127
t=np . l i n s p a c e ( 0 . , 0 . 8 , 6 0 0 0 . )
# Time s p a c i n g
129
131
from s c i p y . i n t e g r a t e i m p o r t o d e i n t
s o l 2 = o d e i n t ( mukhanov , F0 , t )
133
phi = s o l 2 [ : , 0 ]
135
dphi = s o l 2 [ : , 1 ]
a=s o l 2 [ : , 2 ]
137
n=np . l o g ( a )
U_r=s o l 2 [ : , 3 ]
139
U_i=s o l 2 [ : , 5 ]
175
# Imaginery part
141
h = np . s q r t ( 1 / 3 . * ( 1 / 2 . * d p h i * * 2 + 1 / 2 . * p h i * * 2 ) )
x = ( a * h ) /k
143
145
modulus=( np . s q r t ( ( U_r * * 2 ) +( U_i * * 2 ) ) ) / a
#Mode f u n c t i o n
147
#More g r a p h s !
149
plt . figure ( )
151
p l t . p l o t ( x , U_r/ a , l a b e l=r ’ $ \ nu_r$ ’ )
p l t . l e g e n d ( l o c= ’ b e s t ’ )
153
p l t . x l a b e l ( ’ $aH/ k$ ’ )
p l t . y l a b e l ( r ’ $ \ nu_r / a$ ’ )
155
p l t . t i t l e ( r ’ The r e a l p a r t o f t h e Mukhanov e q u a t i o n ’ )
plt . grid ( )
157
p l t . s a v e f i g ( ’ U_r ( t ) . j p g ’ )
159
plt . figure ( )
161
p l t . p l o t ( x , U_i / a , l a b e l=r ’ $ \ n u _ i $ ’ )
p l t . l e g e n d ( l o c= ’ b e s t ’ )
163
p l t . x l a b e l ( ’ $aH/ k$ ’ )
p l t . y l a b e l ( r ’ $ \ n u _ i / a$ ’ )
165
p l t . t i t l e ( r ’ The i m a g i n a r y p a r t o f t h e Mukhanov e q u a t i o n ’ )
plt . grid ( )
167
p l t . s a v e f i g ( ’ U_i ( t ) . j p g ’ )
169
plt . figure ( )
171
p l t . p l o t ( k/h , modulus , l a b e l=r ’ $ \ nu$ ’ )
p l t . l e g e n d ( l o c= ’ b e s t ’ )
173
p l t . x l a b e l ( ’ $aH/ k$ ’ )
p l t . y l a b e l ( r ’ $ \nu/ a$ ’ )
175
p l t . t i t l e ( r ’ P l o t o f t h e Mukhanov e q u a t i o n ’ )
plt . grid ( )
177
p l t . s a v e f i g ( ’ modulus . j p g ’ )
176
179
181
p_1 =16. * 3 . 1 4 *H* * 2
# E q u a t i o n t o p l o t t h e o r e t i c a l power spectrum , not
incorporating the p o t e n t i a l .
183
plt . figure ( )
185
p l t . p l o t ( k/H, p_1 , l a b e l=r ’ $ n u m e r i c a l r e s u l t s $ ’ )
p l t . l e g e n d ( l o c= ’ b e s t ’ )
187
p l t . x l a b e l ( ’ $k / h$ ’ )
p l t . y l a b e l ( r ’ $P_t ( k ) $ ’ )
189
p l t . t i t l e ( r ’ $ \ m a t h c a l {P} _ t ( k ) $ a s a f u n c t i o n o f $k$ ’ )
plt . grid ( )
191
p l t . s a v e f i g ( ’ spectrum_p_1 . j pg ’ )
193
195
# L e t s now c a l c u l a t e t h e slow r o l l p a r a m e t e r s f o r t h e s c a l a r p o t e n t i a l
197
199
def slow_roll_parameters ( S , t ) :
201
phi , dphi , a , V, dV , ddV = S
H = np . s q r t ( 1 / 3 . * ( 1 / 2 . * d p h i * * 2 + 1 / 2 . * p h i * * 2 ) )
203
V=1/2. * p h i * * 2 .
dV=p h i
205
ddV=1.
dddV=0.
207
d d p h i=−3*H* d p h i −p h i
209
d S d t= [ dphi , −3 *H* d p h i −p h i , H* a , dV , ddV , dddV ]
211
return dSdt
213
p h i =18.
215
S0 = [ 18. , − np . s q r t ( 6 . ) / 3 . / 1 0 * * 1 8 . , 1 . ,
5 . 5 * 10 * * ( − 5 . ) , 1 8 .
177
,1.]
217
t=np . l i n s p a c e ( 0 , 4 0 , 4 0 0 )
219
from s c i p y . i n t e g r a t e i m p o r t o d e i n t
221
s o l 3 = o d e i n t ( s l o w _ r o l l _ p a r a m e t e r s , S0 , t )
223
p h i=s o l 3 [ : , 0 ]
d p h i=s o l 3 [ : , 1 ]
225
a=s o l 3 [ : , 2 ]
n=np . l o g ( a )
227
V = sol3 [ : ,3 ]
dV = s o l 3 [ : , 4 ]
229
ddV=s o l 3 [ : , 5 ]
231
plt . figure ( )
233
p l t . s e m i l o g y ( n , 1 / ( 1 6 . * 3 . 1 4 ) * ( dV/V ) , ’ b ’ , l a b e l = r ’ $ \ e p s i l o n $ ’ )
p l t . semilogy ( n , 1 / ( 8 . * 3 . 1 4 ) * ( 1 . /V) , ’ g ’ , l a b e l = r ’ $\ e t a $ ’ )
235
plt . xlabel ( ’n ’ )
p l t . t i t l e ( r ’ $ \ e p s i l o n $ and $ \ e t a $ a s a f u n c t i o n o f $n$ ’ )
237
p l t . l e g e n d ( l o c= ’ b e s t ’ )
plt . s a v e f i g ( ’ slow_roll_parameters . jpg ’ )
scalar_potential.py
I
Python Code - Starobinsky Potential
Due to the complexity of the code (compiled from several files) it is avaliable to be viewed/downloaded at [122]. Credit for this code goes to [123, 124, 125]. The user guide gives an
indepth explanation of the code and the parameters used [124].
178
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