Dissertation

Dissertation
Dissertation
submitted to the
Combined Faculties for the Natural Sciences and for Mathematics
of the
Ruperto-Carola University of Heidelberg, Germany
for the degree of
Doctor of Natural Sciences
put forward by
Simon Schettler
born in Poughkeepsie, USA
Oral examination: July 21st, 2015
A Model with
Two Periods of Inflation
Referees: PD Dr. Jürgen Schaffner-Bielich
Prof. Dr. Matthias Bartelmann
Abstract
In this work a scenario with two subsequent periods of inflationary expansion in the
very early Universe is examined. The model is based on a potential motivated by
symmetries being found in field theory at high energy. For various parameter sets of the
potential the spectra of scalar and tensor perturbations that are expected to originate
from this scenario are calculated. Also the beginning of the reheating epoch connecting
the second inflation with thermal equilibrium is studied. Both is done in comparison
with standard potentials leading to accelerated cosmic expansion. Perturbations with
wavelengths leaving the horizon around the transition between the two inflations are
special: It is demonstrated that the power spectrum at such scales deviates significantly
from expectations based on measurements of the cosmic microwave background (CMB).
This supports the conclusion that parameters for which this part of the spectrum leaves
observable traces in the CMB must be excluded. Parameters entailing a very efficient
second inflation correspond to standard small-field inflation and can meet observational
constraints. Particular attention is paid to the case where the second inflation leads
solely to a shift of the observable spectrum from the first inflation. A viable scenario
requires this shift to be small.
In dieser Arbeit wird ein Szenario mit zwei aufeinanderfolgenden Phasen inflationärer
Expansion im sehr frühen Universum untersucht. Das Potential zu diesem Modell stützt
sich auf Symmetrien, die für Feldtheorien bei hoher Energie relevant sind. Für verschiedene das Potential bestimmende Parametersätze werden die Spektren skalarer und
tensorieller Störungen berechnet, die durch die Inflationen in diesem Modell entstehen.
Außerdem wird der Beginn des Reheating untersucht, das den Übergang von der Inflation zur anschließenden Big-Bang-Expansion im thermischen Gleichgewicht bildet. Die
Ergebnisse werden jeweils mit Rechnungen zu den klassischen inflationären Potentialen
verglichen. Störungen auf Längenskalen, die um den Übergang zwischen den zwei Inflationen herum den Horizont verlassen, zeigen ungewöhnliches Verhalten: Die Ergebnisse
zeigen, dass das Leistungsspektrum für diese Wellenlängen signifikant von den Erwartungen abweichen, die sich aus Beobachtungen des kosmischen Mikrowellenhintergrundes
(CMB) ergeben. Dies lässt den Schluss zu, dass Parametersätze, für die dieser Teil des
Spektrums im CMB beobachtbar ist, ausgeschlossen werden müssen. Die Ergebnisse aus
Parametersätzen, die zu sehr starker Expansion während der zweiten Inflation führen,
liegen teils im durch die Beobachtungen erlaubten Bereich. Die Fälle, bei denen die
zweite Inflation ausschließlich zu einer Verschiebung des beobachtbaren Spektrums aus
der ersten Inflation führt, werden eingehender untersucht. Die Beobachtungen fordern
eine möglichst kleine Verschiebung.
Contents
1. Introduction
1
2. Some aspects of the theory of relativity
2.1. Special relativity . . . . . . . . . . . . . . . . . . . .
2.2. General relativity . . . . . . . . . . . . . . . . . . . .
2.2.1. Important quantities . . . . . . . . . . . . . .
2.2.2. The Friedmann–Robertson–Walker spacetime
2.2.3. Basics of metric fluctuations . . . . . . . . . .
2.2.4. Primordial spectra of scalar fluctuations . . .
2.2.5. Primordial spectra of tensor fluctuations . . .
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5
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3. Elements of field theory
3.1. Equations for flat and FRW spacetimes . . . . . . . .
3.2. Symmetries and conserved currents . . . . . . . . .
3.3. Chiral symmetry for two flavors of fermions . . . . .
3.4. Quantum chromodynamics . . . . . . . . . . . . . . .
3.4.1. The Lagrange density of QCD . . . . . . . . .
3.4.2. Symmetries and other properties . . . . . . .
3.5. Effective theories of strong interactions . . . . . . . .
3.5.1. Mesonic states and their chiral transformation
3.5.2. Chiral symmetry breaking . . . . . . . . . . .
3.5.3. Linear sigma model . . . . . . . . . . . . . . .
3.5.4. The dilaton field in string theory . . . . . . .
3.5.5. Linear sigma model with dilaton . . . . . . .
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19
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41
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5. Inflation
5.1. Short history of cosmological inflation . . . . . . . . . . . . . . . . . . .
5.2. Inflation as a solution to cosmological problems . . . . . . . . . . . . . .
5.3. Inflation driven by a homogeneous scalar field . . . . . . . . . . . . . . .
47
47
49
51
4. Hot
4.1.
4.2.
4.3.
Big Bang cosmology
Evolution of the universe after inflation
A cosmic timeline . . . . . . . . . . . . .
Problems of pure Big Bang . . . . . . .
4.3.1. Horizon problem . . . . . . . . .
4.3.2. Flatness problem . . . . . . . . .
4.3.3. Monopole problem . . . . . . . .
4.3.4. Primordial perturbations . . . . .
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5.3.1. Slow-roll inflation . . . . . . . . . . . . . . . . . . .
5.3.2. Large-field inflation . . . . . . . . . . . . . . . . . .
5.3.3. Hilltop inflation . . . . . . . . . . . . . . . . . . . .
5.3.4. Hybrid inflation . . . . . . . . . . . . . . . . . . . .
5.4. Inflation and the origin of fluctuations . . . . . . . . . . .
5.4.1. Scalar perturbations . . . . . . . . . . . . . . . . .
5.4.2. Tensor perturbations . . . . . . . . . . . . . . . . .
5.4.3. The spectra of fluctuations after slow-roll inflation
5.5. Examples and observations . . . . . . . . . . . . . . . . .
5.6. A universe without inflation? . . . . . . . . . . . . . . . .
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52
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77
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7. Fluctuations produced after inflation: Preheating
7.1. Initial vacuum fluctuations . . . . . . . . . . . . . . . . . .
7.2. Periodically changing mass and the Mathieu equation . . .
7.2.1. Broad and narrow parametric resonance . . . . . .
7.2.2. Transmission and reflection coefficients . . . . . . .
7.2.3. Evolution of the particle number . . . . . . . . . .
7.2.4. Particle production in Minkowski spacetime . . . .
7.3. Effects of the Hubble expansion . . . . . . . . . . . . . . .
7.4. Preheating within λφ4 theory . . . . . . . . . . . . . . . .
7.5. Preheating after hilltop inflation . . . . . . . . . . . . . .
7.5.1. Slow roll and oscillations of the homogeneous field
7.5.2. Tachyonic preheating . . . . . . . . . . . . . . . . .
7.5.3. Parametric resonance . . . . . . . . . . . . . . . . .
7.6. Preheating within a model with two periods of inflation .
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89
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6. Fluctuations after two periods of inflation
6.1. The potential and its simplification . .
6.2. Evolution of the homogeneous mode .
6.3. Scalar and tensor perturbations . . . .
6.4. Compatibility with measurements . . .
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8. Conclusion
119
Appendix A. Equations of linearized General Relativity
A.1. Linearized Einstein tensor . . . . . . . . . . . . .
A.2. Conformal transformations . . . . . . . . . . . . .
A.3. Linearization within FRW background . . . . . .
A.4. Linearized evolution equations . . . . . . . . . . .
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125
125
126
127
128
Appendix B. LATTICEEASY
B.1. General lattice issues . . . . . . . . . . . . .
B.2. The implementation of vacuum fluctuations
B.3. Staggered leapfrog method . . . . . . . . . .
B.4. Particle number density of classical fields . .
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131
131
132
134
135
Appendix C. An auxiliary calculation
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137
1. Introduction
After countless attempts to understand our world as a whole and yet many years and centuries after scientific methods have been adopted as the way of learning from nature, in
the early twentieth century Albert Einstein’s invention of the general theory of relativity
laid the foundation of modern cosmology. This chapter lists some of the most important
steps, leading to Einstein’s insight and afterwards to cosmology supported by precision
measurements. The presentation relies on information from Refs. [2, 53, 82, 83, 116].
As other branches of science, cosmology emerged from prehistoric myths. Even among
the early archeological evidence of mankind, such as the paintings in the caves of Lascaux
and Altamira, some have been interpreted as symbols and reproductions of observations
in the night sky. Much later, around 5500 years ago, megalithic cultures left their
monuments to be marveled at by civilizations to come. It is assumed that for example
the arrangements in Stonehenge have been built also in the context of astronomical
observations.
About 4500 year-old, written documents from ancient Egypt and Mesopotamia give
much more details on the belief about the cosmos in these cultures. It is not based on
observation, of course, but on religious traditions. Also in Indian and Greek culture,
cosmology maintained the character of a creation myth, often complemented by the assumption of cyclic processes of generation and destruction. It should be mentioned that
the approach to cosmology is in contrast with a more scientific treatment of astronomy
in ancient Mesopotamia.
A major step towards modern science has been made by Greek philosophers from 600
B.C. onward. They tried to understand natural phenomena in terms of general laws
leaving behind the belief in personal gods being responsible for them. Examples are
the Milesian philosophers Thales and Anaximander, who already made considerations
of symmetry: Taken to be at the center of the universe the Earth should be immobile.
This can be seen as an example that reasoning has been accepted as a possible way
towards insight, regardless of whether the result seems doubtful from a modern view.
A more religious school was established by the Pythagoreans. However, it should be
mentioned that they promoted the assumption of a spherical Earth not being at the
center of the universe. Around 430 B.C., this was a major new concept.
Astronomy had become an important field of observation and theory in ancient
Greece. In contrast, being more elusive to observation, cosmology left the area of interest
for a long time. Let us leave the presentation of ancient ideas of the cosmos by stating
that Plato and Aristotle argued in favor of Earth as the center of the universe. As a
Chapter 1. Introduction
consequence, the observed motions of the planets were not easy to understand. Nevertheless, the picture of the central immobile Earth and of planets moving on complicated
paths was not questioned in the following centuries.
Nicolaus Cusanus (1401–1464) speculated about the infinity of the cosmos and argued
that the Earth is not the center of the universe. Nicolaus Copernicus (1473–1543) is
famous for breaking with this tradition. His assumption of the sun as the center of the
universe can be dated back at least to the time around the year 1512. Giordano Bruno
(1548–1600) went even further and stated for example that from each point in space
the universe looks the same. This bold assertion is known as the cosmological principle
today. At that time it proved difficult for many scientists to accept a heliocentric world.
This changed during the seventeenth century when Galileo Galilei (1564–1642) observed
the phases of Venus and Johannes Kepler (1571–1630) published his discovery of a new
star, contradicting the assumed invariability of the fixed star sphere.
The heliocentric picture has been accepted and the Milky Way has proven to consist
of single stars when Isaac Newton (1642–1727) developed the first theory of gravity.
Published in 1687, his “Philosophiae Naturalis Principia Mathematica” were a first step
towards a unification of forces: Gravity on Earth could be explained by the same equations as Kepler’s laws of planetary motion. It is well-known that Newtonian mechanics
is sufficient to explain many phenomena and is applicable to a large range of scales.
Newton thought of the universe as being static and tried to find the mechanism which
prevents the distribution of stars from collapsing. However, such problems could not be
solved with the methods of the time.
Immanuel Kant (1724–1804) speculated about the cosmos outside our galaxy and
assumed that the known nebulae could be distant galaxies. This was also to back the
cosmological principle of homogeneity on large scales.
Serious doubts about the contemporary view on the universe were raised by Heinrich
Wilhelm Olbers (1758–1840) in 1823 when he considered the darkness of the night sky
contradictory to the assumption that the universe is static and infinite in both space and
time. A few years earlier, Carl Friedrich Gauss (1777–1855) has written to him about
his speculations on alternatives to Euclidean geometry. Bernhard Riemann (1826–1866)
identified three possibilities—flat space, positive curvature and negative curvature—
as candidates for a viable geometry of the universe at a given moment in time. He
pointed out that a three-dimensional space with positive curvature can be finite and
unbounded and that this might be the case in our universe. On this basis Karl Friedrich
Zöllner (1834–1882) could overcome Olbers’ paradox without putting bounds to time
and space of the universe. In 1900 Karl Schwarzschild (1873–1916) put limits on the
spatial curvature using the parallax of distant stars [118]. However, curvature of space
was beyond the interest of physics and astronomy of the time.
In 1913 Albert Einstein (1879–1955) appreciated the benefit of Riemann’s work with
regard to the inclusion of gravity into his special theory of relativity. The general theory
of relativity (GR) from 1915 was successfully tested during the solar eclipse in 1919. It
did not show its value for cosmology because the generally accepted picture of the cosmos
2
still had the dimensions of the Milky Way within a void surrounding. Application of
GR to cosmology was stimulated by Schwarzschild and Willem de Sitter (1872–1934).
In 1917 Einstein’s result included the cosmological constant such that a stationary state
is possible [45].
However, GR did not play an important role for astronomers in the following years.
Most important for many scientists were the size of the Milky Way and the nature of
the nebulae, topics which were discussed at the Great Debate in 1920. This was settled
by Hubble’s observation of Cepheid variables in the Andromeda Nebula. Using Henrietta Leavitt’s [86] relation of the period and the luminosity of those pulsating stars, he
could show that this structure is located far beyond the range of the Milky Way. This
supported the picture of an “island universe” with several similar structures surrounding
our galaxy. A further issue backing this picture was raised by Vesto Melvin Slipher’s
(1875–1969) observation of Doppler shifts in the spectral lines of nebulae receding from
us [119]. In 1922 Alexander Friedmann (1888–1925) realized the theoretical possibility
of an expanding cosmological solution of Einstein’s equation and also considered a beginning of the evolution with zero radius [54]. A similar result was obtained in 1927 by
Georges Lemaître (1894–1966) who ascribed the redshift of the nebulae to an effect of
cosmological expansion [87]. He is considered as the inventor of the Big Bang scenario.
Lemaître’s work became known only after Edwin Hubble (1889–1953), who himself was
unaware of these results, published his empirical studies on the linear relation between
redshifts and distances in 1929, see Ref. [72]. Compared to the modern value, he obtained a far too large proportionality constant between distance and redshift: In the
later interpretation, this means that he obtained much bigger recession velocities than
accepted today. Also Hubble’s work was not seen as groundbreaking at that time. Big
Bang models became more popular in the 1940’s: In 1948, George Gamow (1904–1968),
Ralph Alpher (1921–2007), and, for the sake of naming, Hans Bethe (1906–2005) published their “αβγ theory” of Big Bang nucleosynthesis. In the 1950’s the theory became
less popular because nucleosynthesis in stars and novae was found as an explanation for
the existence of heavier elements which could not be synthesized in the early universe.
A serious competitor was the steady state theory which extended the cosmological
principle to the assumption of homogeneity in time. Thus it avoided the age paradox
being a problem of the Big Bang models. However, starting in 1964 with the discovery of the cosmic microwave background (CMB) by Arno Penzias (*1933) and Robert
Woodrow Wilson (*1936), the steady state theory became disfavored by most scientists.
Robert Dicke (1916–1997) and James Peebles (*1935) interpreted the radiation as a relic
of the early universe [44, 110].
An important ingredient to Big Bang cosmology was added in the 1990’s, when observations of supernovae suggested a recent acceleration of the universe. For this discovery,
Saul Perlmutter (*1959), Brian Schmidt (*1967), and Adam Riess (*1969) shared the
Nobel Prize in 2011. Their method uses type Ia supernovae as standard candles similar
to the use of Cepheides earlier. The acceleration is attributed to the fact that vacuum
energy gives the dominant contribution to the energy content of the universe today.
3
Chapter 1. Introduction
The latest developments in cosmology were due to the increasingly accurate observations of the CMB fluctuations. Important projects in this respect were the space-based
measurements of COBE (Cosmic Background Explorer), WMAP (Wilkinson Microwave
Anisotropy Probe), and Planck [1]. COBE showed in 1992 the anisotropy of the CMB on
scales larger than 10◦ . The measurements have been improved dramatically by WMAP
since 2001 and by Planck since 2009. The details of the CMB anisotropies permit a
much deeper understanding of the cosmic history.
Today, the CMB fluctuations are explained by various mechanisms. One of them gives
rise to the primeval spectra of density fluctuations and gravitational waves which is later
imprinted on the background radiation of photons. This mechanism is called inflation.
Since the key issue of this work is the inflationary scenario in the early universe its
historical development is given some extra space in Section 5.2.
This thesis has the following structure: In Chapter 2 some basic notions of the theory
of relativity are introduced. Then cosmological perturbations and their characterizing
spectra are presented.
The discussion of cosmological inflation makes use of some concepts of field theory.
For example, with regard to the motivation of the potential, symmetries in field theories
play an important role. These concepts are the main topic of Chapter 3.
In Chapter 4 a timeline of the universe after inflation is given and the observational
problems arising for a scenario without inflation are discussed.
Chapter 5 reviews three classic scenarios of inflation together with their observable
consequences before the details of a model with two subsequent inflationary periods are
described in Chapter 6.
This work also includes calculations of particle spectra originating from the decay of
the inflaton field after inflation. More precisely, Chapter 7 gives a presentation of lattice
computations, which concern the early, non-perturbative phase of this decay. Such a
period of strong particle production is called “preheating”. It precedes (or is sometimes
also included into) the stage of reheating that leads to a state of thermal equilibrium.
When equilibrium is reached, the hot Big Bang evolution of the universe begins. The
calculations on preheating are done for the same potentials that are considered with
respect to the production of fluctuations during inflation.
The presentation ends with some remarks on the conclusions that can be drawn and
with hints to related and possible future work.
Throughout the thesis, units with c = kB = ~ = 1 are used. The Planck mass is
mPl = 1/G ≈ 1.2 · 1019 GeV. The signature of the metric is (+, −, −, −). If not stated
differently, dots denote derivative with respect to cosmic time. Primes are used for
derivatives with respect to conformal time with the exceptions V ′ , V ′′ , and U ′ , where
the derivative of the potentials is taken with respect to a scalar field. In this thesis
our Universe is called universe. A persistent distinction according to the degree of
uniqueness in the corresponding context seemed too contrived to me.
4
2. Some aspects of the theory of
relativity
2.1. Special relativity
In special relativity the notion of a distance in space is extended to spacetime. The
generalized concept is called the spacetime interval and up to an overall sign, which
depends on convention, it is defined as
ds2 = dt2 − dx2 − dy 2 − dz 2 .
(2.1)
With the metric tensor ηµν = diag(1, −1, −1, −1) this equation is conveniently written
as
ds2 = ηµν dxµ dxν .
(2.2)
Here and throughout this thesis the Einstein convention implies summation over each
pair of identical indices. Coordinates (t, x, y, z), in which the spacetime interval can
be written as above, are called Minkowski coordinates. They set up an inertial frame.
Different inertial frames are connected by coordinate transformations. Besides reversal
of coordinates, these transformations are elements of the Poincaré group, which consists
of translations and Lorentz transformations Λ ∈ SO(1, 3). The defining property of the
latter is the invariance of the metric,
Λαµ Λβν ηαβ = ηµν ,
(2.3)
where the entries of the transformation matrix Λ are spacetime independent. The coordinates in the new frame are
x̃µ = Λµν xν ,
(2.4)
and the transformation matrix can be written as
Λµν =
∂ x̃µ
.
∂xν
(2.5)
Under such a transformation a scalar field stays the same:
φ̃(x̃) = φ(x).
(2.6)
Chapter 2. Some aspects of the theory of relativity
The components of any (contravariant) vector field with upper indices, Aµ (x), behave
as in Eq. (2.4):
∂ x̃µ ν
õ (x̃) =
A (x).
(2.7)
∂xν
Upper indices can be lowered by contraction with the metric,
Bµ (x) = ηµν B ν (x),
(2.8)
obtaining a covariant vector. The result of a contraction Aµ Bµ is a scalar and does not
change under Lorentz transformations. So the transformation of Bµ has to be inverse
to that of Aµ ,
∂xν
Bν (x).
(2.9)
B̃µ (x̃) =
∂ x̃µ
The way of a particle through spacetime is called its worldline. It can be parameterized
by the time τ measured by an observer moving with the particle: xµ (τ ). Then the
4-velocity of the particle is
dxµ
uµ =
.
(2.10)
dτ
The 4-momentum is
pµ = muµ = (E, p),
(2.11)
with particle mass m, energy E and momentum p. The second version is valid also
for massless particles. If not single particles but fluids are considered, the velocity
and momentum fields uµ (xµ ) and pµ (xµ ) replace the last two functions. Energy and
momentum of a fluid are more properly accounted for with the help of the energy–
momentum tensor T µν . For a perfect fluid it can be written in terms of uµ , the energy
density ρ and the pressure p,
T µν = (ρ + p)uµ uν − pη µν .
(2.12)
The metric with upper indices η µν can be thought of as the inverse of ηµν , but it has
the same numerical entries in Minkowski spacetime. Energy–momentum conservation
is written as
∂µ T µν = 0.
(2.13)
2.2. General relativity
2.2.1. Important quantities
In combination with quantum mechanics, special relativity is able to treat the dynamics
of the forces in the standard model of particle physics. However it is not suitable
to handle gravity. To include gravitational effects the notion of inertial coordinates
within Minkowski spacetime has to be abandoned. Instead, spacetime is viewed as a
6
2.2. General relativity
more general, curved manifold which is described using a metric gµν (x). The interval
Eq. (2.2) is replaced by
ds2 = gµν (x)dxµ dxν .
(2.14)
Because in general the choice of a global inertial system is not possible, Lorentz transformations lose their specific character. One considers the much bigger class of diffeomorphic transformations of coordinates
xµ −→ x̃µ (xν ).
(2.15)
The transformation laws as written in Eqs. (2.6), (2.7), and (2.9) can be adopted without
change. Tensors with multiple indices transform like products of according numbers of
co- and contravariant vectors.
Raising and lowering indices is done with the metric gµν and its inverse gµν . The
determinant
det gµν = g < 0
(2.16)
√
is used to define the diffeomorphism invariant measure −gd4 x for integration. To
describe dynamics within curved spacetime also an appropriate derivative is needed:
Tensors that are defined on different points of a manifold need to be (parallel) transported to the same point. Only then they can be subtracted from each other in order
to calculate a derivative. The parallel transport is only along a small distance, and so
its effect is taken to be linear. Put together, a derivative transforming a tensor into a
tensor can be defined as
∇ν Aµ (x) = ∂ν Aµ (x) + Γµρν (x)Aρ (x)
(2.17)
and is called covariant derivative. The (non-tensorial) quantities Γµρν are called the
Christoffel symbols. They constitute the connection within general relativity (GR) and
they are calculated as
1
Γµνλ = gµρ (∂ν gρλ + ∂λ gρν − ∂ρ gνλ ).
2
(2.18)
The covariant derivative is defined with the help of parallel transport. It can be used
to quantify curvature because in curved space parallel transport depends non-trivially
on the path which is chosen between two points. The following commutator of two
covariant derivatives (acting on the vector Aρ ) corresponds to the difference of parallel
transport along two paths that are both piecewise along one coordinate:
ρ
[∇µ , ∇ν ]Aρ = (∇µ ∇ν − ∇ν ∇µ )Aρ = Aσ Rσµν
.
(2.19)
It can be checked that it is a linear transformation and all derivatives of the vector field
Aρ cancel. This yields the Riemann tensor,
µ
Rνρσ
= ∂ρ Γµνσ − ∂σ Γµνρ + Γµλρ Γλνσ − Γµλσ Γλνρ ,
(2.20)
7
Chapter 2. Some aspects of the theory of relativity
as a characterization of curvature. The Ricci tensor Rµν and the Ricci scalar R are
defined via contractions of the Riemann tensor:
σ
Rµν = Rµσν
,
R = Rµν gµν .
(2.21)
The action that defines GR is the Hilbert action with Lagrangian L proportional to R,
Z
√
m2Pl
(2.22)
SH = −
d4 x −gR.
16π
Variation with respect to the metric yields the left hand side of the Einstein equation
1
8π
Rµν − Rgµν = 2 Tµν .
2
mPl
(2.23)
From the calculation of the Riemann tensor it is seen that it contains second derivatives
of the metric. However, doing the variation of SH shows that this does not lead to
higher derivatives in the equation of motion: Eq. (2.23) is a differential equation of
second order.
The right hand side of Eq. (2.23) stems from the action of the matter field content of
spacetime. This action should be added to SH . The result of variation with respect to
the metric is found to be the energy–momentum tensor Tµν . For example the Lagrangian
of a scalar field
1
L = gµν ∂µ φ∂ν φ − V (φ)
(2.24)
2
gives
Tµν = ∂µ φ∂ν φ − gµν L.
(2.25)
Because the Einstein tensor,
1
Gµν = Rµν − Rgµν ,
2
(2.26)
is divergenceless, energy–momentum conservation,
∇µ T µν = 0,
(2.27)
is consistent with the Einstein equation (2.23). In terms of energy density ρ, pressure p
and the local velocity field uµ the energy–momentum tensor (2.25) is written as
Tµν = (ρ + p)uµ uν − pgµν .
(2.28)
This is the general form for an ideal fluid.
2.2.2. The Friedmann–Robertson–Walker spacetime
General relativity is the framework which is used to describe the cosmological evolution,
and from observations it is known that our universe is isotropic and homogeneous in
8
2.2. General relativity
space on large scales. Then the theory of maximally symmetric spaces states that spacetime can be described as a 3-dimensional sphere, plane or hyperboloid which evolves in
time. The metric of such a space can be cast into the form
dl2 =
dr 2
+ r 2 (dθ 2 + sin2 θdφ2 ),
1 − κr 2 /R2
(2.29)
where κ is +1 for the sphere, 0 for the plane and −1 for the hyperboloid. R is the radius
of the sphere: Thought of as embedded in a 4-dimensional space with coordinates x(i) ,
the equation for the sphere and the hyperboloid would be
x2(1) + x3(2) + x2(3) + x2(4) = R2
(2.30)
x2(1) − x3(2) − x2(3) − x2(4) = R2 ,
(2.31)
and
respectively. Including the time coordinate, the metric is
ds2 = dt2 − a2 (t)γij dxi dxj ,
(2.32)
where γij dxi dxj stands for the spatial part, Eq. (2.29), and a(t) is the scale parameter.
The right hand side of Eq. (2.32) is called the Friedmann–Robertson–Walker (FRW) or
Friedmann–Lemaître–Robertson–Walker (FLRW) metric. The use of conformal time η
with
dt
(2.33)
dη =
a(t)
simplifies many calculations. Furthermore, spatial curvature can be neglected on large
scales. So the metric is written as
ds2 = a2 (η)(dη 2 − δij dxi dxj ).
(2.34)
The scale-factor a(η) is a non-vanishing function. So, by definition, this metric is related
to the Minkowski metric ηµν by a conformal transformation. For better clarity, let us
list the following properties:
√
gµν = a2 (η)ηµν , gµν = a−2 (η)η µν ,
−g = a4 (η).
(2.35)
The next step is to determine the behavior of a(t) and of the field content of spacetime.
This is done with the help of Eq. (2.23) whose 00-component reads
2
1
ȧ
κ
R00 − g00 R = 3
+
(2.36)
2
a2 a2
(left hand side) and
T00 = (p + ρ)u0 u0 − g00 p = ρ
(2.37)
(right hand side). With the definition of the Hubble parameter,
H(t) =
ȧ(t)
,
a(t)
(2.38)
9
Chapter 2. Some aspects of the theory of relativity
this gives the (first) Friedmann equation,
H2 =
κ
8π
ρ − 2.
2
a
3mPl
(2.39)
For later reference, some further consequences—not always independent from each
other—are listed in the following: Equation (2.23) exists also in a slightly different
variant,
8π
1
(2.40)
Rµν = 2 Tµν − T gµν ,
2
mPl
which is obtained by taking the trace in the original version and using therein the result
again. Its 00-component,
ä
4π
(2.41)
= − 2 (ρ + 3p),
a
3mPl
is sometimes called the second Friedmann equation. Using conformal time, the first
Friedmann equation reads
a′2
8π
κ
=
(2.42)
ρ− 2
2
4
a
a
3mPl
and, together with the second one, the Raychaudhury equation,
2
κ
a′′ a′2
8π
− 4 = − 2 p − 2,
3
a
a
a
mPl
(2.43)
is obtained. The Friedmann equations are complemented by energy–momentum conservation (2.27), which takes on the forms
a′
ρ̇ = −3H(ρ + p) and ρ′ = −3 (ρ + p)
a
(2.44)
when using cosmic and conformal time, respectively. To fix the behavior of the medium
and the scale factor in the homogeneous and isotropic case, the last equation needed is
the equation of state,
p = p(ρ).
(2.45)
Some examples will be discussed in Chapter 4. The rest of this chapter is devoted to
fluctuations in the metric and the fluid.
2.2.3. Basics of metric fluctuations
Fluctuations of the metric (2.34) are included as a spacetime dependent tensor hµν . It
is defined by the equation
gµν = a2 (η)γµν = a2 (ηµν + hµν (x)).
(2.46)
The homogeneous part of the metric is ḡµν = a2 ηµν . The entries of hµν are taken to
be small and calculations are done up to terms of linear order. This means that the
indices of hµν (and of other small quantities as well) are raised and lowered with the
10
2.2. General relativity
unperturbed metric ηµν . In addition this entails a simple expression for the inverse of
γµν ,
−1
γµν
= γ µν = η µν − hµν ,
(2.47)
and likewise for gµν ,
−1
gµν
= gµν =
1 µν
(η − hµν ) .
a2
(2.48)
Gauge transformations. The tensor hµν is not completely fixed by (2.46). Rather,
the invariance of GR under coordinate transformations allows for various splittings of
the metric into background and fluctuations. The corresponding coordinate systems
are linked up by “small” changes of coordinates which are called gauge transformations.
More precisely, gauge transformations are induced by arbitrary infinitesimal functions
ξ µ (x) as
x̃µ = xµ + ξ µ (x),
(2.49)
such that
∂ x̃µ ∂ x̃ν λρ
µλ
νλ
g (x) = g µν (x) + g(0)
∂λ ξ ν ,
(2.50)
∂λ ξ µ + g(0)
∂xλ ∂xρ
where unperturbed quantities are labeled with a zero. The calculation is done to first
order in the perturbations and in ξ µ . On the other hand a Taylor expansion gives
g̃ µν (x̃) =
µν
µν
(x) + δg̃ µν (x̃)
(η) + ξ σ ∂σ g(0)
g̃ µν (x̃) ≈ g(0)
≈ g̃µν (x) +
µν
(x),
ξ σ ∂σ g(0)
(2.51)
(2.52)
where the first two terms of the first step add up to the background metric in the new
coordinate system and the third one is the perturbation. The result is equated to the
right hand side of (2.50), where the metric is split similarly as
µν
gµν (x) = g(0)
(η) + δgµν (x).
(2.53)
µλ
µν
νλ
∂λ ξ ν .
(x) + g(0)
∂λ ξ µ + g(0)
g̃µν (x) ≈ g µν (x) − ξ σ ∂σ g(0)
(2.54)
Then one has
In this line the derivative of the metric can be replaced by Christoffel symbols (and
metrics) because the corresponding covariant derivative vanishes. They are reorganized
into conformal derivatives of ξ µ such that one obtains
g̃µν (x) = g µν (x) + ∇µ ξ ν (x) + ∇ν ξ µ (x)
(2.55)
to first order in the perturbation. Equation (2.50) allows for immediate use of (2.48) to
leave the gauge transformation of the metric fluctuation,
h̃µν (x) = hµν (x) − 2η µν ξ σ
∂σ a
− ∂ µξν − ∂ ν ξµ.
a
(2.56)
It will be used to simplify the calculation of perturbations.
11
Chapter 2. Some aspects of the theory of relativity
Helicity decomposition. Since the FRW metric is spatially homogeneous and fluctuations are considered only to linear order, the treatment of Fourier transformed quantities is adequate. For example the metric fluctuation is written as
Z
hµν (η, x) = d3 keikx hµν (η, k),
(2.57)
where the quantities x, k, and η are all conformal. For each Fourier mode, k sets
a direction in momentum space, which permits a further decomposition according to
the transformation properties under rotations around the axis k. This amounts to
the consideration of irreducible representations of the symmetry group SO(2). They
have definite helicity, which is the eigenvalue with respect to the generating angular
momentum operator. The helicity of quantities being independent from momentum or
with proportionality to ki or ki kj is zero. They are called scalar perturbations. The
helicity eigenvalue of vector perturbations is 1 up to a phase. Under rotations around k
they transform like transverse vectors, i.e. vectors in the plane orthogonal to k. Tensor
perturbations can be constructed from products of two vectors being orthogonal to k
which entails helicity 2. They give rise to gravitational waves.
The Einstein equations do not mix helicity in the linear case: The corresponding
operations comprise time derivatives and, in momentum representation, multiplication
with ki , which both leave the behavior under rotations around k unchanged. Of course,
these considerations lose their validity if nonlinearities need to be considered.
The decomposition of each momentum mode k of the metric can be written as
(2.58)
h00 = 2Φ
h0i = iki Z +
ZiT
(2.59)
hij = −2Ψδij − 2ki kj E +
i(ki WjT
+
kj WiT ) +
hTijT .
(2.60)
In the following, the gauge freedom (2.56) is used to set
h0i = 0,
E = 0.
(2.61)
The first condition can be achieved by choosing ξ 0 appropriately. Then one can still use
gauge transformations that leave h0i = 0 invariant. These are derived from functions
ξ µ with
∂ 0 ξ i + ∂ i ξ 0 = 0.
(2.62)
This equation is for example fulfilled by ξ µ (η, x) obtained from any function σ(η, x)
through
ξ 0 = ∂ 0 σ(η, x) and ξ i = −∂ i σ(η, x)
(2.63)
Such a transformation has the following effect on the spatial part of hµν :
h̃ij = hij − 2∂ i ∂ j σ − 2
12
a′ ij ′
δ σ.
a
(2.64)
2.2. General relativity
The choice σ = E leaves Ẽ = 0 in the new metric.
It can be concluded that the scalar Z and the transverse vector ZiT will not be considered. Likewise the vectorial quantity WiT will be disregarded since vector perturbations follow decaying solutions and are not expected to leave any observable traces. So
it is
h00 = 2Φ, h0i = 0, hij = −2Ψδij + hTijT .
(2.65)
The tensor hTijT is transverse and traceless. So, for each momentum mode one has
ki hTijT (k) = kj hTijT (k) = 0.
(2.66)
Evolution of scalar perturbations. The linearized Einstein tensor in terms of scalar
metric perturbations is calculated in App. A. Then, in order to write down the linearized
Einstein equation, the energy–momentum tensor of the perturbed cosmological medium
is needed. Starting point is the energy–momentum tensor of an ideal fluid with energy
density ρ̂ = ρ̄ + δρ, pressure p̂ = p̄ + δp, and local 4-velocity ûµ = ūµ + δuµ ,
Tνµ = (ρ̂ + p̂)ûµ ûν − δνµ p̂.
(2.67)
As in the case of metric perturbations, the calculation is done up to linear order in the
inhomogeneities δρ, δp, and δuµ . The 4-velocity is characterized by
gµν uµ uν = 1.
(2.68)
Because the spatial components of uµ are small and the same is true for the off-diagonal
entries of the metric, one is left with
giving
g00 u0 u0 = 1 or g 00 u0 u0 = 1,
(2.69)
1
a
1
≈ (1 − Φ) and u0 = √
≈ a(1 + Φ).
u0 = √
a
a 1 + 2Φ
1 − 2Φ
(2.70)
The components of Tνµ are
and
T00 = ρ̂ = ρ̄ + δρ,
(2.71)
Ti0 = −(ρ̄ + p̄)v i = −(ρ̄ + p̄)vi ,
(2.72)
Tji
=
−δji p̂
(2.73)
to first order. In these equations the physical velocity v i = aui and vi = +δij v j have
been defined. For the linearized Einstein equation only the perturbations,
δT00 = δρ,
δTi0 = −(ρ̄ + p̄)vi ,
δTji = −δji δp,
(2.74)
are needed. The perturbed Einstein equation and energy–momentum conservation equation are derived to linear order in App. A. As shown there, the results for the 00-, the 0i-,
13
Chapter 2. Some aspects of the theory of relativity
and the spatial components of the Einstein equation as well as the 0- and i-components
of the conservation equation are given by
a′2
4πa2
a′
∆Φ − 3 Φ′ − 3 2 Φ = 2 δρ,
a
a
mPl
′
a
4π
Φ′ + Φ = − 2 (ρ̄ + p̄)v,
a
mPl
(2.75)
(2.76)
a′
a′′
a′2
4πa2
Φ′′ + 3 Φ′ + 2 Φ − 2 Φ = 2 δρ,
(2.77)
a
a
a
m
′Pl
a
(2.78)
((ρ̄ + p̄)vi )′ + ∂i δp + (ρ̄ + p̄) 4 vi + ∂i Φ = 0,
a
a′
δρ′ + 3 (δρ + δp) + (ρ̄ + p̄)(3Φ′ + ∂i vi ) = 0,
(2.79)
a
respectively. For a = 1, the first equation reduces to the Poisson equation for the
Newtonian potential. Note that the sources in the Einstein equation refer to the total
fluid. By contrast, energy–momentum conservation (Eqs. (2.78) and (2.79)) holds for
each component separately, as long as interactions between the fluids can be neglected.
The master equation for the potential Φ is obtained by multiplying (2.75) with the
sound speed squared, u2s = δp/δρ and adding the result to (2.77). Then it is
′′
a′
a
a′2
′′
2
′
2
Φk + 3 (1 + us )Φk + 2 − 2 1 − 3us Φk + u2s k2 Φk = 0
(2.80)
a
a
a
for the modes with comoving momentum k. This equation can be simplified using
2
8πa2
a′′
=
(ρ − 3p) and
a
3m2Pl
8πa2
a′2
=
ρ,
a2
3m2Pl
(2.81)
which are obtained from Eqs. (2.42) and (2.43). Under the assumption
p = u2s ρ
(2.82)
the expression in curly brackets is then seen to vanish. One is left with the wave equation
a′
Φ′′k + 3 (1 + u2s )Φ′k + u2s k2 Φk = 0.
(2.83)
a
Three scales are important for the behavior of the solutions to this equation: The Hubble
horizon H −1 , which is of the order of the distance traveled by light since the Big Bang;
the sound horizon us H −1 , which is the corresponding distance covered by a sound wave;
and the wavelength λ = 2π/k. If the wavelength is much larger than the sound horizon,
us k ≪ Ha,
(2.84)
the last term in Eq. (2.83) can be neglected and there is a constant solution. Oscillation
sets in when the inequality (2.84) is valid in the opposite direction. This is possible in a
radiation dominated epoch, as is observed for example in the traces of baryon acoustic
oscillations in the CMB. Further discussion of the important issue of structure formation
can be found in the literature.
14
2.2. General relativity
Evolution of tensor perturbations. The equation of motion for the tensor perturbations (2.85) is derived in the Appendix, see Eq. (A.46).
hTijT
′′
+2
16πa2
a′ T T ′
hij
− ∆hTijT = − 2 ΠTijT
a
mPl
(2.85)
In the ideal fluid approximation the right hand side is zero because the anisotropic
stress vanishes. Then this equation describes the free propagation of a gravitational
wave as long as the wavelength is much smaller than the Hubble radius. A spectrum of
gravitational waves is expected to be produced during inflation (Section 5). To calculate
this spectrum it is necessary to know their action ST T which can be obtained from the
Hilbert action (2.22) by expansion to second order in the tensor perturbation hTijT . The
result
Z
n
o
2
m2Pl
ST T =
d4 xa2 ∂η hTijT − ∂k hTijT ∂k hTijT
(2.86)
64π
is valid for ideal fluids. A further decomposition of each mode into the basis of two
×
polarizations e+
ij and eij ,
X (A)
hTijT =
eij h(A) ,
(2.87)
A=+,×
streamlines the notation somewhat for the action,
2
X m2 Z
3
2
(A)
(A)
(A)
Pl
d xdηa
ST T =
∂η h
− ∂i h ∂i h
,
64π
(2.88)
A=+,×
as well as for the resulting equation of motion,
a′
∂η h+,× − ∆h+,× = 0.
(2.89)
a
In contrast to the case of scalar perturbations, the solution to this equation is governed
solely by the Hubble horizon and by the wavelength k of the mode in question. A mode
well outside the horizon fulfills
a′
(2.90)
k ≪ Ha, or, equivalently, k ≪ ,
a
which can be directly used to identify the relevant contributions to Eq. (2.89). Hence,
in the superhorizon regime one solves the equation
∂η2 h+,× + 2
a′
(2.91)
h′′k + 2 h′k = 0
a
and identifies, besides a decaying solution, the behavior of a tensor perturbation outside
the horizon to be time independent. When the universe is not inflating, i.e. when
d
Ha < 0,
(2.92)
dt
then modes that are outside the horizon will at some time enter the horizon. Then
Eq. (2.91) has to be replaced by
ä < 0
⇔
h′′k + k2 hk = 0
(2.93)
and the solution is an oscillating gravitational wave.
15
Chapter 2. Some aspects of the theory of relativity
2.2.4. Primordial spectra of scalar fluctuations
Regarding scalar fluctuations, this work is confined to the discussion of the adiabatic
mode. For a medium in local thermal equilibrium being described with the help of a
metric in conformal Newtonian gauge, the defining property of the adiabatic mode is
∂ρλ
δT (x, η)
∂T
∂pλ
δpλ (x, η) =
δT (x, η),
∂T
δρλ (x, η) =
(2.94)
for fluctuations over scales exceeding the Hubble radius. The index λ identifies the
component in question. Equation (2.94) suggests that the composition of the cosmic
medium is the same throughout spacetime and the only independent fluctuation is in
temperature. This is in contrast to the isocurvature mode in which the fluctuations
are independent for different particle species. Observations do not show evidence for
perturbations in the isocurvature mode.
In order to generalize the definition of the adiabatic mode to a medium out of thermal
equilibrium, δT can be replaced by the function ǫ(x, η)T ′ (η) with the result
δρλ (x, η) = ρ′λ ǫ(x, η),
δpλ (x, η) = p′λ ǫ(x, η).
(2.95)
The purpose of this section is to establish a quantity that characterizes the the strength
of fluctuations before they enter the horizon. It will be seen that there are two equivalent
possibilities, ζk and Rk , that are both constant outside the horizon, i.e. for k ≪ aH.
To obtain a suitable definition, Eq. (2.79) is considered in the superhorizon approximation, ∂i vi = 0. With Eq. (2.95) one has
(ρ′λ ǫ)′ + 3
a′ ′
ρλ + p′λ ǫ − 3Φ′ (ρλ + pλ ) = 0
a
(2.96)
and repeated use of (2.44) leaves
′
Φ =−
a′
ǫ
a
′
.
(2.97)
The conclusion is that the quantity
ζ = −Φ −
δρtot
a′
ǫ = −Φ +
a
3(ρtot + ptot )
(2.98)
is constant as long as the mode has not entered the horizon. The second equality is
obtained with Eqs. (2.44) and (2.95). The subscript refers to the total medium.
The definition of R is equal to that of ζ with −ǫ replaced by
P
(ρλ + pλ )vλ
vtot = P
,
(2.99)
(ρλ + pλ )
16
2.2. General relativity
so,
R = −Φ +
a′
vtot .
a
(2.100)
For negligible k in the superhorizon regime it is ζ = R because the result in the calculation
m2
a′
k2 Φ
δρtot
− vtot = − Pl 2
(2.101)
ζ −R =
3(ρtot + ptot )
a
12πa (ρtot + ptot )
can be taken to vanish as well. So, both R and ζ are adequate and equivalent measures
for scalar fluctuations before horizon entry. Their physical interpretation is given in the
rest of this section. For this purpose the perturbations are considered in the comoving
reference frame, which means vtot = 0. It follows from Eq. (2.74) that switching into
this frame is achieved by the gauge transformation (2.56) when the function ξ(xµ ) obeys
∂0 ξi = −∂i ξ0 = vi :
(2.102)
Applying the transformation (2.54) to the energy–momentum tensor leads to
T̃i0 (x) = Ti0 (x) − ξ σ ∂σ Ti0 (x) + Tiρ ∂ρ (x)ξ 0 − Tλ0 ∂i (x)ξ λ
(2.103)
So the transformation of the perturbation is to first order
δT˜i0 (x) = δTi0 (x) + Tii (x)∂i ξ 0 − T00 (x)∂i ξ 0 ,
(2.104)
where the index i should not be summed over. Now Eq. (2.74) is used to replace the
fluctuations observing
δTi0 = −(ρ̄ + p̄)vi
and δT̃i0 = −(ρ̄ + p̄)ṽi ,
(2.105)
which entails (2.102) as a conclusion. It should be stated that this gauge transformation preserves h0i = 0, which is the condition for Eq. (2.100) to be valid. So, as an
intermediate step towards the interpretation of R, note that R = −Φ in the comoving
gauge, and that “comoving” has the meaning of vtot = 0 in this context.
The next step is to calculate the curvature of a hypersurface of constant time in these
coordinates. First, the calculation is done for the metric γij , which is defined according
to the beginning of Section 2.2.3. So, one has a(η) ≡ 1. From Eq. (A.8) in three
dimensions one has
R(3) (γ) = ∂i ∂j hij + ∂i ∂i hjj .
(2.106)
The only scalar contribution comes from
hij = −2Ψδij + 2∂i ∂j E
(2.107)
R(3) (γ) = 4∂i ∂i Ψ ≡ 4∆Ψ.
(2.108)
and reads
17
Chapter 2. Some aspects of the theory of relativity
Restoring the scale parameter by use of the conformal transformation (A.18) gives an
additional factor 1/a2 and for ideal fluids it is Ψ = −Φ, see Section A.4. Therefore, the
comoving reference frame (vtot = 0) implies equal-time hypersurfaces with curvature
R(3) = −
4
∆R,
a2
(2.109)
representing the physical interpretation of R. Observations do not require higher correlation functions than hR(k1 )R(k2 )i for the description of R(k), which means that
they are consistent with the assumption that R(k) is a random Gaussian field. It is
completely determined by its power spectrum PR (k) through
hR(k)R(k′ )i =
PR (k)
δ(k + k′ ).
4πk3
The fluctuations of R(x) then obey the equation
Z ∞
dk
2
hR (x)i =
PR (k).
k
0
Their amplitudes are more conveniently quantified by
p
∆R (k) = PR (k).
(2.110)
(2.111)
(2.112)
Within the standard picture of structure formation these spectra originate in a period
of accelerated expansion in the early universe, called inflation (see Chapter 5). In
Chapters 5 and 6 the spectra will be computed for different models of inflation.
2.2.5. Primordial spectra of tensor fluctuations
In analogy to the case of scalar perturbations, the tensor modes are quantified by their
power spectra. Despite of lacking observational evidence one can try to describe the
tensor modes as a Gaussian random field with a power spectrum defined by
PT (k)
1
δ(k + k′ ).
hh(A) (k)h(B) (k′ )i = δAB
2
4πk3
As in the last section, the amplitude is characterized by
p
∆T = PT
and the power spectrum is connected to the fluctuations in space through
2 Z ∞ dk
X (A)
h (k)
=
PT (k).
k
0
(2.113)
(2.114)
(2.115)
A
In these equations the notation of Section 2.2.3 has been used. Also the tensor power
spectrum is discussed for different inflationary scenarios in Chapters 5 and 6.
18
3. Elements of field theory
The purpose of this chapter is mainly to prepare the background for the presentation of
the theory of inflation and to motivate the potential which is later used for an inflationary scenario. So, it is far from a well-balanced account on the vast field of (quantum)
field theory but instead concentrates on some indispensable basics needed for this work.
Since the potential that will be used later on is rooted in effective potentials of strong
interactions, the presentation will also include some aspects of quantum chromodynamics. This chapter follows Refs. [79] and [100] with additional reference to [111, 120] and
other sources given in the following.
3.1. Equations for flat and FRW spacetimes
A field theory is defined by its action, which in curved spacetime and for a single scalar
field φ(x, t) may be written as
Z
Z
1 µν
4 √
4 √
S = d x −gL = d x −g
g ∂µ φ∂ν φ − V (φ) ,
(3.1)
2
where the Lagrange density L is the expression in brackets and a canonic kinetic term
is assumed. The equation of motion is obtained by variation with respect to the field
and its derivative ∂µ φ. Using the FRW metric (2.32), the result is
φ̈ + 3H φ̇ +
∂V (φ)
dV (φ)
a′
= 0 or φ′′ + 2 φ′ + a2
=0
∂φ
a
dφ
(3.2)
for a homogeneous field using cosmic and conformal time, respectively. As in chapter 2,
conformal time is defined by dη = dt/a. The energy–momentum tensor is obtained by
varying the action with respect to the metric,
2 δS
= ∂µ φ∂ν φ − gµν L.
Tµν = √
−g δgµν
(3.3)
The energy density ρ = T00 and the pressure p = Tii 1 are
1
ρ = φ̇2 + V (φ),
2
1
No sum over the spatial index i is intended here.
1
p = φ̇2 − V (φ)
2
(3.4)
Chapter 3. Elements of field theory
in the homogeneous case. Inserting the energy density into the Friedmann equation (2.39)
leaves
1 2
8π
2
φ̇ + V (φ) .
(3.5)
H =
3m2P l 2
In flat spacetime the equation of motion is
∂µ ∂ µ φ +
∂V
= 0.
∂φ
(3.6)
The calculations, on which this thesis is built, have been done with real scalar fields.
However, the motivation of the employed model relies on symmetries including spinor
fields, which are used to describe fermions. Therefore, a short glimpse at some of their
basic properties is now included.
A spinor ψ is a four-component object used to describe the dynamics of fermionic
fields. The Lagrange density of a free fermionic field with mass m is
L = ψ̄(iγ µ ∂µ − m)ψ
(3.7)
leading to the equation of motion
(iγ µ ∂µ − m)ψ = 0.
The 4 × 4 Dirac matrices γ µ are defined as
!
1 0
0
γ =
, γi =
0 −1
with the 2 × 2 identity matrix
1=
!
1 0
,
0 1
2
!
0 −i
,
i 0
and the 2 × 2 Pauli matrices
1
τ =
!
0 1
,
1 0
τ =
(3.8)
0 τi
−τ i 0
!
,
(3.9)
(3.10)
3
τ =
!
1 0
.
0 −1
(3.11)
The Pauli matrices obey the (anti-)commutation relation
[τ i , τ j ] = 2iǫijk τ k ,
{τ i , τ j } = 2δij 12×2 .
(3.12)
The Lagrange density (3.7) is invariant under Lorentz transformations if ψ̄ := ψ † γ 0 .
There are other representations of the Dirac matrices as well. The essential property is
the anticommutator relation
{γ µ , γ ν } = γ µ γ ν + γ ν γ µ = 2η µν ,
where η µν is the Minkowski metric Eq. (2.2).
20
(3.13)
3.2. Symmetries and conserved currents
3.2. Symmetries and conserved currents
For the construction of effective models in particle or nuclear physics, symmetries are
often used as a guiding principle. If the microscopic theory exhibits a symmetry, one
can try to mimic its dynamics by transferring the symmetric behavior under a certain transformation to the effective fields that are used to describe the system. This
section presents symmetry transformations that reflect phenomena observed in strong
interactions.
First, it is demonstrated that invariance of the Lagrange density L(φ) under a continuous symmetry transformation, φ → φ + δφ, leads to a conserved current. This is
the statement of Noether’s theorem [106]. The symmetry assumption is written as the
first equality in
0 = L(φ + δφ) − L(φ)
∂L
∂L
=
δφ +
δ(∂µ φ)
∂φ
∂(∂µ φ)
∂L
∂L
∂µ δφ
δφ +
= ∂µ
∂(∂µ φ)
∂(∂µ φ)
∂L
δφ ≡ ∂µ J µ ,
= ∂µ
∂(∂µ φ)
(3.14)
and an expansion is applied in the second one. The third line is obtained using the
equation of motion
∂L
∂L
− ∂µ
=0
(3.15)
∂φ
∂(∂µ φ)
for the first term, and commuting the derivative ∂µ and the variation δ in the second
term. The last line in (3.14) contains the definition of the conserved current,
Jµ =
∂L
δφ.
∂(∂µ φ)
(3.16)
The zero component J 0 is the density of the conserved charge related to the symmetry,
Z
d
Q = 0.
(3.17)
Q = d3 xJ 0 ⇒
dt
In the case of invariance under spacetime translations, the conserved current is the
energy–momentum tensor. However, this presentation concentrates on internal symmetries, which are inherent to specific combinations of fields in the Lagrange density.
Important symmetries of this kind correspond to unitary transformations, as it is the
case for rotations and translations in spacetime. An infinitesimal unitary2 transformation of a number of fields φi can be written as
φi → δij − iϑa Tija φj ≡ Uij φj
(3.18)
2
The group of unitary N × N matrices is called U (N ); it is U ∈ U (N ) ⇔ U † = U −1 . The group of
special unitary N × N matrices is called SU (N ); the additional requirement is det(U ) = +1.
21
Chapter 3. Elements of field theory
with implied summation over the index a of the real infinitesimal ϑa and the traceless
hermitian3 matrices or operators Tija . A transformation with finite ϑ is obtained by
repeated use of (3.18), which results in
~ → exp (−iϑa T a ) φ.
~
φ
(3.19)
The T a form a basis of the generators of the unitary transformations (3.18). The vector
space of these generators is additionally equipped with a map similar to Eq. (3.12),
which makes it an algebra. Each generator comes with a conserved current,
J µ,a = −i
∂L
T a φj ,
∂(∂µ φi ) ij
(3.20)
and the resulting conserved charge. As an aside it should be mentioned that the highly
successful standard model of particle physics is built on the direct product of three
unitary symmetry groups, U (1) × SU (2) × SU (3).
The following gives a short account on the symmetries that are important for effective
theories in nuclear physics. They can be crucial even if they are not exact in nature, as
is the case for chiral symmetry. Therefore, let us write down the effect of a symmetry
breaking contribution Lsb to the Lagrange density on the conservation of J µ : The
calculation (3.14) yields
∂µ J µ = δLsb ,
(3.21)
which gives a direct relation of the symmetry breaking of the Lagrange density to the
non-conservation of the current.
3.3. Chiral symmetry for two flavors of fermions
A system of two sorts (flavors) of massless non-interacting fermions is described by the
Lagrange density
L = iψ̄u γ µ ∂µ ψu + iψ̄d γ µ ∂µ ψd ,
(3.22)
where the indices can be read as “up” and “down”. If both flavors are combined into the
spinor ψ = (u, d), one can consider the “vector transformation” ΛV , which rotates the
two components into each other. Of course, this presentation is alluding to the situation
in nuclear physics, where quarks and mesons are allocated the quantum numbers isospin,
strangeness and others. Then the u- and d-quarks are interpreted as the same particle
within two quantum states with isospin plus and minus one half. The transformation ΛV
and also the axial vector transformation described below act on this space of isovectors.
The action on the two components of ψ is given by
a
a
ΛV
aτ
aτ
ψ −→ exp −iϑ
ψ ≈ 1 − iϑ
ψ
2
2
(3.23)
a
a
ΛV
aτ
aτ
ψ̄ ≈ 1 + iϑ
ψ̄.
⇒ ψ̄ −→ exp +iϑ
2
2
3
22
A Hermitian matrix A fulfills A† = A.
3.3. Chiral symmetry for two flavors of fermions
From this one gathers that the Lagrange density itself is invariant because the exponents
commute with γ µ . The conserved “vector current” V µ,a associated with the transformation is found from Eq. (3.16) to be
V µ,a = ψ̄γ µ
τa
ψ.
2
(3.24)
The axial vector transformation ΛA can be studied when the matrix
!
0
1
2×2
γ 5 := iγ 0 γ 1 γ 2 γ 3 γ 4 =
0
12×2
(3.25)
with
{γ 5 , γ µ } = 0,
γ 5 γ 5 = 14×4
is introduced.4 ΛA is defined as the first line in
a
a
ΛA
5 aτ
5 aτ
ψ ≈ 1 − iγ ϑ
ψ
ψ −−→ exp −iγ ϑ
2
2
a
a
ΛA
5 aτ
5 aτ
⇒ ψ̄ −−→ ψ̄ exp −iγ ϑ
≈ ψ̄ 1 − iγ ϑ
.
2
2
(3.26)
(3.27)
In the second line the order of the terms becomes important, because γ 5 and γ 0 anticommute rather than commute, see (3.26). When calculating the modification of the
Lagrange density, however, a further sign change occurs due to the γ µ in the kinetic
terms in Eq. (3.22). Once more, this gives invariance of the Lagrange density. Therefore,
also the axial current
τa
Aµ,a = ψ̄γ µ γ 5 ψ
(3.28)
2
is conserved. Taken together, the two symmetries constitute the chiral symmetry
SU (2)V × SU (2)A .
However, considering a system of quarks, one deals with particles of finite mass; and
whereas a mass term in the Lagrange density,
Lm = −mu ψ̄u ψu − md ψ̄d ψd ,
(3.29)
respects invariance under ΛV , it breaks the axial symmetry. From Eqs. (3.21) and (3.29)
the symmetry breaking is seen to be proportional to the masses. So, chiral symmetry
can be regarded as a useful guide as long as quark masses are small. For the light quarks
(up and down) with masses of a few MeV compared to the QCD scale ΛQCD ∼ 200 MeV
this is indeed the case. The following linear combination of the generators of vector and
axial transformations,
τ
1
τL = (1 − γ 5 )
2
2
4
and τR =
1
τ
(1 + γ 5 ) ,
2
2
(3.30)
As Eq. (3.9) the result in Eq. (3.25) depends on the representation.
23
Chapter 3. Elements of field theory
can be seen to commute, which means that they act independently of each other. The
generator τL,R belongs to the subgroup SU (2)L,R , respectively, the cartesian product of
which is the full SU (2),
SU (2) = SU (2)L × SU (2)R .
(3.31)
Because the projection operators
1
1
PL = (1 − γ 5 ) and PR = (1 + γ 5 )
2
2
(3.32)
are used in the definitions (3.30), a transformation generated by one of the τL and
τR projects on a state with definite chirality. The indices stand for left- and righthandedness, respectively.
3.4. Quantum chromodynamics
Quantum chromodynamics (QCD) is widely accepted as the theory of strong interactions. It describes interactions between quarks and gluons by introducing color charge
as a new quantum number. In this framework, a quark can carry one of the three colors “red,” “green,” and “blue,” which is another distinguishing property adding to the
quantum number “flavor” that comes in the six different types
up,
down,
charm,
strange,
top,
bottom/beauty.
In correspondence to the leptons, the quarks are organized into three generations, which
are indicated by the column structure of the list above. However, let us stick to color
dynamics here:
The strong interaction is symmetric under the color transformation SU (3)C that
rotates states of different color charge into each other. So, each quark flavor q f can be
represented by a spinor triplet of different color states.5
 f
qr
 f
f
q = qg  .
(3.33)
f
qb
As in the electroweak theory, the interaction is introduced by allowing for spacetime
dependence of the symmetry transformation. This is phrased as turning the global
SU (3) into a local, “gauge” transformation. In order to keep the Lagrange density
invariant, the derivative has to be turned into a covariant one. This is in correspondence
to the covariant derivative of GR and, historically, it has been done earlier in a similar
way for the U (1) symmetry of quantum electrodynamics (QED). Then it is natural to
permit a separate term governing the dynamics of the new field which is also in analogy
5
In the following these are indicated by r, g and b, respectively.
24
3.4. Quantum chromodynamics
to QED. The only difference in the notation is an additional index a. It is needed
because the algebra of the generators of SU (3) is an eight-dimensional vector space.
In the fundamental representation it is the vector space of traceless, hermitian 3 × 3
matrices. A conventional basis is the collection of Gell-Mann matrices λ1 to λ8 ,








0 1 0
0 −i 0
1 0 0
0 0 1








1 0 0 ,  i 0 0 , 0 −1 0 , 0 0 0 ,
0 0 0
0 0 0
0 0 0
1 0 0








(3.34)
1 0 0
0 0 −i
0 0 0
0 0 0
1 







0 0 0  , 0 0 1 , 0 0 −i , √ 0 1 0  .
3
0 0 −2
i 0 0
0 1 0
0 i 0
These represent a three-dimensional generalization of the Pauli matrices, Eq. (3.11).
The normalization of the generators T a is chosen as
Ta =
such that
1 a
λ ,
2
1
tr T a T b = δab .
2
(3.35)
(3.36)
3.4.1. The Lagrange density of QCD
Let us now examine the ingredients out of which the QCD Langrange density,
LQCD =
X
f
1 a a,µν
F
,
q̄f (iγ µ Dµ − mf ) qf − Fµν
4
(3.37)
is built. Note that the comma in the index does not indicate a derivative but only
separates the two types of indices. Then, it should be mentioned that, since they
are fermions, the behavior of the quark fields, qf , under Lorentz transformations is
governed by the spinor representation of the Lorentz group SO(1, 3), which leaves the
combination q̄q invariant. The indices µ and ν are Lorentz indices and their contraction
leads to Lorentz invariance as well. The conformal derivative Dµ is defined as
Dµ = ∂µ + igT a Gaµ ≡ ∂µ + igGµ .
(3.38)
In order to keep the kinetic term of the quarks invariant under the gauge transformation
qf → exp (−iεa (x)T a ) qf ≡ U qf ,
(3.39)
the transformation of the field Gµ is introduced as
i
Gµ → U Gµ U −1 − U ∂µ U −1 .
g
(3.40)
The field Gµ is the gauge field of QCD. It represents the eight gluons. They are bosons, which are the field quanta of the strong interaction. The field strength tensor is
25
Chapter 3. Elements of field theory
constructed from the gauge field Gµ . It is a vector in the algebra and a tensor in color
space, too:
a
Fµν = T a Fµν
(3.41)
where the index a runs from 1 to 8. It is defined as a generalization to the QED case
by adding a component proportional to the commutator, which is necessary in order to
preserve gauge invariance:
Fµν = ∂µ Gν − ∂ν Gµ + ig [Gµ , Gν ] .
(3.42)
From this equation it is seen that the gluons themselves are charged under the strong
interaction. The Lagrange density contains terms with three or four gluons and the
coupling constant g. They show that gluons interact with each other, as opposed to
photons. The components
a
Fµν
= ∂µ Gaν − ∂ν Gaµ − gf abc Gbµ Gcν
contain the structure constants f abc of SU (3), which are given by
i
h
T a , T b = if abc T c .
(3.43)
(3.44)
It can be shown that the field strength tensor (3.42) transforms as
Fµν → U Fµν U −1 ,
(3.45)
such that the last term in Eq. (3.37) is invariant because it is a trace. Altogether, the
conclusion is that the Lagrange density (3.37) is indeed symmetric under SU (3)C gauge
transformations. Arguing the other way round, one can state that the gauge symmetry
guarantees a universal coupling strength to all particle fields with color charge. This is
because the coupling g is not only present in the covariant derivative but also in the
definition of the field strength. Then it is clear, that this argument is only valid for a
non-vanishing commutator in Eq. (3.42), i.e. for a non-Abelian gauge group.
3.4.2. Symmetries and other properties
Quantum chromodynamics is built around the central assumption of the existence of a
local SU (3) symmetry acting on a new degree of freedom called color. However, there
are further symmetries and characteristics of the physics of strong interactions, which
prove important when effective, simplifying theories shall be found, that nevertheless
incorporate the essence of QCD. Some of them will be discussed in this section:
Global U (1) symmetry. Obviously, the QCD Langrange density is invariant under the
global U (1) symmetry
U (1)
q −−−→ exp(−iϕ)q
26
(3.46)
3.4. Quantum chromodynamics
leading to baryon number conservation: The conserved current is
JBµ = q̄(x)γ µ q(x)
and the baryon number
B=
Z
d3 xq † q
(3.47)
(3.48)
is the conserved charge. For massless quarks the Lagrange density is also symmetric
under the axial UA (1) transformation,
UA (1)
q −−−−
→ exp(−iγ 5 ϕ)q,
(3.49)
but it has been shown that this symmetry is broken by quantum effects. This is known
as the Adler-Bell-Jackiw anomaly [8, 9, 74, 111].
Global SU (2) flavor symmetry. In Section 3.3 the chiral transformation of quarks
has been presented. Chiral SU (2) symmetry is a substantial feature of QCD although
it is only approximately realized and in addition spontaneously broken. Nevertheless,
the quark masses mu and md are small, and the corresponding vector current (3.24)
is conserved to good approximation. Also the axial vector current (3.28) is at least
partially conserved. The zero component of the first one is the isospin charge
Z
Z
τi
3
0
(3.50)
Qi = d xVi = d3 xψ † ψ.
2
More on chiral SU (2) symmetry will be said in the Sections to follow.
Global SU (3) flavor symmetry. Including the strange quark qs , one could ask for a
possible SU (3) flavor symmetry. Since the strange quark is considerably heavier than
the up and down quarks [107],
mu ≈ md ≈ 3.4MeV,
ms ≈ 94MeV,
(3.51)
the axial SU (3)A is not a good symmetry. However, the transformation SU (3)V , though
likewise broken explicitly due to the differences in the masses listed above, can still be
successfully utilized in order to structure hadrons containing strange quarks. Their
quantum numbers and their corresponding behavior in nuclear reactions can be understood from their arrangement in multiplets of SU (3)V .
Asymptotic freedom. It is known from QED that the effective coupling constant, experienced by electrically charged particles in scattering experiments, depends on the
momentum transfer q. This can be seen by calculating the contribution from the individual Feynman graphs that are relevant in this process. If q is large enough, the contribution of the exchange of one photon, as represented by the Feynman diagram 3.1(a)
is significantly corrected by terms of higher order in the coupling constant
αQED =
e2
1
∼
.
4π
137
(3.52)
27
Chapter 3. Elements of field theory
(a) Tree-level diagram of photon
exchange by two electrically
charged fermions.
(b) Example for a diagram with
fermion loop.
Figure 3.1.: Two diagrams from the calculation of fermion-fermion scattering in
QED. The fermions (for example electrons or positrons) exchange a
photon represented by the wiggly line (left). To higher order in the
coupling, more complicated diagrams contribute. The example on the
right-hand side is called a vacuum polarization diagram. It reduces the
value if the effective coupling experienced by the particles.
An example is the diagram 3.1(b), which belongs to the terms characterizing the
vacuum polarization. It describes the short-term decay of the exchanged photon into
a fermion–antifermion pair. This leads to a shielding of the charge in analogy to the
shielding by a dielectric medium. It is also phrased as the running of the coupling.
The situation in QCD is different because the gluons do not only mediate the strong
interaction but carry color charge themselves. Also 3- and 4-gluon vertices are possible,
which leads to Feynman diagrams as the one depicted in Fig. 3.2(c). The interaction
of gluons among each other entails an anti-shielding contribution. The effect of the
color charge of the gluons can be interpreted as a distribution of the quark color. If
the momentum exchange between two scattering quarks is large, then they penetrate
each other’s color cloud and, therefore, are subject only to a part of the charge. This
phenomenon is known as asymptotic freedom because it leads to the fact that quarks
inside hadrons behave almost like non-interacting particles. This is seen for example in
deep inelastic scattering experiments using leptons and hadrons.
The value of the coupling constant αs is obtained from scattering experiments and
from lattice calculations, see Ref. [19]. Due to the effects mentioned above, this coupling
varies with energy scale. The cited reference gives the value
αs (mZ ) = 0.1184 ± 0.0007
(3.53)
at the scale of the Z-boson mass mZ ≈ 90 GeV. The running of αs is quantified by
the β-function. Once again referring to the Particle Data Group [17], the functional
dependency can be cited as
µ2
dαs
= β(αs )
dµ2
= −(b0 α2s + b1 α3s + b2 α4s )
33 − 2nf 2
≈−
αs ,
12π
28
(3.54)
3.4. Quantum chromodynamics
(a) Tree-level diagram of quarkquark scattering. The exchanged
gluon is depicted by the curly
line.
(b) Higher order diagram with
quark loop.
(c) This diagram is only possible in the non-Abelian case,
because it implies charged exchange particles.
Figure 3.2.: Feynman diagrams contributing to the quark-quark scattering crosssection. Diagram (c) belongs to the contributions which reduce the
effective coupling for high momentum-transfer.
where the numerical values b1 and b2 from the given reference have been waived in
the last line. To be precise, the variable µ is not exactly the momentum transfer q
between two scattering particles but the renormalization scale. For the purpose of this
thesis let us simply take µ = q. The most important feature is the negative sign of
the derivative (3.54) which is in contrast to the QED case. It leads to both asymptotic
freedom and the confinement of color charge described in the next paragraph.
Confinement. Experiments have not only lead to the finding of asymptotic freedom
but have also revealed its counterpart at large distances, the confinement of quarks and
gluons into color-neutral objects. Quarks and gluons are never observed as separate
particles but instead are always found to be bound in hadrons.
An effective description of this situation is given by the Bag Model [33, 43], which
assigns a higher value of energy density to the inside of hadrons. This is said to contain
the perturbative vacuum as opposed to the non-perturbative QCD vacuum outside the
hadron “bags”. In a first approximation the quarks inside the bags are assumed to
be only subject to the confining boundary of the bag and to behave otherwise as free
particles. The energy difference of both vacua is fixed by the bag constant B. It can be
estimated by comparing the pressure from the kinetic energy of the quarks inside the
bag with the pressure from the energy difference of the two vacua.
29
Chapter 3. Elements of field theory
Within this picture the QCD phase transition to the deconfined phase of the quarkgluon plasma can be understood in the following way: With higher temperature the
kinetic pressure inside the bags rises and the bags expand until they overlap with neighboring ones. Similarly, this merging of bags can occur when the hadron density rises
and a number of bags is forced into a smaller and smaller volume.
Scale invariance. In the limit of massless quarks, there is no dimensionful parameter
left in the QCD Lagrange density. Then the physics stays exactly the same when the
unit of length is changed. However, this invariance is spoiled by the renormalization of
QCD. Then the scale ΛQCD occurs as the energy scale, for which the running coupling
constant grows large: Coming from large energies, at ΛQCD any perturbative treatment
of QCD breaks down and low-energy QCD has to rely on different methods such as
lattice calculations and effective models. The presentation in this paragraph is based
on [36]. For comparison, also [21] has been used.
First, we must take a short detour and include a few lines about some basics of
scale symmetry: The change of the physical scale, which is also called a dilatation or a
conformal transformation, can be written as
Λϕ : xµ → e2ϕ xµ ,
ϕ ∈ R.
(3.55)
Note, that this is no coordinate transformation, leaving invariant any reasonable field
theory, but a true transformation of spacetime. A more general type of conformal
transformation with spacetime dependent parameter ϕ is introduced in Appendix A.
Assuming that a spacetime dependent field φ(x) transforms linearly under Λϕ , one
writes
Λϕ : φ(x) → e2ϕD φ e2ϕ x
≈ (1 + 2ϕD) (φ(x) + ∂µ φ 2ϕ xµ )
(3.56)
µ
≈ φ(x) + 2ϕ (D + x ∂µ ) φ(x)
with some operator D. From this, one defines the change of the field under a transformation as
δφ = (D + xµ ∂µ )φ(x).
(3.57)
The following choice of D renders many theories of massless fields invariant under Λϕ ,
as long as only dimensionless couplings are contained:
Dφ = φ
for bosonic fields φ and
3
Dψ = ψ for fermionic fields ψ.
2
4
For example the potentials φ , and ψ̄ψφ transform by
δ φ4 = 4φ3 δφ = 4φ3 (1 + xµ ∂µ ) φ = (4 + xµ ∂µ ) φ4 and
3
3
µ
µ
+ x ∂µ ψ̄ ψφ + ψ̄
+ x ∂µ ψ φ + ψ̄ψ (1 + xµ ∂µ ) φ
δ(ψ̄ψφ) =
2
2
= (4 + xµ ∂µ ) ψ̄ψφ,
30
(3.58)
(3.59)
(3.60)
3.4. Quantum chromodynamics
both of which are seen vanish upon partial integration. One may conclude that products
of fields are invariant as long as their scaling dimensions D add up to four. Kinetic terms
as ∂µ φ∂ µ φ/2 or ψ̄iγ µ ∂µ ψ are found to be invariant as well:
1
µ
∂µ φ∂ φ = ∂µ {(1 + xν ∂ν )φ} ∂ µ φ
(3.61)
δ
2
= ∂µ φ∂ µ φ + {(1 + xν ∂ν )∂µ φ} ∂ µ φ
1 ν
µ
= ∂µ φ∂ φ + 1 + x ∂ν ∂µ φ∂ µ φ = 0
2
3
3
δ(ψ̄iγ µ ∂µ ψ) =
+ xν ∂ν ψ̄iγ µ ∂µ ψ + ψ̄iγ µ ∂µ
+ xν ∂ν ψ = 0.
2
2
(3.62)
(3.63)
(3.64)
In the fermion case the result is obtained from partial integration of one of the terms.
However the scaling dimension is not always equal to the mass dimension of dimensional
analysis. Indeed, masses and dimensionful couplings themselves do not transform under
the scaling, which is the reason for the scale symmetry breaking of mass terms like
1 2 2
m φ
2
or mψ̄ψ.
(3.65)
Finishing the detour on scale transformations, let us now have a look on the QCD
Lagrange density in terms of its scaling behavior. It is clear from the analysis above
that the gluon part is invariant because every term contains altogether four derivatives
or gluon fields. Only the quark mass terms transform non-trivially.
Then one could assume that QCD is still approximately symmetric under scale transformations because the quark masses are small. However this is not the case because
the approximate symmetry is broken by an anomaly in quantum dynamics [37]. This
does not mean that the use of considering scale transformations is lost: Inside hadrons
for example, scale symmetry is approximately restored and is used in effective descriptions [27, 76].
Trace anomaly. Scale symmetry is connected to the trace of the energy–momentum
tensor. A prominent example is QED, which is traceless and symmetric under scale
transformations. For strongly interacting matter this trace is non-zero because of an
anomaly [39]. The Bag Model, which has been mentioned earlier in this Section, ascribes
the energy density B to the perturbative vacuum inside hadrons. Since the equation of
state in these volumes is that of vacuum, one obtains 4B as the trace of the energy–
momentum tensor. On the other hand, QCD calculations yield a proportionality to the
gluon condensate [39],
Tµµ =
βQCD a µν,a
hFµν F
i,
2g
with αs =
g2
.
4π
(3.66)
Again, the comma is only meant to separate the two types of indices and does not denote
a derivative. Furthermore, one can define the energy–momentum tensor such that it is
31
Chapter 3. Elements of field theory
µ
by
connected to the Noether current of dilatation symmetry JD
µ
JD
= xν T νµ :
(3.67)
Because of ∂µ xν = ηµν , one then obtains that the trace of the energy–momentum tensor
is zero if the scale current is conserved.
So, it is seen that the dilatation symmetry is connected with the gluon condensate
and the trace of the energy momentum tensor. This motivates the introduction of a
new field, the dilaton χ(x) as the supposed Goldstone boson of this symmetry. As
already mentioned above, quantum effects break scale symmetry and the dilaton is not
expected to be massless or light [76]. Nevertheless, the dilaton has been considered as
a pseudo-Goldstone boson and identifications with observed mesonic resonances have
been made [67]. The dilaton field can be used to match the scaling properties of the
linear sigma model terms to the corresponding ones in QCD [32, 99, 109]. Thus the
dilaton field is associated with the gluon condensate, which breaks scale invariance and
leaves the rest of the theory approximately scale-free when a phase transition leads to
its disappearance. Likewise, the dilaton is expected to undergo a phase transition from
a finite value at low temperatures to χ = 0 in the high-temperature phase. An effective
model of QCD including a dilaton field will be introduced in Section 3.5.5.
3.5. Effective theories of strong interactions
3.5.1. Mesonic states and their chiral transformation
The first experimental evidences supporting the chiral nature of nuclear forces came
from the nuclear beta decay. Although it is a process due to the weak interaction, it
allows for the conclusion that chiral symmetry is approximately realized in the strong
interaction. Therefore, effective nuclear theories are constructed such that their behavior
under chiral transformations can be easily controlled. It will be seen that these models
permit perfectly symmetric behavior under chiral transformations as well as spontaneous
and explicit symmetry breaking. In the following the linear sigma model will serve as
example. Its potential couples nucleons or quarks with the mesonic pion and sigma
fields π a and σ. Guided by their transformation behavior they can be built from quark
fields as
π a = iψ̄τ a γ 5 ψ,
and σ = ψ̄ψ.
(3.68)
The τi makes the pion transform as a vector under isospin rotations and the γ 5 contributes negative parity. Vector mesons with Lorentz index can be constructed by including
γ µ . The chiral transformation of the mesons arises from the transformation of the quark
32
3.5. Effective theories of strong interactions
spinors: Equations (3.23) and (3.27) give
τb
τc
ΛV
π a = iψ̄τ a γ 5 ψ −−→
iψ̄ 1 + iϑb
τ a γ 5 1 + iϑc
ψ
2
2
1
= iψ̄τ a γ 5 ψ + ψ̄γ 5 [τ a , τ b ]ϑb ψ
2
= iψ̄τ a γ 5 ψ + iϑb ǫabc ψ̄γ 5 τ c ψ
(3.69)
to linear order in ϑa . This can be written as an isospin rotation of the pion by the angle
ϑa ,
ΛV
~ × ~π .
~π −−→
~π + ϑ
(3.70)
The axial transformation,
Λ
A
π a = iψ̄τ a γ 5 ψ −−→
iψ̄τ a γ 5 ψ + ϑa ψ̄ψ,
(3.71)
is obtained using the anticommutation relation in (3.12). Again, one can also write
Λ
A
~
~π −−→
~π + ϑσ.
(3.72)
The transformation of the sigma meson is obtained similarly. Like the mass term in
the example of Section 3.3, it transforms trivially under vector transformations. Axial
transformations rotate the σ and the π a fields into each other, see also Eq. (3.72),
Λ
V
σ −−→
σ,
Λ
A
~π.
σ −−→
σ − ϑ~
(3.73)
The results on mesonic transformations need to be borne in mind when the dynamics
of the effective theory is to respect the desired chiral behavior.
3.5.2. Chiral symmetry breaking
As stated in Section 3.4, chiral SU (2) symmetry is approximately realized in the QCD
Lagrange density. However, this symmetry does not become manifest in the mesonic
mass spectrum: Chiral partners as the σ and the π mesons or the vector mesons ρµ
and a1µ do not have approximately the same masses—as they should, following the
naive expectation. The solution to this apparent problem is a spontaneous breaking of
chiral symmetry in addition to the explicit one that comes from the comparatively small
current quark masses.
A symmetry is termed spontaneously broken if it is obeyed by the Lagrange density
but not respected by the ground state. So, violation of chiral symmetry need not pose a
problem in the low energy hadronic regime as long as a phase transition occurs at higher
energies, where chiral symmetry is restored. In the chirally restored phase mesons and
also nucleons should be massless.
There is the following specialty about spontaneously broken symmetries; it is formulated by the Goldstone theorem [56, 57]: It states that any spontaneously broken
33
Potential V (σ, π a = 0)
Chapter 3. Elements of field theory
−fπ
0
Scalar field σ
fπ
Figure 3.3.: Potential of the linear sigma model in the π a = 0 plane. The potential
should be thought of as rotationally symmetric around the σ = 0 axis.
Whereas the potential is chirally symmetric, the ground state, σ = fπ ,
π a = 0, is not. At high temperatures the effective potential (dashed
line) does not allow symmetry breaking.
continuous symmetry leads to a massless bosonic mode carrying the same quantum
numbers as the generator of the symmetry. They are called Goldstone bosons. In the
case of chiral symmetry this part is attributed to the pions. For an exact symmetry the
pions should be massless. This is obviously not the case. However, since chiral symmetry is subject to an explicit breaking by the non-zero quark masses, one should not
expect zero mass but a comparatively small one. This is satisfied by the pion masses.
From experiment it is known that pions are pseudo-scalar particles, which means that
they obtain a minus sign under parity transformations. This is consistent with their
origin from axial symmetry breaking because the factor of γ 5 in Eq. (3.28) makes the
accordant charge a pseudo-scalar as well.
3.5.3. Linear sigma model
An important example of an effective theory of strong interactions is the linear sigma
model [55] with the Lagrangian
1
1
L = ∂µ π a ∂ µ π a + ∂µ σ∂ µ σ + iψ̄γ µ ∂µ ψ
2
2
2
λ
+ gπ ψ̄(σ + iγ 5 τ a π a )ψ −
(~π 2 + σ 2 ) − fπ2 ,
4
(3.74)
which is examined in the following: It is composed of standard kinetic terms for the
(pseudo) scalar fields π a and σ and the fermion ψ in the first line, and of interaction
terms in the second one. The π a and the σ are thought of as the meson fields from
Section 3.5.1. Depending on the phase, the spinor ψ can be interpreted either as nucleon
34
Potential VXSB (σ, π a = 0)
3.5. Effective theories of strong interactions
−fπ
0
Scalar field σ
fπ
Figure 3.4.: Potential of the linear sigma model in the π a = 0 plane as in Fig. 3.4, but
here with small symmetry breaking term linear to σ. In the minimum
the potential has non-vanishing curvature in π direction (vertical to the
drawing plane), which generates the pion mass. The effective potential
above the critical temperature is indicated by the dashed red line.
or as quark field. For SU (2) it consists of the two components for proton and neutron
or for up and down quark, as the case may be.
Now, the invariance under chiral transformations should be checked: Equations (3.70),
(3.72), and (3.73) give the invariance of the last term,
Λ ,Λ
V
A
(~π 2 + σ 2 ) −−
−−→
(~π 2 + σ 2 ).
(3.75)
The first term in the second line transforms in the same manner because of the identification (3.68). The kinetic terms of the mesons share the same structure (3.75) and
the free fermion Lagrange density has already been seen to be invariant.
The minimum of the potential lies in a circle of radius fπ in field space, see Fig. 3.4.
In the ground state this value is assumed by the scalar σ because as opposed to the pion
field, its quantum numbers are the same as of vacuum. So, the chirally symmetric circle
of the potential, ~π 2 + σ 2 = fπ2 , is replaced by the choice σ = fπ , ~π = 0 for the ground
state. Of course, there will be fluctuations around these values. In accordance with
the Goldstone theorem, there is a direction in field space with vanishing curvature of
the potential. The fluctuations into these directions are interpreted as (massless) pions,
whereas the radial fluctuations correspond to massive sigma particles.
The small masses of the pions have their origin in the small masses of the light
quarks, which explicitly break chiral symmetry. From Eqs. (3.29) and (3.68), the quark
mass terms are seen to be equivalent to a term linear in the sigma field. This suggests
to introduce such a linear potential in order to take explicit symmetry breaking into
consideration. Taking this into account, the potential of the model,
2
λ
V =
(~π 2 + σ 2 ) − fπ2 ,
(3.76)
4
35
Chapter 3. Elements of field theory
is replaced by
Vsb =
where the choice of
2
λ
(~π 2 + σ 2 ) − v02 − fπ m2π σ,
4
v0 = f π
m2π
1−
2λfπ2
(3.77)
(3.78)
and of the symmetry breaking term leaves the minimum at σ = fπ to leading order in
the parameter fπ m2π . The pion mass calculated from this potential is mπ . The nucleon
mass in the ground state is read from Eq. (3.74),
mN = gπ σ0 = gπ fπ
(3.79)
and, as for the pion mass, the value of mN is retained here.
3.5.4. The dilaton field in string theory
The most prominent occurrence of a dilaton field is not in effective theories of the
strong interaction but in string theory. The following short description of its role in
this field of physics is based on information gathered from the textbooks and reviews
[15, 16, 108, 115, 128].
String theory is an approach to describe the physical world based on one-dimensional
objects called strings. As opposed to the case of point particles, the motion of a string
through spacetime is not described by a world-line but by a two-dimensional worldsheet. Hence, the action of a free string is not the length of the world-line, but the area
of the world-sheet. It is called the Nambu–Goto action SN G . Depending on whether the
string in question is open or closed, its world-sheet takes the form of a ribbon or a tube.
The degrees of freedom of a string are allocated to translation, rotation and various
modes of vibration. The fluctuations of a vibrating string correspond to ripples of the
world-sheet. For both types of string, open and closed, these excitations are subject
to boundary conditions, which lead to discrete spectra of energy states. The goal is to
identify these string excitations with particle spectra known from the standard model
and its extensions. While in general there are many obstacles on the way towards a
stringy basis of testable field theoretical models, string theory shows some qualities that
have stimulated continuous research interest over the last decades.
A nice property of perturbative calculations in string theory is that the number of
Feynman diagrams to each order is reduced with respect to field theory. In addition,
the spreading of point-like vertices in field theory to areas of meeting world-sheets in
string theory avoids divergences in the calculation of the diagrams.
As will be indicated in the following, a hint at a major plus is already seen when
quantizing the excitations of a closed string: In this case the creation and annihilation
operators come in pairs, corresponding to right-moving and left-moving waves,
i†
ai†
n , ān
36
and ain , āin .
(3.80)
3.5. Effective theories of strong interactions
The index n labels the possible wavelengths and the index i specifies the direction of the
embedding space into which the mode is oscillating. The various ground states differ by
their non-vibrational degrees of freedom. They are annihilated by any annihilation operator, and all states of the string are obtained by repeated use of the creation operators.
Then, a more thorough analysis yields a matching condition connecting the occupation numbers of right-movers and left-movers. Altogether, this gives the spectrum of
particles that have their origin in the vibration of the string.
The vacuum state corresponds to a particle with negative mass squared, a tachyon.
This entails an unstable vacuum. While also arising in the theory of open strings, it can
be avoided when fermionic states are included.
The first excited state is obtained by the action of two creation operators because of
the matching condition mentioned above. It can be written as
X
j†
Rij ai†
(3.81)
1 ā1 |string vacuumi ,
i,j
where the indices i and j now both label the spatial direction of the embedding. The
state, on which the creators act, is the vacuum state in the sense that it shows no
excitation of the vibration modes, but it is not constrained with respect to translation
and rotation of the string. The arbitrary quadratic matrix Rij can be decomposed into
three contributions:
1. The symmetric-traceless part is a massless spin-two state. It is interpreted as the
graviton giving rise to Einstein gravity in certain limits of the theory. This is part
of the reason why string theory is called a viable way to a quantum theory of
gravitation.
2. The antisymmetric part yields the Kalb–Ramond field Bµν . It is a generalization of
the vector potential of electrodynamics. For bosonic field theory without fermions
it turns out that this field is massless only in 25+1 dimensions. This is a necessary
condition for Poincaré invariance. It can be shown that the particle content also
includes exchange bosons of non-Abelian gauge theories.
3. The trace of Rij leads to one single state. This massless scalar state is called the
dilaton. It occurs in all perturbative string theories. It will be discussed below.
The inclusion of fermions is achieved in superstring theory in the framework of supersymmetry: The coordinates of the world-sheet are interpreted as bosonic fields, and
corresponding fermion fields need to be introduced. For example this is possible with
two-dimensional spinor fields on the world-sheet, as it is done within the Ramond–
Neveu–Schwarz formalism.
In superstring theory there are no tachyonic states in the particle spectrum, and the
number of required spacetime dimensions is reduced to 10. However, also in this case a
compactification of 6 dimensions to small and therefore unobservable scales is necessary.
37
Chapter 3. Elements of field theory
The dilaton field plays an important role for string interactions. To see this we leave
string theory as the generalization of ordinary quantum mechanics of particles to onedimensional objects and we take a glimpse on what could yield a string field theory,
after a second quantization. This is no fully developed theory yet, but at least there
is a recipe for the computation of Feynman diagrams [108]. On this basis it can be
shown that the dilaton field occurs in the string scattering amplitudes. The simplest
example is the vacuum amplitude, which is calculated as a path integral over world-sheet
coordinates x and world-sheet metrics γij . The γij are introduced as auxiliary fields in
order to facilitate quantization, which is difficult in the original Nambu–Goto action.
The resulting action is the Polyakov action SP . The result for the vacuum amplitude is
the path integral
X −χ(t) Z
−SP Pt
.
(3.82)
Z=
eφ0
P DxDγe
{
}
t
t
As usual, the integrand is the exponential of an action. The integral is split into a
sum over different topologies t and a remaining integral over all world-sheets with the
topology in question. The Euler characteristic χ(t) counts holes and other topological
features. Its negative value can be interpreted as the loop order of a diagram and
the exponential in front is seen to play the role of a coupling constant. The vacuum
expectation value of the dilaton field, φ0 , occurs here because it is present as a prefactor
of the world-sheet metric. As such, it controls the extent, the dilatation, of the world
sheet. The apparent dilatation symmetry of the Polyakov action is broken by anomalies,
just as in the case of QCD. As far as string theory is concerned it should be stressed,
that the string coupling
g = e φ0
(3.83)
is not a constant given as an input. Instead, it is the result of a dynamical process,
namely of the dilaton’s rolling into its minimum. It is assumed that the dilaton acquires
a mass through spontaneous supersymmetry breaking in certain vacua. From the QCD
perspective this looks quite familiar: It has been shown that the gluon condensate gives
rise to an anomalous trace of the energy–momentum tensor, thereby breaking scale invariance. In an effective approach the gluon condensate is described by a scalar field with
corresponding behavior. This field should undergo a symmetry-breaking phase transition somewhere around the QCD energy scale mimicking the condensation of gluons.
We will return to this point in the next section.
3.5.5. Linear sigma model with dilaton
For the calculations to this thesis a potential has been used that is derived from the linear
sigma model for strong interactions (3.74). When explicit symmetry breaking is taken
into account, its potential includes the term linear in the sigma field, as in Eq. (3.77).
In the discussion of scale invariance in Section 3.4.2, it has been argued, that the highenergy dynamics of any theory is expected to be scale invariant. From QCD, however,
we know that this is an approximation which is valid only for the classical theory. For the
38
3.5. Effective theories of strong interactions
effective description of strongly interacting matter this has the following consequence:
Effective theories of QCD try to keep as many properties of QCD as possible while being
as simple as possible. So, one should try to mimic the behavior of QCD under scale
transformations as closely as possible by simple means. Such an approach is followed
by the authors of [22, 32, 99, 109], which has been mentioned at the end of Section 3.4.2.
Here the scalar dilaton field is used to confer the expected transformation property on
the individual terms of the action. Therefore, the original potential (3.77) is written as
Vsb =
χ 2
χ 2
2
λ 2
− fπ m2π σ
~π + σ 2 − v02 (~π 2 + σ 2 )
4
v
v
4
χ 4 1
χ
+ χ4 ln
,
+k
v
4
v4
(3.84)
where the constant k is introduced and v is the minimum of the dilaton potential.
From comparison with the above discussion of scale symmetry, it is clear that not every
contribution to this potential is invariant. Indeed, the first two terms are, whereas the
third transforms non-trivially, as expected from a mass term. (Recall that σ is treated as
ψ̄ψ.) The second line contains a pure dilaton potential. Up to a constant, it is the same
as the one displayed in Fig. 5.3. The logarithm of χ is not scale symmetric and gives a
non-zero trace of the energy–momentum tensor. It becomes important when the dilaton
evolves from the flat maximum at χ = 0 to one of the two minima at |χ| = v. This
implies spontaneous scale symmetry breaking during the cooling of the system, which
is similar to the breaking of chiral symmetry: When the dilaton potential is included,
the part that breaks scale symmetry is small for high temperatures6 because the value
of χ is small compared to v. It gets large for low temperatures where the logarithm
gives a sizeable contribution. This is the effective description of gluon condensation
at low temperatures. It imitates the formation of the scale-symmetry breaking gluon
condensate.
To make use of this potential in the context of cosmological inflation, it has been
simplified: The third, symmetry breaking term has been omitted and the chiral fields ~π
and σ have been reduced to one, φ. In that sense, the computations have been done on
the chiral circle. Finally, a vacuum contribution has been included to keep the potential
positive. After renaming the constants, the potential now reads
1
1
1
χ 1
4
V (φ) = V0 + λ0 χ ln −
(3.85)
+ λ1 φ4 − λ2 χ2 φ2
4
v
4
4
4
with
1
(3.86)
λ0 v 4 exp λ22 /λ0 λ1 .
16
We will return to it in Eq. (6.1). As it will be shown, within such a potential two
inflationary periods are possible.
V0 =
6
It is the pion mass term.
39
4. Hot Big Bang cosmology
This chapter deals with some important fundamentals of cosmology. It is based on
Refs. [60] and [101]. After the presentation of a few basic equations on redshift and the
time dependence of energy density, Section 4.2 will introduce the most important epochs
in the evolution of the universe after inflation and reheating. This is the history of the
hot Big Bang. Then it will be shown that there are observations that are difficult to
explain within this scenario. In the next chapter, an elegant solution to these problems
is considered. It is the aforementioned scenario of cosmological inflation.
4.1. Evolution of the universe after inflation
The rate of expansion of the universe can be characterized by the Hubble parameter,
which is defined in terms of the scale parameter a(t),
H=
ȧ
.
a
(4.1)
Its value today is approximately H0 ≈ 70 km/s Mpc. It gives the timescale of the
universe, t = 1/H0 ≈ 1.4 · 1010 yrs, and the lengthscale l = 1/H0 ≈ 4.3 · 103 Mpc. The
physical wavelength λph of a photon is redshifted as the universe expands: Starting from
an initial value λi at time ti it grows like
λphys (t) =
a(t)
λi .
a(ti )
(4.2)
The redshift z is then defined to fulfill
λphys
= 1 + z.
λi
(4.3)
Frequency and wavenumber decrease correspondingly. The majority of the photons in
the universe today belongs to the CMB. They are not in thermal equilibrium anymore
but they nevertheless show an almost perfect Planck spectrum. It is not entirely perfect
because of small fluctuations of relative size 10−5 –10−4 , which will be important for
the following chapters. However, it still can be seen as a Planck spectrum to a very
good approximation. This is because these photons once were in thermal equilibrium
and redshift preserves the Planck distribution. The temperature that corresponds to
this distribution is subject to redshift as well. According to this statement, a high
Chapter 4. Hot Big Bang cosmology
temperature should be expected immediately after reheating, gradually decreasing to
the present value T0 ≈ 2.73 K.
After the decay of the inflaton field, the energy content of the universe is dominated
by radiation. This period is followed by a matter dominated epoch and finally by today’s
era of vacuum domination. This sequence results from the different equation of state
of the corresponding media. To see this let us collect the equations connecting ρ, t, η,
H, and a for each type of medium.7 Energy–momentum conservation and the second
Friedmann equation, (2.41, 2.43, 2.44), are used to calculate the entries of the three last
columns in
Radiation
Matter
Vacuum
p
1
3ρ
0
−ρ
a
a0 t1/2
a0 t2/3
a0 eHt
a
a0 η
a0 η 2
1
− ηH
H
1
2t
2
3t
const.
ρ
a−4
a−3
const.,
where the entry in each column equals the quantity in the same line of the first column.
So, it is seen that the energy density of radiation decays most rapidly with growing
scale parameter. Vacuum energy density does not decay at all. Then one expects the
primordial radiation dominated era to be followed by matter domination, finally ending
with the vacuum dominated universe of today.
4.2. A cosmic timeline
More details with respect to the evolution of the universe are given by the following list
of important timescales, for which information from Ref. [101] is used:
• t ∼ 10−43 , T ∼ 1019 GeV: GR is not valid at this scale and must be replaced
by some quantum theory of gravity, maybe string theory. The particles or field
content of the universe is unknown.
• t ∼ (10−43 − 10−14 ) s, T ∼ (1019 − 104 ) GeV: GR might be used below the Planck
scale. The particle content is possibly very different than at energies accessible
to accelerators. Maybe there is a supersymmetric phase. This is also the energy
scale, where “Grand Unification” of strong and electroweak interactions is assumed.
Probably, baryon asymmetry is produced and inflation takes place.
7
Curvature is set to zero in the calculations.
42
4.3. Problems of pure Big Bang
• t ∼ (10−14 − 10−10 ) s, T ∼ (104 − 102 ) GeV: All gauge bosons of the electroweak
interaction, Z, W ± , and γ, are massless. Electroweak symmetry is broken at the
end of this period and the Z and W ± become massive.
• t ∼ 10−5 s, T ∼ 200 MeV: This is the scale of the strong interaction. Scale
symmetry and chiral symmetry of QCD are broken. The quarks and gluons of the
quark–gluon plasma are confined into hadrons.
• t ∼ 0.2 s, T ∼ (1 − 2) MeV: Weak interactions become inefficient. Hence, the ratio
of neutrons to protons stays constant and neutrinos decouple from the cosmic
medium.
• t ∼ 1 s, T ∼ 0.5 MeV: The typical photon energy drops below the electron mass.
Electron–positron pairs annihilate and are not produced anymore. A few electrons
are left over. The annihilation increases the photon temperature with respect to
the neutrino temperature because the neutrino interaction with the medium has
already come to an end.
• t ∼ (200 − 300) s, T ∼ 0.05 MeV: Epoch of Big Bang Nucleosynthesis (BBN). Helium and traces of other light elements form. Observations of the helium abundance in the universe confirm the BBN scenario at thermal equilibrium. Therefore,
inflaton decay and thermalization have to be finished.
• t ∼ 1011 s, T ∼ 1 eV: Matter–radiation equality. Up to now, the contribution
of radiation to the energy content has surmounted the matter part. Because
energy density of radiation decays faster with a than that of matter, the radiative
contribution becomes more and more negligible.
• t ∼ (1012 − 1013 ) s: Recombination of electrons and protons to hydrogen atoms
leaves freely streaming photons. They form the CMB today, the observation of
which allows a conclusion to be drawn about the state of the universe at decoupling. CMB measurements also yield important information about inflation, see
Chapter 5.
• t ∼ (1016 − 1017 ) s: Age of structure formation. Gravitational instabilities of overdense regions leads to collapse resulting in galaxies and galaxy clusters, filaments,
and voids in between.
4.3. Problems of pure Big Bang
Besides its great successes, the cosmological hot Big Bang theory leaves some open
questions concerning the initial conditions of the scenario. They are described in the
following sections and the mechanism of inflation as a possible solution will be presented
in the next chapter.
43
Chapter 4. Hot Big Bang cosmology
4.3.1. Horizon problem
The Big Bang theory without an initial period of inflation fails to explain why the CMB
is isotropic to a very high degree. Within this theory the photons of the CMB originate
from regions of space that have never been in causal contact, which suggests that there
should be no correlations between them. Correlations and even isotropy seem plausible
only for radiation that is detected within small angles around a given direction. To
see this, the size of a patch of sky, for which causal contact at decoupling of the CMB
photons is expected, can be estimated by calculating the particle horizon of that time.
This is the distance a photon may have travelled since the Big Bang. First, note that a
light-like geodesic is characterized by ds2 = 0, which gives
(4.4)
|dx| = dη
in a comoving coordinate system. Then the comoving particle horizon at time t is
Z t
dt′
.
(4.5)
lp.h. (t) = η(t) =
′
0 a(t )
The ratio of this quantity at different times is estimated as in Eq. (4.6). Being a ratio
of two lengths both measured at the same time, it makes no difference whether they are
taken as comoving or physical:
R a0
R t0 ′
′)
da/Ha2
dt
/a(t
lp.h. (t0 )
a1 H(a1 )
= R0a1
≈
= R0t1
.
(4.6)
2
′
′
lp.h. (t1 )
a0 H(a0 )
dt /a(t )
0 da/Ha
0
For the second equality dt is replaced by da/Ha. The approximation is exact under the
assumption a(t) ∝ tα , α < 1 because then
H ∝ t−1 ∝ a−1/α
and therefore
lp.h. (t0 ) ∝
Z
0
a0
da
a2−1/α
1/α−1
= a0
(4.7)
∝
1
.
a0 H(a0 )
(4.8)
Apart from the power law a(t) ∝ tα with constant α, the calculation relies on a second
simplification: The lower boundary of the integrals is taken as zero although the laws
of physics are unknown at least until the Planck time tPl . So nothing can be said about
the evolution of the particle horizon before that time. However, it seems plausible that
at tPl the particle horizon is of the order of the Planck scale, too. It could be taken as
a lower value for the integrals in Eq. (4.5) and (4.6) but it is neglected in comparison
to the much larger upper values. This is done by taking the initial time as zero because
within radiation (or matter) domination the lower value of the integral vanishes.
Now the redshift at photon decoupling z ≈ 1100 is inserted in Eq. (4.6). This yields
an angle of order 1/30 within which causal connection can be assumed. So the number
of patches of the sky which seem to be causally distinct is of order 1000. Therefore
within a pure hot Big Bang scenario very homogeneous initial conditions have to be
assumed without motivation.
44
4.3. Problems of pure Big Bang
4.3.2. Flatness problem
The horizon problem arises because the Planck length lPl seems to be a natural guess for
the horizon size at Planck time tPl . A similar guess for the scale of the spatial curvature
can be made from the structure of the Friedmann equation (2.39): Written as
H2 =
8π
(ρm + ρr + ρΛ + ρc ),
3m2Pl
(4.9)
the curvature term is interpreted as a contribution to the energy content as well. Normalization of the four energy densities by H 2 and appropriate definition of Ω for each
contribution leaves
(4.10)
1 = Ωm + Ωr + ΩΛ + Ωc ,
the most natural initial contributions to the Hubble parameter coming from matter
(ρm ), radiation (ρr ), the cosmological constant (ρΛ ), and curvature (ρc ) are suggested
to be of the same order for all sources. Comparison with Eq. (2.39) gives
Ωc (t0 )
=
Ωc (t1 )
a(t1 )H(t1 )
a(t0 )H(t0 )
2
,
(4.11)
similarly to Eq. (4.8). If the two moments of time are chosen as tP l and today, one
estimates a(tPl )/a(t0 ) ∼ T0 /mPl and H(tPl )/H(t0 ) ∼ mPl /H0 . So Eq. (4.11) gives the
value
H2
(4.12)
|Ωc (tP l )| ∼ |Ωc (t0 )| 20 ∼ 10−60 .
T0
for the contribution of curvature to the energy content at the Planck epoch. This is
many orders of magnitude smaller than what should be assumed from Eq. (4.10).
In other words the flatness problem is the following: For any cosmological evolution
with ä < 0, Eq. (4.11) shows that Ωc grows with time. Why then is it still so small?
4.3.3. Monopole problem
The monopole problem is not concerned with the initial conditions of Big Bang cosmology. However, it is included here because it can be solved by inflation. At very high
energies magnetic monopoles or topological defects may be produced as predicted by
theories beyond the Standard Model of particle physics. Possible monopoles from a GUT
phase transition played an important role in motivating early models of inflation [64].
This problem is different from the ones mentioned before because it stems from particle
physics and not from Big Bang cosmology itself. Nevertheless, if such relics are to be
expected from the very early mechanism and if they contradict observation there should
be a mechanism which dilute them down to a tolerable concentration.
45
Chapter 4. Hot Big Bang cosmology
4.3.4. Primordial perturbations
The horizon problem discussed above raises the question whether the observed homogeneity and isotropy of the universe seem natural or not. But in addition to the homogeneity on large scales, the sky shows obvious perturbations like planets, stars and
galaxy clusters. Cosmology can explain their formation from small initial perturbations
upon a homogeneous background. However, in order to understand the creation of these
initial perturbations, an additional mechanism is needed: inflation.
46
5. Inflation
This section presents the basics of the theory of cosmological inflation. It is based on
the introductions in [59,94,95]. Assuming an inflationary epoch before the hot Big Bang
scenario is a possible solution to several problems cosmology, which are discussed at the
end of the preceding chapter. By definition, a period of time in the early universe is called
inflation when the scale parameter a(t) is accelerating. With the Hubble parameter
H = ȧ/a the following equivalent defining conditions for inflation are found:
ä > 0
⇔
d 1
<0
dt Ha
⇔
−
Ḣ
<1
H2
(5.1)
The second one stresses the fact that the comoving Hubble length 1/Ha becomes smaller
during inflation. The third inequality in Eq. (5.1) sets a limit on the change of the Hubble
parameter with time. The decrease of H has to be small within one Hubble time 1/H.
In Chapter 4 it was found that the Hubble length is a measure for the observable part
of the cosmos, which is now seen to shrink in an accelerating universe. On the other
hand, the comoving wavelength of a fluctuation is constant during any cosmological
epoch. Then it should be expected that there are fluctuations whose wavelengths grow
with respect to the Hubble scale and finally exceed it. Then they are said to leave the
Hubble horizon. Such a horizon crossing changes the character of the time evolution of
the mode and prevents it from extinction.
Independently from the mechanism and the dynamics of inflation, these two aspects
of accelerated expansion, namely decrease of the observable part of the universe and
freezing of oscillations during horizon crossing, solve problems posed by standard hot
Big Bang cosmology.
The second section of this chapter discusses the solutions to some of these open
questions more thoroughly. Before that, however‚ we should make a short detour to
take a glimpse on the historic evolution of the inflationary theory:
5.1. Short history of cosmological inflation
In the 1970s the cosmological role of phase transitions in gauge theories was examined.
As discussed by Andrei Linde in Ref. [88], the hot stage of the universe could be preceded
by an era with the dominating part of energy and momentum coming from a metastable
vacuum state of gauge theories. Within this scenario the cold era ends with the decay
Potential V (φ)
Chapter 5. Inflation
tunneling
Scalar field φ
Figure 5.1.: Sketch of the potential in “old inflation.” The exponential expansion is
driven by the potential energy of the false vacuum state. The scalar field
during inflation is illustrated with the black dot. Inflation ends when φ
tunnels through the potential barrier and decays while oscillating around
the absolute minimum.
of this “false vacuum” into the true one. This phase transition is accompanied with a
strong rise of entropy. However, the universe after such a scenario was found to be too
inhomogeneous to match with observations.
An approach within quantum gravity was followed by Alexei Starobinsky [121] who
obtained a spectrum of primordial gravitational waves that are produced before the
classical Friedmann expansion starts. Also the presence of scalar perturbations can be
explained within this model as was found later in Ref. [102]. Here the perturbations
were viewed as possible seeds for structure formation.
In 1981, Alan Guth suggested a scenario which was based on a phase transition again.
He called it the “inflationary universe” and showed that it could solve the three major
problems concerning flatness, the horizon size, and the amount of entropy within the
observable universe [64]. He also pointed out what was later called the “graceful exit
problem”: The exponential expansion in this model is driven by the potential energy
of a field trapped in a false vacuum state, see also Fig. 5.1. The phase transition
takes place when bubbles of the new phase, the true vacuum, form. Bubble nucleation
occurs through a tunneling process. If the state of acceleration is to last long enough
to solve the cosmological puzzles, the maximal nucleation rate is determined by the
Hubble expansion. Nucleation rates that are allowed by this condition entail too strong
inhomogeneities to be acceptable. This is because there are too few bubble collisions to
thermalize the medium before nucleosynthesis [70]. The assumption that the observable
universe today has emerged from a single bubble was considered in Ref. [65]. It had to
be discarded because the entropy in such a bubble would be much lower than observed.
48
5.2. Inflation as a solution to cosmological problems
Later this scenario was called “old inflation.”
At the end of the year 1981, Linde suggested an improved inflationary model in [89],
which was based on spontaneous symmetry breaking induced by thermal corrections
(Coleman–Weinberg mechanism [38]). A few months later, a similar way of solving the
cosmological problems was examined in [10]. The graceful exit problem of old inflation is
not encountered here because the barrier between the false and the true vacuum vanishes
when the temperature decreases. The scenario is now called “new inflation” and belongs
to the category of hilltop inflation considered in Sect. 5.3.3. An important difference
to old inflation is the non-zero φ̇ of the inflaton field, which can make the spectrum of
generated perturbations slightly tilted. However, just as old inflation this scenario relies
on the thermodynamics of phase transitions. This is problematic since observations
demand a very small coupling constant for which thermal equilibrium cannot be taken
as given.
In 1983 Linde proposed a scenario that he called “chaotic inflation” [90]. It is based on
the observation that for a large class of potentials V (φ) an inflationary period occurs if
the initial field values are chaotically distributed. The most important condition is that
there are field values for which the potential is flat enough. For sufficiently large φ this is
for example the case in a quadratic or quartic potential. Then an inflationary solution
is obtained. Within a chaotic field distribution it is natural to assume the existence
of regions where inflation takes place. Such a region could evolve into the observable
universe today.
From then on a multitude of potentials and scenarios have been considered which
include periods of accelerated expansion of the early universe. There are many scalar
fields that can be motivated from string theory and supergravity and used for driving
inflation. In models of multifield inflation, various fields can contribute to acceleration
and production of fluctuations.
In 1991 a new version of chaotic inflation was proposed by Linde who called it hybrid
inflation [92]. Hybrid inflation is a multifield model but there is only one inflaton field
that evolves during inflation. A second one, the “waterfall field” ends inflation by rolling
down into its vacuum state.
Some details on how the inflationary universe can reconcile observations with the hot
Big Bang theory and on important examples for potentials leading to inflation will be
provided in the next sections.
5.2. Inflation as a solution to cosmological problems
In Sec. 4.3 some problems are described which cannot be overcome within the standard
hot Big Bang scenario of cosmology. However, if a period of inflation takes place during
the very early universe all of these problems are solved at once. There are arguments
that inflation should start very early within the nascent spacetime because it needs
sufficient homogeneity within the Hubble horizon. This quantity increases with time
49
Chapter 5. Inflation
and fulfillment of the condition of homogeneity seems to become less probable. In
addition, if Ω ≡ 1 − Ωc > 1 the universe is expected to recollapse soon after its creation
unless there is a mechanism that drives this value very close to unity. Inflation has
this effect as can be seen from Eqs. (5.1) and (4.11): The first equation shows that
during inflation the combination H(t)a(t) grows with time. Then Eq. (4.11) gives the
corresponding decrease of Ωc .
A small value of Ωc after inflation resolves the flatness problem of cosmology. Assuming the Hubble parameter to be constant, Eq. (4.12) gives the necessary growth of the
scale parameter from ai at the beginning of accelerated expansion to its end at af :
af
≈ 70.
(5.2)
Ne ≡ ln
ai
In this equation Ne is called the total number of e-foldings of inflation. This value
corresponds to a vast expansion of the universe, but, as will be seen below, it is easily
obtained in simple models of inflation.
The number of e-foldings required to solve the horizon problem is much smaller: A
short calculation of the comoving particle horizon Eq. (4.5) within de Sitter spacetime
(a(t) ∝ exp(Ht)) shows that it stays constant while the comoving Hubble horizon
decreases as 1/a(t). Because the former is a measure for the volume that has been
in causal contact and the latter is a measure for the observable universe, the part of the
CMB sky for which isotropy seems plausible is enlarged by a factor exp(Ne ). So only a
few e-foldings are enough to cover the full sky instead of an angle ∼ 1/30 and to explain
observations.
The exponential dilution of any structures that may have existed before inflation
solves the problem of missing magnetic monopoles described in Sec. 4.3.3. After phase
transitions with spontaneous symmetry breaking topological defects such as domain
walls or cosmic strings may occur. They are likewise attenuated if they are produced
before inflation. However, especially if the preheating temperature after inflaton decay
is high, production of such unwanted relics after inflation can arise and may pose a
problem.
The last and for this work the most important effect of inflation is the production of
small density perturbations as seeds for structure formation. As will be demonstrated
later, they are generated from quantum fluctuations in one or more scalar fields on each
scale at the time of its horizon exit. Direct observation of these fluctuations today is
possible for a spectral range from 10−3 Mpc to 104 Mpc, where the latter is the length
scale of the observable universe 1/H0 . Most probably they require a period of about
16 e-folds of almost exponential inflation. The total amount of inflation must be larger
because the largest observable scale had to be driven out far enough as to reenter
the horizon not earlier than now. The corresponding value of Ne is estimated as in
Eq. (5.124). The value q0 = 0.002 Mpc−1 should only be replaced by the current Hubble
scale H0 ∼ 10−4 Mpc−1 , which makes a difference of 3 in the value Ne . So the necessary
inflationary expansion is in the range Ne ∈ [50, 65], mainly depending on the models of
50
5.3. Inflation driven by a homogeneous scalar field
inflation and preheating.
Put together, the total number of e-folds necessary to solve the presented problems
is around 70. Generically inflation leads to an expansion much larger than this, as will
be seen in Sec. 5.3.
5.3. Inflation driven by a homogeneous scalar field
If the energy density of the universe is dominated by vacuum energy, the scale parameter
a(t) will accelerate. This is seen in the following way:
Because of coordinate invariance, the energy–momentum tensor of vacuum has the
form
Tµν = ρvac gµν
(5.3)
with a constant energy density ρvac . Comparison with the energy–momentum tensor of
an ideal fluid in the local rest frame,
Tµν = diag (ρ, p, p, p),
(5.4)
reveals that the pressure is negative and has the same absolute value as the energy
density. This gives the equation of state
pvac = −ρvac .
(5.5)
Since ρ + 3p < 0, the positive acceleration of a is seen from the second Friedmann
equation Eq. (2.41). The Hubble parameter has the constant value
Hvac =
s
8π ρvac
,
3 m2Pl
(5.6)
yielding the exponential growth of the scale parameter
a(t) ∝ eHvac t .
(5.7)
This is a de Sitter solution. Of course, it does not constitute a viable model for inflation because there is no mechanism of a transition to a radiation dominated universe.
An evolution into the universe today seems more plausible for an inflation driven by
dynamical fields. If there are suitable interactions, a decay into the observed particle
content is conceivable.
In this section the simplest case of a single, homogeneous scalar “inflaton” field is
presented. The potentials discussed in the following correspond to the scenarios for
which the historic evolution has been outlined in Sec. 5.2. But first let us collect the
basic equations for the description of a simplified setting: Inflation driven by a slowly
rolling scalar field.
51
Chapter 5. Inflation
5.3.1. Slow-roll inflation
The evolution of a homogeneous scalar field φ(t) within a spatially flat FRW universe
obeys the equation of motion (3.2) and the Friedmann equation (3.5). Pressure and
energy density can be read off from the energy momentum tensor and are given by
Eq. (3.4). This equation for ρ and p yields a vacuum-like equation of state if the kinetic
energy is negligible compared to the potential energy,
φ̇2
≪ 1.
2V (φ)
(5.8)
To ensure that this situation continues for some time, the acceleration φ̈ should be small,
too. As for a classical particle moving in a potential, this means that it is negligible
compared to the friction term:
φ̈ (5.9)
≪ 1.
3H φ̇ The last two equations constitute a possible definition for slow-roll evolution of a scalar
field. If they are satisfied, Eqs. (3.2) and (3.5) can be approximated as
φ̇ = −
and
H=
1
mPl
1 ′
V (φ)
3H
8πV
3
1/2
(5.10)
.
(5.11)
The two conditions (5.8) and (5.9) also prevent the Hubble parameter from rapid relative
changes within the scale of a Hubble time 1/H:
Ḣ (5.12)
2 ≪ 1.
H This inequality shows that a(t) grows almost exponentially, as could be expected from
the similarity of the equation of state to that of vacuum. Equation (5.12) is obtained
from the time derivative of Eq. (5.11): After replacing V ′ with the help of Eq. (5.10)
one has
3 φ̇2
Ḣ
=−
H
(5.13)
H
2V
and (5.12) follows from the slow-roll condition (5.8).
Now that a slowly evolving field has been shown to entail an accelerated expansion,
the question arises, in what kind of potential slow roll occurs. Using the conditions (5.8)
and (5.9) and the slow-roll versions of the equations of motion, Eqs. (5.10) and (5.11),
the following requirements can be deduced:
ǫ ≪ 1,
52
η ≪ 1,
(5.14)
5.3. Inflation driven by a homogeneous scalar field
where the two slow-roll parameters are defined as
m2
ǫ = Pl
16π
V′
V
2
m2
and η = Pl
8π
V ′′
V
.
(5.15)
A potential with small slow-roll parameters is called flat. The two inequalities (5.14)
were obtained under the assumption of slow roll and quasi-exponential expansion. However, inflation as defined in Eq. (5.1) poses a weaker requirement. So there can be
accelerated expansion without slow roll. Scenarios of fast-roll inflation before and after
a slow-roll stage are considered for example in [93] and [41], respectively. On the other
hand, also a flat part of a potential can fail to sustain inflation if the initial field velocity
φ̇ is too large.
For slow-roll inflation the expansion taking place between two given field values can
be calculated as in Eq. (5.16). The function Ne (φ0 ) denotes the number of e-foldings
from the time when the field takes on the value φ0 = φ(t0 ) to the end of inflation at
φf = φ(tf ). It is calculated as
Ne (φ0 ) = ln
=
Z
af
a(φ0 )
φ0
φf
=
Z
tf
H(t)dt =
t0
3H 2
8π
dφ = 2
′
V
mPl
Z
Z
φf
φ0
φ0
φf
H(φ)
dφ
φ̇
V
dφ.
V′
(5.16)
Through Eqs. (5.10) and (5.11) the slow-roll assumption enters in the second line. The
same two equations are used for calculating the time a slowly rolling field needs to evolve
from φ0 to φf :
∆t =
Z
tf
dt =
t0
√
24π
=
mPl
Z
Z
φf
φ0
φ0
φf
dφ
=
φ̇
√
V
dφ.
V′
Z
φ0
φf
3H
dφ
V′
(5.17)
(5.18)
In the sections to follow examples for slow-roll inflation are discussed.
5.3.2. Large-field inflation
In this section some properties of inflation within the potentials
V =
1
gn φn
n
(5.19)
are presented. In the following, the coupling gn is also written as λ and m2 in the cases
n = 2 and n = 4, respectively. For these potentials inflation is usually assumed to start
from chaotic initial conditions. This was suggested in Ref. [90] where the scenario was
named chaotic inflation. When the initial field values are randomly distributed, there
53
Chapter 5. Inflation
0.6
Potential V (φ)/m4P l
quantum gravity
0.5
0.4
eternal inflation
0.3
0.2
slow-roll inflation
0.1
oscillation
0
− m1
− √1m −1
0
1
Scalar field φ/mP l
√1
m
1
m
Figure 5.2.: Sketch of the quadratic potential m2 φ2 /2 for large-field inflation. Above
φ ∼ 1/m the description within classical general relativity breaks down.
Eternal inflation occurs when quantum fluctuations dominate over the
slow-roll evolution of the field. If φ starts in this regime, then in the
greatest part of the universe inflation continues forever. Standard slowroll inflation takes place at lower energies. After inflation a period of
field oscillations is expected, which lead to the decay of the homogeneous
mode. The situation is similar for other large-field models.
should be regions of space where the system is in the slow-roll regime but below energies
where quantum gravity effects become important:
4 1/n
mPl
n
√ mPl ≪ φ ≪
.
(5.20)
4 π
gn
The first inequality is the slow-roll condition. It requires φ to be super-Planckian,
φ > mPl . At the upper boundary also the energy density is of order the Planck scale
m4Pl . The inequalities in Eq. (5.20) can be fulfilled for small coupling gn . This is
consistent with the requirement for small couplings from observation.
In this context one should also mention the possibility of “eternal inflation” [58, 91].
This concept is based on the fact that for very high field values in chaotic inflation the
quantum fluctuations in φ can be larger than the classical evolution of the homogeneous
part. Then the field can also move upwards with time. The accelerated expansion gives
rise to more and more regions within the universe which are not interacting anymore
since they are outside of each other’s horizon. So the quantum fluctuations of φ lead to
strong inhomogeneities on large scales. In fact it should be expected that large parts
of the universe remain in the inflationary state.8 A similar mechanism exists for hilltop
inflation, see [124].
8
54
In connection with string theory this might alleviate problems with the anthropic principle: Eternal
inflation gives birth to ever new universes and to infinitely many of them with possibly different
5.3. Inflation driven by a homogeneous scalar field
Let us now continue with the amount and the duration of large-field inflation: Equation (5.16) leads in this case to
4π φ20
n m2Pl
Ne (φ0 ) =
(5.21)
if φ0 is taken as the value at begin of inflation and the field value at the end is assumed
to be much smaller. Similarly, Eq. (5.18) gives

 φ0
for V = m2 φ2 /2
∆t ∼ m·m1 Pl
(5.22)
√
ln φφ0f for V = λφ4 /4.
λ·m
Pl
For a universe starting to inflate at
V ∼ m4Pl
⇔
4/n
φ0 ∼ gn−1/n mPl
(5.23)
and for coupling constants m = 10−6 and λ = 10−13 one gets
Netot ∼ 1013 ,
Netot
7
∼ 10 ,
∆ttot ∼ 10−31 s for V = m2 φ2 /2
∆t
tot
−35
∼ 10
4
s for V = λφ /4.
(5.24)
(5.25)
for the total number of e-foldings and the total duration of inflation, respectively. When
eternal inflation is taken into account these numbers corresponding to standard inflation
will be smaller. But in either case the required number of e-foldings is surpassed by a
large amount.
5.3.3. Hilltop inflation
The slow-roll conditions can also be fulfilled around a flat maximum of a potential. An
example that was discussed early in this context is “new inflation” within the Coleman–
Weinberg potential [38],
φ 1
1 4
(5.26)
V (φ) = V0 + λφ ln −
4
v
4
with
V0 =
1 4
λv ,
16
(5.27)
where v is the symmetry-breaking scale. This potential complies with the slow-roll condition not only for large field values but also around φ = 0. The beginning of inflation
in this region can be motivated by assuming symmetry restoration for high temperature and a symmetry breaking phase transition when the medium cools. Breaking the
compactification of extra-dimensions and different vacuum state. So the existence of a (region of
the) universe with life-sustaining conditions could be seen as less improbable.
55
Chapter 5. Inflation
ǫ ≪ η < 0.5
ǫ < 0.5, |η| ≥ 0.5
tachyonic, not inflating
Potential V (φ)/V (0)
1
0.8
0.6
0.4
0.2
-1
-0.5
0
0.5
1
Field φ/v
Figure 5.3.: The logarithmic potential Eq. (5.26) for new inflation. The tachyonic region is shaded in light grey. The field values where the universe is inflating but already preheating (fixed at ǫ < 0.5 < η) are marked with darker
grey. The strip with the darkest shading tags the range of slow-roll inflation (ǫ < η < 0.5). The shadings at negative field values correspond
to the case v = 10−1 mPl and the ones on the right to v = 10−3 mPl .
symmetry the field evolves slowly down into a minimum at φ = ±v. During small-field
inflation the potential can be simplified to
V (φ) = V0 −
1
λφ4 .
16
(5.28)
Then the slow-roll parameters are
ǫ=
1
π
mPl λφ3
16 V0
2
(5.29)
and
m2Pl 3λφ2
,
(5.30)
32π V0
Having the larger absolute value, slow roll is broken by η first. Then the energy density is
typically still approximately V0 . This value, however, can be considerably smaller than
the Planck scale. Then also for rapid decay of the homogeneous mode after inflation
the reheating temperature is many orders of magnitude below TPl . So the hot Big Bang
evolution might start with energy densities for which physics is well known. A problem
might rather arise from thermalization taking place below the temperature of Big Bang
Nucleosynthesis, see Sec. 4.1.
The total amount of accelerated expansion is calculated as in Eq. (5.16): For this
purpose inflation is taken to start at
η=−
φ0 = H
56
(5.31)
5.3. Inflation driven by a homogeneous scalar field
because this is the value expected from quantum fluctuations around φ = 0. The
contribution from the upper boundary φf of the integral for Ne can be neglected because
of φ0 ≪ φf . So the computation is
Netot
=
≈
Z
φ0
3H 2
dφ
V′
(5.32)
6H 2 1
1
∼ ≫ 1.
2
λ
φ0 λ
(5.33)
φf
Also in this case the number of e-foldings is far beyond the necessary amount. The
duration of inflation is
Z φ0
1 1
3H
tot
∆t
.
(5.34)
dφ ∼
′
V
λH
φf
In the small-field case the Hubble parameter does not decrease substantially during slow
roll. So one can read off from this result that inflation lasts much more than one Hubble
time.
5.3.4. Hybrid inflation
There are many inflationary models with more complicated potentials. A broad class is
hybrid or multifield inflation. In this section the first model of this kind is presented,
see [59, 92]. The corresponding potential,
V (φ, χ) = V0 +
with inflaton potential
λ
1 2 2
g φ − µ2 χ2 + χ4 + U (φ),
2
4
1
U (φ) = m2 φ2
2
and offset
V0 =
µ4
4λ
(5.35)
(5.36)
(5.37)
is shown in Fig. 5.4. The parameters in [92],
g2 = λ = 0.1,
m = 102 GeV,
µ = 1.3 · 1011 GeV,
(5.38)
are used. The evolution of the fields is taken to start at large φ and small or zero χ. The
initial stage resembles large field inflation with chaotic spatial distribution of φ. When
the critical value
µ
φc =
(5.39)
g
is reached the waterfall field χ breaks slow roll and terminates inflation. During the
subsequent reheating the fields oscillate around a minimum at φ = 0 and χ = ±v with
the symmetry breaking scale
µ
(5.40)
v=√ .
λ
57
Chapter 5. Inflation
6
4
3
4
Field φ/φc
2
2
1
0
0
-1
-2
-2
-4
-3
-6
-4
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Field χ/v
Figure 5.4.: Potential for hybrid inflation Eq. (??). The universe is inflating when φ
is in one of the valleys at χ = 0 and |φ| > φc . When φ reaches the critical
value ±φc inflation ends and the system reheats during field oscillations
around one of the minima. The parameters v for this figure were taken
from Ref. [92].
The end of the accelerating period in hybrid inflation bears more analogy with the hilltop
case. If the offset V0 dominates over the inflaton potential at φc during inflation, the
mechanism leading to acceleration is different from the large field case. The additional
potential energy extends slow roll down to field values well below the Planck scale.
Assuming a quick drop of the waterfall field into the minimum, the number of efoldings before the end of inflation can be calculated as
Ne (φ) ≈
8π
m2Pl
Z
φ
φc
V0
.
U′
(5.41)
The contribution of U to the numerator has been neglected. For a quadratic and a
quartic inflaton potential, integration gives
Ne (φ) ≈
φ0
8πV0
ln
2
2
φc
mPl m
and
4πV0
Ne (φ) ≈ 2
mPl λ
1
1
− 2
2
φ0 φi
(5.42)
(5.43)
respectively.
For large φ the offset V0 is negligible compared to U (φ). So the total amount of
inflation is again beyond the required Ne ≈ 70, as obtained from the computation
within large-field inflation.
58
5.4. Inflation and the origin of fluctuations
5.4. Inflation and the origin of fluctuations
In the early 1970’s Edward R. Harrison and Yakov B. Zeldovich proposed an initial
flat spectrum of scalar metric fluctuations to explain the origin of structure in the
universe [68,127]. Ten years later the production of these primordial inhomogeneities was
attributed to an inflationary period [69,102]. As already mentioned, tensor perturbations
from inflation were considered already very early [121]. Later their possible traces in
the CMB were studied [3, 48, 117]. Up to now tensor perturbations from inflation could
not be detected [5–7].The topic of this section is the theory of fluctuations produced in
slow-roll inflation. The presentation follows [59], see also [95] and [103].
5.4.1. Scalar perturbations
This section concentrates on scalar perturbations during inflation. More precisely the
goal is to obtain the equation of motion for fluctuations of a scalar field and for the
scalar perturbations in the metric. The latter are characterized by R, which quantifies
the curvature of 3-dimensional equal-time hypersurfaces in the comoving reference frame,
see Section 2.2.4. The velocity v, which is zero by definition in this coordinate system,
will be found shortly. From R the curvature scalar R(3) is calculated as
R(3) = −
4
∆R.
a2
(5.44)
Metric and energy–momentum tensor. The system is described using a metric with
scalar perturbations in conformal Newtonian gauge,
ds2 = a2 (η) (1 + 2Φ)dη 2 − (1 − 2Φ)dx2 ,
(5.45)
see Section A.4. The field φ is separated into a homogeneous, classical part and fluctuations,
φ(x, t) = φc (t) + ϕ(x, t).
(5.46)
Then the energy–momentum tensor,
Tνµ
µρ
= g ∂ν φ∂ρ φ −
δνµ
1 ρσ
g ∂ρ φ∂σ φ − V (φ) ,
2
(5.47)
allows to read off the energy density
ρ = T00 =
1
1
1
1
φ′2 + 2
∂i φ∂i φ + V (φ).
2
2a 1 + 2Φ
2a 1 − 2Φ
(5.48)
Primes denote a derivative with respect to conformal time. The first order in the small
fluctuations Φ and ϕ is
dV
1
δρ = 2 ϕ′ φ′c − Φφ′2
ϕ
(5.49)
c +
a
dφc
59
Chapter 5. Inflation
and using the equation of motion (3.2) results in
a′ ′
1
′ ′
′2
′′
δρ = 2 ϕ φc − Φφc − φc + 2 φc ϕ .
a
a
(5.50)
The (0, i)-component of the perturbation is
δTi0 = g 00 ∂i φ∂0 φ =
1
∂i ϕφ′c
a2
(5.51)
to first order. For the following the velocity potential defined in Eq. (2.72) is needed.
Comparing this equation,
δTi0 = −(p + ρ)∂i v,
(5.52)
with Eq. (5.51) gives
v=−
ϕ
1 φ′c
ϕ=− ′,
2
a ρ+p
φc
(5.53)
where the zero mode of both v and ϕ are taken to vanish and the second equality uses
Eq. (3.4). The field is unperturbed in the comoving reference frame defined by v = 0.
Note that Eq. (5.53) gives also
φ′2
(5.54)
ρ + p = c2 .
a
Equations of motion. The following two equations are components of the perturbed
Einstein equation obtained from Eqs. (2.75) and (2.76). The contributions from the energy momentum tensor on the right hand sides stem from the corresponding expressions
in Eqs. (5.50) and (5.51):
a′2
4π
a′ ′
a′ ′
′2
′ ′
′′
∆Φ − 3 Φ − 3 2 Φ = 2 −φc Φ + φc ϕ − φc + 2 φc ϕ ,
(5.55)
a
a
a
mPl
a′
4π
(5.56)
Φ′ + Φ = 2 φ′c ϕ.
a
mPl
These equations are now transformed into the wave equation (5.63): Using the Friedmann equations (2.42) and (2.43) one rewrites the term proportional Φ in Eq. (5.55)
with the result
′′
a
a′2
4π
a′
a′
+ 2 Φ = 2 φ′c ϕ′ − φ′′c + 2 φ′c ϕ .
(5.57)
∆Φ − 3 Φ′ −
a
a
a
a
mPl
This can be simplified by replacing the second Φ on the left by the corresponding
expression obtained from (5.56). Then one arrives at the more convenient equation
4π a ′2 d
a′ ϕ
∆Φ =
φ
(5.58)
Φ+ ′ .
mPl a′ c dη
aφc
With the additional definitions
u = ϕa + Φ
60
φ′c a2
a′
(5.59)
5.4. Inflation and the origin of fluctuations
and
z=
it is further abbreviated to
φ′c a2
a′
4π ′ z d u .
φ
mPl c a dη z
∆Φ =
Using the definition (5.59) in Eq. (5.56) yields
a′ d a3
4π
Φ = 2 φ′c u.
a2 dη a′
mPl
(5.60)
(5.61)
(5.62)
From the system (5.61) and (5.62) all variables but u and z can be eliminated: After
sorting the derivatives one arrives at a wave equation for u(x, η),
u′′ −
z ′′
u − ∆u = 0.
z
(5.63)
This equation governs the behavior of the scalar field and of the metric. The oscillation
of fluctuations stops in the superhorizon regime because then the spatial derivative of
u is negligible in the last equation. In this regime there is a decaying and a growing
solution. The former is not important for cosmology and the latter is
u(x, η) ∝ z,
(5.64)
which will be referred to later. During slow roll (5.63) is simplified to
u′′ −
a′′
u − ∆u = 0
a
(5.65)
1
ηH
(5.66)
with
a=−
because then H and φ̇c vary only slowly with time and so
z=
φ̇c
φ′c a2
= a ∝ a.
′
a
H
(5.67)
Spatial curvature outside the Hubble horizon. Up to now the behavior of small
scalar fluctuations during inflation has been studied. The following concentrates on the
resulting spectrum of fluctuations of R. From Eq. (2.100) it is
R=
a′
v − Φ.
a
(5.68)
Taking the velocity potential v from Eq. (5.53) and remembering the definitions of u,
(5.59), and z, (5.60), the curvature R is seen to be proportional to u,
R(x, η) = −
u(x, η)
,
z(η)
(5.69)
61
Chapter 5. Inflation
where the dependence on the spacetime coordinates has been restored. With Eq. (5.64)
it is now seen that for modes larger than the Hubble horizon the curvature R stays
constant in time.
For slow-roll inflation the following simplifications can be made: When definition (5.59)
is written for cosmic time t,
!
φ̇c
u= ϕ+Φ
a,
(5.70)
H
the metric perturbation is seen to be negligible because φ̇c ≪ H. Then the curvature is
R=−
Hϕ
ϕa
=−
z
φ̇c
(5.71)
where the last equality makes use of Eq. (5.67).
Scalar field perturbations on unperturbed background. Equation (5.71) relates the
curvature perturbations to field fluctuations ϕ. These are created by the spacetime
curvature arising from accelerated expansion. They are calculated without considering
inhomogeneities of the metric. The derivation is presented in the following: The action
S of a scalar field within a curved background is given in Eq. (3.1). The action of a
small perturbation ϕ, defined by
φ(x, η) = φc (η) + ϕ(x, η),
then reads
Sϕ =
1
2
Z
√
d4 x −g gµν ∂µ ϕ∂ν ϕ − V ′′ (φc )ϕ2 ,
(5.72)
(5.73)
where the prime indicates a derivative with respect to the classical field φc . The action
is taken to second order in ϕ because the aim is a linear equation of motion. For
subhorizon modes during slow roll the second term is small as is shown by
2
8π
k
′′
2
2
V ≈ 2 V η ≈ 3H η ≪ H ≪
,
(5.74)
a
mPl
where η is the slow-roll parameter. The following equations will use η as letter for
conformal time again. The dominant part of the action is conveniently written using
this coordinate:
Z
1
Sϕ =
d4 xa2 (η) (∂η ϕ)2 − (∂i ϕ)2 .
(5.75)
2
The equation of motion is noted for later reference:
a′
ϕ′′ + 2 ϕ′ − ∆ϕ = 0.
a
(5.76)
The expansion of the universe is better accounted for when the field
χ(x, η) = a(η)ϕ(x, η)
62
(5.77)
5.4. Inflation and the origin of fluctuations
is taken as variable. The corresponding replacement in Eq. (5.75) is done in
Z
′2
′
1
2
2 ′ a
4
′2
2a
− (∂i χ)
d x χ +χ 2 − χ
Sχ =
2
a
a
Z
′′
1
2
4
′2
2a
=
− (∂i χ) ,
d x χ +χ
2
a
(5.78)
(5.79)
where partial integration of the third term leads to the second line. Variation with
respect to χ gives
a′′
(5.80)
χ′′k − χk + k2 χk = 0
a
for a mode χk . From Eq. (5.66) the second term in this equation is seen to be small
for large negative η, i.e. at early times. So it is concluded that the action and the
equation of motion are those of a massless, non-interacting field in flat spacetime. The
regime described above coincides with the sub-horizon condition k > 1/aH for a mode
of momentum k. The corresponding field operator is written as
Z
d3 k
√
χ(x, η) =
eikη−ikx A†k + e−ikη+ikx Ak .
(5.81)
(2π)3/2 2k
Inside the horizon the modes of χ oscillate freely. This means that the oscillation of the
physical field ϕ encounters a damping proportional to 1/a. Later the second term in
Eq. (5.80) becomes more important until the third one can be neglected. Apart from a
possible decaying contribution, the solution of χk is then proportional to a. So we are
left with constant modes ϕk outside the horizon.
In order to calculate the constant result outside the horizon, it has to be matched
with the oscillating solution inside the horizon. For this purpose the solution around
horizon crossing has to be obtained: The equation of motion Eq. (5.80) can be simplified
under the assumption a = −1/ηH. This is a good approximation also if the accelerated
expansion is not exactly exponential because the time span in question is not much
larger than a Hubble time. Thus, Eq. (5.80) is written as
χ′′k −
being solved by
χ±
k
2
χk + k 2 χk = 0
η2
1 ±ikη
i
=√ e
.
1±
kη
2k
(5.82)
(5.83)
Here the upper (lower) signs refer to a solution with positive (negative) frequency.
Well inside the horizon the solutions correspond to a non-interacting field because then
the second term in brackets is small. So the field operator χk evolves from the noninteracting oscillatory form to a growing one. The mode function of the inflaton operator
is then
1 ±ikη
i
±
.
(5.84)
ϕk = − √ e
H η±
k
2k
63
Chapter 5. Inflation
Late times correspond to η approaching zero from below. According to Eq. (5.84)
the oscillation stops in this case and a constant solution develops. With some timeindependent phase shift αk the field operator then reads
Z
H −ikx+iαk †
d3 k
√
e
Ak + eikx−iαk Ak .
(5.85)
ϕ(x, η) =
(2π)3/2 2k k
The last equation is only valid for modes that have left the horizon.
The final goal of this section is to arrive at the formula for the power spectrum PR (k)
of R after inflation. The connection between R and ϕ has been obtained in Eq. (5.71).
So the next step is the power spectrum of scalar field fluctuations ϕk after inflation. The
power spectrum of a field ϕ quantifies the strength of its fluctuations, as will be seen in
the following: First it is assumed that they are Gaussian random fluctuations, such that
they are fully characterized by their two-point correlation function. This statement is
also valid the Fourier transformed quantities. The calculation is done for homogeneous
and isotropic space. From the two-point function in real space, hϕ(x)ϕ(y)i, the twopoint function in Fourier space is obtained as
Z 3 3
d xd y −ikx−k′ y
′
e
hϕ(x)ϕ(y)i
(5.86)
hϕ(k)ϕ(k )i =
(2π)6
Z 3 3
d zd y −i(k+k′ )y −izk
=
e
e
hϕ(x)ϕ(y)i
(5.87)
(2π)6
Pϕ (k)
δ(k, k′ ),
(5.88)
=
(2π)3
where for the second line the variable z = x − y is introduced. The function Pϕ (k) in
the third line is defined as correlation function in Fourier space. In the homogeneous
and isotropic case, the correlator can be written as a function of |z| = |x − y|: D(|z|) =
hϕ(x)ϕ(y)i. Then the Fourier transformation is
Z
Pϕ (k) = d3 ze−izk D(|z|),
(5.89)
which has been used in Eq. (5.88). Later on, an alternative definition of the power
spectrum will be used,
k3
(5.90)
P(k) = 2 P (k).
2π
In either case, the fluctuations are quantified as
Z
′
hϕ2 (x)i = d3 kd3 k′ ei(k+k )x hϕ(k)ϕ(k′ )i
(5.91)
Z
Z
dk
3 Pϕ (k)
= d k
=
Pϕ (k).
(5.92)
(2π)3
k
It can be calculated using Eq. (5.85) and the commutation relation
[Ak , A†k′ ] = δ(k − k′ ).
64
(5.93)
5.4. Inflation and the origin of fluctuations
As a result the power spectrum of scalar field fluctuations after horizon exit in slow-roll
inflation is obtained:
Hk 2
.
(5.94)
Pϕ =
2π
Hk denotes the Hubble parameter during horizon exit of the mode k:
Hk =
a
.
k
(5.95)
Curvature power spectrum. Equation (5.94) and the connection to spatial curvature
in Eq. (5.71) result in the formula for the curvature on equal-time hypersurfaces within
the comoving reference frame:
2 2
H
PR (k) =
(5.96)
2π φ̇c tk
As indicated by tk the Hubble parameter and the velocity of the classical field are
evaluated at horizon crossing. The spectrum is also characterized by the square root of
the power spectrum,
p
H2
,
(5.97)
∆R (k) = PR (k) =
2π|φ̇c | tk
which is a direct measure for the amplitude of the fluctuations.
5.4.2. Tensor perturbations
In this section the production of gravitational waves during inflation is examined. From
comparison of the action of a scalar field (5.75) to the action of tensor perturbations
(2.88) it is seen that the fluctuations behave identically when the factor m2Pl /32π is
taken into account: The action of one degree of freedom of GWs, h(A) = h(+) or h(×) ,
is equal to the action for ϕ times this factor. A redefinition
mPl (A)
h̃(A) = √
h
32π
(5.98)
allows for a literal repetition of the scalar field case. Then the power spectrum for each
h̃(A) is given by Eq. (5.94). Restoring the factor in Eq. (5.98) and summing over both
polarizations gives the power spectrum of gravitational waves outside the horizon:
PT (k) =
16 Hk2
.
π m2Pl
(5.99)
5.4.3. The spectra of fluctuations after slow-roll inflation
This section contains a discussion of the spectra of scalar and tensor metric fluctuations
after slow-roll inflation. The link between properties of the inflaton potential and of the
resulting spectra is established. The equations derived in this section are valid for both
large-field and small-field potentials. Specific examples are worked out in Section 5.5.
65
Chapter 5. Inflation
Using Eqs. (5.10) and (5.97) the amplitude of scalar perturbations is seen to be
r
3Hk3
8π V 3/2
=
4
,
∆R (k) =
2πV ′
3 m3Pl V ′
(5.100)
where in the last step the potential is evaluated at the time of horizon exit of the mode
k. The tensor power spectrum is connected to the potential by
PT =
128 V
,
3 m4Pl
(5.101)
see Eq. (5.99). With Eqs. (5.100) and (5.101) the scalar-to-tensor ratio r in the slow-roll
regime can be calculated as
m2
PT
= Pl
r=
PR
π
V′
V
2
,
(5.102)
where also Eqs. (5.10) and (5.11) have been used. The spectral scalar and tensor indices
are defined as
d ln PR (k)
d ln PT (k)
ns =
+ 1 and nT =
.
(5.103)
d ln k
d ln k
If they are constant, this is equivalent to
PR (k) ∝ kns −1
and PT (k) ∝ knT .
(5.104)
Both spectral indices are connected to the parameters defined so far: It is
ns − 1 = 2η − 6ǫ
(5.105)
r
nT = − .
8
(5.106)
and
To derive these equations the power spectra are expanded around arbitrary k0 in the
following way:
d ln PR
(φ(tk ) − φ(tk0 ))
(5.107)
PR (k) = PR (k0 ) 1 +
dφ
′
V
V ′′
= PR (k0 ) 1 + 3
− 2 ′ (φ(tk ) − φ(tk0 ))
(5.108)
V
V
and
d ln PT
(φ(tk ) − φ(tk0 ))
PT (k) = PT (k0 ) 1 +
dφ
V′
(φ(tk ) − φ(tk0 )) .
= PT (k0 ) 1 +
V
66
(5.109)
(5.110)
5.5. Examples and observations
Both results depend on field values fixed by the horizon exit of the arbitrary modes k
and k0 . The slow-roll approximation allows for a formula connecting k with φ(tk ): First
remember Eq. (5.16) and use it in the Taylor expansion
Ne (φ(tk )) − Ne (φ(tk0 )) =
=
dNe
(φ(tk ) − φ(tk0 ))
dφ
8π V
(φ(tk ) − φ(tk0 ))
m2Pl V ′
(5.111)
(5.112)
to obtain the second line. Then it is observed that the left hand side of Eq. (5.111) can
also be written in terms of the wavenumbers in question:
k0
.
(5.113)
Ne (φ(tk )) − Ne (φ(tk0 )) = ln
k
The combination of both results,
m2 V ′
ln
φ(tk ) − φ(tk0 ) = Pl
8π V
k0
k
,
(5.114)
is now used in the expansions of the power spectra above. The resulting combinations
of the potential and its derivative match the slow-roll parameters defined in Eq. (5.14).
More precisely, it is
k
(5.115)
PR (k) = PR (k0 ) 1 + (2η − ǫ) ln
k0
and
k
PT (k) = PT (k0 ) 1 − 2ǫ ln
.
k0
(5.116)
Equations (5.105) and (5.106) follow from comparison with the definitions in (5.103).
Only relying on slow roll, Eq. (5.106) connects the two observables r and nT . This offers
a verification of inflation that is independent of specific models. However, measurements
of nT are not possible in the foreseeable future.
5.5. Examples and observations
Let us now turn to the behavior of the homogeneous field and the fluctuations in specific
examples of slow-roll inflation. The large-field models discussed here have a potential
of the form
1
V (φ) = gφn .
(5.117)
n
The time evolution of a free massive scalar field rolling down from a large value is
displayed in Fig. 5.5. The two slow-roll parameters ǫ and η (defined in Eq. (5.15)) are
equal for V = m2 φ2 /2 and well below one down to φ = 1mPl . Assuming Eqs. (5.10)
and (5.11) to be valid in this regime, a constant velocity of the field is expected. This
is displayed in Fig. 5.5, as well as the corresponding decay of the Hubble parameter H
67
Chapter 5. Inflation
4
1040
scale parameter a(t)
φ(t)/mP l
ǫ(t)
3
2
1
1030
1020
1010
0
0
2
4
6
8 10 12 14
time t/106m−1
Pl
16
18
20
12
1
0
2
4
6
8 10 12 14
time t/106m−1
Pl
16
18
20
10−10
10−11
8
ä /m2
Pl
a
H(t)/10−6mP l
10
10−12
6
4
10−13
2
10−14
0
2
4
6
8 10 12 14
time t/106m−1
Pl
16
18
20
0
2
4
6
8 10 12 14 16 18 20
time t/106m−1
Pl
Figure 5.5.: Time evolution of the scalar field φ, the slow-roll parameter ǫ, and of the
scale parameter and its derivatives during inflation within the potential
V = m2 φ2 /2, m = 1.5 · 10−6 . Hubble damping leads to a linear decrease
of φ(t) and of the Hubble parameter H(t). Accelerating expansion ends
when the slow-roll parameter ǫ becomes bigger than one (top left and
bottom right panels).
(left panels). The latter is linear too, because the kinetic term in Eq. (3.5) is negligible.
The right panels illustrate the strong, almost exponential growth of the scale parameter
a(t) and its acceleration, which ends shortly after ǫ and η acquire values greater than
one.
The evolution of these quantities in λφ4 /4 is similar and is therefore not displayed here.
Let us instead go on to the fluctuations within the scalar field, which are produced at the
horizon exit of each mode. In order to compare theoretical predictions with observation,
it is necessary to connect the length scales during inflation with corresponding ones
today. The calculations in this thesis will be compared with WMAP and Planck data
obtained for modes with wavenumber q0 = 0.002 Mpc−1 today. For example, this will
be done for the power spectrum PR and its spectral index ns .
During inflation a comoving length scale λ = 2π/k can be identified by the number of
e-foldings Ne that follow after its horizon crossing but before the end of inflation. This
is defined by
aend
.
(5.118)
exp(Ne (k)) =
a(tk )
68
5.5. Examples and observations
Here and in the following the label “end” is used for quantities evaluated at the end
of inflation. a(tk ) is the scale parameter at the time when a wave with comoving
wavenumber k crosses the Hubble horizon: The time tk is fixed by
(5.119)
k = H(tk )a(tk ).
So for a present-day wavenumber q0 = k/a0 , the exponent in Eq. (5.118) can be calculated as (see the following text for explanations or Ref. [59]):
aend H(tk )
aend areh H(tk )
(5.120)
Ne (k) = ln
= ln
k
areh a0 q0
!
1/β
ρreh T0 H(tk )
≈ ln
(5.121)
1/β
ρend Treh q0
!
4/β−1 1−2/β
T0 Treh Hend H(tk )
≈ ln
,
(5.122)
2/β
q0
Hend
m
Pl
where the following considerations are used: For the second line the energy density
during the reheating epoch is assumed to decay like a−β (with, for example, β = 4 for
radiation and β = 3 for matter) with a constant β. The ensuing radiation dominated
era is taken to last until today without changes of the degrees of freedom: So the present
scale factor is written as
Treh
.
(5.123)
a0 = areh
T0
4 and ρ
2
2
The third line uses the simplifications ρreh ≈ Treh
end ≈ Hend mPl . The final form
2
is obtained after approximating Hend ≈ Tend /mPl (where Tend is the temperature that
would result after instantaneous reheating) and splitting the logarithm:
4
Tend 1
mPl
H(tk )
T0
− 1 ln
−
− ln
+ ln
.
(5.124)
Ne (k) ≈ ln
q0
β
Treh
2 Hend
Hend
The largest contribution to Ne (k) comes from the first term: ln(T0 /q0 ) ≈ 65 for T0 =
2.7K and q0 = 0.002 Mpc−1 . The subsequent terms are smaller and model dependent:
The second term is determined by the mechanisms of preheating and reheating after
inflation: The prefactor lies between 0 and 1/3; the temperature Tend in the logarithm
can be estimated as
√
−3
2 2


 mφend ≈ 10 mPl for V = m φ /2
p
Tend ≈ Hend mPl ≈ λ1/4 φ ≈ 10−3 mPl
for V = λφ4 /4


(λv 4 /16)1/4 ≈ 10−4 v for small-field inflation, Eq. (5.26).
(5.125)
The range of Ne (k) is now computed with Treh ∈
and, for the case of
−3
small-field inflation, with v ∈ (10 , 10)mPl . The last term in Eq. (5.26) is neglected. So
for each inflationary scenario, any q0 today is to be connected with this range in Ne (k),
stemming from our ignorance about the history of the universe after inflation. For the
(108 GeV, Tend )
69
10−8
1.0
1.0
1.3
1.3
1.5
1.5
3.0
3.0
Scalar and tensor power spectra
Scalar and tensor power spectra
Chapter 5. Inflation
10−9
10−10
10
20
30
40
50
Ne(k)
60
70
80
10−8
1.0
1.0
1.5
1.5
2.2
2.2
10.0
10.0
10−9
10−10
10
20
30
40
50
Ne (k)
60
70
80
Figure 5.6.: Scalar (solid lines) and tensor (dashed lines) power spectra after inflation
within the potentials m2 φ2 /2 (left panel) and λφ4 /4 (right panel). The
couplings m and λ are given in units of 10−6 mPl and 10−13 , respectively.
The grey vertical band between Ne = 50 and 60 corresponds to the
wavenumber q0 = 0.002 Mpc−1 today. The horizontal band indicates
the measurements of WMAP [71] in this region, which fix the couplings
to m2 ≈ (1.4 ± 0.1) · 10−6 mPl for the quadratic potential and to λ ≈
(1.8 ± 0.4) · 10−13 in the φ4 case.
choice of parameters in the potentials this leaves a greater freedom that is illustrated by
the grey vertical band in the figures to follow. Some effort has been made to match the
borders of this band to the potential and its parameters but the uncertainty of these
values should be kept in mind.
In Fig. 5.6 the scalar and tensor power spectra are shown for the quadratic and the
quartic inflaton potentials with various values of the couplings. The reasoning and the
calculations are based on Ref. [59]: The function Ne (φ) is special to each model but in
the present cases it is independent of the coupling. This can be read from the result of
the calculation (5.16). For V = gn φn /n one gets
Ne (φ0 ) =
4π φ20
.
n m2Pl
(5.126)
Then the sole dependence of the tensor power spectrum on φ in Eq. (5.101) explains, why
for different couplings the dashed lines in Fig. 5.6 are simply shifted in the y-direction.
As already discussed, the mode with wavenumber q0 = 0.002 Mpc−1 , for which the
WMAP measurements obtained the value
PR (k) = ∆2R (k) = (2.46 ± 0.09) · 10−9
(5.127)
can correspond to a range of values of Ne (k) or φ(tk ). This results in the uncertainty
in the couplings m2 ≈ (1.4 ± 0.1) · 10−6 mPl and λ ≈ (1.8 ± 0.4) · 10−13 , as suggested by
the results displayed in Fig. 5.6. The dependence of the scalar power spectrum on the
coupling constant is calculated from the square of Eq. (5.100). With
V 3/2
=
V′
70
√
gn n/2+1
φ
n
(5.128)
Scalar and tensor power spectra
5.5. Examples and observations
10−7
λ = 4 · 10−14,
λ = 2.5 · 10−14,
λ = 5 · 10−14, v
λ = 3 · 10−14, v
10−8
v = 10
v = 10
= 10−3
= 10−3
10−9
10−10
10−11
10
20
30
40
50
Ne(k)
60
70
80
Figure 5.7.: Scalar (solid lines) and tensor (dashed lines) power spectra after inflation
within the small-field model Eq. (5.26). For two values of v, which is
given in units of the Planck mass, the allowed range for the coupling
λ is found. The shaded regions again mark the part of the spectrum
corresponding to q0 = 0.002 Mpc−1 and the measurement of WMAP at
this wavenumber [71].
and Eq. (5.126) the result is
16 8π
PR (k) = 2
n 3
nNe
4π
(n+2)/2
gn
m4−n
Pl
(5.129)
giving the order of magnitude of the estimations noted above.
Fig. 5.7 shows the scalar power spectra after inflation in this potential for several
parameter sets. For a wide range of the energy scale v the coupling λ has to be of the
order 10−14 . As in the cases above, the power spectrum decays with smaller Ne because
this corresponds to later times leading to smaller amounts of energy in the potential and
larger field velocities. For equal symmetry breaking scale v, a similar argument as the
one after Eq. (5.16) predicts the simple proportionality between spectra with different
coupling λ. This is verified by the two pairs of solid lines in Fig. 5.7. From Eq. (5.101) a
very small tensor perturbation in the low-v case is expected. The result is many orders
of magnitude below the range of this figure.
The tensor-to-scalar ratio calculated in Eq. (5.102),
m2
r = Pl
π
V′
V
2
= 16ǫ,
(5.130)
is displayed in Fig. 5.8. As expected, when plotted against Ne or φ, it is independent of
the coupling in the case of quartic and of quadratic potentials. The same is true for the
hilltop potential if v is not varied. The analytic estimate for slow roll in V = λφn /n is
m2
r(Ne ) = Pl
π
2
n
4n
=
,
φ
Ne
(5.131)
71
Tensor to scalar ratio r and −8nT
Chapter 5. Inflation
1.0
λ = 1.0, 1.5, 2.2, 10.0
mφ = 1.0, 1.3, 1.5, 3.0
v = 10mP l , λ0 = 4 · 10−14
0.8
0.6
0.4
0.2
0.0
10
20
30
40
50
Ne (k)
60
70
80
Figure 5.8.: Tensor-to-scalar ratio resulting from inflation within a quartic, a quadratic, and a hilltop potential illustrated by the red, green, and blue
lines, respectively. Shaded horizontal regions correspond to the analytic
estimation (5.132) based on the corresponding potential. The units of
the couplings in the two former cases are chosen as in Fig. 5.6.
The tensor spectral index nT is plotted in the form −8nT in dashed
lines. Within the slow-roll approximation this expression is predicted to
be equal to r.
where Eq. (5.16) is used for the second equality. Inserting Ne = 50 − 60 gives
r = 0.13 − 0.16,
r = 0.27 − 0.32,
n = 2,
n = 4.
(5.132)
This is not within the region allowed by Planck measurements [113] which give r(q0 =
0.002 Mpc−1 ) < 0.12 at 95% confidence level. In contrast, the results for r in the hilltop
model are compatible with this bound. The second quantity displayed in Fig. 5.8 is the
tensor spectral index defined in Eq. (5.103). Assuming slow roll the proportionality
r
nT = −
(5.133)
8
has been shown in Eq. (5.106). In Fig. 5.8 the quantity −8nT is indicated in dashed
lines. The agreement with the solid lines confirms the slow roll prediction for large
enough Ne .
The scalar spectral index ns as a function of Ne is displayed in Fig. 5.9. As with r and
nT , the function ns (Ne ) does not change when the potential is multiplied by an overall
factor. So for the simple quadratic and quartic potentials the outcome is independent
of the coupling. The numerical result for V = m2 φ2 /2 and λφ4 /4 closely follows the
analytical formula Eq. (5.134)
n+2
(5.134)
ns − 1 = −
2Ne
72
5.5. Examples and observations
which is derived from Eq. (5.105) (ns − 1 = 2η − 6ǫ) and using Eq. (5.16) for potentials
of the form V = λφn /n. Inflation with a massive free scalar inflaton field yields results
within the range allowed by Planck measurements and so does hilltop inflation with a
large enough energy scale.
In the above discussion the example for a potential leading to hilltop inflation was (5.26).
It has been argued that during slow roll the simplified form
V (φ) = V0 −
λ 4
φ
16
(5.135)
can be used. The coupling, for which the power spectrum (5.127) is obtained, is calculated as follows: Using Eq. (5.16) gives
32π
Ne (φ0 ) = − 2
mPl
Z
φ0
φf
φ
V0
−
3
λφ
16
dφ
(5.136)
The second term and the term proportional to φ−2
are neglected. Then one has
f
φ2 =
16π V0 1
.
m2Pl λ Ne (φ)
(5.137)
On the other hand, evaluating Eq. (5.100) for the present case leads to
PR =
∆2R
8π
=
3
16
λφ3 m3Pl
2
V03 .
(5.138)
So the formula connecting the fluctuations with the coupling is
λ = 6π 2
PR
.
Ne3
(5.139)
Inserting the required value for the spectrum at Ne = 60 gives λ ≈ 7 · 10−13 . In
accordance with the different prefactors in Eq. (5.117) and (5.135) this value is somewhat
larger than the one found in Eq. (5.129). In hilltop and hybrid inflation the slow-roll
parameter ǫ is generically very small, ǫ ≪ |η| ≪ 1. Therefore, by virtue of Eq. (5.130),
a small tensor-to-scalar ratio is expected.
In the hybrid inflation scenario Eq. (5.35) the inhomogeneities are produced during
slow roll of φ within the potential
1
U (φ) = V0 + m2 φ2 .
2
(5.140)
A significant difference to inflation driven by a free massive field requires the offset V0
to be the dominant contribution. Approximating φ ≈ φc at Ne = 60 and using the
slow-roll parameter
m2 m2
(5.141)
η = Pl
8π V0
73
Chapter 5. Inflation
Scalar spectral index ns
1.00
V ∝ φ4 + slow-roll
λ = 1.0, 2.2, 10.0
V ∝ φ2 + slow-roll
mφ = 1.0, 1.3, 1.5, 3.0
λ = 4, v = 10
λ = 5, v = 10−3
0.98
0.96
0.94
0.92
0.90
30
40
50
60
70
80
Ne (k)
Figure 5.9.: The scalar spectral index ns as a function of Ne . The black lines illustrate
the slow-roll formula Eq. (5.134) for the quartic (solid line) and quadratic
(dashed line) potentials. The numerical results in the large-field models
only slightly deviate from the corresponding part of Eq. (5.134) and
coincide for different values of the couplings, which are given in units of
10−14 (λ) and 10−6 mPl (mφ ). The units within the small-field examples
(blue and magenta) are 10−14 (λ) and mPl (v).
gives
3/2
V 3/2
V
≈ 02
′
V
m φ
(5.142)
and, with Eq. (5.100),
PR =
m2
.
12π 2 φ2c η 3
(5.143)
Within this model the appropriate coupling m depends on the critical field and η. The
former is expected to be below the Planck scale and the latter should be much smaller
than one. Therefore also for this potential measurements require m ∼ 10−6 or smaller.
A similar model will be discussed in greater detail in Chapter 6.
To assess the viability of an inflationary potential it is convenient to plot the resulting
tensor-to-scalar ratios r and scalar spectral indices ns for the values of Ne that are
accessible to measurements. This is done for several models in Figs. 6.10, 6.11, and 6.12.
5.6. A universe without inflation?
Of course, neither is inflation a scenario without conceptual problems nor is it the only
theory trying to solve the open questions which are posed by hot Big Bang cosmology.
In this Section some alternative scenarios are mentioned. The presentation relies on
Ref. [24, 26].
74
5.6. A universe without inflation?
Inflation renders the initial conditions of cosmological evolution more plausible. The
strong requirements arising from the spatial flatness and the homogeneity of the universe
can be reduced to the less restrictive necessary conditions for a successful inflationary
stage.
However, the latter makes its claims too. For inflation driven by scalar fields the initial
cosmological medium needs to be dominated by flat potentials. The initial condition
for the fields must allow for the necessary amount of accelerated inflation. In addition,
the field content of the inflationary system must be appropriately coupled to the fields
of the known subsequent evolution, see Chapter 7.
Then it has been shown that for scalar field inflation there is still the unsolved problem
of the initial singularity [23]. Inflation relies on the vast dilatation of very small scales.
Since physics at scales smaller than the Planck length lPl is not known, it is not clear if
the predictions for cosmological scales can be trusted. Reference [75] presents conditions
under which this is the case, while Ref. [96] shows deviating results when the assumptions
about high energy physics are varied. This is called the trans-Planckian problem of
inflation. It can be concluded that it is reasonable to look for alternative scenarios and
to check how promising they are.
One such a scenario is called the matter bounce where the initial singularity is removed and time extends to the infinite past [25,81]. The observable universe is causally
connected because the Big Bang scenario is assumed to be preceded by a period of contraction that is symmetric to later expansion. The trans-Planckian problem does not
arise if the bounce is slowed down before the energy density penetrates the realm of
unknown physics. Matter domination during the contraction leads to a scale-invariant
spectrum [112]. On the other hand one should mention as a severe drawback that the
flatness problem is not solved in this scenario. Another problem is that during the
bounce strongly growing anisotropy may spoil a smooth transition to the expanding
phase. A non-singular bounce within General Relativity is only possible when fields
with exotic properties are introduced [14].
For example this is done in the scenario of the ekpyrotic9 universe [31, 78], which
solves the horizon and flatness problems and can provide a slightly tilted spectrum. The
scenario of a bouncing universe can be extended to a cyclic history of cosmic crunches
and bangs [122].
A spectrum of scalar perturbations can also be produced during a period of constant
scale-factor. This scenario is called the “emergent universe” [46,47]. It solves the horizon
problem and can provide a flat spectrum of perturbations. An important difference to
standard inflation is that the emergent universe avoids the initial singularity. A possible
realization is string gas cosmology [105].
Many scenarios and theories were proposed as an alternative to cosmological inflation.
However, it seems that in spite of the open questions of inflation, up to now there is no
strong competitor to the elegance of inflationary theory.
9
The greek word ἐκπύρωσις (ekpyrosis) has been used by ancient philosophers and denotes the cyclic
destruction of the world by fire.
75
6. Fluctuations after two periods of
inflation
6.1. The potential and its simplification
In this section the results of calculations within the potential
1
1
1
χ 1
4
V (φ) = V0 + λ0 χ ln −
+ λ1 φ4 − λ2 χ2 φ2
4
v
4
4
4
with
(6.1)
1
(6.2)
λ0 v 4 exp λ22 /λ0 λ1
16
are presented. The potential Eq. (6.1) is displayed in Fig. 6.1 along with a possible
path (χ(t), φ(t)) during and after inflation. The color code reflects the value of the
potential V (χ, φ) in units of V0 . The constants are chosen as λ0 = λ1 = λ2 = 10−14
and v = 0.1mPl and the calculation of the path is done without consideration of field
inhomogeneities. The starting point of the fields is at large φ and small positive χ with
zero initial field velocities. Then φ(t) starts to evolve down the quartic potential while
the value of χ is still close to zero, i.e. much smaller than the symmetry breaking scale
v. During this period of time the evolution can be described as large field inflation with
an offset V0 in the potential. After one and a half oscillations the field φ stays close to
zero due to Hubble damping. Now the main evolution in field space is a slow roll along
the χ direction. Accordingly, this period of time can be characterized as a small-field
hilltop inflation. Between these two inflationary stages the universe undergoes a short
phase of decelerated expansion.
This model is related to the standard potential for hybrid inflation discussed in Section 5.3.4. One difference is that in the potential Eq. (6.1) there is no channel the field
χ could be confined to during the first inflationary period. In order not to terminate
inflation too early, the initial values of χ and χ̇ therefore have to be very close to zero,
as mentioned above. Then the field χ does not act as a waterfall field that rapidly
rolls down to the minimum of the potential after a critical value of φ has been passed.
Instead, it could lead to the second period of inflation mentioned in the last paragraph.
This second inflation is due to the symmetry breaking logarithmic term in the potential. If λ2 6= 0, there is a second symmetry breaking term in the potential Eq. (6.1).
Because it can considerably enlarge the offset V0 in Eq. (6.2), the inflationary range in
V0 =
Chapter 6. Fluctuations after two periods of inflation
14
0.15
12
Field φ/mP l
0.1
10
0.05
8
0
6
-0.05
4
-0.1
2
-0.15
0
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
Field χ/mP l
Figure 6.1.: Possible way of the fields φ and χ through the potential Eq. (6.1). In
this case the energy scale and the couplings are chosen as v = 0.1mPl ,
λ0 = λ1 = λ2 = 10−12 . The white dot marks the end of accelerated expansion. The color of the background encodes the value of the potential,
V (φ, χ)/V0 .
field space is extended to χ ≈ v for large enough λ2 . The scenario ends with oscillations
of the scalar fields around one of the minima in Fig. 6.1. The particle production that
accompanies the oscillations reheats the universe and the standard hot Big Bang history
sets in.
The potential discussed above includes two scalar fields both acting as inflaton fields
which by definition drive the inflation. This means that the resulting fluctuations should
in general be calculated within the framework of multifield inflation [61, 73, 125]. Then
one has to care for adiabatic modes which correspond to fluctuations along the inflaton
field and for entropy modes which describe fluctuations between the energy contributions of different fields without changing the total energy density ρ. Even after horizon
exit, entropy modes can source adiabatic fluctuations with corresponding wavelengths.
As discussed in Ref. [61] however, entropy fluctuations are only produced when the
background fields follow a curved trajectory in field space. Thus, only during the time
when the main field evolution changes its direction from ±φ to χ, the development of
entropy fluctuations is expected. But since at this stage the field φ is not evolving
slowly anymore and no accelerated expansion takes place, there will be no production of
non-adiabatic fluctuations even in this moment. So within this scenario no complication
due to multifield dynamics will occur. Therefore, in this work the calculations are done
within a simplified setting with single field inflation. The two inflationary epochs are
both ascribed to the same scalar field φ evolving within an adapted potential. More
78
6.2. Evolution of the homogeneous mode
6
combination of λ1 φ4 /4 and
logarithmic potential
alternative version with m2 φ2 /2
Potential V (φ)
5
4
3
2
1
0
-1
-0.5
0
0.5
1
1.5
Scalar field φ
Figure 6.2.: The two potentials Eqs. (6.3) and (6.4) that are used in the following
calculations of fluctuations produced during inflation.
precisely the computation of fluctuations is done using the two alternative potentials
(
φ<0
V0 + 14 λ1 φ4
(6.3)
V4 (φ) =
χ
φ≥0
V0 + 14 λ0 φ4 ln v − 14
and
V2 (φ) =
(
V0 + 12 m2 φ2
V0 +
1
4
4 λ0 φ
φ<0
χ 1
ln v − 4
φ ≥ 0,
(6.4)
see Fig. 6.2. In both cases V0 is given by Eq. (6.2).
6.2. Evolution of the homogeneous mode
Within the simplified scenario described in the last section, the field starts at a super–
Planckian negative value and moves slowly down to the flat region around φ = 0. In
spite of vanishing potential gradient, for small field values slow roll ends for a short time
within many parameter sets. The second part of the evolution uses the same hilltop
potential for both variants. The solution φ(t) is sketched in Fig. 6.3 for three different
sets of couplings and the same symmetry breaking scale v = 1mPl in each case. For
most of the evolution of the homogeneous mode down to φ = 0, the contribution of the
offset V0 is negligible. So Eq. (5.18) yields the proportionality
m(2)
∆t(1)
=
∆t(2)
m(1)
(6.5)
for the time the field φ needs to evolve down a fixed interval ∆φ within different parameter sets 1 and 2. The delay of the blue and green curves when arriving at φ = 0 is a
79
Chapter 6. Fluctuations after two periods of inflation
2
field φ(t)/mP l
0
λ0 = 5 · 10−14
, m22 = 5 · 10−13
λ0 = 5 · 10−14
, m = 5 · 10−14
λ0 = 5 · 10−15 , m2 = 5 · 10−14
-2
-4
-6
-8
-10 6
10
107
108
time t/m−1
Pl
109
Figure 6.3.: The scalar field φ rolling down the potential Eq. (6.4) for three different
parameter sets with v = 1mPl in each case. Too large λ0 /m2 retains the
field for a very long time at values around φ = 0 (green curve).
consequence of this result. Also the duration of the second inflation can be estimated
with Eq. (5.18) to be
v
u (2)
(1)
uλ
∆t
(6.6)
= t 0(1) ,
(2)
∆t
λ0
which matches the numerical result. The latter can be read from the duration of the
√
almost flat regions of the red and blue curves in Fig. 6.3: Being delayed by a factor 10,
the second inflation appears to have the same length in the third (“blue”) parameter set
√
because it lasts longer by a factor of 10. For the parameter set λ0 = 5 · 10−14 ,
m2 = 5 · 10−14 m2Pl , the evolution of φ(t) is stopped at φ = 0. This happens when the
inflaton is slowly rolling until it reaches this value. Due to the negligible kinetic energy,
φ is stopped by the Hubble drag and does not overcome the region with V ′ ≈ 0.
Corresponding phenomena are found in the field motion within the potential Eq. (6.3),
and are not displayed here. This is also the case with the content of the following figures.
For the same parameter sets as in the previous figure, in Fig. 6.4 the Hubble parameter
is plotted as a function of time. As for the evolution of the field φ(t), also here the
successive cosmological periods can be observed. Reflecting the energy content of the
field, H(t) starts from a large value and evolves downward, and for fixed φ the coupling
m2 =
b λ1 determines the energy density. As expected, the value of H during the second
inflation is the same for equal λ0 because v is left unchanged and the kinetic energy
of the field is negligible again. Fig. 6.5 shows the absolute value of the acceleration
parameter ä/a for the previous cases as a function of time. Knowing that the evolution
starts with a period of inflation, time intervals of decelerated expansion can be identified
from this plot. They correspond to sign changes in ä which appear as steep dips. There
80
6.3. Scalar and tensor perturbations
Hubble parameter H(t)/mP l
10−4
10−5
λ0 = 5 · 10−14, m2 = 5 · 10−13
λ0 = 5 · 10−14, m2 = 5 · 10−14
λ0 = 5 · 10−15, m2 = 5 · 10−14
10−6
10−7
10−8
106
107
108
time t/m−1
Pl
109
Figure 6.4.: Evolution of the Hubble parameter H(t) in the same three cases as in
Fig. 6.3. Periods of inflation correspond to (almost) constant H. The
different potential energies determined by the couplings can be clearly
distinguished.
is no interruption of accelerated expansion in the second case. As already mentioned,
this is the reason for the field to be trapped at φ ≈ 0 entailing a sustained accelerated
expansion.
6.3. Scalar and tensor perturbations
The spectra being produced in this scenario are displayed in the left panel of Fig. 6.6.
The parameter sets one and three lead to potentials that are proportional to each other,
which is the reason why the function Ne (φ) is identical in both cases as long as the
slow-roll assumption is valid, see Eq. (5.16). So the transition from the first to the
second period of inflation takes place at the same value of Ne ≈ 50. Even for larger
Ne this proportionality holds, which suggests that the short time interval of slow-roll
breaking is not important here. Then it follows from Eq. (5.100) that also the two scalar
power spectra should be proportional to each other. The same equation explains the
diverging behavior of the scalar power spectrum in the second case, where φ̇ vanishes
at the transition to the second inflation. The green curve is given an arbitrary offset in
Ne (k)-direction. Because there is no end of inflation, it cannot be uniquely fixed in this
case.
The right panel of Fig. 6.6 shows the corresponding tensor power spectra. Being
proportional to the Hubble parameter at horizon exit of the mode in question, the curves
just illustrate the loss of (mostly) potential energy of the field. The scalar and the tensor
power spectra arising from the potential V4 , Eq. (6.3), are displayed in Fig. 6.7. As far
as the period of inflation is concerned, the non-zero coupling λ2 used for this figure
81
Chapter 6. Fluctuations after two periods of inflation
10−7
10
λ0 = 5 · 10−14, m2 = 5 · 10−13
λ0 = 5 · 10−14, m2 = 5 · 10−14
λ0 = 5 · 10−15, m2 = 5 · 10−14
−8
10−9
ä/a(t)
10−10
10−11
10−12
10−13
10−14
10−15
10−16
106
107
108
time t/m−1
Pl
109
Figure 6.5.: Absolute value of the acceleration of the scale parameter a(t) for the
same three parameter sets as in Fig. 6.3. During the steep dips in the
cases one and three, ä changes its sign twice. During these short intervals, inflation stops and the expansion of the universe is decelerated.
The green line shows no dip, which corresponds to a single, uninterrupted inflation.
10−7
10−9
Tensor power spectra
10−8
Scalar power spectra
10−8
λ0 = 5 · 10−14, m2 = 5 · 10−13
λ0 = 5 · 10−14, m2 = 5 · 10−14
λ0 = 5 · 10−15, m2 = 5 · 10−14
10−10
10−11
10−12
10−13
10−14
10−15
20
30
40
50
Ne (k)
60
70
80
λ0 = 5 · 10−14, m2 = 5 · 10−13
λ0 = 5 · 10−14, m2 = 5 · 10−14
λ0 = 5 · 10−15, m2 = 5 · 10−14
10−9
10−10
10−11
10−12
10−13
10−14
10−15
20
30
40
50
Ne (k)
60
70
Figure 6.6.: Resulting scalar and tensor power spectra PR (k) and PT (k) in the same
three cases as in Figs. 6.3 and 6.4. As can be seen from the cases one and
three, shifting the potential by a constant factor leads to the same shift
in the spectra. In the right panel, the green line interpolates between
the two other cases because the respective potentials are (almost) the
same at the relevant instances of time.
82
80
6.3. Scalar and tensor perturbations
10−5
10−7
10−8
10−9
10−10
10−11
10−12
10
20
10−8
10−9
10−10
10−11
10−12
10−13
10−13
−14
λ0 = 5, λ1 = 5, λ2 = 2.4
λ0 = 0.5, λ1 = 5, λ2 = 2.4
10−7
Tensor power spectra
Scalar power spectra
10
10−6
λ0 = 5, λ1 = 5, λ2 = 2.4
λ0 = 0.5, λ1 = 5, λ2 = 2.4
−6
30
40
50
Ne (k)
60
70
80
10−14
20
30
40
50
Ne (k)
60
70
80
Figure 6.7.: Results of a similar calculation as in Fig. 6.6 but within the model
Eq. (6.3). The couplings are given in units of 10−12 . As seen in the
text, a larger coupling λ0 entails less e-foldings of the second inflation
and thus shifts the spectra of this figure to the left.
just means an additional offset of the potential and the new offset is not proportional
to λ0 , see Eq. (6.2). Then the potentials corresponding to the two parameter sets
are not proportional to each other anymore and also a shift of the spectra in Ne (k)–
direction is possible. This shift can be seen in Fig. 6.7. It is reproduced analytically
with the help of Eq. (5.16). The parameters for Figs. 6.3 to 6.9 are chosen such that the
wavenumber q0 today corresponds to a mode that exits the Hubble horizon around the
transition between the two inflationary periods. As already stated, the correspondence
to q0 cannot be narrowed down to a single value of Ne or k because the physics of the
inflaton decay is not settled yet. Altogether, CMB measurements cover an Ne –range
of roughly ten [29]. For these scales the primordial fluctuations have been obtained to
high accuracy. So, for the presented scenarios to be viable, there should be a range
of at least ten e-foldings in which the computational results match with observation.
This range should overlap with the grey band in Figs. 6.6 and 6.7. The observational
result10 is represented by the narrow horizontal stripe at 2.4 · 10−9 . While it is possible
to tune the parameters such that the results match for a fixed wavelength, the strong
variation of PR (k) on the scale of a few Ne (k) renders it impossible to fit the curves
to the measurements over the whole interval. This leads to the conclusion that the
transition between large-field and small-field inflation should occur when the Hubble
parameter is either much smaller or much larger than the wavenumber q0 today. This
problem will be discussed further in the following.
The tensor-to-scalar ratio resulting from the previously discussed parameter sets
within the potentials V2 and V4 is depicted in Fig. 6.8. During the transition to smallfield inflation, r drops by several orders of magnitude. This reflects the drop of the field
velocity φ̇. It occurs because the Hubble drag is the only force acting on the field in an
almost flat region of the potential. The cases one and three in Fig. 6.6 yield the same
10
While observation does not yield exactly the same value for the whole interval, compared to the large
gradients of the calculated curves it can be regarded as constant.
83
Chapter 6. Fluctuations after two periods of inflation
1
100
Tensor to scalar ratio
Tensor to scalar ratio
101
10−1
10−2
10−3
10−4
10
λ0 = 5 · 10−14, m2 = 5 · 10−13
λ0 = 5 · 10−14, m2 = 5 · 10−14
0.12
−5
10−6
20
30
40
50
Ne (k)
60
70
80
10−2
10−4
10−6
10−8
20
30
λ0 = 5, λ1 = 5, λ2 = 2.4
λ0 = 0.5, λ1 = 5, λ2 = 2.4
0.12
40
50
60
70
Ne (k)
80
Figure 6.8.: The tensor-to-scalar ratio strongly drops down at the transition to smallfield inflation. If φ̇ becomes very small, also r vanishes at corresponding
values of Ne (k) (left panel, green curve).
(red) curve here. The green curve drops to very small numbers when φ is practically
fixed at a constant value. Also for r, the maximally allowed value of r = 0.12, shows
that compatibility is only possible when φ = 0 is reached outside the range observable in
the CMB. The same is true for the allowed range of the scalar spectral index ns , which
can be seen in Fig. 6.9. During the transition it takes on values far away from the ones
being measured. It is clear from the definition of ns in Eq. (5.103) that multiplying the
scalar spectrum with a constant factor does not alter the value of the spectral index.
So parameter set three yields the same result as set one and is not displayed in the left
panel of the figure. The right panel shows a similar behavior of ns within the potential
V4 .
6.4. Compatibility with measurements
Figures 6.10 to 6.12 summarize the results for the tensor-to-scalar ratio and for the
scalar spectral index within the models considered so far. As noted in the keys of these
figures, the black dashed lines sketch the dependence of r on ns as obtained from an
analytical estimation for the two potentials m2 φ2 /2 and λφ4 /4. The result varies along
these lines when different values of Ne are chosen. Including the possibility of an offset
V0 , the slow-roll equations Eq. (5.102) for the tensor-to-scalar ratio and Eq. (5.105) for
the scalar spectral index combine to
ns − 1
,
V0 /m2 φ2 − 1
(6.7)
16
ns − 1
,
3 4V0 /λφ4 − 1
(6.8)
r2 = 4
and
r4 =
where r2 and r4 are the tensor-to-scalar ratio in the squared and the quartic case,
respectively. The black dashed lines give the results of Eqs. (6.7) and (6.8) with V0
84
6.4. Compatibility with measurements
Scalar spectral index ns
Scalar spectral index ns
1
0.8
0.6
λ0 = 5 · 10−14, m2 = 5 · 10−13
λ0 = 5 · 10−14, m2 = 5 · 10−14
0.4
0.2
20
λ0 = 5, λ1 = 5, λ2 = 2.4
λ0 = 0.5, λ1 = 5, λ2 = 2.4
1.4
1.2
30
40
50
Ne (k)
60
70
1.2
1
0.8
0.6
0.4
0.2
80
20
30
40
50
Ne (k)
60
70
80
Figure 6.9.: Evolution of the scalar spectral index for the previously discussed cases.
The transition region between the two periods of inflation should be far
outside the sensitivity range of the Planck measurements. The scalar
spectrum within the third case in Fig. 6.6 was just shifted by a factor
with respect to the first one. As expected, the logarithmic derivative of these functions is the same, and therefore omitted in this figure.
By choosing a suitable offset in Ne , the result of the analytic formula
Eq. (5.134) can be matched to the red curve.
set to zero. Numerical calculations within the same cases are represented by the five
large dots which are obtained at three different stages of inflation. These are labelled
by the number Ne which is large (Ne = 50 and 60) for early production well within
the slow-roll regime and small (Ne = 10) for late production, i.e. for fluctuations
in modes crossing the horizon shortly before the end of inflation. Then the slow-roll
approximation is expected to lose its validity. So, the deviation of the large dot in dark
red from the long-dashed black line is no surprise since also the dots are calculated
assuming slow roll. The position of the large dots is independent from the coupling
when values λ ∈ [10−15 , 10−11 ] and m2 ∈ [10−15 , 10−11 ] are chosen. The small dots
represent results computed within the combined potentials V2 and V4 . They arrange
along curves starting on the corresponding straight line (for the monomic quadratic and
quartic potential, respectively) and more and more deviate when evolving to larger r and
smaller ns . Since this deviation also occurs for the monomic potential, it is attributed
to the breaking of slow roll. Corrections due to V0 seem to be small in comparison.
They tend to enlarge the value of r at given ns .11
The color of the small dots encodes the number of e-foldings of the second inflation,
which varies according to different parameter choices. It is clearly visible that this
(2)
number Ne grows monotonically when following the dots from the lower right to the
upper left. This suggests the following interpretation: The fluctuations being measured
within the present scenario at Ne = 60 resemble those that would be produced in the
quartic and quadratic potentials at later times. The main effect of the second period of
inflation is a shift of the remaining e-foldings connected to the fluctuations.
11
Note that both the numerator and the denominator in Eqs. (6.7) and (6.8) are negative.
85
Chapter 6. Fluctuations after two periods of inflation
2
70
0.25
Tensor to scalar ratio r
0.2
60
0.15
1.5
50
0.1
0.05
40
0.94
1
0.96
0.98
1
30
20
0.5
φ4
Analytic result for
Analytic result for φ2
Numerical results for combined potentials
Numerical results for the potentials φ4 and φ2
0
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
10
0
1
Scalar spectral index ns
Figure 6.10.: This figure shows the result of different calculations of the tensor-toscalar ratio r and the scalar spectral index ns . These quantities are
computed for modes leaving the horizon a fixed number of e-foldings
before inflation ends. The two dashed lines follow the analytic formulae
(6.7) and (6.8) for slow-roll inflation within the potentials m2 φ2 /2 and
λφ4 /4, respectively. The large dots mark the results of numerical calculations within the same two potentials. The number of e-foldings Ne
before the end of inflation are 60, 50, and 10, respectively. Along the
dashed lines (and more and more departing from them) the small dots
represent results from the combined potentials V2 and V4 described in
the main text. The color code shows the number of e-foldings of the
second inflation, whereas the total Ne is 60 for each of the small dots.
All results were obtained within slow-roll approximation.
86
6.4. Compatibility with measurements
-1
Tensor to scalar ratio r
2
-1.5
1.5
-2
1
0.5
-2.5
Analytic result for φ4
Analytic result for φ2
Numerical results for combined potentials
Numerical results for the potentials φ4 and φ2
0
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
-3
1
Scalar spectral index ns
Figure 6.11.: The same as in Fig. 6.10 but with the colors representing the decimal
logarithm of the slow-roll condition φ̇2 /2V (φ). This figure shows that
the deviation of the points from the dashed lines can be explained from
the lack of slow roll.
The small figure in Fig. 6.10 is a detail of the lower right region of the large one.
This is the interesting range as CMB measurements are concerned. The values allowed
by Planck lie below 0.15 for r and between 0.94 and 0.98 for ns . So one can see that
the second inflation drives the computed values of r and ns further away from the ones
measured in the CMB. The more effective it is in terms of e-foldings the more severe it
renders the discrepancy.
Figs. 6.11 and 6.12 show the same results as Fig. 6.10 but with a different choice
of the color code. This is used to illustrate the reason why the lines formed by the
small dots deviate from the analytic estimate (dashed lines). In Fig. 6.11 the colors
reflect the value of log10 (φ̇2 /2V (φ)) which is a measure for slow roll, see Eq. (5.8). The
largest deviations are observed where the kinetic energy amounts to a few percent of
the potential and the results agree better for a lower proportion.
The values of V0 /m2 φ2 and 4V0 /λ1 φ4 are encoded in the colors of the dots in Fig. 6.12.
These quantities are proportional to the ratio of the offset to the field dependent part
of the potential, and they are computed at the time the mode of interest leaves the
horizon. They also occur in Eqs. (6.7) and (6.8). As stated above, they are expected
to give corrections enlarging r for given ns . So it can be concluded that their influence
on the result is less important than that of missing slow roll. This is confirmed by the
fact that the deviations are considerably smaller for the massive field, where V /m2 φ2 is
largest and slow-roll violation is quite small. For the pure squared and quartic potentials
the quantity reflected by the color code is zero and they are plotted in black.
The parameter sets that yield the small dots are chosen in the following way: First a
sweep over the parameter range v ∈ [0.01, 1]·mPl , λ0 ∈ [10−15 , 10−12 ], λ1 ∈ [10−15 , 10−12 ]
87
Chapter 6. Fluctuations after two periods of inflation
-2
Tensor to scalar ratio r
2
-3
1.5
-4
1
-5
0.5
Analytic result for φ4
Analytic result for φ2
Numerical results for combined potentials
Numerical results for the potentials φ4 and φ2
0
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
-6
-7
1
Scalar spectral index ns
Figure 6.12.: The same as in Figs. 6.10 and 6.11 but with the colors representing
the decimal logarithm of the quantities V0 /m2 φ2 and 4V0 /λ1 φ4 for
the combined potentials with m2 φ2 /2 and with λ1 φ4 /4, respectively.
In combination with Fig. 6.10 this figure suggests that the deviation
of the numerical calculation from the analytic result stems from the
violation of slow roll rather than from the offset V0 in the potential
V (φ).
is done. The parameter λ2 is restricted to λ2 = 0 and the interval λ2 ∈ [10−15 , 10−14 ].
Stronger couplings are not computed with because they lead to larger values of V0 which
stop the evolution of φ before reaching the minimum.
(1)
For each parameter set the e-foldings of the first and the second inflation, Ne and
(2)
(2)
Ne , are determined. The interesting cases are those with enough Ne , such that the
expected spectra are discernible from standard large field inflation. On the other hand
(2)
the value should not exceed Ne = 60 because then only the second, small-field inflation is visible today. As exemplified in Figs. 6.6 to 6.9 the spectra produced near the
transition between the two inflationary periods deviate strongly from the CMB measurements which set tight bounds on the magnitude of fluctuations. These arguments
(2)
lead to the approach that parameter sets entailing Ne ∈ [10, 60] are selected and the
fluctuations are calculated for modes leaving the horizon at Ne = 60 e-foldings before
the end of the second inflation.
88
7. Fluctuations produced after inflation:
Preheating
At the end of inflation, the universe is assumed to be cold and empty, with the exception of the inflaton field. The inflaton pervades space almost homogeneously before it
decays into its excitations and other particles. The first, rapid decay is characterized
by non-perturbative effects. It is crucial for accomplishing thermal equilibrium at the
temperature, where Big Bang nucleosynthesis takes place. This is decisive for a scenario
to be viable. This work follows the convention to give the name “preheating” to the
early non-perturbative stage, while “reheating” is reserved for the subsequent period of
perturbative evolution. The following will only be concerned with preheating.
7.1. Initial vacuum fluctuations
A typical calculation of preheating after cosmological inflation starts when all particles
that might have filled the universe before are extremely dilute. During preheating,
the fluctuations discussed in the last section have wavelengths much larger than the
Hubble horizon. Therefore, interactions with the fluctuations and particles produced
during the preheating epoch are neglected. So, a vacuum state is assumed as initial
condition for the computations. In this section the spectrum of vacuum fluctuations for
a non-interacting massless scalar field in a spatially flat FRW universe is derived. The
presentation follows Refs. [51, 77, 114]. The evolution of the field φ is governed by the
action S and the Lagrange density L,
Z
Z
√
1
µ
4 √
d x −g∂ φ∂µ φ = d4 x −gL,
S=
(7.1)
2
√
in conformal coordinates. It is gµν = a2 ηµν and −g = a4 . The definition ϕ := aφ, and
the prime as shorthand for ∂0 simplify the Lagrange density, the field momentum, and
the Hamiltonian density to
a′ 2 1
1
1√
′
µ
(7.2)
+ (∇ϕ)2 ,
ϕ −ϕ
−g∂ φ∂µ φ =
L=
2
2
a
2
∂L(ϕ, ϕ′ )
a′
′
π=
, and
(7.3)
=
ϕ
−
ϕ
∂ϕ′
a
)
(
′ 2
1
a
2
′
′2
H = πϕ − L =
,
(7.4)
ϕ − (∇ϕ) −
ϕ
2
a
Chapter 7. Fluctuations produced after inflation: Preheating
respectively. Normalizing the Fourier transform of any function f (r) as
Z
1
d3 k exp(−ikr)f (r)
fk =
3/2
(2π)
and using the shorthand k2 := |k|2 , the Hamiltonian reads
)
(
′ 2
Z
1
a
H=
ϕk ϕ−k ,
dηd3 k ϕ′k ϕ′−k + k2 ϕk ϕ−k −
2
a
(7.5)
(7.6)
where for the second term the collection of one minus sign from Eq. (7.4), one of an i2 ,
one from the opposite directions of ±k, and one from the metric ends up with a plus.
Assuming that ϕ(x, η) vanishes for large η, the Lagrange density Eq. (7.2) can also be
written as
′′ 1
1
′2
2a
L=
(7.7)
ϕ +ϕ
+ (∇ϕ)2
2
a
2
′′ Z
a
1
3
′ ′
2
dηd k ϕk ϕ−k + k ϕk ϕ−k −
ϕk ϕ−k .
(7.8)
⇒H=
2
a
When quantizing the system, creation and annihilation operators, â+ (k), â− (k), are
introduced, in terms of which the field and momentum operators are
r
k −
1 −
+
â (k) + â (−k)
and π̂(k) = −i
â (k) − â+ (−k) ,
(7.9)
ϕ̂(k) = √
2
2k
respectively. The vacuum state at conformal time η0 , |0, η0 i, for every k fulfills
â− (k) |0, η0 i = 0.
(7.10)
From this condition one can deduce the coordinate representation of this state: It is
i
â− (k) = ϕ̂(k) + π̂(k),
k
(7.11)
and in the coordinate representation one gets
π̂(x) = −i
δ
δϕ(x)
and π̂(k) = −i
δ
.
δϕ(−k)
(7.12)
So the wave functional of the vacuum state must be a product of
Ψ [ϕ(k, η0 ), ϕ(−k, η0 )] ∝ exp {−kϕ(k, η0 )ϕ(−k, η0 )} .
(7.13)
The corresponding probability distribution for the Fourier modes of the real field ϕ is
P [ϕ(k, η0 )] ∝ exp −2k|ϕ(k, η0 )|2 ,
(7.14)
out of which the initial conditions after inflation are generated. They are used for the
lattice calculations of preheating being presented in the following. Further issues of
implementing the initial conditions will be dealt with in Appendix B.2.
90
7.2. Periodically changing mass and the Mathieu equation
7.2. Periodically changing mass and the Mathieu equation
As the first example for particle production during preheating, the setup of a scalar
field φ with mass m coupled to a massless second scalar X is chosen. This mimics the
situation after chaotic single-field inflation with a massive inflaton φ, see Section 5.5.
The potential is taken to be
V =
1 2 2 1 2 2 2
m φ + g φ X .
2
2
(7.15)
In a first attempt, φ(x, t) = φ0 sin(mt) is assumed and a possible expansion of the
background is neglected. Within a simulation of a preheating universe this will be
included later. The simplifications lead to a periodically changing mass term of X: The
equation of motion for a mode Xk is
Ẍk + k2 + g2 φ20 sin2 (mt) Xk = 0.
(7.16)
Since the potential (7.15) is only quadratic in the field X, the evolution of any mode Xk
does not depend on the other modes. Equation (7.16) is usually written as a Mathieu
equation [84],
Xk′′ + (Ak − 2qcos(2z)) Xk = 0,
(7.17)
where the quantities
Ak =
g2 φ20
k2
+
2q,
q
=
,
m2
4m2
and z = mt
(7.18)
are introduced. Equation (7.17) was first analyzed in [97] in the context of vibrating
membranes. The features of its solutions are well-known. Their stability properties can
be read from Fig. 1 in Ref. [11], which is also used for this presentation. Unstable solutions are characterized by their Floquet exponent µk which determines the exponential
growth of a mode through
1
1
′
2
2
nk =
|X (z)| ∼ |Xk |2 ∝ e2µk z .
ωk |Xk (z)| +
(7.19)
2
2ωk k
A derivation of the Floquet exponent for Minkowski and FRW background is presented
below. Interactions between different modes Xk are not accounted for in this model
and, for now, backreaction of the modes onto the homogeneous field φ is neglected.
Therefore, one can solve the problem numerically mode by mode and obtain the results
shown in Figs. 7.1 and 7.2.
7.2.1. Broad and narrow parametric resonance
There are two regimes in parameter space where the modes Xk behave differently from
each other: One is the regime of narrow resonance where only a few separate modes show
91
Chapter 7. Fluctuations produced after inflation: Preheating
1.5
nX
k (t)
Xk (t)
φ(t)
1
0.5
0
-0.5
-1
-1.5
0
0.5
1
1.5
2
2.5
time t/
2π
mφ
3
3.5
4
4.5
5
√
Figure 7.1.: The mode function Xk starting with a small amplitude 1/ 2ωk . φ(t)
is displayed in units of 10−3 mPl and Xk in units of m−1
φ . Its equation
of motion is that of a harmonic oscillator with a small time-dependent
perturbation which comes from the interaction with the inflaton φ(t).
The additional restoring force exerted by φ(t) is proportional to φ2 (t).
The mode Xk (t) experiences a small amplification when returning from
each maximal displacement because then φ2 is slightly bigger as compared to moments of time when Xk (t) is growing. The case shown in this
figure is an example for narrow parametric resonance because if at all,
neighboring modes are much less amplified. The parameters corresponding to Eq. (7.17) are chosen as Ak = 1 and q = 1/16 being consistent
with Φ = 10−3 mPl , mφ = 10−6 mPl , g = 5 · 10−4 and the wave number
k ≈ 0.968mφ. See also Fig. 7.2.
92
7.2. Periodically changing mass and the Mathieu equation
100
nX
k (t)
10
1
0.1
1
2
3
4
5
6
7 8 9
time t/
10
11
12
13
14
15
2π
mφ
Figure 7.2.: The same setting as in Fig. 7.1 for a longer time period. As in Figure 7.1
and many others to come, time is given in units of an oscillation period
of the inflaton. Up to nX
k ≈ 1 the classical formula Eq. (7.19), which
is displayed here, should not be interpreted as a particle density. The
small wiggles come from superposing the extrema of the mode function
and its derivative.
93
Chapter 7. Fluctuations produced after inflation: Preheating
unstable behavior. This entails a continuous exponential growth of the mode function
and of the corresponding particle density. Both Xk and Xk′ oscillate with the frequency of
φ. In the time dependence of nk this leaves a small oscillating structure, which, because
of squaring and superimposing, shows four times the frequency of the φ oscillations.
Within the more realistic setting of an FRW background, this regime could possibly
occur at the end of preheating when the amplitude φ0 has become small enough to give
q < 1. However, for a successful preheating of the universe a non-negligible density
of produced particles should be expected at this stage. So, the calculation within the
regime of narrow resonance that is presented here is probably not directly applicable to
the evolution of the early universe.
The other regime in parameter space of Eq. (7.17) is that of large q which corresponds
to a broad resonance. This means that a broad range of modes undergoes exponential
growth entailing very efficient particle production. A large parameter q is equivalent
to gφ0 ≫ m and so the oscillation of φ is much slower than that of Xk . The sudden,
stepwise growth of the X particle number occurs in moments of time when the inflaton
field φ and therefore the oscillating X mass squared,
m2X = g2 φ20 sin2 (mt) = g2 φ(t)2 ,
(7.20)
is close to zero. So, one observes
change in nk ⇒ φ(t) ≪ φ0 ,
(7.21)
which is put into a more precise statement below. In these time intervals the frequency
ωk corresponding to the mode k changes non-adiabatically. This means that the adiabaticity condition,
ωk2 ≥ ω̇k ,
(7.22)
is violated, allowing the adiabatically invariant quantity nk to change its value. As a
simple rule one can state that this value grows if |Xk | is growing while φ(t) is small. This
happens for consecutive φ oscillations when Xk is in resonance with them. In contrast,
when the evolution is adiabatic, nX is constant while the frequency of the X-modes,
ωk , follows the sinusoidal oscillation of the field φ. From Eq. (7.19) it is clear that then
the amplitude of the oscillating Xk (z) has a minimum at each extremum of φ.
Again referring to [80], an estimation of the spectral range of produced X particles
and of the corresponding time period is now presented: Being expected to occur at small
φ ≈ φ0 mt, non-adiabaticity of
p
(7.23)
ω ≈ k2 + (gφ0 mt)2
is present for momenta k with
k2 + (gφ)2 ≤ g2 φmφ0
2/3
.
The corresponding function k(φ) assumes its maximal value
p
kmax ≈ gmφ0
94
(7.24)
(7.25)
7.2. Periodically changing mass and the Mathieu equation
30
Xk (t)
φ(t)
20
10
0
-10
-20
0
0.5
1
1.5
2 2.5
2π
time t/ m
φ
3
3.5
4
Figure 7.3.: Broad resonance regime of solutions to the Mathieu equation (7.17).
The oscillatory inflaton field φ(t) couples to the mode function Xk (t)
as in the solution displayed in Figs. 7.1, 7.2. The parameters chosen
for this figure are q = 250 and Ak = 501 corresponding to the coupling
g = 5 · 10−4 and mφ = 10−6 mPl as for Figs. 7.1, 7.2, and k = mφ and
φ0 ≈ 6.3 · 10−2 . The dips of the amplitude of Xk (t) in the adiabatic
regions around the extrema of φ(t) can be explained from Eq. (7.19)
and the adiabatic invariance of nX
k .
95
Chapter 7. Fluctuations produced after inflation: Preheating
10000
nX
k (t)
1000
100
10
1
0.1
0
0.5
1
1.5
2 2.5
2π
time t/ m
φ
3
3.5
4
Figure 7.4.: The occupation number or particle density nX
k in the same setting as in
Fig. 7.3. Being adiabatically invariant, nX
changes
only when φ2 (t) ∝
k
2
mX is close to zero because then ω̇k ≤ ωk is not fulfilled.
at
φ(kmax ) =
1
30.75
s
mφ0
,
g
(7.26)
which, in turn, corresponds to a time span of
T ∼
1
1
2φ(kmax )
1
∼√
∼
.
∼
k
ω(k
gmφ
φ̇
max
max )
0
(7.27)
In this time interval particles with momenta up to the order of kmax are produced. For
the last step, mX (kmax ) ∼ gφ(kmax ) ∼ kmax has been used. It is included in order to
show that the process respects the Heisenberg uncertainty principle.
Before discussing the specific effects of the FRW expansion let us review a more
thorough theoretical treatment of broad parametric resonance. Only for later use the
scale parameter a(t) is already included. During most of the time, the evolution is
adiabatic and the solution to Eq. (7.16) can be written in the WKB approximation,
αk (t) −i R t ωdt βk (t) +i Rtt ωdt
0
+ √ e
χk (t) := Xk (t)a3/2 ≡ √ e t0
.
2ω
2ω
(7.28)
Note that in this equation the comoving mode function χk has been defined with an
√
additional factor a with respect to Sec. 7.1 in order to keep the dependence of αk and
βk on the scale factor as small as possible. (This dependence is not removed completely,
however, because ω is not exactly proportional to 1/a.)
96
7.2. Periodically changing mass and the Mathieu equation
In the non-adiabatic region the sine in the mass term of χ can be linearly approximated
such that the equation of motion reads
2
k
d2 χk
2 2 2
2
+
+ g m φ0 (t − tj ) χk = 0.
(7.29)
dt2
a2
After choosing convenient variables τ = kmax (t − tj ) for time and κ = k/(akmax ) for
momentum, the mode equation can be written as
d2 χk
+ κ2 + τ 2 χk = 0.
2
dτ
(7.30)
Comparison with the time independent Schrödinger equation shows that Eq. (7.30)
describes the behavior of a wave scattered on a parabolic potential barrier V (τ ) = −τ 2 .
This is studied by various authors, for example in [18,20,52,126]. In addition to [80], for
this presentation especially [12] and the chapter on parabolic cylinder functions in [4]
have been used.
7.2.2. Transmission and reflection coefficients
In order to find a formula for the growth of occupation number at each scattering off the
parabolic potential Eq. (7.30) the starting point is to calculate the transmission and the
reflection coefficients for the parabolic barrier. In doing so it is not important whether
it is a space or a time coordinate along which the scattering takes place. So one can just
as well think of the process in terms of an incoming and two outgoing solutions as in the
most typical quantum mechanical scattering problems in one dimension. In order to set
up a situation of scattering, solutions with definite flux direction are necessary. These
are identified in the following. Solutions to Eq. (7.30) can be written as a superposition,
2
2
2
√
κ √
κ √
1
κ
E − , 2τ = W − , 2τ + iξW − , − 2τ ,
(7.31)
2
ξ
2
2
with a real solution W and the constant ξ > 0 defined to fulfill
p
κ2
2
2
ξ = 1 + exp(−πκ ) − exp −π
2
(7.32)
p
κ2
1
2
= 1 + exp(−πκ ) + exp −π
.
ξ2
2
(7.33)
and, consequently,
The validity of the last equation can be checked by multiplication with the preceeding
one. The functional form of W is only needed far away from the potential maximum
(τ ≫ κ2 ). There the following approximations are obtained [4]:
2
2
√
τ
κ2 √
κ
21/4 ξ
Φ π
√
cos
+ ln 2τ − +
,
(7.34)
W − , + 2τ =
2
τ
2
2
2
4
2
2
√
τ
κ2 √
κ
21/4
Φ π
+ ln 2τ − +
.
(7.35)
W − , − 2τ = √ sin
2
τξ
2
2
2
4
97
Chapter 7. Fluctuations produced after inflation: Preheating
In order to save writing, for the rest of this paragraph the shorthand
ζ :=
Φ π
τ 2 κ2 √
+ ln 2τ − +
2
2
2
4
is defined. Put together, for the complex solution at τ ≫ κ2 this gives:
2
κ √
21/4
E − , 2τ = √ exp iζ.
2
τ
(7.36)
(7.37)
A more exact formula would include a power series in τ −2 which, in the limit τ → ∞,
gives 1. The present calculation is done in this limit. The angle Φ is given by the
complex gamma function as
1 + iκ2
Φ = argΓ
.
(7.38)
2
The solution at negative times is obtained to be
2
2
2
√
√
√
κ
1
κ
κ
E − , − 2τ = W − , − 2τ + iξW − , + 2τ
2
ξ
2
2
1/4
21/4 ξ 2
τ ≫κ2 2
−−−→ √ 2 sinζ + i √ cosζ
τξ
τ
1/4
p
2
2
2
=i√
1 + e−πκ e−iζ − e−πκ /2 eiζ ,
τ
(7.39)
where ζ is defined in Eq. (7.36). To identify the incoming, the reflected, and the transmitted part of the solution, which are named Ei , Er , and Et , respectively, one has to
calculate the flux
∂E ∗
∗ ∂E
j = −i E
−
E
(7.40)
∂τ
∂τ
of the three components in Eqs. (7.37) and (7.39). From the sign of this quantity the first
component of Eq. (7.39) is identified as the incoming wave Ei , the second component
of Eq. (7.39) as Er , and Eq. (7.37) as Et . The ratios of the respective solutions give the
reflection coefficient
2
e−πκ /2
Er
2ζi
= −e p
(7.41)
RPCF =
2
Ei
1 + e−πκ
and the transmission coefficient
TPCF =
1
Et
.
= −ie2ζi p
Ei
1 + e−πκ2
(7.42)
The index is to reflect their origin from parabolic cylinder functions. Note that in this
form the coefficients depend on the length of the interval that is considered, e.g. 2τ0 .
This should be expected because, through the time dependent quantity ζ, they include
the phase evolution of the oscillating solution E.
However, applied to the problem of particle production by an oscillating field, the
solution to the equation of motion Eq. (7.16) is a parabolic cylinder function only in
98
7.2. Periodically changing mass and the Mathieu equation
the vicinity of the scattering times tj . Elsewhere it is written within the WKB approximation. Therefore, the correct phase of R and T is obtained by subtracting the WKB
phase shift from the phase shift of the parabolic cylinder functions. From Eq. (7.30) the
first one is
Z τ0
Z τ0 /κ p
p
2
2
2
dτ τ + κ = κ
ΦWKB =
(7.43)
dτ τ 2 + 1
=
−τ0
κ2 h
2
τ
p
−τ0 /κ
iτ0 /κ
p
τ 2 + 1 + ln τ + τ 2 + 1
(7.44)
−τ0 /κ
τ0 /κ
κ2
1
1
≈
τ |τ | 1 + 2 + ln τ + |τ | 1 + 2
2
2τ
2τ
−τ0 /κ
2
2τ
κ
0
+ κ2 ln
,
≈ τ02 +
2
κ
in the limit τ0 ≫ κ2 . This gives
π
∆φk := 2ζ − ΦWKB − = −argΓ
2
1 + iκ2
2
κ2
−
2
2
1 + ln 2
κ
.
(7.45)
(7.46)
(7.47)
In this definition a phase π/2 is subtracted. It comes from the last term in the definition of ζ, Eq. (7.36), and changes factors of the imaginary unit i in the reflection and
transmission coefficients. Then the result for the reflection and transmission coefficient
based on incoming and outgoing waves within the WKB solution is
2
R = −ie
and
i∆φk
e−πκ /2
p
1 + e−πκ2
1
T = ei∆φk p
.
1 + e−πκ2
(7.48)
(7.49)
7.2.3. Evolution of the particle number
With the help of the reflection and the transmission coefficients, the transfer matrix of
the quantities αk and βk in Eq. (7.28) for a single scattering off the inverse parabolic
potential of Sec. 7.2.2 is written as
!
!
R∗k !
1
j −iθ j
−iθkj
k
∗
α
e
αj+1
e
Tk
Tk
k
k
.
(7.50)
= R
j
j +iθ j
1
k
k
β
βkj+1 e+iθk
∗
Tk
T
ke
k
This can be verified by checking the case of a single incoming wave and the complex
conjugate solution. From now on, the notation accounts for the k-dependence of R and
D. Following Ref. [80] the phase θkj is defined similarly to the WKB phase as
Z tj
j
dt ω(t).
(7.51)
θk =
0
99
Chapter 7. Fluctuations produced after inflation: Preheating
nX
k
104
k
k
k
k
= 0mφ
= 2mφ
= 4mφ
= 6mφ
103
V (τ )
102
-10
-5
0
5
10
τ = kmax t
Figure 7.5.: Behavior of a wave scattered off a parabolic potential barrier. The upper
panel shows the occupation number of some modes, nX
k . Away from the
maximum of the potential, which is sketched in the lower panel, the
WKB solution is valid. The phase of the mode function decides whether
the value of nX
k belonging to this solution is bigger or smaller after the
potential barrier.
100
7.2. Periodically changing mass and the Mathieu equation
Using Eqs. (7.48) and (7.49), each zero of the field φ is seen to have the following effect
on the modes of field χ:
!
!
!
p
j
π 2
αjk
ie− 2 κ +2iθk
1 + e−πκ2 e−i∆φk
αj+1
k
p
.
(7.52)
=
j
π 2
βkj
βkj+1
1 + e−πκ2 ei∆φk
−ie− 2 κ −2iθk
Given the normalization |αk |2 − |βk |2 = 1, the occupation number after the (j + 1)th
scattering is then calculated as
j+1 2
nj+1
=
β
k
k r (7.53)
p
2
j
− π2 κ2
−πκ2
−πκ2
−πκ
nk − 2e
+ 1 + 2e
=e
1+e
njk 1 + njk sin γkj ,
where growth or decay of particle number density is determined by k through the rescaled
momentum κ and through the angle
γ = 2θkj − ∆φk + argβkj − argαjk .
For large nk Eq. (7.53) can be approximated as
p
j − π2 κ2
−πκ2
−πκ2 nj ,
nj+1
≈
1
+
2e
−
2sin
γ
e
1
+
e
k
k
k
(7.54)
(7.55)
which sets the growth parameter for the mode k in
j (k)
nj+1
= njk e2πµ
k
to be
µj (k) =
p
π 2
2
1 ln 1 + 2e−πκ − 2e− 2 κ 1 + e−πκ2 sinγkj .
2π
(7.56)
(7.57)
So, depending on γkj , the occupation number can also decrease during the non-adiabatic
evolution at small φ(t). Starting with a small number of particles nk (t0 ) with momentum
k, after N zeros of φ the order of magnitude of nk (t) is


N
X
(7.58)
µj (k) .
exp 2π
j=0
Assuming the case of a time-independent µj , the total number density in χ particles is
then
Z
Z ∞
1
1
3
nχ (t) =
dk k2 e2mµ(k)t ,
(7.59)
d k nk (t) = 2 3
(2πa)3
4π a −∞
which gives
nχ (t) ≈
1 k̃2 e2µ(k̃)mt
q
4πa3
mt|µ′′ (k̃)|
(7.60)
when approximated within the saddle-point method. In the last formula k̃ denotes
the mode with the strongest growth and µ′′ (k̃) stands for the second derivative of the
101
Chapter 7. Fluctuations produced after inflation: Preheating
growth factor at this point. From comparison with a Gauss function, the curvature can
be estimated as |µ′′ (k̃)| ∼ 2µ(k̃)/∆k2 as suggested by Ref. [80]. The authors of Ref. [80]
estimate both the range of unstable modes in the first band (∆k) and the wavenumber
k̃ to be comparable to kmax /2. Then one is led to the following formula predicting
exponential growth of the number of χ particles,
s
3
2
kmax
e2µ(k̃)mt .
(7.61)
nχ (t) ∼
64πa3 πµ(k̃)mt
7.2.4. Particle production in Minkowski spacetime
Before discussing the case of particle production within an expanding universe, in this
section the scale factor is set to unity. Additionally, the formulae describing the mechanism of broad parametric resonance are derived assuming a constant amplitude φ0 of the
oscillating inflaton field φ(t). Then also the rescaled momentum κ in the last section is
time-independent and the phase shift between two successive scattering events is always
the same. This means that Eq. (7.51) simplifies to
θkj = jθk = j
Z
π/m
dt ωk ,
(7.62)
0
and θk , which is an important quantity for particle production, can be calculated as
Z π/m
p
(7.63)
dt t k2 + g 2 φ2 (t)
θk =
0
2gφ0
κ2
gφ0
≈
+
+ 4ln2 + 1
(7.64)
ln
m
2
mκ2
k2
4qm2
√
(7.65)
= 4 q + √ 2 ln 2 + 4ln2 + 1 ,
4 qm
k
if k ≪ gφ0 . The detailed steps from the first to the second line can be found in App. C,
and the last line simply replaces the model quantities g, φ0 , and κ by the resonance
parameter q and k as defined above. The following paragraph will show the dependence
of the growth index µk on the angles θk and φk : We know that for large nk the absolute
values of αk and βk have to be the same. Then from Eqs. (7.61) and (7.52) the ansatzes
(±1)j
αjk,± = √ e(πµk +iθk )j ,
2
(±1)j
j
βk,±
= √ eiϑ± e(πµk +iθk )j
2
(7.66)
seem reasonable. Here ϑ± are two constant phases that will not be important in this
context. Application in Eq. (7.52) leads to the equations
p
π 2
2
± e(πµk +iθk ) = 1 + e−πκ e−i∆φk + ie− 2 κ −iϑ± ,
(7.67)
which determine the growth index: After isolating and removing ϑ± , the solutions to
a quadratic equation in exp(πµk ) suggest a distinction of the cases cos(θk + ∆φk ) ≷ 0
102
7.3. Effects of the Hubble expansion
such that for both versions of Eq. (7.66) the stronger growth is given by
q
p
2
πµk
−πκ
+ (1 + e−πκ2 )cos2 (θk + ∆φk ) − 1.
= |cos(θk + ∆φk )| 1 + e
e
From this it is seen that real µk additionally requires
2
1 + e−πκ cos2 (θk + ∆φk ) ≥ 1
π
2
⇒ |tan(θk + ∆φk )| ≤ e− 2 κ .
(7.68)
(7.69)
(7.70)
The last equation is the condition for a mode k to grow exponentially. The periodicity
of the tangent leads to alternating regions of stability and instability. Equation (7.68)
together with Eqs. (7.47) and (7.65) for ∆φk and θk , respectively, then yields the growth
parameter as a function of Ak and q. In order to calculate this it is convenient to follow
an approximation suggested in [52] which amounts to
1 + iκ2
≈ −0.982κ2 ,
(7.71)
argΓ
2
yielding
k2
√
θk + ∆φk ≈ 4 q + √ 2 (ln q + 9.474).
8 qm
(7.72)
In Ref. [80] a comparison with numerical results shows good agreement up to the first
two resonance bands.
7.3. Effects of the Hubble expansion
The Hubble expansion of an FRW universe leads to a redshift of momenta and a damping
of the inflaton oscillation. So the resonance parameter q = (gφ0 /2m)2 decreases with
time. The result is, that for an efficient decay of the inflaton field the initial value of q
must be very large. Only then the system spends enough time in the parameter region
of broad parametric resonance where strong particle production takes place. This also
means that within a realistic setting, narrow parametric resonance with q ∼ 1 occurs
only when already a large number of particles has been produced. So in contrast to the
approach presented in Sec. 7.2, interaction of these particles with the zero mode and
other modes should then be taken into account. In this section the considerations in [80]
concerning the model
1
1
(7.73)
V = m2 φ2 + g 2 φ2 X 2
2
2
within an expanding background are reviewed. The results of some corresponding calculations are displayed. Following Ref. [101], the first step is to see how Hubble damping
influences the oscillation of φ(t) without interaction with X: The Friedmann equation
for φ(t),
4π 2
(7.74)
φ̇ + m2 φ2 ,
H2 =
3mPl
103
Chapter 7. Fluctuations produced after inflation: Preheating
suggests a parameterization in H and in an angle α which is defined by
r
r
3
3
HmPl cos α,
mφ =
HmPl sin α.
φ̇ =
4π
4π
(7.75)
This together with the equation of motion,
φ̈ + 3H φ̇ + m2 φ = 0,
(7.76)
α̇ = m + 3Hsin(α) cos(α)
(7.77)
Ḣ = −3H 2 cos2 (mt).
(7.78)
combines to
and then gives
Solving for the Hubble parameter yields
H=
2
3t
1+
sin(2mt)
2mt
−1
,
(7.79)
which together with Eq. (7.75) also leaves
sin(2mt) −1
φ(t) = φ0 (t)sin(mt) 1 +
≈ φ0 (t)sin(mt),
2mt
(7.80)
where further terms are at least of order (mt)−3 and the amplitude of the oscillating
inflaton field, φ0 (t), decays with 1/t for large t,
φ0 (t) = √
mPl
.
3π mt
(7.81)
This result can be used for a computation that includes interaction with the second
scalar X within an expanding background. Compared to the calculation presented in
Sec. 7.2 the behavior of the inflaton field is now taken to follow a damped oscillation.
The approximation is valid after a few oscillations of the inflaton field. The calculation
starts at t = π/2m. The resulting behavior of χ modes is exemplified in Figs. 7.6
and 7.7. Another conclusion one can draw from Eq. (7.80) concerns the applicability
of an analysis based on the stability/instability chart of parameter space: Applying the
coupling constant g = 5·10−4 and the inflaton mass mφ = 10−6 mPl as in the calculations
above, the value of the parameter A ≈ 2q and its continuous decrease can be calculated.
Furthermore, from the theory of Mathieu functions [84] it is known that for small q
the instability bands are located at A = n2 with n being a natural number. A short
calculation shows that during a single oscillation of the inflaton the system sweeps over
dozens of instability bands. This renders the Mathieu theory of separate instable modes
inapplicable. Only later, when A has become small enough, the time a mode needs to
cross a single instability band becomes longer. This enables resonant modes to grow
continuously, see Fig. 7.7. Around t/(2π/mφ ) ∼ 20 the behavior lies between broad and
104
7.3. Effects of the Hubble expansion
105
10
nχk (t)
4
103
102
10
1
10−1
0
1
2
3
4
5 6
2π
time t/ m
φ
7
8
9
10
Figure 7.6.: Stochastic resonance in the model (7.73) for a comoving mode χk (t)
within an expanding background. Equation (7.80) gives the time evolution of the inflaton φ(t) whose decreasing amplitude is reflected by the
slowdown of the small oscillations in nχk during the adiabatic intervals.
Redshift prevents the mode from staying in the resonance band. Therefore, the growth parameter µj is different at each step and can also be
negative.
105
Chapter 7. Fluctuations produced after inflation: Preheating
1016
k = 10−2 mφ
k = 5mφ
k = 8mφ
k = 10mφ
k = 50mφ
1014
1012
nχk (t)
1010
108
106
104
102
1
10−2
0
5
10
15
20 25
2π
time t/ m
φ
30
35
40
Figure 7.7.: The same setting as in Fig. 7.6 calculated for a longer time interval and
for different modes. For large k the parameters do not stay in instability regions long enough for the occupation number to be considerably
amplified. The strong growth of nk which occurs for a broad region
in parameter space around t/(2π/mφ ) ∼ 20 can be identified with the
crossing of the last instability band. To the left of this ascent one observes a time interval of a few φ-oscillations with almost constant nk . At
this time the modes pass the stability region before the last instability
band.
106
7.4. Preheating within λφ4 theory
narrow resonance in Minkowski space. After t/(2π/mφ ) ∼ 30 the particle production
has stopped and the stability region at small A and q is reached.
Since the parameter q is time dependent, this is also the case for the angles θk and
∆φk . Similarly to the case in Minkowski space one can calculate
Z tj+1 r 2
k
j
+ g2 φ2 (t)
(7.82)
dt
θk =
a
tj
2gφ0 (t) κ2
gφ0 (t)
≈
+
+
4ln2
+
1
.
(7.83)
ln
m
2
mκ2
This equation shows that if q is large, θkj can be treated as a random number: For a
rough estimate it is enough to keep the first term. In Eq. (7.81) mt is replaced by jπ
where j numbers the zeros of φ(t). Taking the derivative of
gmPl
θkj ≈
+ O(κ2 )
(7.84)
5mj
gives an estimate for the variation of subsequent θkj ’s:
√
2 q0
gmPl
j
∼ 2 ,
δθk ∼
5mj 2
j
(7.85)
where the value of the parameter q0 after the end of inflation is estimated as
q0 :=
φ(t0 )
gmPl
∼
2m
10m
(7.86)
when t0 is chosen to be of order 1/m. Therefore, δθkj is bigger than π for the first
r
1/4
q
1 5πm
≈ √0
nrandom ∼
(7.87)
2 gmPl
2π
inflaton oscillations. During this time the phase shifts θk for consecutive zeros of φ(t)
can be treated as random. Because of Eq. (7.54) the same is done with γ. Then,
from Eq. (7.55) it is seen that this renders the non-adiabatic particle number change a
stochastic process. Depending on the value of γ the particle number can also decrease
but the first two terms in Eq. (7.55) make this case less probable.
7.4. Preheating within λφ4 theory
As a second example for preheating after inflation, a short section on λφ4 theory is now
included. A prominent feature of the evolution of particle number spectra immediately
after inflation is a region in momentum space which shows strong particle production.
In a numerical calculation this resonance peak is seen at k ∼ 1.27, see Fig. 7.8. This
result is analytically explained in the following way [63]: Transforming the equation of
motion for the zero mode φ(t) = hΦ(x, t)i,
φ̈(t) + 3H φ̇ + λφ3 = 0,
(7.88)
107
Chapter 7. Fluctuations produced after inflation: Preheating
from time t to conformal time η =
R
ϕ′′ (η) −
dt/a and defining ϕ = aφ one has
a′′
ϕ(η) + λϕ(η)3 = 0.
a
(7.89)
First, it is now shown that the second term is negligible in λφ4 theory following the
calculation in [123]: Let us assume that the field φ oscillates with frequency ω around
the minimum of a potential V (φ) at φ = 0. The timescale ω −1 of this oscillation is taken
to be much smaller than the Hubble time, ω ≫ H, and so is the oscillation period of
φ̇2 = ρ + p,
(7.90)
see Eq. (3.4). The slowly varying contribution to pressure and energy density is separated from the oscillating part,
φ̇2 = (γs + γf )ρ.
(7.91)
The labels s and f stand for slow and fast, respectively. Energy–momentum conservation
is then expressed as
ρ̇ = −3H(ρ + p) = −3H(γs + γf )ρ
(7.92)
(7.93)
and one finds
ln
ρ
ρ0
= −3
Z
da
−3
γs
a
Z
Hγf dt.
The last integral is negligible,
Z
Z
H
γf
γf Ḣdt = O
≪ 1,
Hγf dt = H −
ω
ω
ω
(7.94)
allowing to write
ρ ∝ a−3γs .
(7.95)
Neglecting curvature, the Friedmann equation (2.39) gives
a ∝ t2/(3γs ) .
(7.96)
The remaining step is to calculate γs for the potentials in question: Taking ∆t as the
oscillation period of φ one calculates
γ=
1
∆t
Z
0
∆t
φ̇2
ρ
R φmax
dt = R0φmax
0
(φ̇2 /ρ)1/2 dφ
(ρ/φ̇2 )1/2 dφ
R φmax = 2R
0
1−
φmax
1−
0
V
1/2
dφ
−1/2 ,
V
dφ
Vmax
Vmax
(7.97)
where the first equality follows from Eq. (7.91). The second equality relies on the fact
that ρ is constant on this timescale. φmax is the maximal displacement of the field
108
7.5. Preheating after hilltop inflation
and Vmax is the corresponding potential. Concentrating on potentials V (φ) = λφn one
computes the result [30, 62]
γ=
2n
n+2
⇒
a(t) ∝ t(n+2)/3n ,
(7.98)
which for n = 2 gives the expansion of a universe filled with massive particles and for
√
n = 4 gives a ∝ t as it is known from a radiation dominated universe. In this latter
case Eq. (??) gives a′′ = 0, and so (7.89) reads
ϕ′′ + λϕ3 = 0.
(7.99)
The solution will be an oscillating function f whose amplitude is for the moment taken to
be normalized to one. f is written as a function of x = λη. The stronger condition 2f ′2 =
√
1−f 4 is fulfilled by the elliptic cosine cn(x, 1/ 2). The solution of the homogeneous case
is now used to find the evolution of the modes ϕk which correspondingly to Eq. (7.99)
follow the equation
ϕ′′k + k2 + 3λϕ2 ϕk = 0
(7.100)
or, after inserting the homogeneous solution,
√ ϕ′′k + k2 + 3cn2 x, 1/ 2 ϕk = 0.
(7.101)
Note that in this equation the amplitude ϕmax is not confined to unity anymore and time
√
and momentum are measured in units of ( λϕmax )∓1 , respectively. Equation (7.101)
is a Lamé equation having solutions with well-known stability properties, see [13]. The
first resonance band at 1.22 . k . 1.32 can be confirmed by a lattice computation.
Figure 7.8 is obtained from calculations done with LATTICEEASY [50].
7.5. Preheating after hilltop inflation
In this section preheating after hilltop inflation within the potential Eq. (5.26) is discussed. Besides parametric resonance, within this potential a second mechanism of
inflaton decay is observed. It stems from the negative curvature of the potential around
small field values and is called tachyonic preheating. The curvature of the potential is
negative during inflation and remains so also for a part of the subsequent evolution, see
Fig. 5.3. The results presented in this section were obtained in [42] for the first time.
The reasoning also relies on [28].
7.5.1. Slow roll and oscillations of the homogeneous field
Around the maximum of the potential at φ = 0 the slow-roll parameters ǫ and η, which
are defined in Eq. (5.15), are both much smaller than one. During inflation the potential
Eq. (5.26) can be approximated by
V (φ) =
1
λ(v 4 − φ4 ).
16
(7.102)
109
Chapter 7. Fluctuations produced after inflation: Preheating
1010
t = 1.5 · 102
t = 3.5 · 102
t = 7.1 · 102
t = 1.7 · 103
particle spectra nk
108
106
104
102
1
1
√
wavenumber k/ λφmax
10
4
Figure 7.8.: Early evolution of the particle spectrum in λφ√
theory. Correspondingly
to the wavenumber, time is given in units of ( λφmax )−1 , where φmax is
the√initial amplitude of inflaton oscillations. The shaded region around
k/ λφmax = 1.27 indicates the (only) resonance band of the theory.
Further peaks, which appear later, form due to scattering processes.
Because the field amplitude decays with time, the peaks tend to move
to the left.
Then the slow-roll parameters are
1 mPl 2 φ 6
ǫ≈
π
v
v
and η = −
3 mPl 2 φ 2
,
4π
v
v
(7.103)
identifying η as the one which will break the slow-roll condition first. So, preheating
starts around the field value φsp ∼ v 2 /mPl , where here and in the following the label
has the obvious meaning. The condensate hφ(t)i continues rolling down the potential
and starts a damped oscillation around the minimum at v. The decay of the amplitude
can be estimated from energy conservation: During one oscillation period from tj to
tj+1 the Hubble damping amounts to
Z φj+1
Z tj+1
dφH φ̇.
(7.104)
dtH φ̇2 = 3
∆E = 3
φj
tj
Approximating the Hubble parameter, the field velocity, and the distance covered by
the field during one oscillation as
2
≈
H 2 ≈ Hsp
λv 4
,
16
∆φ ≈ 2v,
φ̇2 ≈
110
V
λv 4
≈
,
2
3mPl
48m2Pl
(7.105)
and
(7.106)
(7.107)
7.5. Preheating after hilltop inflation
respectively, allows for an estimate of the integral. When the boundaries of the integrals
in Eq. (7.104) are evaluated at minimal values of φ, only the potential contributes to
∆E. With Eq. (7.102), successive minima can be evaluated as
φn
v
4
=
v
mPl
4
√
v
v
+ 2 3n
≈n
,
mPl
mPl
(7.108)
where n ∈ N is introduced as a numeration of the minima. For small enough v the
oscillatory motion of the field condensate partly goes through the tachyonic region of
the potential, see Fig. 7.9. A higher energy scale v though, leads to stronger Hubble
damping which prevents a recurrence to such small field values. The more important
role of Hubble friction in this case is reflected in the broader regions of slow roll and
inflation on the left-hand side of Fig. 5.3.
7.5.2. Tachyonic preheating
A negative curvature of the effective potential is reflected by a negative effective mass
squared m2eff = d2 V /dφ2 < 0 in the equation of motion for the modes,
φ̈k + (k2 + m2eff )φk = 0.
(7.109)
Here, interactions between the modes have been neglected. The solution for modes with
small k is tachyonic. It does not oscillate but is an exponential function. To obtain an
estimate of the fluctuations during preheating, their magnitude at the end of slow roll
should be taken as an initial condition, see also Section 7.1. The modes with wavelengths
around the Hubble scale will experience the strongest growth. This is because above
the Hubble scale spacetime curvature prevents decay. So they have the smallest k for
which particle production takes place. Because of the uncertainty principle their initial
amplitude is expected to be δφ ∼ 1/L ∼ H, where L is the wavelength of the modes.
Hence, the Hubble parameter at the end of inflation can be taken as an initial condition
for the fluctuations.
The growth of long wavelength inhomogeneities at the beginning of preheating can
be estimated in the following way: Taking Hubble friction to be negligible, the equation
of motion of the field condensate φ(t) is
with the time derivative
φ̈ = −V ′
(7.110)
...
φ = −V ′′ φ̇ ≡ −m2eff φ̇.
(7.111)
So, the derivative of the field follows the same equation of motion as the fluctuations
with small k, see Eq. (7.109), yielding
φk ∝ φ̇ and φ̇k ∝ φ̈ ∝ V ′ .
(7.112)
111
Chapter 7. Fluctuations produced after inflation: Preheating
1
1.2
10−4
0.8
tachyonic regime
0.4
10−8
hφi/v
hδφ2 i/v2
0
0
hφi/v
hδφ2 i/v2
10−12
10 20 30 40 50 60 70 80
√ −1
λv
time t/
700 750 800 850 900 950 1000
√ −1
λv
time t/
Figure 7.9.: This figure illustrates direct recalculations of Ref. [42]. The coupling
constant is λ = 10−12 . The energy scale of the potential is v = 0.1mPl
(left panel) and v = 10−3 mPl (right panel). For high energies the inflaton
field does not reenter the tachyonic regime below the dashed line and
no particle production is observed. For lower energies a combination of
tachyonic preheating and parametric resonance leads to a rapid decay of
the zero mode.
On the other hand, in the potential Eq. (7.102) the field velocity φ̇ grows as φ2 when
rolling downhill. Therefore, also the long-wave fluctuations should show this proportionality. As mentioned above, most of the growth occurs for small k which justifies
δφ ∝ φ̇ ∝ φ2 . Put together, the fluctuation at the start of preheating is
p
√ 2
V (φ)
λv
δφsp ∼ H ∼
∼
,
(7.113)
mPl
mPl
and when φ = v for the first time, the fluctuation is
δφv ∼ δφsp
√
v2
∼ λ mPl .
2
φsp
(7.114)
If this fluctuation fulfills δφv ≥ v ≈ hφi, then preheating can be considered as finished
already at this stage. If it is still smaller than the condensate, then the latter starts
an oscillation around the potential minimum v. Of course, this goes with temporarily
negative φ̇ and also a sign change in the mode functions φk for small k. In the upper
right panel of Fig. 7.10 this is reflected by the sudden downward jumps of |φk |2 . The
asymmetry of the potential around v leads to a more rapid change of the low k mode
functions when φ is on the right-hand side of the minimum and a slower change when φ
is on the left-hand side. This is also the reason for the oscillations in nk for early times
and small k and in the variance of φ, see Fig. 7.9: When the field condensate is at the
steeper (right-hand) side of the minimum, φ̈ and φ̇k are large for small k, whereas on the
flat (left-hand) side φ̈ and φ̇k are small. The mode φk is small at both the maximal and
the minimal value of φ(t). Having this in mind, a look at the equation for the number
112
1015
k ≈ 4 · 10−3
k ≈ 0.5
1010
|φk (t)|2
1018
k ≈ 4 · 10−3
k ≈ 0.1
1016
k ≈ 0.5
1014
1012
1010
108
106
104
10
1
800
820
840
860
880
√
time t/( λv)−1
105
1
900
1020
Occupation number nk
nk (t)
7.5. Preheating after hilltop inflation
920
800
820
840
860
880
√
time t/( λv)−1
900
920
t=0
t ≈ 860
t ≈ 880
t ≈ 900
t ≈ 920
1015
1010
105
1
10−5
10−3
10−2
1
10
10−1
√
Wavenumber k/( λv)
102
Figure 7.10.: Some results of lattice calculations within the model (5.26) at the
√ energy scale v = 10−3 mPl . The wavenumber k is given in units of λv.
Details are provided in the text.
113
Chapter 7. Fluctuations produced after inflation: Preheating
density Eq. (B.46),
1
1
|φ̇k (t)|2 ,
ωk |φk (t)|2 +
2
2ωk
shows that also the number density of long-wavelength modes oscillates at the beginning
of preheating. This is verified by the lattice results displayed in the upper left panel
of Fig. 7.10. The small peaks before and after the maxima of nk (t) form because the
frequency ωk is zero there. For larger values of k the mode functions φk follow the
behavior of φ̇ less closely.
nk =
7.5.3. Parametric resonance
The momentum range of tachyonic production in the UV depends on the maximal
negative curvature of the potential which in the case
|φ| 1
V ′′ (φ) = 3λφ2 ln
+
(7.115)
v
3
is
3
(7.116)
V ′′ (φmax ) = − λv 2 e−5/3 .
2
√
So the highest momenta excited tachyonically are those around k = 0.5 λv. The
exponential growth of these modes is weaker than for smaller k and thus there should
√
be a different explanation for the peak at k ≈ 0.5 λv and t ≈ 890: The growth of these
modes can be attributed to a period of non-adiabatic evolution around the inflection
point of the potential. This is demonstrated following an estimate done in Ref. [42]: At
the inflection point
φ̃ = v exp(−1/3)
(7.117)
the frequency squared of fluctuations is just k2 which leaves the low momentum modes
prone to violation of adiabaticity. The corresponding condition
ω̇ < ω 2
(7.118)
is indeed violated: Comparison of
ω̇ =
with ω 2 = k2 gives
3 φφ̇
dp 2
k + V ′′ (φ(t)) |φ=φ̃ = λ |φ=φ̃
dt
2 k
(7.119)
2
k ≤ λv
(7.120)
3
as an approximative condition for non-adiabatic evolution around φ̃. To arrive at the
last inequality the field velocity is estimated from energy conservation neglecting Hubble
damping. The conclusion is, that the modes around 0.5λv can undergo strong growth
as seen from Fig. 7.10. Different mechanisms leading to a peak at this momentum scale
are described in Ref. [28]. Additionally, in this reference the computation of particle
production is done similarly to the case in Sect. 7.2, namely by matching WKB solutions
to the exact solutions of a simplified potential.
114
7.6. Preheating within a model with two periods of inflation
7.6. Preheating within a model with two periods of inflation
In Chapter 6 the fluctuations from inflation in the model Eq. (6.1) are discussed. The
present section gives a short account on preheating within this scenario: The second
inflationary period ends somewhere not too far from the origin in field space, the fields
being on their way to one of the minima of the potential. This situation is similar to
preheating in hilltop inflation with an additional coupling to a second field taken into
account. So the evolution of the χ particle spectra is expected to resemble the one
seen in Fig. 7.10. Figs. 7.11 to 7.13 depict the results of a lattice calculation using this
potential.
The calculation is done within the parameter set λ0 = λ1 = λ2 = 10−13 and v =
10−3 mPl . As for the single field case in the preceding section, the lattice parameters
are chosen following Ref. [42]: In order to cover the relevant range of momenta the
√
necessary spatial extent of the lattice is L ≈ 1500/ λv. The expected momenta of
growing modes range up to the scale of the curvature at the minimum of the potential,
√
λv. Therefore, N = 214 = 16384 lattice points12 are necessary to resolve the whole
spectrum. Assuming that the number of dimensions does not play a decisive role for
the mechanisms of preheating, the computation has been done in one spatial dimension.
Of course, this makes the calculation much less expensive. However, one then has to
dispense with realistic statements on the late time properties of spectra being shaped
by turbulence [98].
The initial values of the fields are both set to 10−3 v. Presuming slow roll, the initial
field velocities are set to zero. From Fig. 7.11 it is seen that the mean value of χ starts to
grow whereas hφi stays at zero almost until χ reaches the minimum. Since all couplings
λi are the same, the absolute values of the fields at the four minima
r
2 λ2
λ2
χmin = ±v exp
χmin ,
(7.121)
,
φmin = ±
4λ0 λ1
2λ1
√
differ by a factor of 2 for the two fields. Comparison with Fig. 7.12 shows that the
oscillation around these values ends when the variances have grown to almost unity.
The increase of the variance is mainly because of the growth of low momentum modes,
which are governed by the same equation of motion as the field velocity φ̇, see Sec. 7.5.
Fig. 7.13 contains the spectra of inhomogeneities of the field χ at some moments of
√
time. The time unit is again taken as ( λv)−1 being the time scale of oscillations around
the minimum. As has been done for Fig. 7.10, the noise of the spectra is reduced by
averaging over some sets of different initial vacuum fluctuations, see App. B.2.
The shape of the spectra in Figs. 7.10 and 7.13 differs only in details. The characteristic momentum scale of the transient maximum of the spectrum shows the same
dependence on the parameters of the potential. In both cases the scattering of particles
out of the maximum appears as a small amplification of the modes with the approximately double momentum. Scattering processes drive the front of the spectrum to higher
12
LATTICEEASY requires the number of points to be of the form 2n .
115
Chapter 7. Fluctuations produced after inflation: Preheating
Mean fields hχi and hφi
10
1
hχi/v
hφi/v
10−1
10−2
10−3
10−4
800
850
900√
950
Time t/( λv)−1
1000
Figure 7.11.: Evolution of the means hχi and hφi after the end of inflation. In this
calculation both fields start at 10−3 v with zero initial velocity. As
already observed in the case of one field, Fig. 7.9, after a few oscillations
the zero mode of χ is at rest in the potential minimum. The mean of φ
can also switch between different values of minimal energy. Here and
for the following Figures 7.12 and 7.13, the couplings are λ0 = λ1 =
λ2 = 10−13 and the energy scale is v = 10−3 mPl . Field fluctuations are
included.
1
1
hδχ2 i/v 2
10−2
Variance of φ
Variance of χ
10−2
10−4
10−6
10−8
10−4
10−6
10−8
10−10
10−10
10−12
10−12
800
hδφ2 i/v 2
850
900√
950
Time t/( λv)−1
1000
800
850
900√
950
Time t/( λv)−1
1000
Figure 7.12.: Time dependence of the field variances in the same case as in Figs. 7.11
and 7.13. As in the single field problem discussed in Sec. 7.5, the
variance of χ follows the growth of the zero mode. Inhomogeneities of
φ evolve with a short delay, which is also observed for the mean field.
116
7.6. Preheating within a model with two periods of inflation
Occupation number nk
1025
10
t=0
t ≈ 890
t ≈ 930
t ≈ 900
t ≈ 950
20
1015
1010
105
1
10−5 −3
10
10−2
10−1
1
√
Wavenumber k/( λv)
10
102
Figure 7.13.: Spectra of χ particles from the same calculation as discussed in
Figs. 7.11 and 7.12. The spectra of the field φ are very similar and
are therefore omitted. As in the case without coupling to a second
scalar,
a temporary maximum forms at slightly lower momenta than
√
λv, which is the effective mass squared of χ in the minimum.
and higher momenta, finally leading to thermal equilibrium. For the field φ no spectra
are displayed because they show no significant differences to the corresponding spectra
of χ. As expected, by varying the couplings λi the location of the resonance scale can
be moved to other values.
The calculation of preheating in the potential Eq. (6.1) has also been done for higher
energy scale v = 10−1 mPl . As in the case without additional field, no efficient particle
production could be observed. The variances did not grow from very small values for
various combinations of the couplings.
117
8. Conclusion
In this thesis, calculations on the behavior of fields and particles during and after cosmological inflation have been presented. Besides a review of the results in standard
inflationary models, a potential leading to two subsequent stages of inflation has been
discussed. These two epochs of accelerated expansion need to correspond to two separate regions in field space where inflation is possible. In this work, the calculations
concerning inflation make use of the slow-roll approximation, thus depending on the
existence of two such domains, where the slow-roll conditions are fulfilled.
It has been shown that this situation is given in the potential (6.1), which is the
simplified form of one that has been used to describe strongly interacting matter with
effective fields. Effective field theories are motivated by their similarity to the original
theory if it comes together with a significant simplification. The similarity may be
measured in terms of common symmetries and other properties.
So, as it has been retraced for the example of the strong interaction, an examination
of the original theory and its physics is required. Then, one can try to construct a
simpler model, mimicking as much of the starting point as possible.
A lot of the symmetries of QCD are, in a similar fashion, common to other theories as
well. Especially at high energies, nature seems to rediscover her inclination to unification
and simplicity. Thus one might draw the conclusion that at inflationary energy scales,
symmetries should play a vital role. Of course, still many possibilities remain.
For this work, the potential has been modeled on properties of QCD. Based on the
linear sigma model, which incorporates chiral symmetry, an additional scalar field has
been introduced to imitate the QCD-behavior under scale transformations. This has
been done following the references given in Chapter 3. In one of these, Ref. [22], a
dilaton-extended linear sigma model has already been used for an inflationary scenario
during the cosmological QCD phase transition. In this work, the consequences of a
similar potential during primordial inflation well before the QCD phase transition has
been examined.
In this potential, the early stages of inflation are very similar to the corresponding ones
in the quartic and quadratic potentials of large-field inflation. It has been shown that
also the primordial spectrum of fluctuations that originates from this epoch does not
differ significantly from the spectra produced by these inflationary models. However,
as in other models of hybrid inflation, at the end of this period the fields have not
reached the minimum of the potential. This occurs only after an additional waterfall
Chapter 8. Conclusion
stage that can bring about a considerable number of e-foldings: In Ref. [34] such a
scenario has been discussed for large amounts of inflation during the waterfall period.
Then all observational inflation occurs in a small-field setting. By contrast, in Ref. [35]
the second inflation is assumed to be short or absent, such that no additional e-foldings
of a second inflation have to be taken into account:
Then, the first possibility is a result equivalent to chaotic inflation. This may happen
when the offset of the potential has negligible effect, as it has been seen for the calculations in this thesis. It corresponds to the simplified picture of an inflaton rolling down
a potential V0 + λφn /n, where the potential is immediately set to zero when φ = 0 is
reached.
A second possibility is the following: The slow roll of φ becomes unstable much earlier.
Then inflation stops at a position in the potential φend for which Ne (φend ) > 1 when
computed in the respective monomic potential. The resulting spectrum resembles that
from λφn /n but its momentum scale today is shifted.
The spectra resulting from the calculations of this work are shifted, too, but towards
the opposite direction: The scenario with two inflationary epochs, which is described
in Chapter 6, prolongs inflation after its natural end. Instead of being interrupted
early, inflation continues (after a short pause) and thus shifts the fluctuations to larger
wavelengths than expected. So, the fluctuations at some physical q0 today are similar
to those which are expected from monomic inflation at some smaller momentum scale.
They will enter the horizon at some time in the future. This is equivalent to the
following statement: The fluctuations obtained from CMB measurements and being
linked to a certain number of e-foldings subsequent to their horizon exit, exhibit features,
which are expected from fluctuations connected to some significantly smaller value of
Ne , when a monomic potential is assumed to drive inflation. The case with negligible
(2)
second inflation (Ne ≪ 10) gives spectra indistinguishable from the λφn /n standard
scenario and is therefore omitted from further discussion. It has already been justified
(2)
in Chapter 6 that the discussion concentrates on parameter sets with Ne ∈ (10, 60):
Larger values would imply pure small-field inflation in the observable range, belonging to
the standard scenarios of Section 5.5. Altogether, this means that the setting introduced
here includes a transition from large-field inflation to small-field inflation after the modes
constrained by CMB observations have left the horizon. This might cause a problem if it
cannot be excluded that preheating sets in after the end of the first inflation, frustrating
a subsequent restart of accelerated expansion.
For this thesis it has been assumed that particle production is overcome by Hubble
dilution, and inflaton decay around φ = 0 need not be accounted for. Taken into the
extreme, one could consider the alternative of an uninterrupted inflation. However,
while it proves difficult to rule out a viable scenario with one uninterrupted inflation,
no parameter set has been found that allows for a smooth transition from large-field to
small-field inflation such that both types of inflation with their typical features would
show up in the observable range. Typically, inflation is either interrupted, where the
120
interruption needs to take place outside the CMB range in order not to be ruled out
by measurements; or inflation continues down to φ = 0 giving rise to many e-foldings
for small φ. The evolution either gets stuck around φ = 0 or there is at least such
a strong expansion during this stage that there remains no possibility of observable
large-field inflation. As already stated, this leads to the conclusion that parameter sets
(2)
with Ne ∈
/ [10, 60] should be discarded. Also within this range the resulting spectra
(2)
coincide better with measurements for smaller Ne .
It has been argued in this thesis that the resulting spectra should not be altered
significantly when the computation includes multifield dynamics. This is because multifield effects such as entropy modes are only expected when the path in field space is
curved. This curvature, however, typically occurs only at one point in the evolution and
also coincides with the aforementioned interruption of inflation. For this thesis both,
multifield dynamics and possible preheating during the interruption are neglected. This
should be treated more thoroughly in future work.
121
APPENDIX
.
A. Equations of linearized General
Relativity
In this appendix the Einstein equations for fluctuations on a FRW background are
derived. In addition the derivation of the energy-momentum conservation equation for
fluctuations is presented. The calculations are done within the gauge h0i = 0. The
presentation relies on [59, 60].
A.1. Linearized Einstein tensor
In this section the Einstein tensor for the metric
(A.1)
γµν = ηµν + hµν
is obtained. Only the linear order in hµν is accounted for. Then the Christoffel symbols
are
1 λρ
γ (∂µ γρν + ∂ν γρµ − ∂ρ γµν )
2
1
= η λρ (∂µ hρν + ∂ν hρµ − ∂ρ hµν )
2
1
= (∂µ hλν + ∂ν hλµ − ∂ λ hµν )
2
λ
γµν
=
(A.2)
(A.3)
(A.4)
λ prevents confusion with the metric.
where the different arrangement of indices of γµν
λ
Here the following γµν are needed:
0
γ00
=
1 ′
h ,
2 00
0
γ0i
=
1
∂i h00 ,
2
1
0
γij
= − h′ij .
2
(A.5)
Note that they are calculated with the simplification h0i = 0. From the formula for the
Riemann tensor (2.20) the Ricci tensor is seen to be
λ ρ
λ ρ
λ
λ
γµν − γρµ
γνλ .
+ γρλ
Rµν (γ) = ∂λ γµν
− ∂µ γλν
(A.6)
Only the first two terms contribute to first order. Then it is
1
λ
λ
λ
λ
Rµν (γ) =
∂µ ∂λ hν + ∂ν ∂λ hµ − ∂λ ∂ hµν − ∂µ ∂ν hλ
2
(A.7)
Appendix A
and finally
R(γ) = Rµµ (γ) = ∂µ ∂ν hµν − ∂µ ∂ µ hνν
(A.8)
for the Ricci scalar. Note that also index contraction must not entail higher order in
hµν . Let us finally note down the Einstein tensor:
1
Gµν (γ) = Rµν (γ) − ηµν R(γ).
2
(A.9)
In order to make use of the helicity decomposition described in Section 2.2.3, it is useful
to write out the sum in Eq. (A.7). With the definitions h = hii and ∆ = ∂i ∂i it is
1
R00 (γ) = (h′′ + ∆h00 ),
2
1
Ri0 (γ) = (∂i h′ − ∂j h′ij ),
2
1
Rji (γ) = (∂i ∂k hjk + ∂j ∂k hik + h′′ij − ∆hij + ∂i ∂j h00 − ∂i ∂j h),
2
(A.10)
(A.11)
(A.12)
where the Latin indices are summed over irrespectively of being on the same level. Recall
that lowering a spatial index gives a sign change. The result for the Ricci scalar is
R(γ) = h′′ + ∆h00 + ∂i ∂j hij − ∆h.
(A.13)
A.2. Conformal transformations
In the last section the calculation of geometric quantities corresponding to a perturbed
metric γµν has been presented. The next step is to connect them to their counterparts
corresponding to the metric gµν that results from γµν after a conformal transformation.
By convention this is phrased as
gµν = e2ϕ(x) γµν ,
(A.14)
where ϕ(x) is an arbitrary smooth function. Below this will be applied to ϕ = ln a. The
Christoffel symbols transform as
Γµνλ (g) = Γµνλ (γ) + δλµ ∂ν ϕ + δνµ ∂λ ϕ − g µρ gνλ ∂ρ ϕ.
(A.15)
The Ricci tensor is obtained from the contraction of Eq. (2.20) as
Rµν (g) = ∂λ Γλµν (g) − ∂µ Γλλν (g) + Γλρλ (g)Γρµν (g) − Γλρµ (g)Γρνλ (g).
(A.16)
Here Eq. (A.15) has to be inserted. After reorganizing the expression the result is
Rµν (g) = Rµν − 2∇µ ∂ν ϕ − γµν γ λρ ∇λ ∂ρ ϕ + 2∂µ ϕ∂ν ϕ − 2γµν γ λρ ∂λ ϕ∂ρ ϕ,
(A.17)
where all expressions on the right hand side are evaluated with the help of the metric
γµν . Contraction with g µν yields the Ricci scalar R(g) in terms of γµν and the function
ϕ,
R(g) = e−2ϕ (R − 6γ µν ∇µ ∂ν ϕ − 6γ µν ∂µ ϕ∂ν ϕ)
(A.18)
126
A.3. Linearization within FRW background
Again, the right hand side is calculated in terms of γµν . This holds also for the following
formula for the Einstein tensor,
Gµν (g) = Gµν (γ) − 2∇µ ∂ν ϕ + 2∂µ ϕ∂ν ϕ + γµν γ λρ (2∇λ ∂ρ ϕ + ∂λ ϕ∂ρ ϕ) .
(A.19)
Applied to the situation in cosmology, ϕ = ln a, this reads
2
4
1
λρ 2
Gµν (g) = Gµν (γ) − ∇µ ∂ν a + 2 ∂µ a∂ν a + γµν γ
∇λ ∂ρ a − 2 ∂λ a∂ρ a . (A.20)
a
a
a
a
Index raising of Gµν (g) is done with the metric gµν and results in
∇λ ∂ρ a ∂λ a∂ρ a
µλ ∂λ a∂ν a
µ λρ
2 µ
µ
µλ ∇λ ∂ν a
+ 4γ
+ δν γ
−
. (A.21)
2
a Gν (g) = Gν (γ) − 2γ
a
a2
a
a2
A.3. Linearization within FRW background
The Einstein tensor after the conformal transformation gµν = a(η)2 γµν has been obtained in Eq. (A.21). Linearization gives
∇λ ∂ν a
∂λ a∂ν a
∂λ ∂ν a
− 2η µλ
− 4hµλ
a2 δGµν (g) = Gµν (γ) + 2hµλ
a
a2
a
∂λ ∂ρ a ∂λ a∂ρ a
∇λ ∂ρ a
− δνµ hλρ 2
−
+ 2δνµ η λρ
2
a
a
a
∂
a
∂λ a∂ν a
∂
∂
a
λ ν
σ σ
+ 2η µλ γλν
− 4hµλ
= Rνµ (γ) + 2hµλ
a
a
a2
λρ ∂λ ∂ρ a
λρ ∂λ a∂ρ a
λρ σ ∂σ a
µ 1
R(γ) + 2h
−h
+ 2η γλρ
,
− δν
2
a
a2
a
(A.22)
(A.23)
where Gµν (γ), Rνµ (γ), and R(γ) are already of first order in the perturbation. The
components are obtained with the help of the previous results in Eqs. (A.10) to (A.13):
Plugging in, reorganizing, and making use of h0i = 0 yields
′
a′2 1
1
′a
−
∂
∂
h
+
∆h
−
h
,
(A.24)
i
j
ij
a2
2
2
a
1
1
a′
a2 δG0i = ∂i h′ − ∂j h′ij + ∂i h00 ,
(A.25)
2
2
a
1
1
1
1
a′
1
a2 δGij = ∂i ∂k hjk + ∂j ∂k hik + h′′ij − ∆hij + ∂i ∂j (h00 − h) + h′ij
2
2
2
2
a
2
′′
′2
′
1 ′′ 1
1
1
a
a
a
− δji
h + ∆h00 + ∂l ∂k hlk − ∆h + 2 h00 − 2 h00 + (h′00 + h′ ) . (A.26)
2
2
2
2
a
a
a
a2 δG00 = −3h00
Now the decomposition of the metric perturbation, Eq. (2.60) is used. As in Section 2.2.3
the coordinate system is chosen such that E = 0 additionally to h0i = 0 (conformal
Newtonian gauge). For the scalar perturbations this means
h00 = 2Φ,
h0i = 0,
hij = −2Ψδij ,
(A.27)
127
Appendix A
which simplifies Eqs. (A.24) to (A.26):
′
a′2
′a
,
−
2∆Ψ
+
6Ψ
a2
a
a′
a2 δG0i = − 2∂i Ψ′ + 2 ∂i Φ,
a
a2 δGij = ∂i ∂j (Ψ + Φ)
a′′
a′2
a′
+ δij 2Ψ′′ − ∆Ψ − ∆Φ + 2 (2Ψ′ − Φ′ ) − 4 Φ + 2 2 Φ .
a
a
a
a2 δG00 = − 6Φ
(A.28)
(A.29)
(A.30)
A.4. Linearized evolution equations
The components of the linearized energy-momentum tensor have been derived in Eq. (2.74),
δT00 = δρ,
(A.31)
δTi0
δTji
(A.32)
= −(ρ̄ + p̄)vi ,
=
−δji δp.
(A.33)
Comparison of Eqs. (A.30) and (A.33) yields Ψ = −Φ for an ideal fluid because the first
term in Eq. (A.30), ki kj (Ψk + Φk ), is independent from δij δpk .
Put together, the 00- and the 0i-components of the Einstein equation for scalar perturbations read
a′2
4πa2
a′
∆Φ − 3 Φ′ − 3 2 Φ = 2 δρ
a
a
mPl
′
a
4π
and Φ′ + Φ = − 2 (ρ̄ + p̄)v,
a
mPl
(A.34)
(A.35)
respectively. The remaining part of Eq. (A.30) is the one in curly brackets. Together
with Eq. (A.33) it gives
a′′
a′2
4πa2
2 Φ − 2 Φ = 2 δρ
(A.36)
a
a
mPl
The formulation of energy-momentum conservation makes use of covariant derivatives.
They include the Christoffel symbols of the metric gµν = a2 (ηµν + hµν ):
a′ 1 ′
+ h00 ,
a
2
1
Γ00i = Γi00 = ∂i h00 ,
2
′
1
a
Γi0j = δij − h′ij ,
a
2
′
a
a′
1
Γ0ij = (1 − h00 )δij − hij − h′ij ,
a
a
2
1
i
Γjk = − (∂j hik + ∂k hij − ∂i hjk ).
2
Γ000 =
128
(A.37)
(A.38)
(A.39)
(A.40)
(A.41)
A.4. Linearized evolution equations
Then the zero component of energy-momentum conservation is
∇µ T0µ = δρ′ + 3
a′
(δρ + δp) + (ρ̄ + p̄)(3Ψ′ − ∆E ′ + ∂i vi ) = 0
a
(A.42)
and the spatial components are
∇µ Tiµ
a′
= −((ρ̄ + p̄)vi ) − ∂i δp − (ρ̄ + p̄) 4 vi + ∂i Φ
a
′
=0
(A.43)
for an ideal fluid. In the gauge E = 0 and with Φ = Ψ, Eq. (A.42) simplifies to
a′
δρ′ + 3 (δρ + δp) + (ρ̄ + p̄)(3Φ′ + ∂i vi ) = 0.
a
(A.44)
In the case of a non-ideal fluid one has to account for anisotropic stress which is added
to the spatial components of the energy-momentum tensor:
δTji = −δji δp − Πij .
(A.45)
The additional component is traceless and symmetric and it is treated as a small perturbation. Being transverse, ∂i Πij = 0, it respects the energy-momentum conservation,
which is also true for the corresponding part of the Einstein tensor, hTijT in Eq. (2.60).
The equation of motion for this piece of the metric is obtained by inserting the transverse
traceless part of this equation, i.e. hTijT , into Eq. (A.30) and equating it to Eq. (A.45).
The result is
′′
a′ T T ′
16πa2
(A.46)
hTijT + 2
hij
− ∆hTijT = − 2 ΠTijT
a
mPl
Its solution is a gravitational wave. Ideal fluids do not source gravitational waves.
Instead, within such a medium they propagate freely through spacetime as long as their
scale is much smaller than the Hubble scale.
129
B. A short reference to LATTICEEASY
Many calculations being used in this work have been done with LATTICEEASY [50],
an open source lattice code for C++ written by Gary Felder and Igor Tkachev. Here
a short account on its properties is given using the documentation on [51] by the same
authors. LATTICEEASY and also its parallel computing version CLUSTEREASY [49]
can be downloaded from [51].
LATTICEEASY solves the classical Euler-Lagrange equations of scalar fields (??)
on a rectangular lattice in real space with N lattice points for each spatial dimension.
If needed the simulation can be done within an expanding background and with the
number of dimensions reduced to d = 2 or 1.
Before doing a calculation LATTICEEASY must be provided with the number of
scalar fields in the problem, with their potential, and with derivatives of the potential
with respect to the fields. Suitable values for the number of lattice points in real space,
N , and the length of the lattice, L, have to be chosen. They determine the spectrum
which can be resolved within the calculation. 1/L sets the smallest value of momentum
k that can be resolved and N/L fixes the largest one.
Possible output includes the field values on the lattice points, means and variances of
the fields, as well as spectra of energy or occupation numbers.
B.1. General lattice issues
When doing a calculation on a lattice one has to care for its finite size L and the finite
spacing ∆x = L/N . To do so let us start from a quantity like hf 2 (x)i, where the brackets
h...i stand for taking the mean over all lattice points. hf 2 (x)i may well depend on ∆x,
because the spacing sets the UV cutoff for Fourier modes, but should be independent of
L. With the following definition of the Fourier transformation for infinitely large spatial
Volumes,
∞
F (k) :=
Z
dxf (x)e−ikx ,
(B.1)
Appendix B
this leads to
1
hf (x)i = d
L
2
Z
f 2 (x)dx
Vol. Z
1
= d
dxdydkf (x)f (y)eik(x−y)
L (2π)d
Z
Z
1
∞
2
= d dk|F (k)| = dk|F (k)|2 ,
L
(B.2)
(B.3)
(B.4)
where in the last line the finite space Fourier transformed F (k) := L−d/2 F ∞ (k) has been
defined. So in order to keep hf 2 (x)i independent of the spatial extent L, for example
the initial fluctuations should be set in terms of F (k) which is the relevant quantity in
our context. The implementation of the initial conditions will be discussed below. Let
us now turn to the finite lattice spacing ∆x: Since LATTICEEASY evolves the fields
in discretized real space, the Fourier transform to be dealt with is not exactly F (k) but
X
f (k) =
f (x)e−2πikx/N
(B.5)
x
Z
1
→
dxf (x)e−2πikx/N
(B.6)
∆x3
Ld/2
1
∞
F
(k)
=
F (k),
(B.7)
=
∆x3
∆x3
where, correspondingly to the discreteness of x-space, f (k) has a periodicity N . For
Eq. (B.6) the sum has been turned into the corresponding integral in the limit ∆x → 0.
While computing the time evolution the time-steps ∆t have to be small enough in
order to make a stable behavior possible. More precisely, ∆t should meet the Courant
stability condition [40],
∆x
,
(B.8)
∆t < √
dim
which for dim dimensions ensures that the physical dependence region of any spacetime point x(t) (i.e. the region in space-time which can possibly influence the solution
at x(t)) lies inside the numerical dependence region of that point (i.e. within the numerical problem each point on the lattice can be reached by information of at least all
lattice points within the corresponding physical dependence region). This is a necessary
condition for convergence to the analytical solution.
B.2. The implementation of vacuum fluctuations
In Section 7.1 it is described how the spectrum of vacuum fluctuations in an expanding
space can be calculated. An account on the implementation as initial conditions of
some of the calculations is given here. The probability distribution of modes of a scalar
field follows Eq. (7.14) which is written down again with k replaced by ω(k) in order to
generalize for a massive field:
P [ϕ(k, η0 )] ∝ exp −2ω(k)|ϕ(k, η0 )|2 .
(B.9)
132
B.2. The implementation of vacuum fluctuations
The corresponding normalized distribution of the absolute value is
P [|ϕ(k, η0 )|] = 4ω(k)|ϕ(k, η0 )|exp −2ω(k)|ϕ(k, η0 )|2 .
(B.10)
To obtain numbers which are distributed according to this Rayleigh function LATTICEEASY first generates a random number on the basis of a constant probability
distribution in the interval (0, 1)
(
dx if 0 < x < 1
Puni [x]dx =
(B.11)
0
otherwise.
From the transformation law to an arbitrary probability distribution,
|P [|ϕk |] dϕk | = |Puni [x]dx|,
one sees
dx ,
P [|ϕk |] = Puni [x] dϕk where ϕk is shorthand for ϕ(k, η0 ). The last equation leads to
Z ϕk
P [|ϕ̃k |] dϕ̃k .
x=
(B.12)
(B.13)
(B.14)
0
This is to be resolved for ϕk (x) which sets the absolute value of the mode function
following the distribution (B.10). The phase of each mode function is drawn from a
uniformly distributed sample.
There are some further issues connected with the time dependence of field fluctuations:
So, in order to generate isotropic initial conditions one should initialize each mode as a
superposition of waves moving in opposite directions, for example as a standing wave.
Another point concerns calculations on lattices with one or two spatial dimensions.
These can be done for different reasons: First, in order to check the theoretical predictions for a reduced number of dimensions numerically, and second in order to save
computation time. If it is for the latter reason, one should correct the resulting spectra
by a factor which renders the contribution of each mode to the variance equal to the
amount in the three-dimensional case. In the rest of this section the corresponding
calculation is presented:
From the definition of a discrete Fourier transform in Eq. (B.5) it is seen
1 X
f (x) = d
f (k)e2πikx/N
(B.15)
N
k
1 X
2
|f (k)|2 .
(B.16)
⇒ h|f (x)| i = 2d
N
k
The isotropic system allows for a reduction to a one-dimensional sum. In three dimensions this leads to the Jacobian 2πn2 in the following expression,
1 X
2πn2 |f (k)|2 ,
(B.17)
h|f (x)|2 i ≈ 6
N
133
Appendix B
where n is the momentum k in the units of the Fourier transformed lattice, k = 2π
L n,
and the sum is taken over both positive and negative k. Replacing n by k then gives
h|f (x)|2 i ≈
L2 X 2
k |f (k)3d |2 ,
2πN 6
(B.18)
where the dimension of the Fourier transform has been labelled. In one and two dimensions the corresponding expressions are
h|f (x)|2 i =
1 X
|f (k)1d |2
N2
and h|f (x)|2 i ≈
respectively. Comparison shows
L X
|k||f (k)2d |2 ,
2N 4
L2 2
k |f (k)3d |2
2πN 4
L
|k||f (k)3d |2 .
|f (k)2d |2 ≈
πN 2
|f (k)1d |2 ≈
and
(B.19)
(B.20)
(B.21)
So, in order to approximate the three-dimensional occupation number one should multiply the one- or two-dimensional result with a factor
2πN 4
L2 k 2
and
πN 2
,
Lk2
(B.22)
respectively.
B.3. Staggered leapfrog method
LATTICEEASY calculates the time evolution of the fields by use of the staggered
leapfrog method which is discussed in [66]. The name refers to the fact, that within
this integration scheme the function and its second derivative are evaluated at different
times compared to the first derivative. This is done alternately. To see how it works let
us consider a second-order differential equation
J[f (t)] = f¨(t)
(B.23)
and the discretized version
J[f (t)] ≈
f˙(t + ∆t/2) − f˙(t − ∆t/2)
f (t + ∆t) − 2f (t) + f (t − ∆t)
≈
,
∆t
∆t2
(B.24)
where ∆t is the step-size in time. A rather stable numeric solution to this problem is
to first desynchronize the time-steps for f and f¨ on the one hand and of f˙ on the other
hand,
∆t
∆t
˙
(B.25)
= f˙(t) + J[f (t)] ,
f t+
2
2
134
B.4. Particle number density of classical fields
to evolve f (t) and its derivatives by the same time steps ∆t,
∆t
f (t + ∆t) = f (t) + f˙ t +
∆t,
2
∆t
3
˙
˙
+ J[f (t + ∆t)]∆t,
f t + ∆t = f t +
2
2
and to resynchronize after N steps,
1
∆t
˙
˙
f (t + N ∆t) = f t + N −
∆t + J[f (t + N ∆t)] .
2
2
(B.26)
(B.27)
(B.28)
However, the advantage in stability is lost, when f¨ depends also on f˙. This does not
occur if there is no expansion of space. But within an FRW background time and fields
need to be rescaled with appropriate powers of the scale factor a in order to eliminate
the contribution proportional to f˙ in the equation of motion.
B.4. Particle number density of classical fields
This section gives an account on how particle densities or occupation numbers can
be defined within a classical calculation. For highly occupied modes this definition
coincides with the quantum mechanical approach based on Bogolyubov transformations.
In order to be applicable to all calculations done in this work, the derivation includes
an expanding background. Let us first write down the equation of motion of a classical
scalar field with potential V within a spatially flat FRW universe, see Eq. (??),
φ̈ + 3Hφ −
∂V
1
∆φ +
= 0.
a2
∂φ
(B.29)
Here, ∆ refers to comoving spatial coordinates x. To write down the conformal version
of this equation, one again defines adη ≡ dt and ϕ ≡ aφ and sees
1
a′
1 ′
φ , φ̈ = 2 φ′′ − 3 φ′
a
a
a
′
a
∂V
⇒ φ′′ + 2 φ′ − ∆φ + a2
= 0.
a
∂φ
φ̇ =
(B.30)
(B.31)
Written in terms of ϕ, the equation does not depend on the first derivative of the field
with respect to time anymore:
φ′ =
1 ′ a′
ϕ − 2 ϕ,
a
a
1 ′′
a′
a′2
a′′
ϕ − 2 2 ϕ′ + 2 3 ϕ − 2 ϕ
a
a
a
a
a′′
′′
4 ∂V
⇒ ϕ − 2 ϕ − ∆ϕ + a
= 0.
a
∂ϕ
φ′′ =
(B.32)
(B.33)
Now let ϕk be the Fourier transform of the conformal field, ϕk = aφk , and approximate
the derivative of the potential in Fourier space by
2 ∂ V
∂V
ϕk .
(B.34)
≈
∂ϕ k
∂ϕ2
135
Appendix B
After defining the mode energy
2
4
ωk := k + a
this leads to the oscillator equation
∂2V
∂ϕ2
−
a′′
,
a
ϕ′′k + ωk2 (η)ϕk = 0.
(B.35)
(B.36)
Following [104], now the occupation numbers nk are derived. From the theory of Bogolyubov transformations it is known that
nk = |βk |2 ,
W (uk , vk∗ )
|αk |2 − |βk |2 = 1,
(B.37)
W (vk , uk )
, W (v, u) := v ′ u − vu′ ,
(B.38)
2i
2i
where the Bogolyubov coefficients αk and βk are written in terms of the Wronskian W
of mode functions. These are defined for example in Ref. [104], which is recommended
as a more detailed presentation. Assuming large occupation numbers one can also write
αk =
,
βk =
nk ≈
|αk |2 + |βk |2
2
(B.39)
and then calculate
1
|W (uk , vk∗ )|2 + |W (vk , uk )|2
8
1
1 ′ 2
1
∗
2
v
)
=
v
|
|u
ω
|u
|
+
,
= (uk u∗k vk′ vk′∗ + u′k u′∗
k k
k k k
4
4
ωk k
nk ≈
(B.40)
(B.41)
where for the last equality the initial condition
1
vk (η) = √ eiωk η
ωk
(B.42)
has been used. Writing |Â|2 = ÂÂ+ for a general operator, expression (B.41) can in
terms of operators be rewritten as
1
1
1
+
+
−
−
2
2
′
∗
′∗
|b̂k uk (η) + b̂−k uk (η)| 0, b , (B.43)
0, b ωk |b̂k uk (η) + b̂−k uk (η)| +
Volume
4
4ωk
which is arranged such that one can make use of the general solution of (B.36),
1 +
∗
u
(η)
.
(B.44)
(η)
+
b̂
u
ϕ̂k (η) = √ b̂−
k
−k k
2 k
Doing so we arrive at
1
nk =
Volume
1
1
2
′
2
|ϕ̂k (η)| 0, b ,
0, b ωk |ϕ̂k (η)| +
2
2ωk
and finally write down the formula which is used in the lattice calculations:
1
1
2
′
2
nk =
|ϕ (η)|
ωk |ϕk (η)| +
2
2ωk k
(B.45)
(B.46)
where as above in Eq. (B.4) the volume has been absorbed into the definition of ϕk .
136
C. An auxiliary calculation
In this section the calculation of the integral
Z π p
m
dt k2 + g2 φ2 (t)
θk =
(C.1)
0
in Sect. 7.2.4 is presented: Inserting φ(t) = φ0 sin(mt) and defining z = mt and ε =
k/(gφ0 ) gives
Z rε p
Z π−rε p
Z π p
m
2
2
2
2
dz ε + sin z +
dz ε2 + sin2 z, (C.2)
dz ε + sin z = 2
θk =
φ0 g
0
rε
0
where the integral is split such that 1 ≫ r ≫ ε in the equation above. After approximating sinz ≈ z the first term is now expanded up to O(ε2 ) which yields
h p
i
p
ε2 r r 2 + 1 + ln r + r 2 + 1
(C.3)
as in Eq. (7.44). The second term is expanded in terms of the small quantity ε2 /sin2 z
leading to a sum over an alternating sequence that converges to zero:
Z π−rε
∞ X
1
2n
(−1)n
2n
(C.4)
ε
dz 2n−1 .
n
4
(1
−
2n)
n
sin
(z)
rε
n=0
For n = 0 the integral results in
2cos(rε) ≈ 2 − r 2 ε2 ,
for n = 1 there is
1 2
ε
2
Z
π−rε
rε
tan(π/2 − rε/2)
1 2
1
= ε ln
dz
sin(z)
2
tan(rε/2)
1
4
≈ ε2 ln 2 2 ,
2
r ε
and for n > 1 the result up to O(ε2 ) amounts to
∞ X
2n
(−1)n
2
r 2−2n ,
ε
n 4n (1 − 2n)(n − 1)
(C.5)
(C.6)
(C.7)
(C.8)
n=2
which is negligibly small compared to the other contributions. Adding the results of
Eqs. (C.3), (C.5), and (C.7), rearranging the logarithm and taking the limit r → ∞
gives the result displayed in Sect. 7.2.4:
Z π p
m
2gφ0
κ2
gφ0
2
2
2
θk =
dt k + g φ (t) ≈
+
+ 4ln2 + 1 .
(C.9)
ln
m
2
mκ2
0
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Dank
Ich möchte mich bei allen bedanken, die zu dieser Arbeit beigetragen haben. Dabei
geht der erste, große Dank an Jürgen Schaffner-Bielich, in dessen Gruppe ich die letzten
Jahre gearbeitet habe. Jürgen hat mir die Möglichkeit gegeben, unter seiner Betreuung
ein spannendes Forschungsgebiet kennenzulernen. Seine motivierende und inspirierende
Begleitung haben mir sehr geholfen.
Die entspannte und freundschaftliche Atmosphäre in Jürgens Arbeitsgruppe habe
ich sehr genossen. Für diese Erfahrung bedanke ich mich bei Kreso Baotic, Thomas
Beisitzer, Till Boeckel, Debarati Chatterjee, Thorben Graf, Margit Maly, Giuseppe
Pagliara, Rainer Stiele, Simon Weissenborn und Andreas Zacchi.
Gemeinsam mit Jürgen haben Jan-Martin Pawlowski und Matthias Bartelmann meine
Arbeit betreut. Ich bin dankbar für ihre hilfreichen und freundlichen Hinweise und
Vorschläge.
Für die Begutachtung dieser Arbeit bedanke ich mich besonders bei Jürgen und Matthias. Ich habe mich außerdem sehr darüber gefreut, dass ich mit Werner AeschbachHertig und Klaus Reygers meine Prüfungskommission vervollständigen konnte. Ihre
große Bereitschaft zur Terminfindung hat die Organisation der Disputation sehr erleichtert.
Mein Dank gilt den Mitgliedern des Instituts für theoretische Physik der Universität
Heidelberg: Den Mitarbeiterinnen des Sekretariats und der Bibliothek, Franziska Binder,
Tina Birke, Jeannette Bloch-Ditzinger, Isolde Dobhan, Angela Haag, Irene Illi, Anja
Kamp und Manuela Wirschke, und Elmar Bittner für die Betreuung des Computernetzwerks. Organisatorische Fragen fanden menschliche Antworten bei Jürgen Berges und
Eduard Thommes. Vielen Dank dafür.
Den Mitarbeiterinnen des HGSFP-Prüfungssekretariats, Gesine Heinzelmann, Elisabeth Miller und Karina Schönfeld, danke ich für die freundliche Besprechung vieler
organisatorischer Fragen.
Während meiner Zeit als Doktorand in Heidelberg war ich Mitglied der IMPRS-PTFS
(International Max Planck Research School for Precision Tests of Fundamental Symmetries in Particle Physics, Nuclear Physics, Atomic Physics and Astroparticle Physics
at the University of Heidelberg). Für die freundliche Aufnahme danke ich Manfred
Lindner, Klaus Blaum und Werner Rodejohann. Besonderer Dank gilt Britta Schwarz
für ihre Herzlichkeit, die ihre Arbeit am MPIK prägt.
Bei Philipp Merkel möchte ich mich herzlich für seine Freundschaft und Hilfsbereitschaft, die gemeinsame Zeit und viele neue Ideen bedanken.
Meine Familie war mit ihrer bedingungslosen Unterstützung in vielen Situationen ein
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