Dissertation
submitted to the
Combined Faculties for the Natural Sciences and for Mathematics
of the RupertoCarola University of Heidelberg, Germany
for the degree of
Doctor of Natural Sciences
Put forward by
Dipl.Phys. Steffen Stern
born in Giessen, Germany
Oral examination: November 19, 2008
Dynamical dark energy and variation of fundamental “constants”
Referees:  Prof. Dr. Christof Wetterich 
Prof. Dr. Carlo Ewerz 
Dynamische Dunkle Energie und Variation fundamentaler “Konstanten”
Diese Arbeit beschäftigt sich mit den Auswirkungen möglicher Variationen fundamentaler “Konstanten” auf den Prozess der primordialen Nukleosynthese (BBN). Die gewonnenen Ergebnisse zur Nukleosynthese werden mit Untersuchungen zu variierenden Konstanten in anderen physikalischen Prozessen kombiniert, um Modelle der großen Vereinheitlichung (GUT) und Quintessence zu überprüfen. Unsere Untersuchungen ergeben, dass das LiProblem der Nukleosynthese stark gemildert werden kann, sofern man Variationen von Konstanten zulässt, wobei insbesondere eine Variation der leichten Quarkmassen einen starken Einfluss hat. Weiterhin finden wir, dass aktuelle Messungen zu variablen Konstanten im Rahmen von sechs exemplarischen GUT Modellen nicht miteinander und mit BBN in Einklang gebracht werden können, sofern eine monotone zeitliche Variation angenommen wird. Wir folgern, dass aktuelle Messungen nichtverschwindender Variationen in starkem Widerspruch zueinander stehen und entweder selbst revidiert werden müssen, oder in der Natur erheblich komplexere GUTZusammenhänge (und/oder nichtmonotone Variationen) vorliegen. Die im Rahmen dieser Dissertation vorgestellten Methoden erweisen sich hierbei als mächtige Werkzeuge, um per Experiment unzugängliche Bereiche weit jenseits des Standardmodells der Teilchenphysik bzw. des concordance Modells der Kosmologie auf ihre intrinsische Konsistenz sowie auch Vereinbarkeit miteinander zu überprüfen, sofern einmal erste unumstößliche Beweise für Variationen von Naturkonstanten vorliegen sollten.
Dynamical dark energy and variation of fundamental “constants”
In this thesis we study the influence of a possible variation of fundamental “constants” on the process of Big Bang Nucleosynthesis (BBN). Our findings are combined with further studies on variations of constants in other physical processes to constrain models of grand unification (GUT) and quintessence. We will find that the Li problem of BBN can be ameliorated if one allows for varying constants, where especially varying light quark masses show a strong influence. Furthermore, we show that recent studies of varying constants are in contradiction with each other and BBN in the framework of six exemplary GUT scenarios, if one assumes monotonic variation with time. We conclude that there is strong tension between recent claims of varying constants, hence either some claims have to be revised, or there are much more sophisticated GUT relations (and/or nonmonotonic variations) realized in nature. The methods introduced in this thesis prove to be powerful tools to probe regimes well beyond the Standard Model of particle physics or the concordance model of cosmology, which are currently inaccessible by experiments. Once the first irrefutable proofs of varying constants are available, our method will allow for probing the consistency of models beyond the standard theories like GUT or quintessence and also the compatibility between these models.
Contents

I Introduction and prerequisites
 1 Introduction

2 Variation of “constants”
 2.1 The laws of physics and the constants of nature
 2.2 The question of constancy
 2.3 Theoretical arguments for variation of constants
 2.4 Equivalence principles and possible violations
 2.5 Variation of dimensionful parameters
 2.6 Probes of varying constants
 2.7 Finetuning of constants and the anthropic principle
 3 Cosmology
 4 The Standard Model and beyond
 5 Models of quintessence

II Big Bang Nucleosynthesis
 6 Big Bang Nucleosynthesis
 7 BBN with varying constants
 8 From nuclear to fundamental parameters
 9 Constraints on variations

III Unifying cosmological and latetime variations
 10 Experimental tests of variations

11 Variations from BBN to today in GUTs
 11.1 GUT relations

11.2 Variations in six different unified scenarios
 11.2.1 Varying alone
 11.2.2 Scenario 1: Varying gravitational coupling
 11.2.3 Scenario 2: Varying unified coupling
 11.2.4 Scenario 3: Varying Fermi scale
 11.2.5 Scenario 4: Varying Fermi scale and SUSYbreaking scale
 11.2.6 Scenario 5: Varying unified coupling and Fermi scale
 11.2.7 Scenario 6: Varying unified coupling and Fermi scale with SUSY
 11.3 Epochs and evolution factors
 12 Probing quintessence models
 13 Conclusion and outlook
 A Conventions
Part I Introduction and prerequisites
Chapter 1 Introduction
The constants of nature
Since the time of Newton, the constancy of the fundamental laws of nature has been undoubted. Comparing and reproducing experiments have been at the root of the scientific approach: A physical experiment which we perform today will have the same outcome as the same experiment performed tomorrow^{1}^{1}1Neglecting experiments which incorporate probabilities, for instance quantum mechanical effects.. Neglecting local gravitational effects, it should also not matter where we perform the experiment. Hence, it has been unquestionable for a long time that the laws of nature are constant over space and time. Moreover, Einstein formulated this space and time independence of physics in his strong equivalence principle, making it an essential part of his theory of general relativity.
Today’s view of this question is somewhat different, at least from theoretical aspects. Even though compelling evidence for changes in the laws of physics has up to now not been found, we have to admit that we are still lacking a profound test of this constancy. In the past, the laws of physics have only been thoroughly tested on time and length scales accessible by mankind, i.e. on timescales of years and on length scales that do not go beyond the size of our solar system^{2}^{2}2Note that general relativity has furthermore not been tested on length scales smaller than about 1mm.. Only recently astrophysics and cosmology have opened a door to test physics on immensely broader scales, reaching out to unimaginable length scales of several gigaparsecs and going back in time to the very beginning of our Universe.
This thesis will deal with probes of possible variations of constants throughout the whole accessible history of the Universe. In a first part, we will study one of the most distant (in time and space) events where physics can be applied and tested, primordial nucleosynthesis. It is the process during which the light elements of our Universe were formed and which happened when our Universe was only one minute old, extremely hot and dense. If physics was really subject to variations, primordial nucleosynthesis is a prime candidate for any studies of this kind. In a second step the obtained results will be combined together with further tests of varying constants at later times to derive a “history of variations”. Finally, we will show how these results can be used to test models beyond standard physics which currently cannot be accessed directly by experiments.
Outline
In this thesis I will work out the influence of varying “constants” on the process of primordial nucleosynthesis and implied constraints to theories beyond standard physics. In the next chapter, I will give a short introduction to variations of physical constants, some historical remarks and theoretical motivations. Chapter 3 will lay the theoretical framework for our understanding of the Universe as a whole, explaining general relativity and the main laws of cosmology. In Chapter 4 I will introduce the concepts of supersymmetry and grand unified theories (GUTs) which are widely accepted as extensions of the Standard Model of particle physics. Chapter 5 will introduce quintessence models which can yield variations of constants.
Part II will focus on the details of one particular process in the history of our Universe, Big Bang Nucleosynthesis (BBN). Chapter 6 will explain the standard process of BBN and the physics behind. Chapter 7 will introduce the possibility of varying constants in the process of BBN, and Chapter 8 will demonstrate how one can relate the results to variations of the Standard Model parameters. Finally, in Chapter 9, the observed element abundances will be used to derive constraints on variations of fundamental parameters.
In Part III I will study relevant tests of varying constants from the Big Bang until today. Chapter 10 gives an overview over tests of varying constants, and in Chapter 11 I will combine these tests within six different GUT models, showing how variations of constants can in principle be used to probe models of grand unification. Using the six GUT models, Chapter 12 shows how models of quintessence can be probed under the assumption of grand unification.
Finally, in Chapter 13, I will sum up the findings of this thesis and give some final conclusions and outlook.
The work on this thesis has led to four main publications:

Michael Doran, Steffen Stern, Eduard Thommes,
Baryon Acoustic Oscillations and Dynamical Dark Energy,
JCAP 0704:015 (2007) [DST06] (not in focus of this thesis) 
Thomas Dent, Steffen Stern, Christof Wetterich,
Primordial nucleosynthesis as a probe of fundamental physics parameters,
Phys. Rev. D 76, 063513 (2007) [DSW07] 
Thomas Dent, Steffen Stern, Christof Wetterich,
Unifying cosmological and recent time variations of fundamental couplings,
Phys. Rev. D 78, 103518 (2008) [DSW08.1] 
Thomas Dent, Steffen Stern, Christof Wetterich,
Time variation of fundamental couplings and dynamical dark energy,
Preprint arXiv:0809.4628, accepted by JCAP [DSW08.2]
Chapter 2 Variation of “constants”
2.1 The laws of physics and the constants of nature
The fundamental laws of physics, represented by the Standard Model of particle physics and Einstein’s theory of general relativity, consist of two parts. One part is the mathematical form of the laws (e.g. the behavior of Newton’s theory of gravity), the other part is the actual strength of the interactions relative to each other. Whilst the first part, the mathematical form of the laws of nature, can be derived from considerations of fundamental symmetries of nature^{1}^{1}1For example, the Standard Model of particle physics is obtained when demanding a local symmetry. See Sec. 4.1., the second part has to be put into the theories “by hand” in form of about 27  from a theory standpoint a priori absolutely arbitrary  numerical values, the constants of nature. Tab. 2.1 gives a list of these fundamental constants^{2}^{2}2As will be explained in Sec. 2.5, only ratios of masses are measurable fundamental parameters. Hence, in fact one can get rid of one the mass terms in Tab. 2.1, for instance by defining all masses with respect to the Planck mass. This would reduce the number of fundamental parameters by one. for the Standard Model of particle physics and general relativity^{3}^{3}3In cosmology some more free parameters turn up which have to be determined by observations, for instance those describing the composition of our Universe. However, it is assumed that these parameters can in principle be obtained from some fundamental laws of physics once the processes in the very early stage of the Universe are better understood.. Note that the list of fundamental parameters gets much larger when going to theories beyond the Standard Model, e.g. supersymmetry (see Sec. 4.4). Up to now it is unclear where these constants come from and if they are “real” constants in the sense that their numerical values are fixed once and for all.
Number of  

Type of constant  parameters 
3 coupling constants  3 
masses of 6 quarks  6 
CKM matrix (3 angles + 1 complex phase)  4 
masses of 3 leptons  3 
Higgs mechanism  2 
strong CP phase  1 
masses of 3 neutrinos  3 
PMNS mixing matrix for neutrinos  4 
gravitational constant  1 
in summa  27 
2.2 The question of constancy
The question if the constants of nature are actually constant was probably first raised by Dirac [Dirac37, Dirac38, Dirac79]. In his “large numbers hypothesis”, he argues that very large (or small) dimensionless constants must not enter in basic laws of physics. Based on his numerological principle, he suggests that very large numbers rather characterize the state of the Universe, specifically the time which has passed since the Big Bang. For instance, he finds that the age of the Universe in atomic time, , is of the same order of magnitude as the ratio of electrostatic to gravitational force between proton and electron, . Consequently, he suggests that also the latter quantity should vary with cosmic time. Attributing the variation to the gravitational sector, the intensity of all gravitational effects would then decrease with a rate of about y. It was quickly found that this would lead to astrophysical effects [Chandrasekhar37] which could not be detected in the following time. Hence, Dirac’s theory was finally abandoned, but the discussion on varying constants had started^{4}^{4}4See for instance [Uzan02] for a more complete review of the history of varying constant theories..
In 1961, Brans and Dicke [BransDicke61] used Mach’s principle^{5}^{5}5There are different formulations of Mach’s principle. In Brans’ and Dicke’s argument it states [Brans05] that the gravitational constant should be a function of the mass distribution in the universe. to derive what we now call a “scalartensor theory”. In their model, the gravitational constant is replaced by a scalar field which can vary in space and time. Besides others, models of this kind are still being considered as theoretical arguments for variation of constants.
2.3 Theoretical arguments for variation of constants
In highenergy theories such as string theory, which unifies gravity with the Standard Model of particle physics, our lowenergy laws of physics appear as an effective theory whose parameters are set dynamically by vacuum expectation values which break the “higher” symmetry. In particular string theory offers a plethora of possibilities to introduce variations of constants, for instance due to the fact that it is formulated with 10 (or 11) spacetime dimensions which need to be compactified in order to arrive at the 4 spacetime dimensions of the Standard Model (we will give some more details in Sec. 5.5). Similar considerations also apply to other theories with extra dimensions, for instance the possibility of varying constants in KaluzaKlein theories has been studied in [Marciano83]. Hence, both temporal and spatial variations of constants are from a theoretical standpoint well founded, even though those highenergy theories mostly do not give any hint on the actual size of the variations.
Also, “lowenergy” theories, for instance theories which extend the concordance model of cosmology by introducing a cosmological scalar field, allow variations of constants. In this thesis we will concentrate on theories of coupled quintessence in which constants can depend on cosmic time and the environment.
This thesis will examine the possibility of variations of constants from today back to the time of Big Bang Nucleosynthesis (BBN). During BBN, the composition of the Universe was quite different from today’s composition (concerning temperature and pressure). Hence, composition dependent effects which might cause spatial variations today might have caused variations at BBN time. However, since the Universe was almost homogeneous at BBN, these variations can effectively be treated as a timedependent effect. This thesis will not evoke the question of spacedependence of constants but treat possible variations at BBN as purely temporal effects.
2.4 Equivalence principles and possible violations
Particle theory is based on Poincaré covariance. In quantum field theory (QFT), we demand that each of the fundamental fields is a representation of the Poincaré group. Hence, amongst others, invariance under spacetime translations is automatically built in. However, we can still implement spacetime variations by introducing additional dynamical fields, whose values are determined by the fields’ own actions and their couplings to the rest of the theory. While the theory as a whole remains Poincaré invariant, variations in measurable quantities can still arise if the solution for the additional fields has a nontrivial spacetime dependence.
This discussion can be extended to general relativity (GR), which is also based on symmetry principles that are apparently violated by variations of constants. In particular, GR is based on the strong equivalence principle, which can be decomposed into the following symmetries

Weak equivalence principle: The trajectory of a freely falling test body only depends on its initial position and velocity and is independent of its composition.

Local Lorentz invariance: The outcomes of any experiments (whether gravitationally or not) in a laboratory moving in an inertial frame of reference are independent of the velocity of the laboratory.

Local position invariance: Outcomes of experiments (whether gravitationally or not) do not depend on their position in space and time.
A space or time variation of fundamental constants obviously violates local position invariance. Also, as the gradient of any varying fundamental parameter defines a direction in spacetime, local Lorentz invariance is violated. Finally, it has been shown (see e.g. [Nordtvedt02]) that any spacetime variation of fundamental constants will necessarily lead to an additional gravitational force, hence also the weak equivalence principle will be violated^{6}^{6}6We will work out the relation between violation of the weak equivalence principle and variation of constants in Sec. 12.3.. As probes of general relativity so far do not find any violation of the theory, we can immediately conclude that variations of constants must be extremely tiny. Note, however, that GR has only been tested on relatively small time scales and also only on length scales from 1mm to the size of our solar system.
2.5 Variation of dimensionful parameters
When measuring or estimating possible variations of constants, one always has to keep in mind that the variation of any dimensionful quantity is not physically welldefined, as one always has to specify how the dimension (e.g. the unit [Energy]) is defined. In general, a dimensionful quantity can only be measured by comparison with another dimensionful quantity, so in fact only dimensionless ratios are measurable. For example, measurements of variations of the electron mass are only welldefined if one states how the mass unit is defined, for instance by choosing a system of units where the mass scale is kept fixed. Popular system of units are the “Einstein frame” where the Planck mass is kept constant, or the “Jordan frame” where some particular particle physics scale is kept fixed. Considering the electron mass in the Einstein frame, the actually measured varying quantity (without system of units ambiguities) is then rather .
In the part of this thesis which deals with Big Bang Nucleosynthesis, we use a system of units where the QCD invariant scale is kept constant. This is convenient for dealing with nuclear reactions, where the energy scales are mainly determined by the strong interaction. Thus the variations of dimensionful parameters include implicitly some power of . For example, if we take the electron mass as a varying parameter we are implicitly considering a variation of . In the last part of this thesis we will work with theories of grand unification. There, the grand unified scale enters as natural scale which we choose to be constant. The appropriate conversion from a constant to a constant system of units is explained in Sec. 4.6.3.
2.5.1 The chiral limit
Many studies on varying constants work with the chiral limit, i.e. they assume that all quarks are massless [Epelbaum02, BeaneSavage02, Donoghue06]. Then all dimensionful QCD parameters are simply proportional to a power of , which ameliorates their treatment considerably. For instance, QCD masses like the proton mass simply scale like
(2.1) 
and any other dimensionful QCD parameter according to its mass scale (for instance, cross sections with = [Energy] scale like ). Switching on the quark masses, one obtains a finite range for pionmediated interactions, which may greatly affect static and dynamical properties of nuclei. Also, the masses of all hadrons are affected at some order in chiral perturbation theory [Gasser82]. In this thesis we will work with the full quark contributions, which are for most QCD parameters known at least in first order chiral perturbation theory, i.e. to terms linear in the quark masses.
2.6 Probes of varying constants
A possible variation of constants can be tested in various ways. Common tests are laboratory based measurements, for instance of atomic transitions. Also, a multitude of astrophysical and cosmological effects can be studied under the question of constancy, which allow to probe physics over a timescale unreachable with laboratory measurements. In recent years probes of variations in the constants of nature have been performed with increasingly high accuracy. Whilst direct laboratory measurements do not point towards any variation, some astrophysical tests yield slight variations. In part III (Sec. 10) we will list all recent relevant probes of varying constants, followed by detailed studies on how one can combine the different outcomes in unified scenarios. BBN as a probe of varying constants will be examined in detail in part II of this thesis.
2.7 Finetuning of constants and the anthropic
principle
Connected to the question of constancy of fundamental constants is the question of finetuning of these constants. Even though this question can be seen as a rather philosophical one, we will shortly comment on it.
As far as we know today, the value of most of the 27 fundamental constants is extremely finetuned in order to allow life to appear. It has been argued [Tegmark97] that even small deviations (less than or order of 1%) will make the appearance of any life impossible. For example, if the strong force was slightly weaker, multiproton nuclei would not be stable, and if it was slightly stronger, hydrogen could fuse into helium2. Similar arguments can be found for the electromagnetic and weak force and for many other natural constants.
This finetuning problem can be ameliorated, like all problems of this kind, by evoking the anthropic principle^{7}^{7}7The concept of the anthropic principle was systematically introduced by Brandon Carter in a contribution to a symposium honoring Copernicus’ 500th birthday in 1973 [Carter74], even though the idea of the anthropic principle has already been used long before.. In short, this principle states that the Universe which we observe has to be capable to develop intelligent life like us. Otherwise we would not be here and could not ask the question why the Universe has exactly the laws of nature which it has. The final outcome is that the question why we are living in such a highly finetuned, i.e. extremely improbable, universe has simply disappeared, because the actual probability we have to discuss is rather the probability under the condition of our existence, which is no longer vanishingly small.
In recent years scientists have come up with the idea of “multiverses”, stating that universes with many different kinds of physical properties are constantly formed [Linde86]. This is supported by candidate theories of everything (like string theory) which ab initio do not seem to have hard constraints which would exclusively select our physics. Rather, they allow an extremely high number of different physical configurations. In the framework of those theories, universes with many different physical configurations bubble out constantly, and the anthropic principle states that our universe is the one of these many universes which allowed us to appear.
These considerations are not directly connected to the investigations which are subject of this thesis, except the fact that varying constants would lead to an even more finetuned universe: Not only the values of the constants today, but also their whole time evolution would need to be tuned such that we could appear. We will not comment on the point of finetuning in the following, but it has become clear that the problems we are tackling have some deeper connection to philosophy and the question of why we are actually here.
Chapter 3 Cosmology
In this thesis we will consider probes for varying constants from today back to the first minute after the Big Bang. Hence it is essential to understand the evolution of our Universe from the Big Bang until today. This chapter gives a short review of our current picture of the Universe, its history and present status, and the important equations that govern its evolution.
3.1 General relativity and the basics of cosmology
3.1.1 General relativity
General relativity is an extension of the theory of special relativity, which states that gravity is a purely geometric effect, generated by the curvature of spacetime. The relation between curvature and stressenergy is given by the Einstein field equations
(3.1) 
where is the Ricci tensor, the Ricci scalar, the metric tensor and the stressenergy tensor. Equation (3.1) is a complicated differential equation which can in general only be solved if one makes simplifying assumptions and/or uses numeric techniques.
3.1.2 The basics of cosmology
In cosmology one is interested in the evolution of the Universe as a whole. Thus, one usually confines oneself to physics on large scales which allows to make some simplifying assumptions that dramatically reduce the complexity of equation (3.1).
Assumption 1.
The main assumption of cosmology is that the Universe is homogeneous and isotropic on large scales.
Of course, the existence of objects like the earth, sun etc. contradicts this assumption locally. However, if one averages over distances ( Mpc), it turns out that Assumption 1 is observationally welljustified^{1}^{1}1The biggest known structure is the Sloan great wall which is 1.37 billion lightyears long.. Demanding all quantities to be homogeneous and isotropic, one can show [WeinbergGRT] that the metric takes the simple form^{2}^{2}2There are theories claiming that the averaged Einstein tensor which enters in Eq. (3.1) is not equivalent to the Einstein tensor derived from an averaged metric as given in Eq. (3.2). Since the actual outcome of these considerations is still unclear, we will not consider those theories in this thesis. See [Buchert07] for a recent review.
(3.2) 
where k describes the curvature and is the scale parameter, related to the redshift via
(3.3) 
The metric (3.2) is called FriedmannRobertsonWalker metric (FRW metric). The scale parameter fulfills the Friedmann equations
(3.4) 
(3.5) 
where is the Hubble constant and and denote the total energy and pressure density. These two densities are usually split up into the different components which are assumed to be present in today’s Universe, baryonic and dark matter, dark energy (denoted with the symbol ), photons, neutrinos and curvature^{3}^{3}3In the very early Universe, also electrons will make a substantial contribution to the expansion rate. This applies to the epoch of BBN and will be explained in more detail in chapter 6.,
(3.6) 
The pressure is related to the energy via an equation of state,
(3.7) 
where the equationofstate parameter depends on the composition of the components as shown in Table 3.1.
Composition  

nonrelativistic matter  0 
ultrarelativistic matter (radiation)  1/3 
curvature  1/3 
cosmological constant  1 
With the critical density defined as
(3.8) 
all densities are usually given as fractional densities
(3.9) 
Note that equation (3.4) yields
(3.10) 
at all times.
In the course of the evolution of the Universe, the energy densities scale like
(3.11) 
which means that the values of do not stay constant over time since we have different for different kinds of energy densities (Tab. 3.1). As baryons and dark matter follow the same equation of state, one can combine these to the matter energy density
(3.12) 
Given today’s values , one can combine Eqs. (3.4), (3.8), (3.9) and (3.11) and use Tab. 3.1 to derive the time evolution of the Hubble constant,
(3.13) 
where we have neglected the neutrinos which have no substantial contribution to today’s content of the Universe^{4}^{4}4Further note that due to the tiny but nonvanishing mass of the neutrinos, the neutrino equation of state might change during the evolution of the universe. (see Tab. 3.2). Eq. (3.13) shows that at early times () nonrelativistic and relativistic matter become dominant and any cosmological constant component irrelevant, whilst at late times () dominates. The flow of cosmological components in the CDM concordance model (see Sec. 3.2) is depicted in Fig. 3.1, where the time evolution of the fractional components is given by
(3.14) 
As can be seen in Fig. 3.1, today’s Universe () is dominated by dark energy () but did undergo 2 transitions, from radiation dominated to matter dominated and from matter to dark energy dominated:

In the early Universe, the expansion was almost completely due to relativistic particles radiationdominated era.

At , about 70,000 years after the Big Bang, we have matterradiationequality and the Universe becomes matter dominated.

At , about Gyrs ago, the Universe becomes dominated by dark energy (in a CDM model).
3.2 The concordance model: Our current picture of the Universe
3.2.1 Historical development
Presumably, the question of where we come from and where we will go is as old as mankind. As a first modern physical approach to questions of origin, evolution and fate of the Universe, one usually considers Einstein’s paper “Cosmological Considerations in the General Theory of Relativity” from 1917 [Einstein17]. One might say that highprecision observational cosmology started with the Hubble space mission in 1990. It was followed by further astrophysical and cosmological investigations, and basically all of these (mainly observational) tests point towards a coherent picture of our Universe, which is called the “concordance model”.
3.2.2 Our current picture of the Universe
According to the concordance model, the Universe started in a Big Bang^{5}^{5}5Even though it is hoped that physics will once be able to explain the actual origin of this singular event, one is lacking an accepted theory of quantum gravitation which would allow to go beyond the time of the Planck epoch. and has been expanding since then. All observational evidence points towards a socalled CDM cosmology, stating that the Universe is geometrically flat () and consists besides known baryonic matter, leptons and photons of an unknown “dark matter” component which has the property of nonrelativistic, only gravitationally interacting heavy particles, and a “dark energy” component, in the simplest version described by a cosmological constant . See for instance [Bartelmann06] for a compilation of the major observational evidences for the Big Bang and [WMAP5] for recent parameter determinations including all major probes of the concordance model.
As can be seen in Fig. 3.1, we live in a darkenergy dominated universe just now but have undergone both radiation and matterdominated phases. The question why dark energy is taking over “just now” is unclear; this problem is called the “coincidence problem” or “why now problem” (see Sec. 5.1).
Looking back to the very beginning of our Universe, the concordance model suggests the following history of our Universe which is depicted in Fig. 3.2. Here, is the age of the Universe, i.e. the time after the Big Bang, and is the temperature of the Universe, defined by the photon temperature^{6}^{6}6See Sec. 6.4.1 for details on the photon temperature..

seconds ( GeV):
Planck epoch, needs to be described by a quantum theory of gravity. 
seconds ( GeV):
Inflation (exponentially fast expansion of the Universe),
Baryogenesis (production of matterantimatter asymmetry) 
seconds seconds ( GeV):
Quarkhadron transition: protons and neutrons form 
, ( MeV, ):
Nucleosynthesis: light elements (D, He, Li) form 
years ( eV, ):
Beginning of matter dominated era 
years ( eV, ):
Recombination, the cosmic microwave background (CMB) forms 
After that:
Galaxy/star formation.
Experimental tests of highenergy physics currently do not go beyond the TeV region. The processes which happened during the Planck epoch, inflation and also baryogenesis go beyond the physics of the Standard Model and are hence subject to a lot of speculation. Also the quarkhadron transition is hard to describe since it involves QCD at low energies, in the regime where QCD effects cannot be evaluated perturbatively. Hence, the oldest cosmological events which can be reasonably described quantitatively with known physics are primordial nucleosynthesis and CMB formation.
3.3 Cosmological parameter values
In recent years, important and quite extensive missions have been undertaken to deepen our understanding of cosmological relations. In particular, WMAP, SDSS and Supernovae Ia [WMAP5, HinshawWMAP5](see also references therein) have yielded a coherent set of cosmological parameters of a precision which had been inconceivable 10 years ago. However, compared to the Standard Model of particle physics, the concordance model of cosmology is rather new and by far less tested. The set of cosmological parameters for a CDM cosmology is given in Tab. 3.2. It turns out that only of our Universe is made of “known” ordinary baryonic matter, the rest of the Universe is dark matter and dark energy. The energy composition of our Universe today is shown in Fig. 3.3.
Using the evolution equations of Sec. 3.1, the presentday values of cosmological parameters allow to deduce the content of our Universe in the past and also in the future^{7}^{7}7Cosmological models allow to extrapolate cosmology into the future, however models are not tested sufficiently to allow a definite prediction of what the ultimate fate of the Universe will be.. As observations also allow us to look back in time, the picture for the past is nowadays quite clear and observationally probed. The composition history of our Universe in a CDM model is shown in Fig. 3.1.
Quantity  Symbol  Value 

Hubble expansion rate  
normalized Hubble expansion rate  
baryon density  
dark matter density  
matter density  
dark energy density  
radiation density  
neutrino density  
baryon to photon ratio  
CMB temperature  K 
Chapter 4 The Standard Model and beyond
4.1 The Standard Model of particle physics
The Standard Model (SM) of particle physics describes the elementary particles and three of the four fundamental interactions, the strong, weak and electromagnetic interaction. This section will only give a rough overview over some particular aspects of the Standard Model which will be of relevance later. For a more comprehensive introduction, see for instance [HalzenMartin] or any other textbook on modern particle physics.
The list of the fundamental particles of the Standard Model comprises

six leptons (),

six quarks (),

the gauge bosons as mediators of the fundamental interactions,

a Higgs boson.
The matter particles enter as pointlike massless fermions, and the interactions are introduced by demanding a local gauge symmetry. Here, the group is responsible for quantum chromodynamics (QCD), with 8 massless gauge bosons called gluons as mediators of the strong force. At small momenta, the strong coupling constant becomes large (see Sec. 4.2), which is thought to be the explanation for confinement, i.e. the fact that only colorneutral particles are observed in nature.
The theory of electroweak interaction goes back to the seminal work of Glashow, Salam and Weinberg [Weinberg67, Salam69, Glashow70] (Nobel Prize 1979). It describes the electroweak interaction by a gauge symmetry which is broken by the Higgs mechanism into the weak interaction (with massive gauge bosons and ) and the electromagnetic sector with the photon as massless mediator of the electromagnetic force. In particular, the gauge symmetry implies four massless gauge bosons, written as , and . Additionally, one introduces a scalar Higgs field (as a weak doublet under which has altogether 4 real components) and gives it a potential which results in a vacuum that is not symmetric under the gauge symmetry. This leads to a spontaneous symmetry breakdown of the electroweak symmetry, and the Higgs field obtains a nonzero vacuum expectation value . One of the four Higgs components becomes a massive scalar particle, which is the only particle of the SM which has not yet been observed. The and a combination of and obtain masses proportional to and become the massive mediators of the weak force, and , where the three remaining components of the Higgs form the longitudinal modes of the and . The coupling constant of the remaining electromagnetic symmetry group can be obtained from the coupling constants of the original coupling constants , (see e.g. [HalzenMartin]),
(4.1) 
Also, the SM fermions obtain masses via the Higgs mechanism, their mass is a product of Higgs v.e.v. and a Yukawa coupling , for instance for the electron
(4.2) 
4.2 Running of couplings
The influence of fluctuations with different momenta leads to scale dependent coupling constants. See for instance [Wilson71, Wegner72, Wilson73] or any good textbook on quantum field theory for details of this process. Generally, physical systems at slightly different scales are described by the similar laws of physics, with slightly changed parameters. In quantum field theory, this behavior is described by the famous beta function, which describes the behavior of the coupling parameter under slight changes of the energy scale ,
(4.3) 
Using the coupling constant instead, one can also define a function for ,
(4.4) 
The mathematical apparatus to investigate these changes of physical systems under scale transformations is called the renormalization group (RG). In quantum field theory, the renormalization group equation (4.3) can only be computed perturbatively as the exact RG equation would in principle include an infinite order of loop corrections. For our purpose the firstorder (oneloop) RG equations are sufficient, and these are known for all three interactions of the Standard Model. In particular, the beta function for QED (with only photons and electrons present) at first order is given by
(4.5) 
which is solved by
(4.6) 
The fine structure constant is defined in the limit of zero momentum transfer, i.e. for . For QCD, the beta function is
(4.7) 
solved by
(4.8) 
Here is the number of quark flavors present, i.e. the number of quarks with mass . As in the Standard Model, the beta function is negative.
The running of coupling constants is shown in Fig. 4.1.
As the beta function for QCD is negative, the QCD coupling diverges when going to low energies. This effect, which was found by Wilczek, Politzer and Gross (Nobel price 2004), is thought to be the reason for confinement. As opposed to the electroweak theory, QCD thus has an intrinsic energy scale induced by the RG equation, the scale where becomes formally infinite. Choosing such that remains constant (), this happens when the denominator in equation (4.8) becomes zero, i.e.
(4.9) 
which happens at the QCD invariant scale , defined by
(4.10) 
Hence Eq. (4.8) can be rewritten as ()
(4.11) 
Note that when the energy becomes of the order of , perturbation theory breaks down, and a world of quarks and gluons becomes a world of pions, protons and so on. This is revealed in the fact that is of the order of light meson masses [Berger06],
(4.12) 
The beta functions given in Eqs. (4.5) and (4.7) are simplified versions. In the first case, Eq. (4.5) only holds for one particle present, and in the second case one has to note that is not constant at all energies . In the full functions, particles only contribute when energies are above the particle’s threshold energies, which are typically the corresponding particle masses. Hence every particle contributes one threshold term to the renormalization group equation. We will give the full renormalization group equations, including extra terms coming from additional supersymmetric particles (see Sec. 4.4) in Sec. 4.5.
4.3 The necessity of a “theory beyond”
From the point of a theorist, the established Standard Model of particle physics cannot be the end of the story. One can definitely say that at latest at the Planck energy scale quantum gravity effects will become important, demanding a quantized description of gravity. A further hint that the Standard Model of particle physics might not be the end of the story is the running of the coupling constants. They seem to meet at a energy scale of as depicted in Fig. 4.1. Hence, it appears likely that electroweak and strong interaction can be unified in a grand unified theory (GUT). Within such grand unified theories, it is most likely that if any coupling constant of the Standard Model varies, all coupling constants will vary. In the later chapters, we will assume that some kind of GUT is realized, and hence the electroweak and strong coupling constants are related to each other. Further details on grand unified theories are given in Sec. 4.5.
Actually, one can even go further with the idea of unification. String theory, for instance, implements an unification of the interactions of the Standard Model and also gravity. The unification of the four forces including gravity in a string theory picture is shown in Fig. 4.2. In such theories, parameters related to the mass of the SM particles (e.g. the Yukawa couplings) should also derive from some sort of unified greater theory, whereas at the level of the Standard Model and also at the level of simple GUTs, there is no direct relation to the gauge coupling sector^{1}^{1}1Due to renormalization group effects, also the Yukawa couplings get contributions from coupling parameters. However, we will show in Sec. 11.1 that these effects are small and can be neglected when studying variations of parameters.. In this thesis, we will assume constant Yukawa couplings.
4.4 Supersymmetry and the MSSM
Many highenergy theories (e.g. string theory) contain supersymmetry as an essential part of the theory. Supersymmetry establishes a symmetry between bosons and fermions. Every boson gets a fermionic partner and vice versa with the same quantum numbers. For this thesis the motivation and theoretical framework of supersymmetry are not needed, so I will refrain from going into too much detail. Introductions to supersymmetry can be found in many textbooks and reviews, e.g. [NillesSUSY].
Within the Standard Model of particle physics, no supersymmetric partner can be found, hence in a supersymmetric extension of the Standard Model one has to introduce additional supersymmetric partners for every single particle of the SM. These supersymmetric partners are assumed to be heavier than the current experimentally tested mass region ().
The minimal supersymmetric extension of the Standard Model is called the MSSM (minimal supersymmetric standard model). It contains an additional supersymmetric partner for every SM particle. Furthermore, with supersymmetry, a single Higgs doublet would result in a gauge anomaly, so a second Higgs doublet is introduced. Hence the MSSM contains 2 additional (heavy) neutral Higgs scalars and two charged Higgs scalars, supplemented by the appropriate superpartners.
The complete particle spectrum of the MSSM is given in Tab. 4.1. When working with supersymmetric theories in this thesis, we will always assume that the MSSM particle spectrum is realized.
SM particle  El. charge  Spin  SUSY partner  Spin 

quarks u,c,t  2/3  1/2  squarks  0 
quarks d,s,b  1/3  1/2  squarks  0 
charged leptons e,  1  1/2  sleptons  0 
neutrinos  0  1/2  sneutrinos ()  0 
photon  0  1  photino  1/2 
0  1  Zino ()  1/2  
neutral Higgs scalar  0  0  Zino Higgs  1/2 
1  Wino ()  1/2  
charged Higgs scalar  0  Wino Higgs  1/2  
8 gluons  0  1  8 gluinos ()  1/2 
neutral Higgs  0  0  higgsino ()  1/2 
2 MSSM neutral Higgs  0  0  2 neutral higgsinos  1/2 
2 MSSM charged Higgs  0  2 charged higgsinos  1/2 
4.5 Grand unification
In grand unified theories, the gauge group of the Standard Model with coupling constants is unified into a bigger Lie group (e.g. or ) with a single coupling constant at a certain energy scale,
(4.13) 
which is assumed an independent parameter and can also vary with time. Its actual value depends on the specific form of the grand unified theory (e.g. SUSY/nonSUSY).
It can be shown (see e.g. [WeinbergQFT2]) that for any unification of with couplings , and into a simple Lie group with coupling , one obtains the relation
(4.14) 
which holds at . For the electromagnetic interaction, Eqs. (4.1) and (4.14) yield
(4.15) 
which actually only holds at but will be of relevance when studying the running of in a GUT framework.
The value of the unified coupling can roughly be estimated from the RG running as displayed in Fig. 4.1, showing that is of the order in the nonSUSY case and in the SUSY case [Amaldi91]. We take as representative values [Dent03]
(4.16)  
(4.17) 
At lower energies, the GUT symmetry is broken and the relation (4.14) does not hold any longer. The coupling constants of the SM evolve separately according to the renormalization group equations (see section 4.2). Generalizing the running of couplings as given in equations (4.6) and (4.8) to the full SM/MSSM particle spectrum, we obtain for QCD
(4.18) 
where the first sum goes over all particles with threshold mass and the second sum goes over all particles with . For the and the values from Tab. 4.2 have to be applied.
Type of particle  

quarks  2/3 
gluons  11/8 
squarks  1/3 
gluinos  1/4 
The corresponding expression for the finestructure constant is
(4.19) 
where the factor derives from Eq. (4.15), denotes the electric charge and for the values given in Tab. 4.3 are applied. An analogous equation also holds for the weak coupling, where the electric charge is replaced by the weak isospin^{2}^{2}2For the SU(2) weak interaction, the weak isospin effectively acts like a multiplicative charge factor, hence it can be treated analogously to the electric charge.. When dealing with weak interactions in this thesis, we will only be working with terms that contain the weak coupling in terms of the Fermi constant^{3}^{3}3See for instance the weak decay of the neutron, Sec. 8.1.3.,
(4.20) 
As with the Higgs v.e.v., the weak coupling drops out and the Fermi constant can be expressed only in terms of the Higgs v.e.v.,
(4.21) 
Type of particle  

chiral (or Majorana) fermion  2/3 
complex scalar  1/3 
vector boson  11/3 
4.6 Variations in a GUT framework
The GUT relations which were introduced in the preceding section show that within a GUT framework, the coupling constants are usually related to further fundamental parameters, in particular the GUT coupling and threshold masses. In this section we will derive the equations which relate variations in the GUT coupling constant and particle masses to variations in the SM coupling constants.
4.6.1 Variation of the electromagnetic coupling
For the MSSM particle spectrum, we obtain for the finestructure constant from Eq. (4.19)
(4.22) 
where it has been used that

The light quarks decouple at .

The quarks enter in 3 different colors

The charged leptons enter as both left and righthanded particles (2 chiral fermions)

The massive gauge bosons have to be supplemented by a charged complex Higgs scalar (longitudinal DOFs)

is the mass of the additional charged Higgs scalars which have to be introduced in MSSM.
Taking the linear variation of Eq. (4.22), we obtain for the variation of the fine structure constant, including the MSSM particles,
(4.23) 
Not knowing the actual mass or mass generating mechanism for the superpartners, we assume that the mechanism is the same for all superpartners and define as the average superpartner mass. To obtain the corresponding relation in nonsupersymmetric models, one simply has to leave out the terms with and .^{4}^{4}4Note that the RG equations (4.22), (4.23) and also (4.24) and (4.26) only hold under the condition that all threshold masses are smaller than . Hence, the nonsupersymmetric case is not obtained in the limit . When later dealing with variations in supersymmetric models, we will further assume , so the last two terms can be combined into one.
4.6.2 Variation of the QCD scale
As diverges at , we are interested in the value of in the regime (in this regime, ). For the MSSM particle spectrum, we obtain
(4.24) 
Inserting Eq. (4.24) into Eq. (4.10) (, ) yields
(4.25) 
and the linear variation gives
(4.26) 
When later dealing with variations in supersymmetric models, we will further assume , so the last two terms can be combined into one.
4.6.3 Conversion of units
As has been explained in Sec. 2.5, we will work with two different systems of units. During the discussion of BBN processes, we choose units with as the BBN energy scale is of roughly the same order of magnitude. When applying grand unified theories, is the more appropriate energy scale to keep constant. However, we usually neglect the reference scale and write instead for , for instance. The conversion to a different system is then performed by keeping track of all reference scales,
(4.27) 
Obviously, the term naturally enters when converting from the to the system of units, its explicit dependence on the unified coupling and particle masses is given in Eq. (4.26). Keep in mind, however, that the reference scale has to enter in the correct power, for instance the gravitational constant has units , hence it enters as .
Chapter 5 Models of quintessence
5.1 Problems of the cosmological constant
Recent observations show that roughly 75% of the energy content of our Universe is made from dark energy (see Sec. 3.2). However, the nature of dark energy is still far from being clear. The assumption that the cosmological constant derives from a vacuum energy density sufferes from a severe finetuning problem. In particular, the oberseved dark energy density evaluates to [Copeland06]
(5.1) 
while the vacuum energy density of particle physics which is evaluated by summing up the zeropoint energies of the present quantum fields gives [Copeland06]
(5.2) 
Here we have chosen as a natural cutoff scale where we assume that the known quantum field theory is no longer applicable. Obviously, there is a discrepancy of the order of . Assuming that the dark energy comes from a particle physics origin, one would have to introduce counter terms which have to be extremely finetuned. Hence this problem is called the “finetuning” problem.
A further problem which is related with dark energy can be seen in Fig. 3.1. In a CDM model, the dark energy is only recently becoming important, and the time when the universe switched from a matterdominated to a dark energy dominated epoch is only 4.3 billion years ago. There is no natural reason why these two presumably completely independent constituents of our Universe are of about the same order of magnitude and / or why we live in a period of time where this is the case. This problem is called the “coincidence problem” or “why now problem”, and typically cosmological models with a cosmological constant (like the CDM model) fail to address this issue.
5.2 Basics of quintessence
Models of quintessence [Wetterich88.1, RatraPeebles88] can offer an explanation to the issues mentioned in the previous section. A good review on dark energy models can be found in [Copeland06].
In quintessence theories, one introduces a scalar field (called the cosmon) which is coupled to gravity and, most times, also to matter and gauge fields. A typical Lagrangian for a quintessence theory including couplings to matter and the electromagnetic gauge field looks like [Wetterich02.1, Copeland06]
(5.3) 
with gauge field coupling
(5.4) 
and matter term
(5.5) 
Here denotes the reduced Planck mass,
(5.6) 
If there are only slight changes in the cosmon field, the dependence of the mass, , can be linearized, i.e.
(5.7) 
with some coupling . However, there are also models with significant changes in the masses, for instance in the models of growing neutrinos in Sec. 5.4, where we apply a more advanced nonlinear expression.
In order to derive one of the main properties of quintessence, its capability of producing accelerated expansion, we can neglect couplings to matter and gauge fields and work instead with the action
(5.8) 
In the background of a flat FRW cosmology (Sec. 3.1.2), and assuming that is homogeneous, i.e. it only depends on time, a variation of the action (5.8) with respect to yields the equation of motion
(5.9) 
The corresponding energy momentum tensor
(5.10) 
yields the energy and pressure densities
(5.11) 
(5.12) 
Then Eqs. (3.4) and (3.5) yield the relations
(5.13)  
(5.14) 
showing that we get an accelerating universe when . Introducing the kinetic energy , we define the equation of state parameter of quintessence,
(5.15) 
In the next section we introduce two specific models of quintessence, crossover quintessence which has been introduced 20 years ago [Wetterich88.1] and a very recent model, where quintessence is strongly coupled to neutrinos.
As the cosmon also couples to other fields and matter, one question one might ask is whether the cosmon evolution decouples in a local cluster with high “cosmon charge density” from the cosmological evolution. It has been shown [Wetterich02.1, Mota03, Shaw05] that for a very light field weakly coupled to matter the local perturbations are generally small relative to the cosmological evolution. In other words, the evolution of the scalar field in a cluster of galaxies or on Earth does not decouple from the cosmological evolution (in distinction to the gravitational field), such that its cosmological time evolution is reflected in a universal variation of couplings, both on Earth and in the distant Universe.
5.3 Crossover quintessence models
Our first class of models is “crossover quintessence” [Hebecker00, Doran07, Wetterich02.2]. Here the scalar field follows tracking solutions [Wetterich88.1, RatraPeebles88] at large redshift. In this early epoch the equation of state is equal to that of the dominant energy component (matter or radiation). One particular difference to cosmological constant models is that this type of quintessence models yields a non vanishing amount of early dark energy, ^{1}^{1}1The effects of early dark energy on the measurements of baryon acoustic peaks have been studied by the author in [DST06]. However, these considerations are not subject of this thesis.. Typically, such models have an exponential potential, for instance
(5.16) 
In this specific case, is related to the early dark energy fraction [Wetterich88.1, Amendola08],
(5.17) 
with for the matter (radiation) epoch. Latetime acceleration can be achieved, for instance, by slight modifications in the cosmon potential or, equivalently, in the kinetic term [Hebecker00].
At some intermediate redshift before the onset of acceleration, the time evolution of the cosmon slows down. In consequence, there is a crossover to a negative equation of state and the fraction of energy density due to the scalar begins to grow. In recent epochs the field has an effective equation of state . The aim of this thesis is not building and solving models of this type in detail, but rather estimating general properties of the scalar evolution. Hence we will not start with appropriate potentials for crossover quintessence and evolve the cosmon over time, but rather simulate the behavior of the field by defining the quintessence equation of state by hand. We set the dark energy equation of state to constant at late times with value . Above some given redshift the equation of state crosses over to the scaling condition in the matter dominated era; then for , before matterradiation equality, we again have scaling through , where can be obtained via
(5.18) 
Then the general relation ()
(5.19) 
may be used to find the matter, radiation and dark energy densities over cosmological time. Combining Eq. (5.15) with Eq. (5.11), we can estimate the scalar kinetic energy via
(5.20) 
and thus integrate from the present back to any previous redshift. The initial conditions are set by specifying the present densities of matter, radiation and dark energy and the model parameters and . For illustration, we set and . The resulting equation of state is displayed in Fig. 5.1, the corresponding evolution of energy components in Fig. 5.2 and the dimensionless cosmon field in Fig. 5.3.
As can be seen from Fig. 5.3, in this type of models the scalar field has a monotonic evolution. Assuming a constant coupling to the fundamental varying parameter, usually , the variation is given by
(5.21) 
Hence this ansatz implies a monotonic evolution of variations.
5.4 Growing neutrino mass models
Growing neutrino models [Amendola08, Wetterich08] explain the value of today’s dark energy density by the “principle of cosmological selection”. The present fraction of dark energy, , is set by a dynamical mechanism. The essential ingredient of this class of models is a neutrino mass that depends on the cosmon field and grows in the course of the cosmological evolution. As soon as the neutrinos become nonrelativistic, their coupling to the cosmon triggers an effective stop (or substantial slowing) of the evolution of the cosmon. Before this event, the quintessence field follows the tracking behavior described in the preceding section. In the models which we will study, the cosmon is assumed to have the potential from Eq. (5.16),
The present dark energy density, , can be expressed in terms of the average present neutrino mass, , and a dimensionless parameter of order unity [Amendola08],
(5.22) 
We follow again our simple proportionality assumption, namely that the cosmon coupling to a typical fundamental parameter is given by Eq. (5.21),
with a proportional variation for other couplings according to the unification scenario that we will study. This is the only contribution to the variation of the unified coupling and . However, a new ingredient is an additional variation of the Higgs v.e.v. with respect to , which only becomes relevant at late time [Wetterich08]. It is due to the effect of a changing weak triplet operator on the v.e.v. of the Higgs doublet. If the dominant contribution to the neutrino mass arises from the “cascade mechanism” (or “induced triplet mechanism”) via the expectation value of this triplet, this changing triplet value is directly related to the growing neutrino mass [Wetterich08]. To understand this mechanism, we start with the most general mass matrix for the light neutrinos,
(5.23) 
The first term is responsible for the seesaw mechanism [Minkowski77] with the mass matrix for heavy “right handed” neutrinos and a Dirac mass term . The second term accounts for the “induced triplet mechanism” [Magg80]
(5.24) 
where a heavy triplet field with mass enters the equation (see [Wetterich08] for details). It is assumed that the mass of the triplet depends on the cosmon field, .
The dependence of the Higgs v.e.v. is introduced by assuming a general effective potential . Solving the field equations for the Higgs doublet field and the triplet field , , , the cosmon potential is then obtained as
(5.25) 
In [Wetterich08] the simple potential
(5.26) 
is assumed, with and some coupling parameters. Solving the field equations for the Higgs doublet, it is found [Wetterich08]
(5.27) 
where has to be chosen such that the measured Higgs v.e.v. is obtained today.
In the following we consider two models, with slightly different functional dependence of the Higgs v.e.v. and neutrino mass on the scalar field.
5.4.1 Stopping growing neutrino model
In the first model, the cosmon asymptotically approaches a constant value (“stopping growing neutrino model”) [Wetterich08] and the neutrino mass is given by
(5.28) 
With a triplet mass dependence
(5.29) 
the additional Higgs variation is given according to
(5.30) 
where
(5.31) 
Here, is the asymptotic value (choosing the parameter in the exponential potential [Wetterich08]). For illustration we take the set of parameters given in [Wetterich08], , and is set by demanding the Higgs v.e.v. being consistent with measurements today, GeV. We set , however in general we only require . The resulting variations are shown in Fig. 5.4 and Fig. 5.5.
The stopping growing neutrino model has an oscillation in that grows both in frequency and amplitude at late times as approaches its asymptotic value. Such oscillations must not be too strong as measurements between and today would measure a high rate of change. The oscillation may be made arbitrarily small by choosing small . However, the linear variation (5.21) is independent of .
5.4.2 Scaling growing neutrino model
The second growing neutrino model [Amendola08] does not lead to an asymptotically constant . Now the coupling of the neutrino to the cosmon is given by a constant , according to
(5.32) 
This “scaling growing neutrino model” leads in the future to a scaling solution with a constant ratio between the neutrino and cosmon contributions to the energy density.
With the choice of parameters , and eV [Amendola08], and given the triplet mechanism of [Wetterich08], the Higgs v.e.v. varies as Eq. (5.30), where now is given by