Imperial College London
Department of Physics

Varying Constants and the Brans-Dicke Theory: a new Landscape in Cosmological
Energy Conservation

_Author:_

Paolo Massimo Bassani

_Supervisor:_
Prof João Magueijo

A thesis submitted in partial fulfilment of the requirements for the degree of
Master of Science in Quantum Fields and Fundamental Forces
in Theoretical Physics at Imperial College London

August 30, 2024

###### Abstract

We develop the Brans-Dicke theory of gravity in the context of varying
constants of Nature. Using the unimodular formalism of General Relativity, we
create a platform to provide physical relational times giving the evolution of
physical constants. We therefore review the ideas and experiments behind
varying constants, mostly focusing on the speed of light and the gravitational
constant. Then, we apply this idea to the energy conservation in cosmology,
illustrating the arising patterns. Motivated by a varying gravitational
constant resulting from Mach’s principle, we develop the unimodular formalism
of varying constants in the Brans-Dicke theory. Doing so, we obtain several
original results, some of which can be compared with phenomenological
observation. Finally, we suggest how a varying Brans-Dicke parameter could be
linked to the Cosmological Constant problem.

### Acknowledgments

Firstly, I would like to thank my supervisor, Professor João Magueijo. His
support, guidance, insights and creativity have been invaluable towards the
success of this dissertation. I am greatly beholden for the opportunity of
having interacted and exchanged ideas with a person of his human and
intellectual stature. His example will be an important part of my future life.

Secondly, I would like to express my endearment to all the friends I have
shared this experience with: Giacomo, Farbod, Greg, Frank, Coen, David,
Abdulaziz and Lorenzo. The long discussions on crazy ideas, the support during
this Master and the happy moments together have enriched my life with the gift
of friendship, which I will always carry with me. Also, I would like to thank
my friend Marcel for the great help provided. Honorable mention goes to Ray,
for the important discussions and insights. A special thought goes to Altay,
whose unstoppable enthusiastic optimism and faith in physics have motivated me
to undertake this Master until its completion.

I am grateful to the unique Flower of the Desert that accompanies me through
life: your delicacy, kindness and loving care have been the strength and
inspiration throughout this time. Above all, your hopefulness and support
haven been the emotional pillar of this accomplishment.

Infine, la mia più sconfinata riconoscenza e infinita gratitudine vanno alle
persone più importanti: la mia famiglia. Grazie Druso, per essere sempre stato
giocoso, allegro e sempre disponibile a passare del tempo con me. Grazie
Costanza, per avermi sempre sopportato, ascoltandomi con pazienza nei momenti
più importanti, mostrandomi il giusto valore delle cose. Sopratutto, grazie
per essere sempre stata mia sorella: spero, da grande, di poter essere tuo
fratello.
Grazie Papà, per le importanti riflessioni e lo sprone necessario per
completare questo Master. Il tuo incoraggiamento e il tuo animo originale e
indipendente mi hanno spinto a continuare il mio percorso con spirito critico
e determinazione.
Grazie Mamma, per il tuo sconfinato affetto e la tua celestiale amorevolezza
che mi hanno sempre accopagnato, senza interruzioni, dalla nascita fino ad
ora. Grazie per essere sempre stata al mio fianco, parlandomi, ascoltandomi
nei momenti più difficili. La tua frizzante curiosità e il tuo amore per la
Natura hanno sempre affascinato il mio intelletto, ispirandolo a ricercare la
conoscenza con gioia e purezza.
A tutti voi, per tutto quello che mi avete donato, ma sopratutto, per essere
stati la mia famiglia, esprimo il mio più sincero affetto: vi voglio bene.

> “Humana ante oculos foede cum vita iaceret in terris oppressa gravi sub
> religione quae caput a caeli regionibus ostendebat horribili super aspectu
> mortalibus instans, primum Graius homo mortalis tollere contra est oculos
> ausus primusque obsistere contra, quem neque fama deum nec fulmina nec
> minitanti murmure compressit caelum, sed eo magis acrem irritat animi
> virtutem, effringere ut arta naturae primus portarum claustra cupiret. Ergo
> vivida vis animi pervicit, et extra processit longe flammantia moenia mundi
> atque omne immensum peragravit mente animoque, unde refert nobis victor quid
> possit oriri, quid nequeat, finita potestas denique cuique quanam sit
> ratione atque alte terminus haerens. Quare religio pedibus subiecta vicissim
> obteritur, nos exaequat victoria caelo.”

De Rerum Natura \- Titus Lucretius Carus

###### Contents

  1. 1 Introduction
    1. 1.1 Unimodular Gravity
      1. 1.1.1 Einstein’s pure unimodular gravity
      2. 1.1.2 Henneaux-Teitelboim theory
    2. 1.2 The Constants of Nature
      1. 1.2.1 The Essence of a Constant
      2. 1.2.2 Constants that Vary
      3. 1.2.3 The Speed of Light
    3. 1.3 Brans-Dicke: a scalar-tensor theory of gravity
      1. 1.3.1 Mach’s principle
      2. 1.3.2 Varying $G$ and Mach’s principle
      3. 1.3.3 Brans-Dicke Theory
    4. 1.4 Overview and Aims
  2. 2 Varying Constants: Developments
    1. 2.1 Open problems in Cosmology
      1. 2.1.1 Horizon problem
      2. 2.1.2 Flatness problem
      3. 2.1.3 Cosmological constant problem
      4. 2.1.4 The solution: Varying Speed of Light
    2. 2.2 Minisuperspace Reduction
      1. 2.2.1 Einstein-Cartan action in MSS
      2. 2.2.2 Einstein-Hilbert action in MSS
      3. 2.2.3 From Einstein-Cartan to Einstein-Hilbert in MSS
    3. 2.3 Energy Conservation
      1. 2.3.1 Full action and constants
      2. 2.3.2 The Hamiltonian and the Dynamics
      3. 2.3.3 Varying $\alpha_{2}$ and $\alpha_{3}$ theory
  3. 3 Brans-Dicke and Evolving Laws
    1. 3.1 Brans-Dicke in MSS
      1. 3.1.1 Reducing the Brans-Dicke action to MSS
    2. 3.2 Non-Dynamical Brans-Dicke
      1. 3.2.1 The Equations of Motion for the $\phi_{0}$ scenario
      2. 3.2.2 The Equations of Motion for the full $\phi$ scenario
      3. 3.2.3 Fixed $\phi_{0}$ with $c_{g}^{2}=c_{g}^{2}(T_{\phi})$
      4. 3.2.4 Fixed $\phi_{0}$ with $c_{m}^{2}=c_{m}^{2}(T_{\phi},T_{N})$
    3. 3.3 Dynamical Brans-Dicke
      1. 3.3.1 The Equations of Motion for the $\phi_{0}$ scenario
      2. 3.3.2 The Equations of Motion for the full $\phi$ scenario
      3. 3.3.3 Full $\phi$ with $\omega=\omega(T_{\Lambda})$
      4. 3.3.4 Full $\phi$ with $\omega=\omega(T_{N})$ and $\alpha_{3}=\alpha_{3}(T_{\omega})$
      5. 3.3.5 Fixed $\phi_{0}$ with $\omega=\omega(T_{\phi})$
      6. 3.3.6 Full $\phi$ with $\omega=\omega(T_{\phi})$
      7. 3.3.7 Full $\phi$ for $c_{g}=c_{g}(T_{\omega})$
      8. 3.3.8 Full $\phi$ with $\omega=\omega(T_{c_{g}})$
      9. 3.3.9 Full $\phi$ with $c_{m}^{2}=c_{m}^{2}(T_{\omega})$
      10. 3.3.10 Full $\phi$ with $c_{m}^{2}=c_{m}^{2}(T_{\phi},T_{\omega})$
      11. 3.3.11 Fixed $\phi_{0}$ with $c_{m}^{2}=c_{m}^{2}(T_{\phi},T_{N})$
      12. 3.3.12 Full $\phi$ with $c_{m}^{2}=c_{m}^{2}(T_{\phi},T_{N})$
  4. 4 Conclusion and Discussion

##  Chapter 1 Introduction

In this chapter, we present the three main background theories – Unimodular
Gravity, the Constants of Nature and the Brans-Dicke theories – to propel the
investigation in the next chapters. The first one provides the mathematical
framework to rigorously implement the idea of varying constants in General
Relativity and in Brans-Dicke theory. The second theory explores the
philosophical and physical depths behind constants and motivates why they
could, paradoxically after all, be variable. Furthermore, it provides an
insight into the broad landscape of roles the speed of light plays in physical
theories. Finally, the third section introduces the Brans-Dicke theory of
gravity, from its connection with the idea of varying $G$ to its fields
equations, employing Mach’s principle. The Brans-Dicke theory will be the
basis for the original research presented in Chapter 3, so it is particularly
important towards the results obtained and their interpretation. To conclude,
section (1.4) summarises how the ideas in these three sections empower Varying
Constant Theories (VCT). We also explain the aims of this research which will
be reviewed and assessed in Chapter 4 in light of the obtained results.

###  1.1 Unimodular Gravity

In this section, we will explore the implementation of unimodular gravity in
General Relativity. In doing so, we will distinguish between two main
formalisms: Einstein’s pure unimodular gravity and the Henneaux-Teitelboim
theory. The former is called so because it was originally suggested by
Einstein as a way to simplify calculation in General Relativity by requiring
that the determinant of the metric $\sqrt{-g}=1$. In 1989, the latter was
proposed to generalise Einstein’s idea, allowing for a less restrictive
condition, i.e., $\sqrt{-g}={\partial_{\mu}\mathcal{T}^{\mu}}$, where
$\mathcal{T}^{\mu}$ is a generic vector density. As we will see, this
extension allows to associate different constants with their respective
conjugate relational times, a central property of theory upon which most of
the results in this work are based.

####  1.1.1 Einstein’s pure unimodular gravity

Since its inception, Einstein’s Theory of General Relativity (EFE) has been
the most experimentally successful attempt at describing the phenomenon known
as gravitation [1]. Einstein’s theory victoriously survived experimental tests
ranging from the perihelion procession of Mercury to light deflection due to
the Sun’s gravitational field [1]. More recently, predictions from General
Relativity have been tested in more exotic phenomena such as gravitational
lensing and gravitational waves [2], once again confirming this theory.
However, one observational evidence controverts the Einstein Field Equations
as they were presented originally in the famous 1915 paper titled _The Field
Equations of Gravitation_ [3]. In fact, the first field equations for a vacuum
solution,

$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=0$ (1.1)

predicted a dynamical universe with space evolving in time. Rejecting this
prediction, in 1917 Einstein introduced a parameter, $\Lambda$, called the
cosmological constant. It allowed the community to believe that the universe
could remain static and eternal, completely deprived of any evolving dynamic
(the so-called Einstein static universe solution). It was not long after his
addition of $\Lambda$ that Hubble discovered the relationship between
velocities and redshift of distant galaxies and concluded the opposite: the
universe was expanding. This experimental evidence combined with theoretical
works by Friedmann (the universe is expanding regardless of the value of
$\Lambda$) led Einstein to call $\Lambda$ his _biggest blunder_ , thus
changing his views on the nature of the Universe.
Since then, the necessity for the cosmological constant has been a matter of
debate until 1998, when it was proven that the expansion of the universe is
accelerating, requiring a positive cosmological constant [4]. Interestingly,
Einstein’s _biggest blunder_ turned out to be a necessary addition to the
field equations describing a dynamical universe subject to an expansion
possibly driven by $\Lambda$.

The appearance of the cosmological constant in the field equations has been
since then a matter of active debate from a theoretical point of view. In
fact, it is not clear, _a priori_ , why $\Lambda$ should be constant and how
it actually enters the field equations. Einstein’s unimodular formulation of
General Relativity has been an attempt to better justify its appearance.
Frivolously, before its inception, $\Lambda$ would simply be inserted in the
Einstein-Hilbert action with matter as follows [5]:

$S_{EH}=\frac{c^{4}}{16\pi
G}\int{d^{4}x\>\sqrt{-g}\>R}\>+S_{m}\rightarrow\>S_{EH}=\frac{c^{4}}{16\pi
G}\int{d^{4}x\>\sqrt{-g}\>(R-2\Lambda)}+S_{m}$ (1.2)

where $S_{m}$ is the matter action. It is clear that using the full action
above, when deriving the EFEs, we arrive at a set of field equations where
$\Lambda$, from a purely mathematical point of view, does not need to be
constant. Varying action (1.2), we obtain the field equations:

$\frac{\delta S_{EH}}{\delta g^{\mu\nu}}=0\Leftrightarrow
R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\Lambda g_{\mu\nu}=\frac{8\pi
G}{c^{4}}T_{\mu\nu}$ (1.3)

where we again stress that $\Lambda$ is taken to be constant purely for the
observational reasons [4], but it does not need to be. The idea of unimodular
gravity, first formulated by Einstein himself as a mean to unify gravity and
matter [6], starts from imposing the constraint $\sqrt{-g}=1$ to action (1.2),
a condition that leads to preserving the determinant of the metric when
varying the action,

$\frac{\delta\sqrt{-g}}{\delta g^{\mu\nu}}=0$ (1.4)

Furthermore, consider the infinitesimal variation of the metric

$\delta g_{\mu\nu}(x)=\nabla_{\mu}k_{\nu}+\nabla_{\nu}k_{\mu}$ (1.5)

where $k^{\mu}$ is the transformation’s gauge vector and $x$ are the general
coordinates, under the infinitesimal coordinate transformation

$x^{\mu}\mapsto x^{\prime\mu}=x^{\mu}+k^{\mu}$ (1.6)

Using the chain rule on (1.4), we can get a full expression for the variation
of $\sqrt{-g}$ the same way it is done in [7]:

$\delta\sqrt{-g}=\frac{1}{2}\sqrt{-g}g^{\mu\nu}\delta g_{\mu\nu}=0\implies
g^{\mu\nu}\delta g_{\mu\nu}=0$ (1.7)

where $g=\det g_{\mu\nu}$. Finally, we can plug (1.5) into the last expression
of (1.7), contract the indices with $g^{\mu\nu}$, and arrive at a condition

$\nabla_{\mu}k^{\mu}=0$ (1.8)

for the gauge vector $k_{\mu}$.
The transformations on $k^{\mu}$ are called transverse diffeomorphisms. They
can be interpreted as symmetric properties of the metric and are naturally
arising from the unimodular constraint imposed on $\sqrt{-g}$. It is now
possible to introduce the unimodular constraint by modifying the Einstein-
Hilbert action with the addition of a Lagrange multiplayer $\lambda$

$S_{EH}=\frac{c^{4}}{16\pi
G}\int{d^{4}x\>\sqrt{-g}R}-\int{d^{4}x\>\lambda(x)(\sqrt{-g}-1)}+S_{m}$ (1.9)

Varying this action with respect to the Lagrange multiplier scalar
$\lambda(x)$ gives the unimodular constraint $\sqrt{-g}=1$. On the other hand,
performing the usual variation with respect to $g^{\mu\nu}$ gives the field
equations

$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\frac{8\pi
G}{c^{4}}\lambda(x)g_{\mu\nu}=\frac{8\pi G}{c^{4}}T_{\mu\nu}$ (1.10)

Finally, by expressing (1.10) for $\lambda$ and then taking its divergence we
get

$\nabla_{\mu}\lambda=\nabla^{\nu}T_{\mu\nu}-\frac{c^{4}}{8\pi
G}\biggl{[}\nabla^{\nu}R_{\mu\nu}-\frac{1}{2}\nabla_{\mu}R\biggl{]}=0$ (1.11)

which is equal to zero because of the energy-momentum conservation, i.e.,
$\nabla^{\nu}T_{\mu\nu}=0$ and because of the contracted Bianchi identities
$\nabla^{\nu}R_{\mu\nu}-\frac{1}{2}\nabla_{\mu}R=0$. Given that and that
$\lambda$ is a scalar, we can write

$\partial_{\mu}\lambda=0$ (1.12)

which tells us that $\lambda$ is indeed an integration constant as we wanted.
Therefore, comparing (1.10) with (1.3), we see that the dynamics of both
theories are the same with the cosmological constant of

$\Lambda=\frac{8\pi G}{c^{4}}\lambda$ (1.13)

This result confirms once again that the cosmological constant can be derived
in the field equations by imposing the unimodular condition on the determinant
of the metric. However, as we will see in the next section, it is possible to
go beyond this result, introducing a formalism that will turn out essential
for the implementation of varying constants in cosmology [8].

####  1.1.2 Henneaux-Teitelboim theory

Generalising Einstein’s idea, Henneaux and Teitelboim formulated a fully
diffeomorphism-invariant extension of the unimodular condition by introducing
the divergence of a vector field density as follows [9]:

$\sqrt{-g}={\partial_{\mu}\mathcal{T}^{\mu}_{\Lambda}}$ (1.14)

where the range of the index is $\mu=0...3$.
We will focus now on condition (1.14) as this will become central to our
developments of unimodular time and other times. Therefore, following [9], it
is possible to write action (1.2) as the sum of two actions:

$S=S_{0}+S_{U}$ (1.15)

where $S_{0}$ is, in our case, the EH action, but it could be any base action,
whereas $S_{U}$ is the unimodular term defined as:

$S_{U}=\int{d^{4}x\>\Lambda\>\partial_{\mu}\mathcal{T}^{\mu}_{\Lambda}}$
(1.16)

which, combined with $S_{0}$, and integrated by parts, leads to the final
result

$S_{0}\rightarrow
S=S_{0}+\int{d^{4}x\>\Lambda\>\partial_{\mu}\mathcal{T}^{\mu}_{\Lambda}}=S_{0}-\int{d^{4}x\>(\partial_{\mu}\Lambda)\mathcal{T}^{\mu}_{\Lambda}}$
(1.17)

where $S_{0}$ is a base action which, for the moment, is the $EH$ action as
defined in (1.2), but will be modified to Brans-Dicke in the following
sections. It is also important to note that in (1.17) $\Lambda$ is not
anymore, _a priori_ , the cosmological constant, but just a scalar, which
happens to appear in the EFEs as a result of being an integration constant.
This allows us to transform the cosmological constant from a fixed parameter
to an integration constant, which will be such only in the equations of
motion. In fact, calculating the equations of motion from action (1.17) we
arrive at

$\frac{\delta S}{\delta\Lambda}=\frac{\delta
S_{0}}{\delta\Lambda}+\frac{\delta
S_{U}}{\delta\Lambda}=0\Leftrightarrow\sqrt{-g}=\partial_{\mu}\mathcal{T}^{\mu}_{\Lambda}$
(1.18)
$\frac{\delta S}{\delta\mathcal{T}^{\mu}_{\Lambda}}=\frac{\delta
S_{U}}{\delta\mathcal{T}^{\mu}_{\Lambda}}=0\Leftrightarrow\partial_{\mu}\Lambda\approx
0$ (1.19)

Firstly, equation (1.18) is the Henneaux-Teitelboim condition introduced in
(1.14). This proves that the condition holds on the level of the equations of
motion. On the other hand, (1.19) proves that now $\Lambda$ is a constant only
on the shell, i.e., when it obeys the equation of motion.

For now, we wish to understand the relation between $\Lambda$ and the vector
density $\mathcal{T}^{\mu}_{\Lambda}$, as well as what the interpretation of
this density could be.
An important insight into this question is achieved by considering the $3+1$
foliation of space-time induced by the ADM formalism in $S_{U}$. Upon the
$3+1$ splitting of the time and space components of
$\mathcal{T}^{\mu}_{\Lambda}$ as $\mathcal{T}^{\mu}_{\Lambda}$ =
$\mathcal{T}^{0}+\mathcal{T}^{i}$, where $i=1,2,3$, the unimodular action
(1.16) becomes

$S_{U}=\int{dt\>d^{3}x\>(\Lambda\>\dot{\mathcal{T}^{0}_{\Lambda}}+\Lambda\>\partial_{i}\mathcal{T}^{i}_{\Lambda})}$
(1.20)

This shows that $\mathcal{T}^{0}_{\Lambda}$ is the canonical conjugate of
$\Lambda$ as they form a canonical pair of variables in the Hamiltonian
formalism. Furthermore, following [9], the only propagating mode of
$\mathcal{T}^{\mu}_{\Lambda}$ is the zero-mode of its time component as
defined below:

$T_{\Lambda}=\int{d^{3}x\>\mathcal{T}^{0}_{\Lambda}}$ (1.21)

This zero-mode $T_{\Lambda}$ defines a time [10] that is naturally canonical
to the cosmological constant. Importantly, the notion of a canonical time to a
constant is intimately related with relational quantum mechanics [11] and,
more generally, the problem of time in quantum mechanics. This allows to
create relational times which depend on their canonical pair (the constant)
instead of an absolute parameter time $t$. As we will see, this formalism
provides a powerful implementation of Leibniz’s idea of relational events,
and, therefore, times [12].
Having established this framework where the cosmological constant forms a
conjugate pair with the identified unimodular time $T_{\Lambda}$, we can
generalise this to any constant of nature appearing in our base action
$S_{0}$. In fact, instead of having only one constant in the unimodular term,
we could have a full vector of constants called $\bm{\alpha}$ conjugate to a
set of different corresponding relational times $\bm{T_{\alpha}}$. Since any
$\bm{\alpha}$ is conjugated with its $\bm{T_{\alpha}}$, we start to see here
how some physical constants will become relational clocks, determining the
evolution of other constants. Using this generalisation in action (1.16), we
arrive at an additional unimodular-like term in the action,

$S_{\bm{\alpha}}=\int\>{d^{4}x\>\bm{\alpha}\>\partial_{\mu}{\mathcal{T}^{\mu}_{\alpha}}}$
(1.22)

which, to conclude, will give us the full action

$S=S_{0}-\int{d^{4}x\>\bm{\alpha}\>\partial_{\mu}\mathcal{T}^{\mu}_{\alpha}}$
(1.23)

From this point on, we will be able to use the formalism introduced in action
(1.23) to derive a set of time evolution equations for different constants.
Following this procedure, we will manage to demote any constant appearing in
the base action to mere constants of motion, thus obtaining the set of
relational times canonical to them. The constant and the associated time will
form a canonical conjugate pair as prescribed by the Hamiltonian formalism, as
we will see in the next chapter. These times will then be used to provide
evolution parameters for other constants, as we will see. To conclude, we will
mostly focus our investigation on using the Brans-Dicke action as a base
theory for reasons that will become clearer later. This will be the core of
Chapter 3 and its results.

###  1.2 The Constants of Nature

> “ Tempora Mutantur et Nos Mutamur in Illis ”

What makes the Reality we perceive unique? What is the property, the
characteristic so deeply connected to our Universe to make it one of a kind,
and therefore differentiable from anything else? Something of this sort should
certainly be the pillar of the world, a unique feature, so exclusive to the
Universe to actually make it The Universe. As such, it most definitely would
need to be a fixed property, eternal and equal everywhere and at the heart of
anything else populating the Universe: it should be constant, a constant.

In fact, as the word suggests, constants are values that not only remain the
same everywhere and at any time (in the past and in the future), but also, and
in parts because of their universality, they are the fundamental constituents
of every physical theory. This is because their spacetime invariance
characterises uniquely a set of phenomena (which is the case for some
constants) or even the essence of the Universe (which is the case for three
constants). Bringing this concept to an extreme, imagine a Universe where the
very constant that make it such vary, not being constant anymore. Since these
constant are the essence, the origin, the $A\rho\chi\acute{\eta}$ of the
Universe, changing them would result in changing the Universe. In other words,
the Universe would not be the same one as before, it would become another
Universe, since its most fundamental identifier are subject to change. This
inevitably leads to think that, since the constant are not invariant, then
some other property of the Universe must be the truly fundamental one, and
what is better then the quantity these constant are changing with respect to
for this role. If we assume time-dependent constants (it will turn out that
this is not the only possibility), it is then time the next subject of our
interest.

As we have seen before, the generalised unimodular formalism provides
excellent definitions of times for the evolution of the constants, but it also
does much more then that. It enables us to implement Leibniz’s idea of
relational times, so intimately connected with the problem of time in quantum
mechanics [13]. Finally, it shows us Nature’s subtle irony: certain constants
vary while other provide definition of the relational times, but this
situation can be reversed. This leads to a racy ambiguity, where the role of a
constant and that one of a time can be interchanged and mixed: where the
ultimate truth lies is a question for the future.

Therefore, in this section, we explore the heuristic and the ideas behind the
concept of a constants and their potential variability in time. Firstly, we
investigate the nature of constants, understanding why they arise, what
property makes them similar or differentiate them into categories and how they
relate to the physical laws. Secondly, subsection (1.2.2) presents the
experimental hints and evidences for varying constants. Furthermore, it also
reviews the early theoretical works on the subject, developing an intuition
for the next chapters. Finally, subsection (1.2.3) provides a crucial
specification to the ideas illustrated before: the “different” speeds of
light. It does so by showing how a single constant like $c$ could have
different roles, depending on the phenomenon it describes. This
differentiation will turn out to be very important in Chapter 3, where
“different” speeds of light can lead to different scenarios in the context of
energy conservation.

####  1.2.1 The Essence of a Constant

> “ Multa sunt quae esse concedimus; qualia sunt? Ignoramus. ”

What is a constant? It is a simple question at the heart of our understanding
of Reality, the very basis of the fundamental phenomena constructing this
Universe, that will prove rather deceptive and elusive to grasp. We will,
nevertheless, try our best to succeed. To shed some light on this topic, it is
enlightening to distinguish the domains of constants: mathematical and
physical ones. Our main interest will pivot around the latter, but we believe
it will be useful to briefly mention mathematical constants and their
relevance to this investigation. Mathematical constants are fundamentally
different from physical ones in one property: they cannot be measured, and
therefore, they are all dimensionless numbers without any uncertainty. In this
regard, they are certainly more abstract than physical ones, as they appear as
recurring numbers across mathematics, almost being structures of the model we
use to describe Nature. Another interesting fact about mathematical constants
is that some of them appear in physical laws, but no measured physical
constants appear in mathematics. The most evocative example is, by far, the
most ancient constant known: $\pi$. It is defined as [14]

$\pi:=\frac{C}{d}$ (1.24)

i.e., the ratio of the circumference of a circle (a 2D object, more generally,
a 1-sphere $\bm{\mathcal{S}^{1}}$) to its diameter. It appears in any
classical physical law possessing some periodic property [15]; it plays a role
in General Relativity, being in the EFE constant $\kappa=\frac{8\pi G}{c^{4}}$
[5], to Quantum Mechanics, e.g., in the Uncertainty Principle [16], defining
reduced Planck’s constant as $\hbar=\frac{h}{2\pi}$. The proliferating
appearance of $\pi$ in so many different fields is incredible, but its most
striking property is indeed its constancy: the value of $\pi$ is invariant in
space, time and other dimensions ($\pi$ is the same no matter the dimension in
which a $n$-sphere is considered [17]). This happens because $\pi$ is actually
the ratio of two geometrical quantities that, by the construction of space,
are related exactly by it: no matter what, when one is changed, the other one
will change proportionally, producing always the same value of $\pi$. This
suggests the first intuition on what a constant is: a fixed value linking two
concepts or quantities (which often might not be evidently related, see
subsection (1.2.3)) so defining their absolute and universal nature. From this
definition, we can easily understand why such a profound quantity like a
constant must be invariant in space, time and dimensions: if it was not, it
would not describe the $n$-sphere anymore, but something else. The
relationships between abstract elements of an $n$-sphere define $\pi$ and
$\pi$ defines those relationships.

As we will see for physical constants, equations, numbers and functions
describe how a system evolves and interacts with other systems, but constants
provide the essential information of what that system is, its fundamental
nature that makes it different from anything else and equal to anything
sharing the same constant. This is due to the principle of identity of
indiscernible [18]. This powerful idea, expressed in its more concrete as well
as abstract realisations, has been at the heart of great discoveries such as
the equivalence of mass and energy (where the constant $c$ is really the link
between the two) or the equivalence principle [19]. Now that we have
established the starting point of our constants discussion, we can explore the
more natural topic of physical constants.

What is a constant? A physical one, this time. Expectedly, a physical constant
usually is a dimensional numerical value arising in the description of a
natural phenomenon. As such, it arises in the same way whenever that
phenomenon, its outcomes or other qualitatively relatable phenomena, are
experienced, making it the characteristic and often defining property of the
phenomenon. The crucial point is that physical constants are the result of
measurements and dimensional, meaning they carry information about the
measurement convention. Units intrinsically distinguish these quantities. Some
constants measure velocity (the state of motion of an object), others measure
charge, mass, other temperature and more. We could, more abstractly, define
physical constants, following [20], as _any parameter that cannot be explained
by the theory it describes_. This highlights the first fork in our analysis:
fundamental and non-fundamental constants. In fact, while the first ones
cannot possibly be determined in terms of any other parameter of the theory,
the second ones are often made up of fundamental constants or are kept because
of historical uses. More specifically, fundamental constants are impossible to
compute from the theoretical framework as this would mean they obey some
dynamical equation; resultantly, they would not be constant anymore.
Conversely, from the experimental point of view, the fundamental constants
must and have been measured, thus confirming the validity of the theory and
fixing the value of the constant up to the experiment’s accuracy.

Following the idea behind fundamental constants, Weinberg defined them as “
_constants that appear in the laws of nature at the deepest level that we yet
understand, constants whose value we cannot calculate with precision in terms
of more fundamental constants, not just because the calculation is too
complicated (as for the viscosity of water or the mass of the proton) but
because we do not know of anything more fundamental_ ”[21]. This definition
remarks the nature of fundamental constants: they create the basis upon which
our physics, our understanding of Reality is constructed. They quite literally
define the Universe, giving the _what_ property of objects as described for
mathematical constants. Another crucial point is that fundamental constants
retain their title until another more general theory can explain their origin.
Therefore, when we discuss variable fundamental constants, whether they are
varying with time [22] (as we will consider here), with respect to space [23]
or something else, or they are just dynamical quantities explained by a more
general theory, the constants will lose their fundamental character. This will
expectedly and drastically change their nature, leading to, as an example
above all, a speed of light that is no longer observer-invariant, that breaks
Lorentz invariance [24] and energy conservation.

We conclude by analysing the current set of fundamental physical constants and
by providing a classification in which three constants stand out above all the
others. As aforementioned, physical constants could be divided into
fundamental and non-fundamental ones; the former cannot be expressed in terms
of anything more fundamental but can be combined to give the latter. These are
the constants describing the four fundamental interactions known thus far:
electromagnetism, weak interaction, strong interaction and gravity. Therefore,
considering these theories, we have 22 unknown constants in total[20]: six
Yukawa couplings for the quarks, three Yukawa couplings for the leptons, two
parameters for the Higgs field, four parameters for the CKM matrix, three
couplings for the symmetry group of the Standard Model, a phase for the QCD
vacuum [25] and, finally, the speed of light, the gravitational constant and
Planck’s constant.
Interestingly, most of these fundamental constants come from the Standard
Model and are related to the individual particles, fields, and their
symmetries. They are, moreover, different in nature from $c$, $G$ and $\hbar$,
as these appear to be more universal and less specific to a particular
physical phenomenon as the Standard Model ones. This is hardly a coincidence,
as fundamental constants can be divided into three different subcategories:
constants characteristic of a particular system, constants characteristic of a
class of physical phenomena, and truly universal constants [26]. It is clear
how the Standard Model constants belong predominantly to the second category,
while $c$, $G$ and $h$ are part of the universal constants, as they describe
the very structure of Reality. Furthermore, as observed in [26], [27] and
[28], they have the property of being concept synthesisers and reference
scales for physical theories.
Firstly, equations like $E=c^{2}m$, $E=\hbar\omega$ and $G_{\mu\nu}=\frac{8\pi
G}{c^{4}}T_{\mu\nu}$ show how each of these three constants allows linking
concepts that are apparently unrelated. They are the conversion, the bridge
between different aspects of the same Reality. Secondly, $c$ and $\hbar$ set
the physical scale of applicability of theories: the speed of light determines
an upper bound for causality, establishing when systems become relativistic,
while Planck’s constant defines the scale of quantum phenomena. The same role
is not clearly defined for $G$, even though it could be interpreted as setting
the scale at which gravitational phenomena become relevant [27]. Nevertheless,
we clearly see how these three constants play a more fundamental and
differentiated role than the others, to the point that we might be tempted to
call them, in an ecclesiastic spirit, the Holy Trinity.

####  1.2.2 Constants that Vary

> “ $\Pi\acute{\alpha}\nu\tau\alpha\>\rho\epsilon\tilde{\iota}$ ”

In the previous subsection, we have explored the properties that substantiate
and differentiate constants. As exemplified, an essential property of a
constant is, tautologically, its constancy in time, space, dimensions, etc.
More generally, a fundamental constant, to be such, should not have any
dependency whatsoever, whether it is from other constants or from the
structure of Reality. It is an absolute, a quantity only measurable and, if we
wish, a Deo datum. However, the idea at the heart of this work challenges this
view by asking: what if the fundamental constants of Nature are not actually
constant? That means that among the many mechanisms that could result in
varying constants models, we assume fundamental constants to be varying in the
sense of being subject to evolution, resulting in their dynamics. This
postulate inevitably leads to the second question of evolution with respect to
what. In this work, we will use the idea of relational times as parameters
determining the evolution of the constant, as provided by the unimodular
formalism. Lastly, we will only consider physical constants appearing in the
Einstein-Cartan, and Brans-Dicke actions provided with a perfect fluid matter
content. This is the case because the unimodular formalism presented in
section (1.1) was originally designed to be applied to gravitational types
action. Resultantly, from the 22 fundamental constants mentioned earlier, the
ones associated with the Standard Model will be neglected, even though they
could become the object of future studies.

Firstly, we would like to explore the heuristic behind varying constants, so
as to understand why such a hypothesis could, after all, not be too insane.
Everything started with the possibility that $G$, the gravitational constant,
could vary depending on cosmic time. In 1938, Dirac formulated the Large
Number Hypothesis [29], creating the basis for varying constants. He noticed a
remarkably close coincidence between two ratios, namely that [30]

$N_{1}=\frac{e^{2}}{Gm_{p}m_{e}}\simeq 2\crossproduct 10^{39}$ (1.25)
$N_{2}=\frac{cH_{0}^{-1}}{e^{2}m_{e}^{-1}c^{-2}}\simeq 2\crossproduct 10^{40}$
(1.26)

where $N_{1}$ is the ratio of electrostatic to gravitational force between the
proton and the electron, while $N_{2}$ is the ratio of the Hubble horizon
radius today, $H_{0}^{-1}$, to the classical electron radius. Intrigued by
their closeness, he thought that the two numbers should be the same up to a
factor of order unity such that

$N_{1}=N_{2}\crossproduct\mathcal{O}(1)$ (1.27)

Suppose that this relationship was valid, then we expect it to hold at all
cosmological times, from the beginning of the Universe till our time. However,
$N_{2}$ is not a constant ratio, as it depends on $H_{0}^{-1}$, the current
Hubble radius. Since the Universe is expanding [4], $H_{0}^{-1}$ varies in
time accordingly, making $N_{2}$ varying. But a non-constant $N_{2}$, by
virtue of relation (1.27), requires also $N_{1}$ to vary over time. Dirac
interpreted this fact as a varying $G$ over cosmological time as $G\propto
t^{-1}$. Alternatively, one could also explain a varying $N_{1}$ as a constant
$G$ but a varying fine structure constant according to $\alpha\propto
t^{\frac{1}{2}}$. Interestingly, a third possibility would be that the Hubble
radius in $N_{2}$ is constant but the speed of light is varying similarly to
VSL theories [24]; though fascinating, this scenario would require a whole
discussion itself tangential to this work.
Several experimental tests have been conducted to check the variability of $G$
[31] [32] [33] [34], from Solar System constraints to binary pulsars to
stellar constraints all the way up to cosmological one [20], converging on
similar variations for $G$. For example, the change in $G$ over time was
estimated using binary pulsars to be [35]

$\frac{\dot{G}}{G}=(1.0\pm 2.3)\crossproduct 10^{-11}\>yr^{-1}$ (1.28)

suggesting a minimal, but measurable change in $G$ over time.

On the other hand, we might interpret (1.27) as an equation where $G$ is
constant but $\alpha$ is varying. We now follow this line of reasoning. This
possibility is particularly interesting as it will closely relate to the
variability of $c$, as we will see. Plenty of different experiments have been
performed to test the variability of $\alpha$, ranging from quasar spectra
[36] to atomic clocks [37] and absorption lines spectra [38], including
measurements of CMB in the WMAP mission [39]. However, among all these
different measurements, the most interesting, constraining, and sensitive one
is the Oklo event [38]. Approximately $1.8\crossproduct 10^{9}\>yr$ in Gabon a
Uranium deposit reached the critical phase of fission reaction due to the
extreme coincidence of favourable conditions (abundance of naturally enriched
${}^{235}U$, low concentration of neutron absorbing materials and water
presence that acted as moderator), turning into a natural nuclear reactor
[20]. The fission continued for a few million years consuming part of its
fuel. Imagine what a strange and breathtaking experience it would be to
witness the spontaneous beginning of fission in a primordial Earth, millions
of years before Fermi managed to start the first man-made nuclear reaction!
The reason why the Oklo phenomenon quickly became relevant for testing the
variability of $\alpha$ is the ratio of two isotopes of samarium found in the
mine. While the ratio between ${}^{149}Sm$ and ${}^{147}Sm$ is usually $0.9$,
the same ratio in the Oklo mine was measured to be around $0.02$ [30]. This
discrepancy in the measurements could be attributed to a change in $\alpha$
[40], as the Oklo reaction happened in a past enough time. Several studies
were conducted on the constraints put on the variability of $\alpha$ by the
Oklo phenomenon, leading to the most stringent one of [41]:

$\frac{\dot{\alpha}}{\alpha}=(0.2\pm 0.8)\crossproduct 10^{-17}\>yr^{-1}$
(1.29)

It shows that the fine structure constant might indeed vary with time.
Together with Dirac’s Large Number Hypothesis and the observations made
regarding a varying $G$, this result motivates our work on the possibility
that further physical constants might vary. However, before mentioning the
last notable motivation for varying constants, we will show how a varying
$\alpha$ links particularly well with a varying speed of light.

The fine structure constant is defined as [42]

$\alpha:=\frac{e^{2}}{\hbar c}$ (1.30)

where $e$ is the electron’s charge, $\hbar$ is the reduced Planck’s constant
and $c$ is our beloved speed of light. If, as we have just seen, $\alpha$
varies with time, given its composite structure, a natural question arises:
which component of $\alpha$ is varying? Since $\alpha$ is a ratio, the
electric charge $e$ could be varying, while $\hbar c$ is constant, a
possibility considered by [43]. Equally, $e$ could be perfectly constant,
while either $\hbar$ or $c$ would be varying. Finally, in principle, all three
constants could be varying to give the observed variation in $\alpha$. The
main difference between the first and second scenario, and also the main
reason why someone would choose to fix $\hbar c$ while making $e$ varying, is
that in the first case, Lorentz invariance would be preserved while breaking
charge conservation. In the second case, Lorentz invariance would be clearly
broken. For simplicity, we leave out the case of a varying $\hbar$, being sure
that a varying $c$ would be sufficiently heretic.
Therefore, in the context of Varying Speed of Light theories, we will consider
that a varying $\alpha$ is actually a varying $c$ with $e$ and $\hbar$ both
constants. It is here that a crucial point arises: varying dimensional
constants do not have any meaning physically [24]. This is because dimensional
constants can only be measured as a ratio to a unit of measurement. It only
makes sense to have dimensionless constants varying, as these are pure numbers
without dimensions. However, it is possible to consider varying $c$ models
where the whole dimensionless fine structure constant is varying, while the
ratio of the other dimensional constants is kept fixed. This will be the basis
of our next developments, where we will consider a set of constants, labelled
$\bm{\beta}$ appearing in the Einstein-Cartan action. We will postulate that
these constants vary with respect to physical relational times as defined in
section (1.1), arising from the generalised unimodular term.

To conclude, we briefly mention the last main reason why we believe that
constants could vary. This is more of a heuristic motivation, extensively
based on the more abstract interpretation of the meaning behind the laws of
physics. As pointed out by [44] and [45], the appearance of General Relativity
in physics drastically changed our understanding of Reality. Neither because
of its complete and extended model of gravity nor because of its successful
tests, but because of the fundamental change it provided to our logic. It
allowed us to change from a strictly static and rigid view of spacetime to a
fluid and dynamic one. This freed our view of the world, allowing us to run in
the wild landscape of a mutable Reality that led to so many new insights and
discoveries. Likewise, we believe that there should not be any limitation on
imagining a Universe where constants might change, leading to new and exciting
models of Reality.

####  1.2.3 The Speed of Light

Speed is conventionally a dimensional quantity ($[\frac{L}{T}]$) that relates
perhaps the two most fundamental concepts of Nature: time and space. It is
really a conversion factor between the two. The faster something moves through
space, the more this space is converted into time, keeping track of the
temporal interval required to complete that distance. As such, the concept of
speed is intrinsically a variable one, making it a purely dynamical quantity
associated with objects, rather than a fundamental one determining the nature
of such objects. As an example, it would be illogical to presume that the
speed at which an electron is moving is defining the essence of an electron.
The electron is such no matter its state of motion, as this can change due to
external influences, but its intrinsic properties like charge and mass are not
only unique but also, and most importantly, unchangeable. It would be madness
to define, to substantiate any particle by the speed at which it propagates in
the Universe, as this would be subject to changes, thus not making it an
absolute property unique to that particle.

This is not, however, the case for the photon, or, to be more abstract, for
any particle lacking mass. Any massless particle travels at the speed of
light, and any particle that is travelling at the speed of light must be
massless [46]. As surprising as it might be, what is colloquially called “the
speed of light” is the intrinsic property defining any massless particle,
because of a simple but rather incredible occurrence: the speed of light is
constant. How exactly can something like the speed of a particle be always
fixed, to the point of becoming an invariant between reference frames? How did
it happen that a concept traditionally variable not only became constant but,
as such, it became the fundamental property defining intrinsically the nature
of a particle, thus ascending to the primary role of constant of Nature? To
answer these questions, we need to ask where this speed first appeared, and we
need to explore the deeper idea underlying the commonly known speed of light
as the speed at which a photon, i.e., electromagnetic waves, propagate in
spacetime.

Everything began with the early experiments on magnetostatic and
electrostatic. Electricity and Magnetism are characterised, as most physical
phenomena, by two constants: the electric vacuum permittivity, $\epsilon_{0}$,
and the magnetic vacuum permeability, $\mu_{0}$. The first one describes the
electric field density forming as a result of a charge, while the second one
models the strength of the magnetic field formed by moving charges.
Interestingly, when combined, they have the same dimensions of a velocity
[42]. This remained a coincidence until Maxwell unified electricity and
magnetism. If we consider the Maxwell’s equations [42]

$\nabla\cdot\bm{E}=-\frac{\rho}{\epsilon_{0}}$ (1.31) $\nabla\cdot\bm{B}=0$
(1.32) $\nabla\times\bm{E}=-\frac{\partial\bm{B}}{\partial t}$ (1.33)
$\nabla\times\bm{B}=\mu_{0}\biggl{[}\bm{J}+\epsilon_{0}\frac{\partial\bm{E}}{\partial
t}\biggl{]}$ (1.34)

where $\bm{J}$ is the electric current density, we can take the curl of
equation (1.33) and, using standard vector identities, we obtain

$\nabla^{2}\bm{E}=\frac{\partial}{\partial t}(\nabla\crossproduct\bm{B})$
(1.35)

And, using substituting equation (1.34) inside, we obtain

$\nabla^{2}\bm{E}=\epsilon_{0}\mu_{0}\frac{\partial^{2}\bm{E}}{\partial
t^{2}}$ (1.36)

Following a similar procedure, we have the same equation for the magnetic
field

$\nabla^{2}\bm{B}=\epsilon_{0}\mu_{0}\frac{\partial^{2}\bm{B}}{\partial
t^{2}}$ (1.37)

Conclusion? Both equations above have the form of the wave equation [47],
i.e.,

$\nabla^{2}\bm{x}=\frac{1}{v^{2}}\frac{\partial^{2}\bm{x}}{\partial t^{2}}$
(1.38)

where $v$ is the speed of the propagation of the wave. By analogy with this
equation, it is straightforward to see that

$c_{0}=\frac{1}{\sqrt{\epsilon_{0}\mu_{0}}}$ (1.39)

where $c_{0}$ is now the speed at which every electromagnetic wave propagates
in the vacuum. This value is constant, as it is made itself of two constants
coming from electric and magnetic phenomena. Strangely enough, electric
charges give rise to electromagnetic waves, the speed of which is constant.
This is our first encounter with one of the many roles the speed of light
takes: it is the fundamental constant of electromagnetism. As we will now see,
following [28], there are other roles this concept can take.

The speed of light appears as well in the Lorentz transformations as the
spacetime constant, and, to differentiate it from the other roles, following
[28], we will label it as $c_{ST}$. If we consider the line element for
Minkowski flat space

$\displaystyle ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}$ (1.40)
$\displaystyle=-c_{ST}^{2}dt^{2}+dx^{2}+dy^{2}+dz^{2}$ (1.42)

we immediately understand that here the speed of light acts as a fixed
conversion factor between time and space coordinates, so that they are fully
unified in a 4-vector, forming spacetime coordinates. Furthermore, by fixing
the speed of light and assuming nothing travels faster like Einstein did, we
can define a causal structure in spacetime. In fact, depending on the sign of
the Minkowski line element, two spacetime events can be separated by the
following intervals [48]

$ds^{2}<0\Leftrightarrow\text{Time-like interval}$ (1.43)
$ds^{2}=0\Leftrightarrow\text{Light-like interval}$ (1.44)
$ds^{2}>0\Leftrightarrow\text{Space-like interval}$ (1.45)

This shows how the speed of light, being a universal invariant constant
between reference frames, fixes the causality of events in spacetime: physical
phenomena could only be time-like or light-like, while space-like phenomena
break causality, as they would propagate faster than light. Lastly, $c_{ST}$
also enters the famous $E=mc_{ST}^{2}$ formula, becoming a conversion factor
between energy and mass. In the next chapters, we will see that $c_{ST}$
enters the gravitational metric $g_{\mu\nu}$ with the same role discussed here
(as a causality constant relating space and time), so we will label it $c_{g}$
to differentiate it from the other types.

The third different type of speed of light appears in General Relativity. When
considering the Einstein Field Equations, we have that

$G_{\mu\nu}=\frac{8\pi G}{c_{FE}^{4}}T_{\mu\nu}$ (1.46)

where this time $c_{FE}$ is a conversion factor between the gravity part of
the field equations and the matter content. As discussed in [28], this speed
of light is ultimately equivalent to $c_{ST}$ in the Newtonian limit. This
equivalence is however only valid in the context of General Relativity, and it
will still be important to differentiate it when considering the variability
of constants. Finally, [28] also shows how the speed of light enters the
linearised field equations, being the speed at which gravitational waves
propagate. This turns out to be, once again, the same as $c_{ST}$, confirming
that the speed of light is actually the maximum speed of propagation of
information and thus causality.

We have seen how the speed of light plays different roles in physics, ranging
from the propagation’s speed of electromagnetic waves to the structure
constant of spacetime determining causality. By doing so, we understand that
the concept of the speed of massless particles is fundamentally related to the
maximum speed of propagation of information in general, whether this is via
photons or gravitational waves. Having established its different roles, we can
now properly differentiate them when considering the variation of $c$,
depending on whether this speed of light comes from the Einstein-Hilbert
action, from the gravitational metric or from the energy density $\rho$.

###  1.3 Brans-Dicke: a scalar-tensor theory of gravity

Following our discussion from section (1.1), we now focus on which base action
$S_{0}$ we could use. Thus far [22] [49] [50], $S_{0}$ has always been taken
to be the Einstein-Cartan action, leading to a relatively small number of
parameters that could be used as $\bm{\alpha}$ or $\bm{\beta}$. Furthermore,
there is no reason why, _a priori_ , we should avoid using any other physical
action as a base theory $S_{0}$, ranging from quantum field theory ones to the
full standard model Lagrangian. These will potentially encounter
renormalisation problems related to the constants used (e.g., the distinction
between bare and physical mass), but could still be interesting fields of
research. Nevertheless, here we will focus on purely gravitational theories,
specifically investigating the outcomes of applying the varying constant
formalism to the Brans-Dicke theory. There are two main reasons why we chose
to use the Brans-Dicke action as $S_{0}$: its inclusion, by construction, of a
varying gravitational constant (via Mach’s principle) and the presence of the
dimensionless parameter $\omega$. Therefore, we will review the foundational
ideas and results of this theory, highlighting how its unique feature nicely
connect with the idea of varying constants.

####  1.3.1 Mach’s principle

Imagine to be floating in empty space. You are surrounded by never never-
ending void, and the only available reference points are distant fixed stars
visible in the background. Suddenly, you start to rotate around yourself, thus
experiencing your arms and legs being pulled away from you. This is nothing
but the centrifugal force caused by the rotation: your body’s inertia is
opposing to the rotation, hence the outward force. However, more
interestingly, the fixed stars of the background are also rotating, prompting
the question: why do you experience centrifugal force when the background
stars are rotating? Is there any connection between you and far distant object
that seems incapable of influencing you from a distance? This thought
experiment, inspired by [51], is at the heart of the idea of Mach’s principle
and its consequences for General Relativity and the Brans-Dicke theory.

Another way to formulate this idea is to think about Newton’s famous ”Bucket
Experiment”. If we imagine a bucket containing water rotating on itself, we
would notice that after some time has passed, the surface of the water will
have changed shape. It will go from being completely flat to having a
paraboloidal shape, with the water touching the bucket’s side higher than the
one in the centre. This will be the result of the inertia of the water
opposing the motion induced by the rotation. Once the bucket stops rotating,
the water’s rotation will also slow down, eventually reaching the flat state
it originally started from. If, however, the bucket is still but the entire
universe around is rotating, Newton’s conclusion was that the water on the
surface would still be flat, since it is not moving and the universe’s
rotation would affect the water’s inertia. Disagreeably to Newton, Mach
claimed that the water in the bucket would still change shape [52], and thus
the inertia would be affected, even when the bucket is not moving but the
universe around it is.

More fundamentally [53], this thought experiment becomes crucial in the
context of the dichotomy between absolute and relative space and time. In
fact, considering two buckets filled with water, one rotating and the other
not, how can we appreciate their difference? This would certainly be possible
by noticing that the water’s surface of the rotating bucket is different from
the stationary one. But with respect to what? Only by comparing them to an
absolute fixed space background, with respect to which one bucket is rotating
and the other one is not, we would be able to tell the buckets’ states. This
evidence led Newton to believe that the rotation _with respect to absolute
space_ is why the two buckets differ. However, Mach pointed out that an
essential assumption in Newton’s interpretation of the bucket experiment would
radically change its outcomes. In fact, Mach argued that Newton’s conclusions
were only valid in a hypothetical empty universe, where any source of matter
has been removed. Furthermore, these conclusions still hold true only if it is
assumed that physical systems retain their properties even when put in full
isolation from the rest. However, Mach’s key intuition was that such
conditions of pure empty space are, at best, ideal, and in the actual physical
universe there is always matter present, undermining Newton’s central
assumption. Following this line of thought, the two buckets can be compared
not relative to absolute space, but rather with respect to the background
fixed stars, an example of the matter in the universe. Therefore, this
suggests that the difference in the inertia of the two buckets can be observed
with respect to the masses of the background stars, establishing a link
between the local inertia and the matter content of the universe.
This is exactly what we mentioned in the example before: because the inertial
mass of an object and the matter content of the universe relate to each other,
Mach believed that also the water in the fixed bucket would change its shape
if the universe is rotating.

Having developed an intuition on how the inertia of rotating objects related
to distance references in their background, we can summarise Mach’s principle
more formally: the geometry of space-time, and hence, the inertial properties
of every infinitesimal test particle, is required to be determined by the
distribution of mass-energy throughout the universe [54] (pag. $303$). On more
heuristic and philosophical grounds, a universe subject to Mach’s principle
would not carry any notion of absolute space and time, but rather it would
only allow the notion of quantities such as motion _relatively_ to some other
reference. This idea is precisely the base of Einstein’s theory of General
Relativity. In fact, generalising Mach’s principle, he postulated a theory of
gravity where only relative motion is observable, thus disallowing any
privileged reference frame. Furthermore, he postulated that inertia should be
related to the gravitational interaction with matter only, and that the
geometry and metric structure of the universe is entirely determined by its
matter content [55].

However, Einstein’s implementation of Mach’s principle is only partial in his
theory. It has been pointed out [56] that Mach’s principle is, to some extent,
conflicting with the other founding principle of General Relativity: the
equivalence principle. While the former has somehow a global nature, as it
relates the local inertia of a particle to the global energy-momentum tensor
of the universe, the latter strictly applies locally, as tidal forces are
negligible only locally, since they can differentiate gravitational
acceleration from inertial. Moreover, going beyond this tension in the
founding principles of the theory, the non-fully-Machian nature of General
Relativity arises when considering some particular solutions to the field
equations. Specifically, the Minkowski spacetime solution posses inertia but
it does not contain any matter [57], the Gödel rotating universe [58] breaks
the idea of relative motion introducing a preferred (and hence absolute)
direction of motion, and finally the Taub-NUT model provides a curved solution
which is singularity free, in contradiction with one formulation of Mach’s
principle [59]. Motivated by the desire to fully implement Mach’s principle in
General Relativity, Brans and Dicke decided to develop their theory of scalar-
tensor gravity, as we will see in the next section.

####  1.3.2 Varying $G$ and Mach’s principle

The main aim of the Brans-Dicke theory of gravity is to provide an extension
of General Relativity fully inclusive of Mach’s principle. To do so, a central
observation is necessary: Dirac’s Large Number Hypothesis [29]. As we have
illustrated in section (1.2), Dirac noticed a deep connection between
cosmological numbers and atomic ones, which lead to the following idea. A
striking result is obtained when considering a hollow-mass sphere containing
two particles attracting each other gravitationally [60]. An equation for the
inertial force experienced by one of the two particles that is fixed in a rest
reference frame can be obtained as

$F_{inertia}=bmMaG^{\alpha}r^{\beta}c^{\gamma}$ (1.47)

where $b$ is a dimensionless constant set to one, $m$ is the mass of the
particle, $M$ is the mass of the spherical shell, $r$ is the radius of the
shell and $a$ is the acceleration of the sphere relative to the particle [61].
Furthermore, $\alpha$, $\beta$ and $\gamma$ are constants as well. To match
the dimensions on the LHS and on the RHS of equation (1.47), we must have that
$\alpha=1$, $\beta=-1$ and $\gamma=-2$. Finally, equating equation (1.47) to
Newton’s second law leads to

$\frac{GM}{Rc^{2}}\sim 1$ (1.48)

where $G$ is the gravitational constant, $c^{2}$ is the speed of light in the
vacuum and $M$ and $R$ are the total mass and radius of the observable
universe. Interestingly, expressing (1.48) in terms of $G$ leads to

$G\sim\frac{Rc^{2}}{M}$ (1.49)

which in turn suggests a fundamental relationship between the gravitational
constant $G$ and the total mass $M$ of the observable universe. Indeed,
recasting (1.49) as $G\sim G(M)$, where $G$ becomes effectively a function of
the total mass $M$, we clearly see that the gravitational constant can be
treated as a function of $M$. This result is exactly the desired
implementation of Mach’s principle. In fact, we have managed to express
mathematically how the geometry of space-time should be determined by the mass
distribution of the universe: the inertia experienced by an accelerated
observer relative to distant matter is equivalent to a gravitational force
acting on a fixed observer due to distant accelerated matter [62].

Additionally, relation (1.49) suggests that either the ratio
$\frac{R}{M}=constant$, or that the gravitational constant should be a locally
variable quantity, determined by the mass distribution about the point.
Discarding the first possibility due to an expanding universe which requires a
changing $R$ implying matter-conservation violations to keep the ratio
constant, we have finally arrived at the full implementation of Mach’s
principle which naturally leads to a variable $G$ dependent on the mass
distribution of the Universe.

####  1.3.3 Brans-Dicke Theory

In this subsection, we review the work presented in [62], highlighting the
most relevant results for our future extension. Given the observations in the
previous section, it is now natural to think of $G$ as a variable function,
changing from position to position depending on the matter distribution. Since
it is desirable that this new $G$-field is coordinate independent, it should
be chosen to be a scalar field. Furthermore, considering where $G$ appears in
the Einstein-Hilbert action (1.2) and setting $c=1$, we can see that

$\phi:=\frac{1}{G}\sim\frac{M}{R}$ (1.50)

where the scalar field $\phi$ has the correct dimensions of $G$, i.e.,
$[\phi]=\frac{MT^{2}}{L^{3}}$. We could, in the spirit of varying constants,
not set $c=1$ above and actually see that a varying speed of light might lead
to a varying $G$, while $M$ and $R$ are constant. However, we will not
consider this case here, as it will be the object of future work [63]. It is
now possible to precede and generalise action (1.2) including a scalar field
of the variation of $G$ with respect to the coordinates. We start from the
Einstein-Hilbert action inclusive of a matter Lagrangian term

$S=\int{d^{4}x\>\sqrt{-g}\>\biggl{[}R+\frac{16\pi G}{c^{4}}L_{m}\biggl{]}}$
(1.51)

where $L_{m}$ is the matter Lagrangian density containing all non-
gravitational fields. We then multiply action (1.51) by $G^{-1}$, substitute
in the Ricci scalar term $\phi$ as the varying gravitational constant, and
include a general Lagrangian density for the scalar field, depending both on
the field and its derivatives [62]

$S=\int d^{4}x\>\sqrt{-g}\>\biggl{[}\phi
R+\frac{16\pi}{c^{4}}L_{m}+L_{\phi}(\phi,\partial_{\mu}\phi)\biggl{]}$ (1.52)

This new action contains the scalar field $\phi$ which is coupled to mass and
geometry. This, therefore, leads to a non-purely geometrical theory of
gravity, as part of gravitational phenomena are described by the scalar field
coupled to the Ricci scalar. Additionally, given its scalar nature, we might
expect that $\phi$ obeys some second-order wave equation such as

$\Box{\phi}=\rho_{scalar}$ (1.53)

where $\rho_{scalar}$ acts as a scalar matter source term for (1.53).
Therefore, following this and minimal dimensional consistency, we can arrive
at a form for $L_{\phi}$ which would give us, when varied, the desired wave
equation for $\phi$:

$L_{\phi}=-\frac{\omega}{\phi}g^{\mu\nu}\>\partial_{\mu}\phi\>\partial_{\nu}\phi$
(1.54)

which can then be inserted into (1.52), leading to the full Brans-Dicke action

$S_{BD}=\frac{c^{4}}{16\pi}\int{d^{4}x\>\sqrt{-g}\>\biggl{[}\phi
R-\frac{\omega}{\phi}g^{\mu\nu}\>\partial_{\mu}\phi\>\partial_{\nu}\phi\biggl{]}}+S_{m}$
(1.55)

where $S_{m}$ is a general matter action.
The parameter $\omega$ is a dimensionless coupling constant which can take any
value, made such by including the factor $\phi^{-1}$. Also, it is possible to
recover Einstein’s theory in the limit where $\omega\rightarrow\infty$. The
Lagrangian density term introduced for the scalar field has the form of a
kinetic term for $\phi$. This is actually the case, as this term will provide
the dynamical evolution of the scalar field. Furthermore, the kinetic term’s
introduction assures the full diffeomorphism’s invariance of the action and
the conservation of the energy-momentum tensor via $\nabla_{\mu}T^{\mu\nu}=0$
[64]. It is also possible to include a general potential $V(\phi)$ [65], thus
arriving at an action with full kinetic and potential terms for $\phi$, but,
in the following, we will not include it.

We can now proceed to derive the field equations of Brans-Dicke theory in the
Jordan frame using action (1.55). Firstly, we derive the field equation for
$\phi$, by varying (1.55) with respect to $\phi$ and $\partial_{\mu}\phi$

$\Box\phi=\frac{g^{\mu\nu}\>\partial_{\mu}\phi\>\partial_{\nu}\phi}{2\phi}-\frac{\phi
R}{2\omega}$ (1.56)

where the d’Alambertian operator is defined as

$\Box\phi=\frac{1}{\sqrt{-g}}\partial_{\mu}[\sqrt{-g}\>\partial^{\mu}\phi]$
(1.57)

Secondly, using equation (1.56), we can vary action (1.55) to obtain the
modified field equations as

$\frac{\delta S_{BD}}{\delta g^{\mu\nu}}=0\Leftrightarrow\frac{\delta
S_{g}}{\delta g^{\mu\nu}}+\frac{\delta S_{m}}{\delta g^{\mu\nu}}=0$ (1.58)

where $S_{g}$ is the gravity action inclusive of the Brans-Dicke kinetic term,
while $S_{m}$ is a generic matter action giving the energy-momentum tensor as

$\frac{\delta S_{m}}{\delta g^{\mu\nu}}=-\frac{1}{g}\frac{\delta g}{\delta
g^{\mu\nu}}L_{m}-2\frac{\delta L_{m}}{\delta g^{\mu\nu}}:=T_{\mu\nu}$ (1.59)

where $g=det(g_{\mu\nu})$ and the variations above have been obtained using
the chain rule. Thus, we arrive at the field equations:

$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}-\frac{\omega}{\phi^{2}}\biggl{(}\partial_{\mu}\phi\>\partial_{\nu}\phi-\frac{1}{2}g_{\mu\nu}\>\partial_{\alpha}\phi\>\partial^{\alpha}\phi\biggl{)}-\frac{1}{\phi}\biggl{(}\nabla_{\mu}\partial_{\nu}\phi-
g_{\mu\nu}\Box\phi\biggl{)}=\frac{8\pi}{\phi c^{4}}T_{\mu\nu}$ (1.60)

where the third term on the RHS is obtained by including $g^{\mu\nu}$
explicitly in action (1.55) to account for a more general curved space-time.
Obviously, the field equations (1.60), as expressed here, should be
interpreted as $\phi$ adding geometrical contributions to (1.60) to model the
full gravitational field (metric plus scalar field). Conversely, if the terms
involving $\phi$ are moved to the RHS of the equations, we see that they
present a relation between space-time geometry and a matter content not only
given by the usual $T_{\mu\nu}$ but also by a scalar field $\phi$. Finally, we
can arrive at a more compact and evocative form for equation (1.56) by
considering contraction of the field equations (1.60) with $g^{\mu\nu}$. This
leads to

$R=-\frac{8\pi}{\phi
c^{4}}T_{\mu\nu}-\frac{\omega}{\phi^{2}}+\frac{\omega}{\phi^{2}}\partial_{\alpha}\phi\>\partial^{\alpha}\phi+\frac{3}{\phi}\Box\phi$
(1.61)

which is then used for the Ricci scalar in equation (1.56) to obtain

$\Box\phi=\frac{8\pi}{2\omega+3}T$ (1.62)

where $T=g^{\mu\nu}T_{\mu\nu}$ is the trace of the energy-momentum tensor.
Equation (1.62) clearly shows that the $\phi$ field has as source term the
matter content of the solutions, described by the energy-momentum tensor.
Therefore, it is reasonable to assume that the scalar field acts as an
auxiliary geometrical contribution to the usual field equations, justifying
its appearance on the LHS. Furthermore, equation (1.62) has the desired form
of a wave equation for $\phi$, as we initially hoped for in (1.53), providing
us with an expression for the evolution of $\phi$. Specifically, it is
interesting to consider a solution for (1.62) found for a quasi-flat universe
with matter content $\rho$:

$\phi=8\pi\frac{4+3\omega}{6+4\omega}\biggl{(}\frac{t}{t_{0}}\biggl{)}^{\frac{2}{4+\omega}}\rho_{0}t_{0}$
(1.63)

where $\rho_{0}$ is the matter density at a given time $t=t_{0}$.
Noteworthily, this solution is compatible with (1.48); when the matter density
is small, $G=\phi^{-1}$ becomes large, which physically implies the
disappearance of inertia which is precisely Mach’s principle’s prescription.
Therein the inertia of a particle is directly related to the universe’s matter
content. This result justifies Brans and Dicke’s intentions of including
Mach’s principle in a theory of gravitation by generalising Einstein’s theory.

Finally, efforts have been made to measure the value of the parameter $\omega$
to understand if, physically, the Brans-Dicke theory is distinguishable from
General Relativity. The two theories are equivalent in the limit
$\omega\rightarrow\infty$, so a high value of $\omega$ would suggest that
Brans-Dicke is essentially equivalent to General Relativity. Recent
observations made with the Cassini Saturn Probe [66] set the value of $\omega$
at $\omega>4\times 10^{4}$, much greater than the value close to unity
proposed by Brans and Dicke. This suggests that, at least in the Solar System,
Brans-Dicke is indistinguishable from General Relativity, so the latter is
preferred. However, the question of which theory is correct on cosmological
scales is still open and more observations are required.

###  1.4 Overview and Aims

In this work, we will study three ideas that are, in appearance, separated,
unifying them to develop several varying constants scenarios. To begin, we
consider the speculation that Natural constants might not be that constant
after all. We investigate what the Universe would look like if the value of a
certain physical constant evolved with time. This possibility has been
initially considered by Dirac in his Large Number Hypothesis [29] and has,
ever since, attracted increasing interest, particularly in the cases of a
varying $G$ and $\alpha$. More recently [24], Albrecht and Magueijo have
developed a theory in which the speed of light can vary, called the Varying
Speed of Light (VSL) theory. Besides providing an alternative explanation to
the open cosmological problems, VSL created a new conceptual landscape: if $c$
can vary along with $G$ and $\alpha$, imagination is really the only limit to
which constants vary.

It is precisely in this context that the second idea considered in this work
arises. How can we provide a rigorous mathematical framework describing the
evolution of multiple physical constants and their respective evolutions’
times? The answer is promptly provided by the idea of unimodular gravity.
Initially formalised by Einstein as a first attempt to solve the Cosmological
Constant problem, it was extended by Henneaux and Teitelboim (HT) to include a
more general framework for $\Lambda$ to be a mere constant on-shell. Following
their idea [22], Magueijo generalised the HT formalism to allow any constant
appearing in the Einstein-Cartan to vary. Furthermore, HT formalism provides
excellent definitions of relational times [22] that can be used as a parameter
for the evolution of our constants. Therefore, the generalisation of
unimodular gravity, as presented in section (1.1), turns out to be the perfect
mathematical platform to develop varying constants theories, rigorously
exploring this idea and its consequences.

We have now the heuristic, the why of our work lying in the idea of varying
constants and the method, the machinery, the how built from the unimodular
gravity formalism. All we are missing is the what: what are we applying all of
these ideas and mathematics to? For the idea of varying constants, one of the
earliest attempts at formalising it in a theory of gravity was the Brans-Dicke
theory. Hoping to develop an extension of General Relativity that would fully
incorporate Mach’s principle, Brans and Dicke proposed a varying gravitational
constant [62], formalised as a scalar field called $\phi$. Motivated by their
early insight, this dissertation extends the unimodular formalism to the
Brans-Dicke theory, producing original results in the field of varying
constants. Specifically, the aim of this work is an open-ended exploration of
energy conservation in a Brans-Dicke cosmological model of the Universe.
Therefore, the main results in this dissertation will be various energy
conservation scenarios given by different varying constants or combinations of
them. Furthermore, as part of a future publication [63], some of the results
obtained in this dissertation will be analysed in greater detail, providing
insights and possibly a solution to the Cosmological Constant problem.

To conclude, this dissertation is structured as follows. In Chapter 1, we have
introduced the necessary background to our results. We review the unimodular
formalism and show how its generalisation provides a development platform for
the idea of varying constants. We provide extensive motivations for the
variation of natural constants, ranging from more speculative ones to actual
experimental pieces of evidence. We also review the Brans-Dicke theory,
developing the intuitions behind it, as well as its main results and
observational constraints.
Secondly, in Chapter 2, we review the main applications of VSL and the
cosmological problem it aims to solve. Furthermore, we provide an extensive
derivation of the minisuperspace formalism for both the Einstein-Hilbert and
Einstein-Cartan actions. Also, we present the foundational results obtained in
[22] applying the varying constant idea to energy conservation scenarios in
the Universe.
Finally, in Chapter 3, we present the original results of this research. They
include several scenarios of varying constants applied to Brans-Dicke
cosmologies, leading to different energy conservation laws. We also lay the
foundations of our future developments in the Cosmological Constant problem.
Let the fun begin!

##  Chapter 2 Varying Constants: Developments

In this chapter we present an additional motivation supporting the idea of
varying constants and we provide examples of its implementation in the
conservation of energy on cosmological scales. Firstly, we show how currently
open problems in cosmology can be solved by the Varying Speed of Light theory.
This provides not only a straightforward application of the idea of varying
constants, but it also supports the heuristic behind why the constant of
Nature should be varying. Secondly, we derive the Einstein-Hilbert and the
Einstein-Cartan actions in minisuperspace (MSS), showing their equivalence
when the torsion-free condition is assumed. This will give the Hamiltonian
which will be used to obtain our results. Finally, we combine the MSS form of
the Einstein-Cartan action with the unimodular generalisation presented in
section (1.1) to obtain multiple energy conservation scenarios for different
varying constants. This last section will be predominantly a review of [22],
which will form the basis for our original results in Chapter 3.

###  2.1 Open problems in Cosmology

In this section, we present the main open problems in the standard Big Bang
cosmology. These are the Horizon, the Flatness and the Cosmological Constant
problems. They are normally explained with a period of accelerated expansion
in the early stages of the Universe, called inflation. Inflationary
cosmologies provide solid models to account for the Horizon and Flatness
problems, while they can partially explain the Cosmological Constant one [67].
Among their main shortcomings is the requirement of a physical mechanism to
drive inflation which is, to this date, still experimentally undetermined
[68]. We therefore present an alternative explanation to inflation, the
Varying Speed of Light theory, where the speed of light is assumed to have
transitioned from a high value in the early Universe to the lower one we
observe today. This theory provides solid explanations to these problems,
opening the possibility to alternative cosmological models.

####  2.1.1 Horizon problem

The Cosmological Principle [69] postulates that the spacial distribution of
matter in the Universe is homogeneous, i.e., equally distributed, and that,
when considered from large scales, it appears to be isotropic, meaning that it
should look the same from every direction. Homogeneity and isotropy are at the
heart of modern cosmology and represent the central assumption behind the
Friedmann-Lemaître-Robertson-Walker (FLRW) metric. Besides it being assumed as
a first principle, and therefore not requiring, a priori, any justifications,
observations [70] have confirmed a very isotropic universe. In fact, measuring
the Cosmic Microwave Background Radiation temperature of very distant regions
of the universe, it was possible to establish an outstanding similarity
between them, setting their temperature to $T_{0}=2.726\pm 0.010$ [70]. This
value turns out to be remarkably similar through different regions of the
universe that are causally-disconnected. To illustrate this, lets remember
that, according to the Big Bang cosmology, the Universe has, in this moment, a
finite age, called $t_{0}$. If we consider the cosmological distance defined
by this age, $ct_{0}$, we have the furthest distance some photons could have
travelled to us. Obviously, this maximum distance stretches from Earth into
every direction, but light coming from one direction to the Earth could not
have had enough cosmic time to travel to the opposite direction. In other
words, light from a region of the universe cannot affect (is not in causal
contact) with another directly opposite region. Nevertheless, both regions
appear to be perfectly homogeneous, showing a remarkable equivalence of their
Cosmic Microwave Background Radiation temperature.

Lets illustrate this issue in more detail. We consider a photon originating
from the surface of last scattering, when the phase transition between a very
hot to a colder universe happened, leading to the recombination era which
ultimately allowed photons to start to prorate freely into the Universe. For
simplicity, we will assume a radial motion, and, given our working assumption,
we will use the FLRW metric to model the early homogeneous and isotropic
Universe, which takes the form [5]

$ds^{2}=-c^{2}N(t)^{2}dt^{2}+a(t)^{2}\biggl{[}\>\frac{dr^{2}}{K(r)^{2}}+r^{2}(d\theta^{2}+\sin{\theta}^{2}d\phi^{2})\biggl{]}$
(2.1)

where $a(t)$ is the scale factor, $K(r)=\sqrt{1-kr^{2}}$ and $k$ is the
Universe’s curvature, which we will assume to be flat, giving $K(r)=1$. Given
radial motion, the line element above simplifies to

$ds^{2}=dt^{2}+a^{2}dr^{2}=0$ (2.2)

since for radial motion $d\theta=d\phi=0$ and we have assumed $N(t)=1$ and
$c=1$. Rearranging (2.2) for the distance travelled in a given time interval,
we obtain

$\int^{r_{2}}_{r_{1}}{dr}=\int^{t_{2}}_{t_{1}}{\frac{dt}{a(t)}}$ (2.3)

where $t_{1}$ is the time the photon was emitted and $t_{2}$ is the time when
it is received, which in our case is the current age of the universe.
Furthermore, considering the Robertson-Walker line element for the Universe’s
spacial slice only at a fixed time $t=t_{0}$ [5], we have

$d\sigma^{2}=a_{0}^{2}dr^{2}\Leftrightarrow dr=\frac{d\sigma}{a_{0}}$ (2.4)

where $a_{0}=a(t_{0})$. Substituting this expression into (2.3), we obtain

$\Delta r=a(t_{2})\int^{t_{2}}_{t_{1}}{\frac{dt}{a(t)}}$ (2.5)

where the integration of the 3D spacial line element $d\sigma$ gives the total
distance travelled by the photon $\Delta r$ (i.e., its proper distance) and we
choose the time $t_{0}$ to be the present age of the Universe coinciding with
the photon’s detection time $t_{2}$. The time $t_{1}$ is the emission of the
photon, which could be at $t_{1}=0$, i.e., the Big Bang, or at $t_{1}=t_{l}s$,
i.e., when the photo left the surface of last scattering. In the first case,
this expression gives us the maximum proper distance a photon emitted at the
Big Bang has travelled, and its called the particle horizon. To obtain an
explicit expression for the maximum proper distance travelled, we assume a
radiation dominated universe, as it was in its very early stages until
approximately $47\>000$ years after the Big Bang, for which
$a(t)=Ct^{\frac{1}{2}}$, where $C$ is some proportionality constant.
Therefore, if $t_{2}=t_{0}=t$, the current age of the universe and $t_{1}=0$,
we get

$\Delta r=Ct^{\frac{1}{2}}\int^{t}_{0}{\frac{du}{Cu^{\frac{1}{2}}}}=2t$ (2.6)

On the other hand, if we would like to obtain the same expression for the
particle horizon in the matter dominated era, we need to consider the scale
factor for matter, which is $a(t)=Dt^{\frac{2}{3}}$, where $D$ is once again a
proportionality constant. This era started $47\>000$ years after the Big Bang,
but it was not until $380\>000$ [71] years that photons could finally freely
travel into the rest of the Universe. This event is called the surface of last
scattering, marking the time from which we are receiving the first photons
from the early Universe. Therefore, the maximum distance traveled by a photon
from the surface of last scattering time until now is given by

$\Delta r=Dt^{\frac{2}{3}}\int^{t}_{0}{\frac{du}{Cu^{\frac{2}{3}}}}=3t$ (2.7)

where, in this case, $t_{1}=0$ coincides with the time of last scattering and
not with the Big Bang. Focusing on the surface of last scattering, we can re-
insert the speed of light in expression (2.6) to obtain the particle horizon
for causal interaction in the early universe as [48]

$\Delta r=2ct\approx 0.27Mpc$ (2.8)

This means that at the time of last scattering, causal interaction between
photons could only happen if they were separated by a distance equal or
inferior to $0.27Mpc$. If, at that time, photons had a greater separation,
they could have not possibly interacted, thus evolving independently until our
time. However, as stated above, we observe that causally disconnected regions
of the Universe appear to be extremely homogeneous. This is exactly the
Horizon problem: how could causally disconnected regions of the Universe
evolve independently to the point of having essentially the same radiation’s
temperature? As we will see in the next subsection, inflation will provide an
answer, but not the only one.

####  2.1.2 Flatness problem

Another shortcoming of the current cosmological model is the Flatness problem.
When defining the Universe’s total energy density as [5]

$\Omega_{total}=\Omega+\Omega_{\Lambda}$ (2.9)

we can re-express the first Friedmann equation as

$\Omega_{total}-1=\frac{k}{a^{2}H^{2}}$ (2.10)

where $k$ is the curvature and $H=\frac{\dot{a}}{a}$ is the Hubble’s
parameter. If we have a flat universe (one for which $k=0$) we immediately see
from (2.10) that $\Omega_{total}$ must equal to one at all times for the first
Friedmann equation to hold. This is the case for the current universe, where
since

$\Omega_{m}+\Omega_{\Lambda}\approx 1$ (2.11)

the total energy density is close to the critical one. If, however, the
curvature is not zero, $\Omega_{total}$ evolves with time because of the
dependence of $a^{2}H^{2}$ in (2.10). To understand better this time
evolution, as we have seen above, we consider the dependence of the scale
factor on cosmological time. If we consider either a radiation dominated or
matter dominated universe, we find that

$aH\propto t^{-\frac{1}{2}}\Leftrightarrow\Omega_{total}-1\propto t$ (2.12)
$aH\propto t^{-\frac{1}{3}}\Leftrightarrow\Omega_{total}-1\propto
t^{\frac{2}{3}}$ (2.13)

where the first equation applies to the evolution of $\Omega_{total}$ in a
radiation dominated Universe, while the second equation applies to a matter
dominated one. In both cases, we notice that the total energy density
increases with time. This time dependence implies that, since we have a total
energy density close to unity today, it should have been the same in the early
universe, otherwise, over time, it would have evolved to be a different value.
In fact, estimates on nucleosyntesis constrain the value of the total energy
density at [48]

$|\Omega_{total}-1|\leq 10^{-16}$ (2.14)

which indeed confirms the closeness to one. For earlier times, this value
would be even smaller. This is the flatness problem: this initial condition on
$\Omega_{total}$ so that the universe would evolve to have a curvature $k=0$
is way too specific. If this condition would have been slightly different, the
Universe would have evolved radically different from how we observe it today.
Therefore, we conclude that such fine-tuned initial conditions are hard to
explain in the current cosmological models.

####  2.1.3 Cosmological constant problem

The last and most important open problem in cosmology is the Cosmological
Constant one. It would be restrictive to regard it as a purely cosmological
problem, as it arises from the comparison of measurements and predictions
coming from very different fields of physics. In fact, as we have seen,
General Relativity predicts the existence of a cosmological constant in the
field equations as a geometrical contribution, responsible for the dynamical
nature of the universe. This results in including the cosmological constant on
the LHS of the field equations as

$G_{\mu\nu}+\Lambda g_{\mu\nu}=\frac{8\pi G}{c^{4}}T_{\mu\nu}$ (2.15)

where we can define the energy density of the cosmological constant to be

$\rho=\frac{\Lambda c^{2}}{8\pi G}$ (2.16)

This is regarded as the the purely geometrical cosmological constant problem
[24], and it is distinct from the vacuum energy problem we will see below.

On the other hand, following a different approach, it is possible to consider
the cosmological constant as a source term for the field equations, thus
moving $\Lambda$ on the RHS of equation (2.15): this is the approach were the
cosmological constant represents the vacuum energy density [72]. In this case,
the cosmological constant becomes part of the energy momentum tensor giving
the spacetime geometry due to the Universe’s matter content, and it can be
equivalently treated as the vacuum energy density $\rho_{vac}$ [73]. This
vacuum energy density is made up of three main components: the bare
cosmological constant, the quantum fluctuations and contributions form
particles and interactions that are yet not accounted for in the Standard
Model [73]. In fact, according to quantum field theory [74], the vacuum has a
non-zero energy density due to particle creation and annihilation: this
energy, as any other form of matter, creates curvature, modifying the
geometrical structure of spacetime. Since this energy density contributes to
curvature, it is measurable with cosmological observations, and, given its
appearance in the theory, it is also theoretically predictable. However, there
is a big inconsistency between these two values.

Following [75], since the energy momentum tensor must respect Lorentz
invariance, it must take the form

$T_{\mu\nu}=-g_{\mu\nu}\rho_{vac}$ (2.17)

Comparing this result with (2.15), we see that this requirement is equivalent
to adding a new term to the total vacuum energy density as

$\rho_{total}=\rho_{vac}+\frac{\Lambda c^{2}}{8\pi G}$ (2.18)

According to cosmological measurements [76], this value has an upper bound
give by

$|\rho_{total}|\leq 10^{-48}\>GeV^{4}$ (2.19)

On the other hand, the theoretically predicted vacuum energy is the sum of
different contributions, ranging from potential energies of scalar fields to
zero-point fluctuations of each field theory, as well as a bare cosmological
constant contributions [76]. Assuming that quantum field theory is still valid
at the Planck’s scale, we obtain a theoretical prediction on the vacuum energy
as

$\rho_{total}\sim 10^{72}\>GeV^{4}$ (2.20)

The ratio of (2.20) to (2.19), the theoretical value to the experimental one,
is indeed the cosmological constant problem

$\frac{\rho_{total}^{theory}}{\rho_{total}^{obs}}\sim 10^{120}$ (2.21)

This huge discrepancy of $120$ orders of magnitudes between theory and
observations is the core of the cosmological constant problem as presented
here.

####  2.1.4 The solution: Varying Speed of Light

The problems exposed above have all one aspect in common: they arise from the
standard Big Bang cosmology, where inflation has been an attempt at solving
them. While successful on many grounds, inflation requires a negative pressure
fluid as well as its fundamental mechanism is still unknown [68]. An
alternative solution to the problems exposed above could be the Varying Speed
of Light theory, firstly explored by Moffat [77] and then extended by Albrecht
and Magueijo [24]. Before diving into the detail of how VSL solves the three
problems mentioned above, we will briefly discuss how inflation is modelled
and how it tackles these issues.

The theory of inflation was first proposed by Alan Guth while trying to
explain the absence of observations of magnetic monopoles. Postulating the
idea of inflation, it was quickly realised that it could solve the horizon and
flatness problems [78], as well as explain how a universe dominated by an
attractive force (gravity) could be expanding [79]. Inflation could have
plenty physical origins, but the common feature of all inflationary models is
the presence of an exotic form of energy for which the pressure is negative,
i.e., $p=-\rho$, thus having a repulsive-like force driving expansion against
gravity.
A common example for inflation is a scalar field [5]. The action for a scalar
field with kinetic and potential energy is

$S=\int{d^{4}x\>\sqrt{-g}\>\biggl{[}\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi+V(\phi)\biggl{]}}$
(2.22)

From this we obtain the scalar field’s pressure and energy density as

$p_{\phi}=\frac{1}{2}\dot{\phi}^{2}-V(\phi)$ (2.23)
$\rho_{\phi}=\frac{1}{2}\dot{\phi}^{2}+V(\phi)$ (2.24)

which combined give

$\rho+3p=2(\dot{\phi}^{2}-V(\phi))$ (2.25)

where, if we assume the slow-roll approximation (the kinetic energy is small),
the potential $V(\phi)$ dominates, giving indeed a negative pressure as
$p\approx-\rho$. On the other hand, if we consider the Einstein Fields
Equations, assuming a perfect fluid as matter content, we obtain the second
Friedmann equation

$\ddot{a}=-\frac{4\pi G}{3}\biggl{(}\rho+\frac{3p}{c^{2}}\biggl{)}a$ (2.26)

In standard cosmology, we usually have a positive energy density and pressure,
making the RHS of (2.26) negative, so that the scale factor can only decrease
over time. This scenario is however not the same if, in the slow-roll
approximation, we use equation (2.25) with $V(\phi)\approx\rho_{\phi}$ such
that equation (2.26) becomes [79]

$\ddot{a}\approx\frac{8\pi G\rho}{3}a\equiv H_{I}^{2}a$ (2.27)

where $H_{I}=\frac{8\pi G\rho}{3}$ is the Hubble parameter at inflation’s
epoch. Solving this equation for the scale factor, we immediately get that

$a\propto e^{\pm H_{I}t}$ (2.28)

This solutions shows that the universe’s scale factor is experiencing a period
of exponential evolution, confirming that a matter content with negative
pressure does indeed produce a repulsive force. We now review the three issues
above in the context of VSL, highlighting how its proposed solutions relate to
inflationary models.

##### Solution to the Horizon Problem

The first problem we consider is the Horizon one. Inflation provides a
straightforward solution, assuming the universe underwent a period of
accelerated expansion. If the early universe experienced a period in which
with $\ddot{a}>0$, regions that are now observed to be causally disconnected
were indeed in causal contact, justifying their extreme homogeneity in
temperature we observe today.
Interestingly, either of two mutually exclusive possibilities would perfectly
address the Horizon problem: either spacetime is expanding subliminally while
the speed of light has the same value as now (i.e., inflation), or,
equivalently, spacetime is not accelerating but the speed of light had a value
several orders of magnitudes greater that it does now [24]. We can therefore
imagine that light travelled much faster in the early Universe then it does
now, thus allowing causal contact for regions that now are disconnected
because $c$ travels slower. Furthermore, the speed of light underwent a sharp
phase transition at time $t_{c}$, where its value changed from $c^{-}$, the
early universe value, to $c^{+}$, the value we measure today. If we express
the universe’s horizon size in terms of conformal time $\eta_{c}$, we obtain

$r_{h}=c^{-}\eta_{c}$ (2.29)

Then, if $c^{-}$ is indeed greater then $c^{+}$, their ratio compares to the
one of the conformal time now $\eta_{0}$ and the conformal time at $t_{c}$ as

$\frac{c^{-}}{c^{+}}\gg\frac{\eta_{0}}{\eta_{c}}$ (2.30)

because, since we do not have inflation in the early Universe, $\eta_{0}$ and
$\eta_{c}$ are close in value. Expression (2.30) implies that $r_{h}\gg r$,
meaning that the entire observable Universe today has always been in causal
constant, since its radius $r$ is smaller then the radius of the horizon. We
have therefore shown how a varying speed of light with a phase transition can
explain the Horizon problem.

##### Solution to the Flatness Problem

Our next topic is the Flatness problem. In the inflationary model, if we
define a total density parameter as [79]

$\Omega=\frac{8\pi G\rho}{3H^{2}}$ (2.31)

where $H=\frac{\dot{a}}{a}$ is the Hubble parameter, we immediately see that
as $a$ increases, for fixed $\rho$, $\Omega$ tends to unity, thus explaining
why the current Universe’s curvature is observed to be close to $k=1$. A
different solution can be found by allowing $c$ to vary in the Friedmann
equations and in the conservation equation. These, inclusive of any $c$ factor
usually set to one, are

$\biggl{(}\frac{\dot{a}}{a}\biggl{)}=\frac{8\pi
G\rho}{3}-\frac{kc^{2}}{a^{2}}$ (2.32) $\ddot{a}=-\frac{4\pi
G}{3}\biggl{(}\rho+\frac{3p}{c^{2}}\biggl{)}a$ (2.33)
$\dot{\rho}+3\frac{\dot{a}}{a}\biggl{(}\rho+\frac{p}{c^{2}}\biggl{)}=-\rho\frac{\dot{G}}{G}+3\frac{kc^{2}}{4\pi
Ga^{2}}\frac{\dot{c}}{c}$ (2.34)

where the first two equations are derived from the Einstein field equations in
a FLRW metric, while the third one can either be the result of a combination
of the two Friedmann equations or it can be directly derived from
$\nabla_{\mu}T^{\mu\nu}$ assuming a perfect fluid. Now we can define
$\epsilon=\Omega-1$, where $\Omega=\frac{\rho}{\rho_{c}}$ is the total energy
density as above, but expressed in terms of critical density $\rho_{c}$, which
reads

$\rho_{c}=\frac{3}{8\pi G}\biggl{(}\frac{\dot{a}}{a}\biggl{)}^{2}$ (2.35)

Therefore, considering the evolution of $\epsilon$ we obtain

$\dot{\epsilon}=(1+\epsilon)\biggl{(}\frac{\dot{\rho}}{\rho_{c}}-\frac{\dot{\rho_{c}}}{\rho_{c}}\biggl{)}$
(2.36)

Finally, using the Friedmann equations for $\rho$ and the conservation
equation for $\dot{\rho}$ we can get an explicit expression for $\epsilon$ as

$\dot{\epsilon}=\epsilon\frac{\dot{a}}{a}(1+\epsilon)(1+3w)+2\frac{\dot{c}}{c}\epsilon$
(2.37)

where the equation of state is $p=w\rho c^{2}$. In this formulation of the
Flatness problem, we need $\epsilon$ going to zero, such that $\Omega$ goes to
one. If, as before, we assume a sharp phase transition for the value of the
speed of light from $c^{-}$ to $c^{+}$, we see that

$\left|\frac{\dot{c}}{c}\right|\gg\frac{\dot{a}}{a}$ (2.38)

This allows us to neglect the $\frac{\dot{a}}{a}$ term in equation (2.37),
leading to

$\frac{\dot{\epsilon}}{\epsilon}=2\frac{\dot{c}}{c}\Leftrightarrow\log{\epsilon}=2\log{c}\Leftrightarrow\epsilon\propto
c^{2}$ (2.39)

This shows the dependency between the total energy density and the speed of
light, confirming that a sharp change is $c$ would indeed drive $\epsilon$ to
zero, giving $\Omega=1$. This equivalently solves the Flatness problem in
cosmology.

##### Solution to the Cosmological Constant Problem

Finally, we consider the Cosmological Constant problem. As we have seen in
subsection (2.1.3), this problem is two-fold, meaning that we can discriminate
the geometrical one from the vacuum energy problem. While the first one can be
solved using VSL theory, the second one is more complicated [72] and will
require more thoughts and consideration. Therefore, following [24], we have a
conservation equation for matter and cosmological constant content, of the
form

$\dot{\rho_{m}}+3\frac{\dot{a}}{a}\biggl{(}\rho_{m}+\frac{p_{m}}{c^{2}}\biggl{)}=-\dot{\rho_{\Lambda}}-\rho\frac{\dot{G}}{G}+\frac{3kc^{2}}{4\pi
Ga^{2}}\frac{\dot{c}}{c}$ (2.40)

We can now define

$\epsilon:=\frac{\rho_{\Lambda}}{\rho_{m}}$ (2.41)

for $\Lambda$ and, assuming that $\Lambda$ is actually constant, by taking the
time derivative of (2.16) we obtain

$\frac{\dot{\rho_{\Lambda}}}{\rho_{\Lambda}}=2\frac{\dot{c}}{c}-\frac{\dot{G}}{G}$
(2.42)

We now would like an equation for the evolution of $\epsilon_{\Lambda}$.
Taking the time derivative of (2.41) we have

$\displaystyle\dot{\epsilon_{\Lambda}}=\frac{\dot{\rho_{\Lambda}}}{\rho_{m}}-\frac{\rho_{\Lambda}}{\rho_{m}^{2}}\dot{\rho_{m}}$
$\displaystyle=$
$\displaystyle\>\epsilon_{\Lambda}\biggl{[}2\frac{\dot{c}}{c}-\frac{\dot{G}}{G}-\frac{\dot{\rho_{m}}}{\rho_{m}}\biggl{]}$
(2.43)

where in the second line we have used (2.42) expressed for
$\dot{\rho_{\Lambda}}$. Finally, using the conservation equation (2.40) for
$\dot{\rho_{m}}$, the equation of state and $\epsilon=\Omega-1$ we arrive at

$\dot{\epsilon_{\Lambda}}=\epsilon_{\Lambda}\biggl{[}2\frac{\dot{a}}{a}(1+w)+2\frac{\dot{c}}{c}\frac{1+\epsilon_{\Lambda}}{1+\epsilon}\biggl{]}$
(2.44)

In the standard inflationary cosmology, we of course have $\dot{c}=0$. Current
estimates and observations set $\epsilon_{\Lambda}$ very close to one [80],
but this cannot be explained by inflation. In fact, the solution
$\epsilon_{\Lambda}=0$ whether $w>-1$ or $w=-1$ would still allow
$\epsilon_{\Lambda}$ to grow after inflation, thus not explaining its current
value. However, this is not the case if we assume $\dot{c}\neq 0$. Once again,
assuming a sharp phase change in $c$ such that (2.38) is also valid in this
case, we can see that the second term in equation (2.44) dominates. The
condition that

$\frac{\dot{c}}{c}\ll 0$ (2.45)

would indeed drive $\epsilon_{\Lambda}$ to zero, thus implementing a mechanism
to explain the small value of $\epsilon_{\Lambda}$ today. Therefore, assuming
condition (2.38), we find that

$\frac{\dot{\epsilon_{\Lambda}}}{\epsilon_{\Lambda}(1+\epsilon_{\Lambda})}=2\frac{\dot{c}}{c}\frac{1}{1+\epsilon}$
(2.46)

which, combined with equation (2.39) gives the evolution of
$\epsilon_{\Lambda}$ in terms of $\epsilon$

$\frac{\epsilon_{\Lambda}}{1+\epsilon_{\Lambda}}\propto\frac{\epsilon}{1+\epsilon}$
(2.47)

Finally, if we assume, as initial conditions $\epsilon\approx 1$ and
$\epsilon_{\Lambda}\approx 1$, we obtain

$\epsilon_{\Lambda}\propto c^{2}$ (2.48)

which indeed solves the geometrical cosmological constant problem by showing
that a function $c(t)$ with a steep phase transition between $c^{-}$ and
$c^{+}$ would lead $\epsilon_{\Lambda}$ to one, the present observed value.

###  2.2 Minisuperspace Reduction

In this section, we provide explicit derivations of the Einstein-Cartan action
in minisuperspace (MSS) for the Friedmann-Lemaître-Robertson-Walker (FLRW)
metric. The same is done for the Einstein-Hilbert action. Finally, in the
third subsection, we show how the Einstein-Cartan action is equivalent to the
Einstein-Hilbert one in minisuperspace under the assumption that the torsion
is set to zero. We also show to to derive the on-shell condition that relates
the conjugate variables $a^{2}$ and $b$ in MSS. This section is essential as
it will provide the starting point to construct the entire dynamics of varying
constants. We will then extend these results using the Einstein-Cartan action
for the Brans-Dicke, reducing it to minisuperspace in Chapter 3.

####  2.2.1 Einstein-Cartan action in MSS

The Einstain-Cartan theory is an extension of General Relativity that allows
to include torsion in spacetime, relating it to the density of intrinsic
angular momentum of particles [81]. Furthermore, the Einstein-Cartan formalism
of gravity is equivalent to the Einstein-Hilbert one, assuming that the
tetrads are non-degenerate ($det\>e^{a}_{\mu}\neq 0$) and that the torsion is
zero. Therefore, we will proceed deriving the Einstein-Cartan (EC) action in
minisuperspace. The EC action is [82]

$S_{EC}=\frac{c^{4}}{32\pi G}\int{\epsilon_{abcd}\>e^{a}e^{b}R^{cd}}$ (2.49)

where $\epsilon_{abcd}$ is the Levi-Civita tensor, $e^{a}$ is the non-
coordinate tetrad basis defined as

$e^{a}=e^{a}_{\mu}dx^{\mu}$ (2.50)

and $R^{cd}$ is the Einstein-Cartan curvature given by

$R^{a}_{b}=\frac{1}{2}R^{a}_{b\mu\nu}dx^{\mu}dx^{\nu}$ (2.51)

To proceed, we need to introduce Cartan’s first and second equations, which
will allow us to find explicit expressions for the tertad which, given line
element (2.56), we can plug into action (2.49). Cartan’s first equation, which
gives the torsion 2-form, dropping the bases $dx^{\mu}$, is [82]

$T^{a}=De^{a}=de^{a}+\Gamma^{a}_{b}e^{b}$ (2.52)

where $\Gamma^{a}_{\mu b}dx^{\mu}$ is the connection 1-form coming from the
exterior derivative, satisfying the property $\Gamma_{ab}=-\Gamma_{ba}$.
Assuming the torsion-free condition, i.e., $T^{a}=0$, we get

$de^{a}=-\Gamma^{a}_{b}e^{b}$ (2.53)

On the other hand, we have the curvature 2-form given by Cartan’s second
equation as [82]

$R^{a}_{b}=d\Gamma^{a}_{b}+\Gamma^{a}_{c}\Gamma^{c}_{b}$ (2.54)

Finally, given the tetrads, we have, by definition,

$ds^{2}=\eta_{ab}e^{a}\otimes e^{b}$ (2.55)

With these equations, the EC action can be reduced to minisuperspace for the
Friedmann-Lemaître-Robertson-Walker (FLRW) metric. We start from the FLRW
metric in $(t,r,\theta,\phi)$ coordinates [5]

$ds^{2}=-c^{2}N(t)^{2}dt^{2}+a(t)^{2}\biggl{[}\>\frac{dr^{2}}{K(r)^{2}}+r^{2}(d\theta^{2}+\sin{\theta}^{2}d\phi^{2})\biggl{]}$
(2.56)

where $N(t)$ is the lapse function, $a(t)$ is the scale factor,
$K(r)=\sqrt{1-kr^{2}}$ and $k={-1,0,1}$ is the universe’s curvature, which
could be locally modelled by a 3-sphere ($\bm{\mathcal{S}^{3}},k=+1$), by flat
euclidean space ($\bm{\mathcal{E}^{3}},k=0$) and by hyperbolic space
($\bm{\mathcal{H}^{3}},k=-1$). It is important to note the inclusion of the
speed of light $c^{2}$ in the time component of the line element. This will be
important in the next sections, where we require $c$ to vary. Hence, it is
crucial to have it included in the Hamiltonian.
Therefore, using metric (2.56) and equation (2.55), we can obtain the tetrads
as [83]

$e^{0}=cNdt$ (2.57) $e^{1}=\frac{a}{K}dr$ (2.58) $e^{2}=a\>rd\theta$ (2.59)
$e^{3}=a\>r\sin{\theta}d\phi$ (2.60)

We can then compute the exterior derivatives of these tetrads, giving

$de^{0}=0$ (2.61) $de^{1}=\frac{b}{ac}e^{0}\wedge e^{1}$ (2.62)
$de^{2}=\frac{b}{ac}e^{0}\wedge e^{2}+\frac{K}{ar}e^{1}\wedge e^{2}$ (2.63)
$de^{3}=\frac{b}{ac}e^{0}\wedge e^{3}+\frac{K}{ar}e^{1}\wedge
e^{3}+\frac{\cot{\theta}}{ar}e^{2}\wedge e^{3}$ (2.64)

Then, using equation (2.53), we arrive at the expression for the connections

$\Gamma^{1}_{0}=\frac{b}{ac}e^{1}=\frac{b}{Kc}dr$ (2.65)
$\Gamma^{2}_{0}=\frac{b}{ac}e^{2}=\frac{b}{c}d\theta$ (2.66)
$\Gamma^{3}_{0}=\frac{b}{ac}e^{3}=\frac{b}{c}r\sin{\theta}d\phi$ (2.67)
$\Gamma^{2}_{1}=Kd\theta$ (2.68) $\Gamma^{3}_{1}=K\sin{\theta}d\phi$ (2.69)
$\Gamma^{3}_{2}=\cos{\theta}d\phi$ (2.70)

where $b=\frac{\dot{a}}{N}$ is the connection variable, as we will see later
when we show how the EC action reduces to the EH action. Importantly, the
$de^{0}$ term does not lead to any contribution and the expressions form
(2.65) to (2.66) are given by the $de^{1}$, $de^{2}$ and $de^{3}$ terms which
also give expressions from (2.68) to (2.70). In fact, as an example, if we
take $de^{2}$ we see that

$de^{2}=-\Gamma^{2}_{b}e^{b}=-\Gamma^{2}_{0}e^{0}-\Gamma^{2}_{1}e^{1}-\Gamma^{2}_{2}e^{2}-\Gamma^{2}_{3}e^{3}$
(2.71)

The $\Gamma^{2}_{2}$ and $\Gamma^{2}_{3}$ terms do not contribute, whereas the
$\Gamma^{2}_{0}$ will give

$\Gamma^{2}_{0}=\frac{b}{ac}e^{2}$ (2.72)

as one of the contributions, while the $\Gamma^{2}_{1}$ term will give
expression (2.68) as

$-\Gamma^{2}_{1}e^{1}=\frac{K}{ar}e^{1}\wedge
e^{2}\Leftrightarrow\Gamma^{2}_{1}=Kd\theta$ (2.73)

Now we can use equation (2.54) to obtain the curvature 2-form $R^{a}_{b}$. To
begin, we compute the exterior derivative of expressions (2.65) to (2.68)

$d\Gamma^{1}_{0}=\frac{\dot{b}}{Nac^{2}}e^{0}e^{1}$ (2.74)
$d\Gamma^{2}_{0}=\frac{\dot{b}}{Nac^{2}}e^{0}e^{2}+\frac{bK}{ra^{2}c}e^{1}e^{2}$
(2.75)
$d\Gamma^{3}_{0}=\frac{\dot{b}}{Nac^{2}}e^{0}e^{3}+\frac{bK}{ra^{2}c}e^{1}e^{3}+\frac{b\cot{\theta}}{ra^{2}c}e^{2}e^{3}$
(2.76)
$d\Gamma^{3}_{1}=-\frac{k}{a^{2}}e^{1}e^{3}+\frac{K\cot{\theta}}{r^{2}a^{2}}e^{2}e^{3}$
(2.77) $d\Gamma^{3}_{2}=-\frac{1}{a^{2}r^{2}}e^{2}e^{3}$ (2.78)

where $d\Gamma^{2}_{1}=0$ because $K$ is constant. Given these, we can use
equation (2.54) to obtain the curvature 2-forms

$R^{0}_{i}=\frac{\dot{b}}{Nac}e^{0}e^{i}$ (2.79)
$R^{i}_{j}=\frac{b^{2}+kc^{2}}{a^{2}}e^{i}e^{j}$ (2.80)

Finally, we expand the EC action as

$\epsilon_{abcd}\>e^{a}e^{b}R^{cd}=2\epsilon_{0ijk}e^{0}e^{i}R^{jk}+2\epsilon_{ij0k}e^{i}e^{j}R^{0k}$
(2.81)

and using the equations for the curvature 2-forms we obtain

$\epsilon_{abcd}\>e^{a}e^{b}R^{cd}=2\epsilon_{0ijk}e^{0}e^{i}e^{j}e^{k}\biggl{[}\frac{\dot{b}}{Na}+\frac{b^{2}+kc^{2}}{a^{2}}\biggl{]}$
(2.82)

which, ultimately, plugged into action (2.49) gives the Einstein-Cartan action
reduced to minisuperspace for the FLRW metric [84]

$S_{EC}=\frac{3V_{c}}{8\pi G}\int{dt\>[\dot{b}a^{2}+Na(b^{2}+kc^{2})]}$ (2.83)

where $V_{c}=\int{drd\theta d\phi\frac{r^{2}\sin{\theta}}{K}}$ is the
Universe’s co-moving volume. We now derive the Einstein-Hilbert action in
minisuperspace and show how it is equivalent to the Einstein-Cartan action we
found above.

####  2.2.2 Einstein-Hilbert action in MSS

We now reduce the Einstein-Hilbert action to minisuperspace for the Friedmann-
Lemaître-Robertson-Walker (FLRW) metric. This will allow us to then use the
Hamiltonian formalism to derive the canonical variable’s equations of motion,
as well as any other dynamics of the system. We begin from the Einstein-
Hilbert action with $\Lambda=0$ for simplicity [5]

$S_{EH}=\frac{c^{4}}{16\pi G}\int{d^{4}x\>\sqrt{-g}R}$ (2.84)

We now need to derive expressions for the Ricci scalar and for the determinant
of the metric in action (2.84) using the FLRW metric (2.56). We start form the
Ricci scalar, which can be computed via

$R=g^{\mu\nu}R_{\mu\nu}=g^{tt}R_{tt}+g^{rr}R_{rr}+g^{\theta\theta}R_{\theta\theta}+g^{\phi\phi}R_{\phi\phi}$
(2.85)

where the summations only applies to the diagonal components because the
metric is zero everywhere else and where $g^{\mu\nu}$ are the components of
the inverse metric and $R_{\mu\nu}$ is the Ricci tensor defined [48] as

$R_{\mu\nu}=\partial_{\rho}\Gamma^{\rho}_{\mu\nu}-\partial_{\nu}\Gamma^{\rho}_{\mu\rho}+\Gamma^{\sigma}_{\mu\nu}\Gamma^{\rho}_{\sigma\rho}-\Gamma^{\sigma}_{\mu\rho}\Gamma^{\rho}_{\sigma\nu}$
(2.86)

where the $\Gamma^{\rho}_{\mu\nu}$ are the Christoffel symbols. Following
(2.86), we obtain the four non-zero components of the Ricci tensor as

$R_{tt}=-\frac{3}{N^{2}c^{2}}\frac{\ddot{a}}{a}$ (2.87)
$R_{rr}=\frac{1}{N^{2}}\biggl{[}\frac{a\ddot{a}+2\dot{a}^{2}+kc^{2}N^{2}}{1-kr^{2}}\biggl{]}$
(2.88)
$R_{\theta\theta}=\frac{r^{2}}{N^{2}}[a\ddot{a}+2\dot{a}^{2}+kc^{2}N^{2}]$
(2.89)
$R_{\phi\phi}=\frac{r^{2}\sin^{2}{\theta}}{N^{2}}[a\ddot{a}+2\dot{a}^{2}+kc^{2}N^{2}]$
(2.90)

Adding these components following (2.85), we arrive at the expression for the
Ricci scalar

$R=6\frac{kc^{2}}{a^{2}}+\frac{6}{N^{2}}\biggl{[}\frac{\ddot{a}}{a}+\frac{\dot{a}^{2}}{a^{2}}\biggl{]}$
(2.91)

where in all the above we have set $\dot{N}=0$ assuming a constant lapse
function. This result agrees with [85], including the speed of light factor.
Additionally, we can calculate the determinant of the metric, thus obtaining

$\sqrt{-g}=\frac{r^{2}\sin^{2}{\theta}}{\sqrt{1-kr^{2}}}cNa^{3}$ (2.92)

At this point, it is important to remember that we may write action (2.84) as

$S_{EH}=\frac{\kappa}{2}\int{d^{4}x\>\sqrt{-g}R}$ (2.93)

where $\kappa=\frac{c^{4}}{8\pi G}$. As it is normally done, we can set
$\kappa=1$ and so, using this form of the action, combining (2.91) and (2.92)
we arrive at

$S_{EH}=3V_{c}\int{dt\>\frac{1}{N}\dot{a}^{2}a+\frac{1}{N}\ddot{a}a^{2}+Nakc^{2}}$
(2.94)

where $V_{c}={drd\theta d\phi}\frac{c\>r^{2}\sin^{2}{\theta}}{K}$ is the
spacial co-moving volume expressed in terms of polar coordinates. It will be
convenient, however, to include the factor of $\frac{1}{8}$ from the constant
$k$ into action (2.94). In fact, this factor will become important in the next
section, where we will show how the Einstein-Cartan action can lead to the
Einstein-Hilbert one with the addition of a boundary term. Therefore, the
final form of the EH action in MSS that we will use is

$S_{EH}=\frac{3V_{c}}{8\pi
G}\int{dt\>\biggl{[}\frac{1}{N}\dot{a}^{2}a+\frac{1}{N}\ddot{a}a^{2}+Nakc^{2}\biggl{]}}$
(2.95)

From this, we will compare our results in this section with the ones in
subsection (2.2.1) to show how they are equivalent.

####  2.2.3 From Einstein-Cartan to Einstein-Hilbert in MSS

From the previous subsections, we have shown how to derive the EC and the EH
actions in MSS for the FLRW metric. We now wish to understand mainly two
features of these results: how does the connection $b$ appears in the EC
action and how can we obtain the EH action from it. In fact, having assumed
the torsion-free condition as we did, it is expected that the two actions are
equivalent [82]. We begin by considering the EC action

$S_{EC}=\frac{3V_{c}}{8\pi G}\int{dt\>[\dot{b}a^{2}+Na(b^{2}+kc^{2})]}$ (2.96)

which is now integrated by parts to obtain an $\dot{a}$ variable, hopefully
bringing us closer to the EH action. The results in

$S_{EC}=\frac{3V_{c}}{8\pi
G}\int{dt\>[-\dot{(a^{2})}b+Na(b^{2}+kc^{2})]+boundary\>term}\ $ (2.97)

where we stress the importance of including the boundary term. In fact, this
will turn out to be crucial to obtain the EH action. As it is usually done, we
might simply assume that the boundary term vanishes at infinity. This,
however, especially in the context of quantum cosmology, is not a good choice
[86]. Therefore, instead of this assumption, we add to action (2.96) a
counter-term, such that it is the same as the boundary term but with opposite
sign:

$S_{EC}=\frac{3V_{c}}{8\pi
G}\int{dt\>[\dot{b}a^{2}+Na(b^{2}+kc^{2})]}+\frac{3V_{c}}{8\pi
G}[a^{2}b]^{t_{f}}_{t_{i}}$ (2.98)

This addition will automatically cancel the boundary term once action (2.96)
is integrated by parts, removing the necessity to assume vanishing terms at
infinity. Furthermore, it is a perfectly valid term from a dynamical point of
view, as it does not affect the equations of motion in any way: they are in
fact the same with or without it. This additional term is known as the
Gibbons-Hawking-York (GHY) boundary term [87]. We will further explain how
this procedure is key to obtain the EH action, but we firstly focus on the
first question, i.e., how $b$ appears in the EC action and how is it related
to the other canonical variable $a^{2}$. Therefore, including the GHY boundary
term and integrating by parts, action (2.98) now reads

$S_{EC}=\frac{3V_{c}}{8\pi G}\int{dt\>[-\dot{(a^{2})}b+Na(b^{2}+kc^{2})]}$
(2.99)

Varying this action,

$\frac{\delta S_{EC}}{\delta b}=0\Leftrightarrow-\dot{(a^{2})}+2Nab=0$ (2.100)

and, expressing equation (2.100) for $b$, we obtain

$2a\dot{a}=2Nab\Leftrightarrow b=\frac{\dot{a}}{N}$ (2.101)

which is exactly the expression for $b$ we used when deriving the EC action in
MSS. Firstly, this condition only holds on-shell, meaning that it is relevant
at the level of the equations of motion. Secondly, we can now provide some
intuition on what the conjugate variables $a^{2}$ and $b$ really are. The
scale factor is, in the FLRW, directly linked to the metric $g_{\mu\nu}$ of
our spacetime manifold. On the other hand, the connection $\Gamma$ contains
first derivatives of the metric. Projecting this link in MSS, we see that $b$
is indeed given by first derivatives of the scale factor $a$. Therefore we
conclude that in MSS the scale factor represents the metric and its conjugate
$b$ the connection. Hence the name connection we used in the previous
subsection for $b$.

Finally, we can show the equivalence of the EC and EH actions. To begin, using
the fundamental theorem of calculus and the expression for $b$, we can write
the GHY boundary term as

$\frac{3V_{c}}{8\pi G}[a^{2}b]^{t_{f}}_{t_{i}}=\frac{3V_{c}}{8\pi
G}\frac{\partial}{\partial
t}\int^{t_{f}}_{t_{i}}{dt\>\biggl{[}a^{2}\frac{\dot{a}}{N}\biggl{]}}=\frac{3V_{c}}{8\pi
G}\int^{t_{f}}_{t_{i}}{dt\>\biggl{[}2a\frac{\dot{a}^{2}}{N}+\frac{a^{2}\ddot{a}}{N}\biggl{]}}$
(2.102)

Then, plugging $b=\frac{\dot{a}}{N}$ inside action (2.98) and including
expression (2.102), we get

$S_{EC}=\frac{3V_{c}}{8\pi
G}\int{dt\>\biggl{[}\frac{-\dot{a}\dot{(a^{2})}}{N}+\frac{a\dot{a}^{2}}{N}+Nakc^{2}+2a\frac{\dot{a}^{2}}{N}+\frac{a^{2}\ddot{a}}{N}\biggl{]}}$
(2.103)

which, when simplified, gives

$S_{EC}=\frac{3V_{c}}{8\pi
G}\int{dt\>\biggl{[}\frac{1}{N}\dot{a}^{2}a+\frac{1}{N}\ddot{a}a^{2}+Nakc^{2}\biggl{]}}\equiv
S_{EH}$ (2.104)

which is exactly the Einstein-Hilbert action we found in equation (2.95).
Therefore, we now can appreciate the importance of the GHY boundary term:
without it, it would not be possible to link the EC action to the EH action.
The same case would arise if we were to just assume that the boundary term
vanishes at infinity.
To conclude, we can easily see that, starting from the EH action (2.95), if we
substitute the on shell condition $b=\frac{\dot{a}}{N}$ we get

$S_{EH}=\frac{3V_{c}}{8\pi G}\int{dt\>\dot{b}a^{2}+Na(b^{2}+kc^{2})}\equiv
S_{EC}$ (2.105)

which is indeed the Einstein-Cartan action reduced to minisuperspace for the
FLRW metric. This result shows that the EC and EH actions are essentially
equivalent assuming the torsion-free condition and using the GHY boundary
term.

###  2.3 Energy Conservation

In this section, we present the current developments in the field of varying
constant theories. These arise as a natural extension of VSL when additional
constant appearing in the EC action are assumed to be varying. Furthermore,
unlike VSL, these theories see a full implementation of the unimodular
formalism applied to minisuperspace, which allows a far more general treatment
of the subject. Specifically, after the EC action’s reduction to MSS done in
section (2.2), it is possible to generalise the unimodular formalism form
section (1.1) to account for a vector of constants $\bm{\alpha}$, giving
relational physical times which provide the evolution of another set of
constants, called $\bm{\beta}$. The aim of this extension is, however,
different from VSL. In fact, by postulating multiple varying constants, we
consider the overall energy conservation in the Universe, developing models in
which we can have net energy violation or net energy conservation, depending
on the constants involved. These models allow us to explore more the
consequences of braking Local Lorentz Invariance, as much as they provide
interesting insights on possible solutions to the Cosmological Constant
problem (more will be explained in the last two chapters). Therefore, in the
next subsections, we provide the basic results we will use to extend this
formalism to the Brans-Dicke theory in Chapter 3.

####  2.3.1 Full action and constants

Having the EC action in minisuperspace, we can now include matter in out
treatment and show how to derive the equations of motion from the Hamiltonian
direclty. From before, we have the EC action in MSS, which we now write as

$S_{EC}=V_{c}\int{dt\>\alpha_{2}\biggl{[}\dot{b}a^{2}+Na(b^{2}+kc_{g}^{2})\biggl{]}}$
(2.106)

We now label the speed of light that was appearing in action (2.83) as $c^{2}$
differently, specifically as $c^{2}_{g}$. This is because, as we have
discussed in subsection (1.2.3), we should differentiate the speed of light
originating from the gravity metric (hence $c^{2}_{g}$) from other speeds of
light, like the one from a matter content (labelled $c^{2}_{m}$). Secondly, we
have introduced the constant $\alpha_{2}$ [22], which is defined as

$\alpha_{2}=\frac{3c^{2}_{p}}{8\pi G_{P}}$ (2.107)

where $c^{2}_{P}$ and $G_{P}$ are the speed of light and the gravitational
constant originating from Planck’s mass. In fact, going back to the EC action
in MSS, we can re-write it as

$S_{EC}=\frac{3V_{c}M_{P}^{2}}{\hbar
c_{P}}\int{dt\>[\dot{b}a^{2}+Na(b^{2}+kc^{2}_{g})]}$ (2.108)

where $M_{P}^{2}=\frac{\hbar c_{P}}{8\pi G_{P}}$ is the Planck’s mass. For
dimensional reasons, we defined $\hbar$ such that it gives a $c_{P}^{2}$
factor in the numerator, hence obtaining the squared speed of light in the
expression (2.107) for $\alpha_{2}$. We remark that in action (2.83) we really
have two distinct and different speeds of light. One, $c^{2}_{P}$ is the speed
of light coming from the Planck’s mass, which is itself appearing in front of
the EC action together with $G_{P}$ because of the required dimensions of the
action. This is ultimately the reason why we use the subscript $P$ for the
speed of light and the gravitational constant. On the other hand, we have the
speed of light $c^{2}_{g}$ coming from the FLRW metric, which is clearly a
gravitational-type constant, since it is originating from a gravitational
metric.
Continuing, we include the unimodular action, which, according to the
procedure outlined in section (1.1), should be combined with the base theory
$S_{0}$, which is the EC action in this case. This gives

$S_{U}=V_{c}\int{dt\>\dot{\bm{\alpha}}\>\bm{T_{\alpha}}}$ (2.109)

where we have integrated by parts action (1.23) to have an expression where
there is a dynamical constant multiplying a conjugate time. As we will see
shortly, $\bm{\alpha}$ is a vector of constants appearing in the EC action.
The inclusion of the unimodular action will allow us to create the relational
physical times giving the constants’ evolution. Finally, we wish to include a
generic matter action in our analysis. Considering a perfect fluid Lagrangian
[88] [89] and reducing it to MSS as in [90] we arrive at

$S_{m}=V_{c}\int{dt\>\alpha_{3}[\dot{m_{i}}\psi_{i}-Na^{3}\rho]}$ (2.110)

where $\alpha_{3}$ is another constant appearing in from of the matter action,
taking the form

$\alpha_{3}=\frac{G_{M}}{G_{P}}$ (2.111)

Interestingly, as pointed out in [89], the canonical pair
$\dot{m_{i}}\psi_{i}$ in the matter action above forms already a unimodular-
like term, with $m_{i}$ acting as a constant of motion.
Once again, we differentiate the gravitational constant originating from the
Planck’s mass (and therefore related to the gravity action) from $G_{M}$,
which is defining the gravitational coupling to matter, as explained in
subsection (1.2.3). Here, we allow the matter action to accommodate for
different types of fluids, hence the index $i$ allows to move from one species
to the other. Furthermore, $m_{i}$ is the constant of motion dual to the
canonical variable $\psi_{i}$ defined in [22]. Finally, the total energy
density $\rho$ is defined as

$\rho=\rho_{\Lambda}+\rho_{M}=\rho_{\Lambda}+\sum_{i}\rho_{i}$ (2.112)

where the $\rho_{i}$ are the different fluids’ densities given by

$\rho_{i}=\frac{m_{i}}{a^{3(1+w_{i})}}$ (2.113)

with $w_{i}$ being the constant relating the pressure to the density in the
equation of state.
Having defined all the actions that will contribute to our final theory, it is
important to comment on their constants. In fact, the constants $\alpha_{2}$
and $\alpha_{3}$ we found for the EC and matter actions are all part of the
$\bm{\alpha}$ vector appearing in the unimodular action. This vector is then

$\bm{\alpha}=(\alpha_{1},\alpha_{2},\alpha_{3},...)=\biggl{(}\rho_{\Lambda},\frac{3c^{2}_{P}}{8\pi
G_{P}},\frac{G_{M}}{G_{P}},...\biggl{)}$ (2.114)

where we have included also $\alpha_{1}=\rho_{\Lambda}$ in case we wish to
consider a purely unimodular time dependence. Furthermore, we could have
additional constants appearing in the $\bm{\alpha}$ vector, as we will do in
Chapter 3, so $\bm{\alpha}$ is not limited to these only. These constants will
give rise to their relational times defined above as the $\bm{T_{\alpha}}$. On
the other hand, the set of $\bm{\beta}$ constants will the one varying with
respect to these clocks as we have outlined in section (1.2).
We can now add these three actions together, arriving at the final form of the
full action

$S_{full}=V_{c}\int{dt\>\alpha_{2}\dot{b}a^{2}+\alpha_{3}\dot{m_{i}}\psi_{i}+Na\alpha_{2}(b^{2}+kc_{g}^{2})-Na^{3}\alpha_{3}\rho+\dot{\rho_{\Lambda}}T_{\Lambda}}$
(2.115)

where, to begin with, we will consider the cosmological constant $\Lambda$ as
our relational time. Before deriving the Hamiltonian and the equations of
motion, it is crucial to stress a key point. In the action above, the vacuum
energy $\rho_{\Lambda}$ is included in the matter content together with the
other matter species. Therefore, it appears in the matter Lagrangian together
with $\alpha_{3}$. However, this case would be different if we were to
consider a geometrical Cosmological Constant, where we would have a
$\alpha_{2}$ factor appearing in front of the matter Lagrangian.

####  2.3.2 The Hamiltonian and the Dynamics

Using action (2.115) from the previous subsection, we can directly infer the
Hamiltonian of the system. In fact, action (2.115) is, explicitly, the
integral of the Lagrangian

$S=\int{dtL}$ (2.116)

and we can relate the Lagrangian of a system to its Hamiltonian via a Legendre
transform [91]

$L(q,\dot{q},t)=\sum_{i=1}^{n}p_{i}\dot{q}^{i}-H(p,q,t)$ (2.117)

where the $p_{i}$ are the generalised momenta and the $\dot{q_{i}}$ are the
generalised velocities. The generalised momenta are defined as

$p_{i}=\frac{\partial L}{\partial\dot{q}_{i}}$ (2.118)

so we immediately see that action (2.115) is in the form required by (2.117)
to obtain the Hamiltonian. This therefore leads to

$H=-Na\alpha_{2}(b^{2}+kc_{g}^{2})+\alpha_{3}\rho Na^{3}$ (2.119)

Firstly, due to $\alpha_{2}$ in front of the canonical pair in action (2.115),
the conjugate of $b$ needs to be defined as $A^{2}=a^{2}\alpha_{2}$ instead of
simply $a^{2}$. This distinction is very important when evaluating the
Poisson’s brackets for $H$, unless $\alpha_{2}$ is kept constant in the
canonical pair. Doing so completely brakes Local Lorentz Invariance and
generates two different theories, as we will see in the next subsections and
in Chapter 3 for Brans-Dicke. For now, we keep $\alpha_{2}$ fixed, while still
writing the Hamiltonian (2.119) in terms of $A^{2}$ as

$H=-NA\sqrt{\alpha_{2}}(b^{2}+kc_{g}^{2})+\alpha_{3}\alpha_{2}^{-\frac{3}{2}}\rho
NA^{3}$ (2.120)

We obtain an Hamilton constraint on the dynamics of our variable by varying
action (2.115) with respect to N,

$\frac{\delta S_{full}}{\delta N}=0\Leftrightarrow
b^{2}+kc_{g}^{2}=\frac{\alpha_{3}}{\alpha_{2}}\rho a^{2}$ (2.121)

The Poisson’s brackets for two functions $f(p_{i},q_{i},t)$ and
$g(p_{i},q_{i},t)$ are defined as [91]

$\\{f,g\\}=\sum_{i=1}^{N}\biggl{(}\frac{\partial f}{\partial
q_{i}}\frac{\partial g}{\partial p_{i}}-\frac{\partial f}{\partial
p_{i}}\frac{\partial g}{\partial q_{i}}\biggl{)}$ (2.122)

Applying this definition to Hamiltonian (2.120), we arrive at the first
Hamilton’s equation, the equation of motion for $a$

$\dot{a}=\\{a,H\\}=\frac{\partial a}{\partial b}\frac{\partial H}{\partial
A^{2}}-\frac{\partial a}{\partial A^{2}}\frac{\partial H}{\partial
b}=-\frac{\partial a}{\partial A^{2}}\frac{\partial H}{\partial b}$ (2.123)

where the first term vanishes because the variable $a$ does not depends on
$b$. Furthermore, using the trick

$\frac{\partial a}{\partial A^{2}}=\frac{\partial
a}{\partial(a^{2}\alpha_{2})}=\frac{1}{\alpha_{2}}\frac{1}{2}\frac{1}{a}\frac{\partial
a}{\partial a}$ (2.124)

since, in this case, $\alpha_{2}$ is kept fixed, we arrive at

$\dot{a}=\\{a,H\\}=Nb$ (2.125)

Moreover, using once again (2.122) for $b$ we have

$\dot{b}=\\{b,H\\}=\frac{\partial b}{\partial b}\frac{\partial H}{\partial
A^{2}}-\frac{\partial b}{\partial A^{2}}\frac{\partial H}{\partial
b}=\frac{1}{2a\alpha_{2}}\frac{\partial H}{\partial a}$ (2.126)

where again the second term vanishes because $b$ does not depend on $A^{2}$.
Therefore, using the Hamilton constraint (2.121) we arrive at

$\dot{b}=\\{b,H\\}=-\frac{\alpha_{3}}{2\alpha_{2}}(\rho+3p)Na$ (2.127)

where we have also used (2.113) for $\rho$ in the matter term of the
Hamiltonian, such that we obtain a term of the form

$-(1+3w)\rho=-\rho-3w\rho=-(\rho+3p)$ (2.128)

where we invoke the equation of state of a perfect fluid $p=w\rho$.

It is also possible to obtain the equations of motion following the Lagrangian
approach. This implies using the Euler-Lagrange equation for the $a^{2}$ and
the $b$ variables. Starting from the $a$ equation, we first need to integrate
the $\dot{b}a^{2}\alpha_{2}$ term in action (2.115) by parts. This allows us
to have an explicit $\dot{a}$ term appearing in the action. The integration by
parts leads to

$S_{full}=V_{c}\int{dt\>[-2a\dot{a}b\alpha_{2}+\alpha_{3}\dot{m_{i}}\psi_{i}+Na\alpha_{2}(b^{2}+kc_{g}^{2})-Na^{3}\alpha_{3}\rho+\dot{\rho_{\Lambda}}T_{\Lambda}]}$
(2.129)

Therefore, the Euler-Lagrange equation for $a$ is

$\frac{\partial L}{\partial b}-\frac{d}{dt}\frac{\partial
L}{\partial\dot{b}}=0$ (2.130)

resulting in

$-2a\dot{a}\alpha_{2}+2bNa\alpha_{2}=0\Leftrightarrow\dot{a}=Nb$ (2.131)

which is consistent with equation (2.125) and it also gives the relation
between $\dot{a}$ and $b$ we found when considering the EC and EH actions.
Continuing, we obtain the equation of motion for $b$ applying the Euler-
Lagrange equation to the non-integrated by parts action this time. This is
because we require an explicit $\dot{b}$ dependence for the equation of
motion. Therefore, using

$\frac{\partial L}{\partial a^{2}}-\frac{d}{dt}\frac{\partial
L}{\partial\dot{a^{2}}}=\frac{1}{2a}\frac{\partial L}{\partial
a}-\frac{d}{dt}\biggl{[}\frac{1}{2\dot{a}}\frac{\partial
L}{\partial\dot{a}}\biggl{]}=0$ (2.132)

leads to

$\dot{b}\alpha_{2}+\frac{N\alpha_{2}}{2a}(b^{2}+kc_{g}^{2})-\frac{3}{2}\rho\alpha_{3}Na=0\Leftrightarrow\dot{b}=-\frac{\alpha_{3}}{2\alpha_{2}}(\rho+3p)Na$
(2.133)

where, once again, we have used the Hamilton’s constraint (2.121) and equation
(2.128). This alternative approach proves that the two results are equivalent,
it does not require defining $A^{2}$ when $\alpha_{2}$ is not fixed and it
will be useful in the next chapter to check that results agree.

Having derived the full dynamics of the theory, we can consider the energy