conservation equation. Just like in General Relativity we can obtain the
conservation equation from the Bianchi Identities, in the MSS treatment we
derive the same result combining the equations of motion with the Hamilton’s
constraint. This gives us the energy conservation in a Universe described by
the FLRW metric, where we assumed a matter content modeled by a perfect fluid,
including also the cosmological constant. To begin with, we take the time
derivative of the Hamilton’s constraint. We do so because it describes the
time evolution of any field in the theory, so it is expected that dotting it
will produce the desired time dependence of the energy density. Doing so, we
include the possibility for $c_{g}^{2}$ to vary generically with parameter
time $t$, as it is part of the set of constants $\bm{\beta}$ which can do so.
For the moment, we do not specify which of the $\bm{\alpha}$ are providing the
relational time $c_{g}^{2}$ is varying with respect to, as these can be many
different ones, as we will see later.
Proceeding, we dot Hamilton’s constraint, obtaining

$2b\dot{b}+k\frac{dc_{g}^{2}}{dt}=\frac{\alpha_{3}}{\alpha_{2}}\dot{\rho}a^{2}+2\frac{\alpha_{3}}{\alpha_{2}}\rho
a\dot{a}$ (2.134)

where $\alpha_{2}$ and $\alpha_{3}$ are, for the moment, kept constant, thus
not contributing the the conservation equation. Then, using the equations of
motion (2.125) and (2.127) for $b$ and $\dot{b}$ in the equation above and
rearranging we have

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=\frac{k\alpha_{2}}{\alpha_{3}a^{2}}\frac{dc_{g}^{2}}{dt}$
(2.135)

This energy “conservation” equation seems to imply, instead, energy violation,
as we clearly have a energy source term given by a varying $c_{g}^{2}$. This
is an expected result, as by braking Local Lorentz Invariance, we are braking
the symmetric properties of our equations, hence generating source terms.
However, this is not the full picture, because, depending on the constant
$\bm{\alpha}$ chosen and the parameter $\bm{\beta}$, the clock might absorb
all the source energy.
To see this, we consider a $c_{g}^{2}$ dependent on the clock defined by the
cosmological constant (in the literature called the unimodular time), such
that $c_{g}^{2}=c_{g}^{2}(T_{\Lambda})$, where we have $\bm{\beta}=c_{g}^{2}$
and $\bm{\alpha}=\rho_{\Lambda}$. Assuming this dependence provides us with
two extra Hamilton’s equations which give the dynamics of the unimodular time
and its conjugate variable $\rho_{\Lambda}$

$\dot{T_{\Lambda}}=\\{T_{\Lambda},H\\}=-\frac{\partial
H}{\partial\rho_{\Lambda}}=-\alpha_{3}Na^{3}$ (2.136)
$\dot{\rho_{\Lambda}}=\\{\rho_{\Lambda},H\\}=\frac{\partial H}{\partial
T_{\Lambda}}=-k\alpha_{2}\frac{dc_{g}^{2}}{dT_{\Lambda}}Na$ (2.137)

Using the chain rule, equation (2.137) can be re-expressed as

$\dot{\rho_{\Lambda}}=-k\alpha_{2}\frac{dc_{g}^{2}}{dt}\frac{dt}{dT_{\Lambda}}Na$
(2.138)

where we can now plug in equation (2.136) obtaining

$\dot{\rho_{\Lambda}}=\frac{k\alpha_{2}}{\alpha_{3}a^{2}}\frac{dc_{g}^{2}}{dt}$
(2.139)

Equation (2.139) suggests that the source term due to a varying $c_{g}^{2}$ in
equation (2.135) is entirely given by the vacuum energy of the system.

####  2.3.3 Varying $\alpha_{2}$ and $\alpha_{3}$ theory

In the previous subsection, we have assumed that $\alpha_{3}$ is not varying
and that also $\alpha_{2}$ is fixed. This, however, is not a strict
requirement, and allowing their variability will include new terms arising in
the equations of motion and, as a consequence, also in the conservation
equation. We begin by considering the full action of our theory

$S_{full}=V_{c}\int{dt\>\alpha_{2}\dot{b}a^{2}+\alpha_{3}\dot{m_{i}}\psi_{i}+Na\alpha_{2}(b^{2}+kc_{g}^{2})-\alpha_{3}\rho
Na^{3}+\dot{\alpha_{2}}T_{R}+\dot{\alpha_{3}}T_{N}}$ (2.140)

where we have removed the pure unimodular term since we would like to consider
the variations given by $\alpha_{2}$ and $\alpha_{3}$. Instead, we have
included the Ricci $T_{R}$ and Newton $T_{N}$ times, associated with our two
varying constants. This action is the same as (2.115), but now $\alpha_{2}$ is
not fixed anymore, leading to a different equation of motion for $\dot{a}$. It
is essential now that the Hamiltonian is expressed in terms of the new
canonical variable $A^{2}=a^{2}\alpha_{2}$ as

$H=-NA\sqrt{\alpha_{2}}(b^{2}+kc_{g}^{2})+\alpha_{3}\alpha_{2}^{-\frac{3}{2}}\rho
NA^{3}$ (2.141)

where now the canonical pair is $(b,A^{2})$ and not $(b,a^{2})$ anymore as
before.
Following the Poisson’s brackets, we calculate the equation of motion form
$A^{2}$, given by

$\dot{(A^{2})}=\\{A^{2},H\\}=\frac{\partial A^{2}}{\partial b}\frac{\partial
H}{\partial A^{2}}-\frac{\partial A^{2}}{\partial A^{2}}\frac{\partial
H}{\partial b}=-\frac{\partial H}{\partial b}$ (2.142)

which, computing the LHS and the RHS leads to

$2a\dot{a}\alpha_{2}+a^{2}\dot{\alpha_{2}}=2bNA\sqrt{\alpha_{2}}=2bNa\alpha_{2}$
(2.143)

eventually giving

$\dot{a}+\frac{\dot{\alpha_{2}}}{2\alpha_{2}}=Nb$ (2.144)

On the other hand, the equation of motion for $b$ is

$\dot{b}=\\{b,H\\}=\frac{\partial b}{\partial b}\frac{\partial H}{\partial
A^{2}}-\frac{\partial b}{\partial A^{2}}\frac{\partial H}{\partial
b}=\frac{\partial H}{\partial A^{2}}=\frac{1}{2a\alpha_{2}}\frac{\partial
H}{\partial a}$ (2.145)

where we have used the definition of $A^{2}$ and we are now using the
Hamiltonian expressed in terms of $a$ instead. Furthermore, using equation
(2.128) to substitute $\rho$, we arrive at the final form

$\dot{b}=-\frac{\alpha_{3}}{2\alpha_{2}}(\rho+3p)Na$ (2.146)

It is also possible to derive the same equations following the Lagrangian
approach. To begin with, we integrate action (2.140) by parts, assuming that
$\alpha_{2}$ is not constant, hence obtaining

$S_{full}=V_{c}\int{dt\>-ba^{2}\dot{\alpha_{2}}-2a\dot{a}b\alpha_{2}+\alpha_{3}\dot{m}_{i}\psi_{i}+Na\alpha_{2}(b^{2}+kc_{g}^{2})-\alpha_{3}\rho
Na^{3}+\dot{\alpha_{2}}T_{R}+\dot{\alpha_{3}}T_{N}}$ (2.147)

Then, writing the Euler-Lagrange equation for $a$

$\frac{\partial L}{\partial b}-\frac{d}{dt}\frac{\partial
L}{\partial\dot{b}}=0$ (2.148)

we obtain

$-a^{2}\dot{\alpha_{2}}-2a\dot{a}\alpha_{2}+2Nab\alpha_{2}=0\Leftrightarrow\dot{a}+\frac{\dot{\alpha_{2}}}{2\alpha_{2}}a=Nb$
(2.149)

On the other hand, using the Euler-Lagrange equation for $b$ on the original
action (2.140),

$\frac{\partial L}{\partial a^{2}}-\frac{d}{dt}\frac{\partial
L}{\partial\dot{a}^{2}}=\frac{1}{2a}\frac{\partial L}{\partial
a}-\frac{d}{dt}\biggl{[}\frac{1}{2\dot{a}}\frac{\partial
L}{\partial\dot{a}}\biggl{]}=0$ (2.150)

invoking the Hamilton’s constraint and equation (2.128) for $\rho$, we finally
get

$\alpha_{2}\dot{b}+\frac{N\alpha_{2}}{2a}(b^{2}+kc_{g}^{2})-\frac{3\alpha_{2}}{2}\rho
Na^{3}=0\Leftrightarrow\dot{b}=-\frac{\alpha_{3}}{2\alpha_{2}}(\rho+3p)Na$
(2.151)

Once again, as before, we dot Hamilton’s constraint, assuming $\alpha_{2}$ and
$\alpha_{3}$ to be varying. This leads to

$2b\dot{b}+k\frac{dc_{g}^{2}}{dt}=\frac{\dot{\alpha_{2}}}{\alpha_{2}}\rho
a^{2}-\frac{\alpha_{3}\rho
a^{2}}{\alpha_{2}^{2}}\dot{\alpha_{2}}+\frac{\alpha_{3}}{\alpha_{2}}a^{2}\dot{\rho}+2\frac{\alpha_{3}}{\alpha_{2}}\rho
a\dot{a}$ (2.152)

Finally, using the equations of motion above for $b$ and $\dot{b}$, we find
the conservation equation

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=-\frac{\dot{\alpha_{3}}}{\alpha_{3}}\rho+\frac{\dot{\alpha_{2}}}{\alpha_{2}}\frac{\rho-3p}{2}+\frac{k\alpha_{2}}{\alpha_{3}a^{2}}\frac{dc_{g}^{2}}{dt}$
(2.153)

We notice that, besides the $\dot{\alpha_{3}}$ term which we would have
expected having allowed $\alpha_{3}$ to vary, we also get the second term on
the RHS for a varying $\alpha_{2}$. As before, we have additional equations
coming from the Hamiltonian which give the dynamics of the clock and its
constant. In this case, since we assume $\alpha_{2}$ and $\alpha_{3}$ to vary,
we have four equations in total: one pair for the Ricci time, and the second
pair for the Newton time. These are the times $c_{g}^{2}$ depends upon, as

$\dot{T_{R}}=\\{T_{R},H\\}=-\frac{\partial
H}{\partial\alpha_{2}}=\frac{\alpha_{3}}{\alpha_{2}}\frac{\rho-3p}{2}Na^{3}$
(2.154) $\dot{\alpha_{2}}=\\{\alpha_{2},H\\}=\frac{\partial H}{\partial
T_{R}}=-\alpha_{2}k\frac{\partial c_{g}^{2}}{\partial T_{R}}Na$ (2.155)

where in the Ricci time equation we have used the Hamilton’s constraint. On
the other hand, the equations for $T_{N}$ and $\alpha_{3}$ are

$\dot{T_{N}}=\\{T_{N},H\\}=-\frac{\partial H}{\partial\alpha_{3}}=-\rho
Na^{3}$ (2.156) $\dot{\alpha_{3}}=\\{\alpha_{3},H\\}=\frac{\partial
H}{\partial T_{N}}=-\alpha_{2}k\frac{\partial c_{g}^{2}}{\partial T_{N}}Na$
(2.157)

where we stress once again that, taking these derivatives, we have used
Hamiltonian (2.141) expressed in terms of $A^{2}$. To show that equation
(2.154) is indeed a conservation equation, we consider that $c_{g}^{2}$ has a
double dependence on $T_{R}$ and $T_{N}$, which implies that

$\frac{dc_{g}^{2}}{dt}=\frac{\partial c_{g}^{2}}{\partial
T_{R}}\dot{T_{R}}+\frac{\partial c_{g}^{2}}{\partial T_{N}}\dot{T_{N}}$
(2.158)

which, using the Ricci and the Newton times equations, becomes

$\frac{dc_{g}^{2}}{dt}=\frac{\alpha_{2}}{\alpha_{3}}\frac{\rho-3p}{2}\frac{\partial
c_{g}^{2}}{\partial T_{R}}Na^{3}-\rho\frac{\partial c_{g}^{2}}{\partial
T_{N}}Na^{3}$ (2.159)

Therefore, plugging the equations for $\alpha_{2}$ and $\alpha_{3}$ into
equation (2.154) and also using equation (2.159), we can show that

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=0$ (2.160)

This result shows how a gravitational parameter, $c_{g}^{2}$, depending on a
gravitational clock does not lead to any net energy violation. It occurs
because, if the clock is gravitational, there is no matter component absorbing
the constant’s variation, so energy is fully conserved. We will investigate
this pattern further once a more complete picture is build, extending these
results to Brans-Dicke theory.

##  Chapter 3 Brans-Dicke and Evolving Laws

In our final chapter, we present the original results achieved in this
research by applying varying constants to energy conservation in cosmology. To
do so, we use the Brans-Dicke action coupled with the unimodular one. The
results are constructed around two main bifurcations: a non-dynamical Brans-
Dicke theory and a dynamical one. In the first case, our results mimic the
ones obtained in [22], with the substitution $\alpha_{2}\rightarrow\phi$.
However, new results are obtained when considering a fixed scalar field
$\phi_{0}$ in the canonical term, such that the $\phi$ multiplying the
Hamiltonian is still varying. On the other hand, when including the Brans-
Dicke kinetic term, we develop several scenarios of varying constants. All of
these produce new results, with particular interest given by the
$\omega=\omega(T_{\Lambda})$ and $c_{g}^{2}=c_{g}^{2}(T_{\omega})$ scenarios.
These are specifically relevant for phenomenology, also providing an insight
in the Cosmological Constant problem, as we will study in [63]. Therefore, to
summarise, in section (3.1) we reduce the Brans-Dicke action to
minisuperspace, showing how to include the kinetic term in the full
Hamiltonian. In section (3.2) we derive the equations of motion for a non-
dynamical Brans-Dicke theory, focusing on the $\phi_{0}$ cases. Finally,
section (3.3) develops the multiple scenarios arising form the dynamical
Brans-Dicke theory, highlighting the main phenomenological results.

###  3.1 Brans-Dicke in MSS

In this short section, we derive the form of the Kinetic Brans-Dicke action in
minisuperspace. As we will see, we will have two cases, one where we set
$\omega$, the Brans-Dicke parameter, to zero and the other one where we use
the full Brans-Dicke action. It is therefore necessary, for the ladder case,
to have the full Brans-Dicke action in MSS. This will allow us to infer the
Brans-Dicke kinetic Hamiltonian, which will be at the heart of our original
results. Finally, we will highlight how including or excluding the $c$ factor
appearing in the Brans-Dicke Hamiltonian can lead to different results,
providing an additional parameter that could be varying, as we will explore in
our future work [63].

####  3.1.1 Reducing the Brans-Dicke action to MSS

Our starting point is the Brans-Dicke action, which should be reduced to
minisuperspace to implement the formalism outlined in the previous chapter.
Firstly, in the Brans-Dicke action, the Ricci scalar term is essentially
equivalent to the Einstein-Hilbert action, except for the constant multiplying
$R$. In fact, if we compare the EH action with the Brans-Dicke one for
$\omega=0$ (non-dynamical Brans-Dicke)

$S_{EH}=\alpha_{2}\int{d^{4}x\>\sqrt{-g}R}$ (3.1)
$S_{\omega=0}=\int{d^{4}x\sqrt{-g}\>\phi R}$ (3.2)

we immediately notice that the scalar field $\phi$ plays the role of
$\alpha_{2}$ once we do not assume a dynamical scalar field. Of course, the
scalar field is nothing but a variable gravitational constant, while
$\alpha_{2}$ includes also the speed of light, but up to a constant (including
the numerical factor and $\pi$) the two theories are equivalent. Therefore, it
is straightforward to reduce the Brans-Dicke action to MSS assuming
$\omega=0$. We simply take action (2.106), replace $\alpha_{2}\rightarrow\phi$
and adjust the constants to get

$S_{\omega=0}=\frac{3V_{c}}{8\pi}\int{dt\>\phi\biggl{[}\dot{b}a^{2}+Na(b^{2}+kc_{g}^{2})\biggl{]}}$
(3.3)

or, more simply, by including the pre-factor $\frac{3}{8\pi}$ in the
definition of $\phi$, we get the final form of the Brans-Dicke action

$S_{\omega=0}=V_{c}\int{dt\>\phi\biggl{[}\dot{b}a^{2}+Na(b^{2}+kc_{g}^{2})\biggl{]}}$
(3.4)

Action (3.4) will be the starting point of the original results we will
introduce in this chapter. Importantly, like in the varying $\alpha_{2}$ case,
when deriving the Hamilton’s equations, it is $A^{2}=a^{2}\phi$ to be used as
a canonical variable. This is even more relevant here because $\phi$, unlike
$\alpha_{2}$, is inherently a variable in the Brans-Dicke theory, modelling
the change of the gravitational constant from spacetime position to spacetime
position.
Since the only difference between action (3.4) and the full Brans-Dicke is the
kinetic term, we might just consider it and understand how to obtain the
Brans-Dicke Hamiltonian in MSS from it. We begin by considering the kinetic
term as

$-\frac{\omega}{\phi}\partial_{\mu}\phi\>\partial^{\mu}\phi=-\frac{\omega}{\phi}[g^{tt}\partial_{t}\phi\>\partial_{t}\phi+g^{ij}\partial_{i}\phi\>\partial_{j}\phi]=-\frac{\omega}{\phi}g^{tt}\dot{\phi}^{2}$
(3.5)

because the scalar field $\phi$ is exclusively time-dependent, so that all its
spacial derivatives vanish. Including the usual $\sqrt{-g}$ gives the full
kinetic term as

$S_{K.T.}=\int{d^{4}x\>\sqrt{-g}\>\biggl{[}-\frac{\omega}{\phi}g^{tt}\dot{\phi}^{2}\biggl{]}}$
(3.6)

Furthermore, from the FLRW metric, we have that
$g^{tt}=-\frac{1}{c^{2}N^{2}}$, and that

$\sqrt{-g}=\frac{r^{2}\sin^{2}{\theta}}{\sqrt{1-kr^{2}}}cNa^{3}$ (3.7)

These two expressions combined lead to the standard form [92]

$S_{K.T.}=V_{c}\int{dt\>Na^{3}\frac{\omega}{\phi}\frac{\dot{\phi}^{2}}{N^{2}}}=V_{c}\int{dt\>\dot{\phi}^{2}\frac{\omega}{\phi}\frac{a^{3}}{Nc^{2}}}$
(3.8)

where the remaining factors coming from $\sqrt{-g}$ have been absorbed into
$V_{c}$. Following the procedure outlined in [93], we obtain the canonical
momentum conjugate to $\phi$ as

$\pi_{\phi}=\frac{\delta
S_{K.T.}}{\delta\dot{\phi}}=2\omega\frac{\dot{\phi}}{\phi Nc}a^{3}$ (3.9)

At this point it is important to notice the extra factor of $c$ multiplying
$N$ in the conjugate momentum expression and in the kinetic action in MSS. If
we leave it in the kinetic term, it adds another constant that can be varying.
However, in this work, for simplicity, we absorb the $c$ factor into the
parameter $\omega$, while its full inclusion in the theory will be the object
of next research [63]. Using the Fourier transform on the Lagrangian as
described in (2.117), we can write

$S_{K.T.}=V_{c}\int{dt\>\omega\frac{\dot{\phi}^{2}}{\phi}\frac{a^{3}}{N}}\overset{!}{=}V_{c}\int{dt\>\dot{\phi}\pi_{\phi}-H_{BD}}$
(3.10)

such that, re-expressing in terms of $H_{BD}$ we obtain

$H_{BD}(\phi,\pi_{\phi})=\dot{\phi}\pi_{\phi}-\omega\frac{\dot{\phi}^{2}}{\phi}\frac{a^{3}}{N}$
(3.11)

Now, following again the prescriptions given by [93], we find an expression
for $\dot{\phi}$ by inverting the conjugate momentum equation, such that

$\pi_{\phi}=\frac{2\omega}{\phi}\frac{\dot{\phi}}{N}a^{3}\Leftrightarrow\dot{\phi}=\frac{\phi\pi_{\phi}N}{2\omega
a^{3}}$ (3.12)

We can use the expression above for $\dot{\phi}$ into the Hamiltonian (3.11),
leading to

$H_{BD}(\phi,\pi_{\phi})=\frac{\phi\pi_{\phi}^{2}N}{2\omega
a^{3}}-\frac{\omega}{\phi}\frac{\phi^{2}\pi_{\phi}^{2}N^{2}}{4\omega^{2}a^{6}}\frac{a^{3}}{N}=\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\omega
a^{3}}N$ (3.13)

This is finally the Hamiltonian coming from the Brans-Dicke kinetic term
which, which, together with the canonical pair $\dot{\phi}\pi_{\phi}$, creates
the kinetic Brans-Dicke Lagrangian. Therefore, combining it with action (3.4),
we arrive at the full Brans-Dicke action in minisuperspace

$S_{BD}=V_{c}\int{dt\>\biggl{[}\phi\dot{b}a^{2}+\dot{\phi}\pi_{\phi}+\phi
Na(b^{2}+kc_{g}^{2})-\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\omega
a^{3}}N\biggl{]}}$ (3.14)

Concluding, we can extend the purely gravitational action to include a
coupling to matter and the generalised unimodular term, as we did in Chapter
2. The unimodular term will allow us to have relational times for the
constants evolution, while the matter action will provide the additional
$\alpha_{3}$ parameter. Doing so, we arrive at the complete unimodular Brans-
Dicke action coupled to matter (assuming a perfect fluid as before):

$S=V_{c}\int{dt\>\biggl{[}\phi\dot{b}a^{2}+\dot{\phi}\pi_{\phi}+\alpha_{3}\dot{m}_{i}\psi_{i}+\phi
Na(b^{2}+kc_{g}^{2})-\alpha_{3}\rho
Na^{3}-\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\omega
a^{3}}N+\bm{\dot{\alpha}}\bm{T_{\alpha}}\biggl{]}}$ (3.15)

This theory will be the starting point of all the following results. It is
also the most general action we will use for Brans-Dicke, so each specific
scenario will differ mostly because of the unimodular term, depending on which
constant we are varying and with respect to which time. These could be very
simple or also very intricate combinations as we will see in the following.
Also, we will explore a bifurcation in the theory given by only fixing $\phi$
in the conjugate pair term. This will be the object of study in the next
sections and subsections.

###  3.2 Non-Dynamical Brans-Dicke

In this section, we present the simplified case of a non-dynamical scalar
field in the Brans-Dicke theory. Accordingly, $\omega=0$, so we cannot invoke
any dependence on this parameter. Importantly, this scenario is essentially
equivalent to the one covered in section (2.3), with the substitution
$\alpha_{2}\rightarrow\phi$, such that we include a dynamical $G$, as required
by the Brans-Dicke theory. However, we investigate a new theory where Local
Lorentz Invariance is fully broken by a constant scalar field $\phi_{0}$
multiplying the canonical term while the scalar field appearing in the
Hamiltonian is left varying. This theory is different from the fixed
$\alpha_{2}$ one analysed in subsection (2.3.2), as it leads to different
equations of motion. Therefore, we derive the equations of motion and the most
general conservation equations for the $\phi_{0}$ and full $\phi$ cases. We
then present some varying constants scenarios for the $\phi_{0}$ case,
excluding any full $\phi$ scenarios, since they are equivalent to previous
sections.

####  3.2.1 The Equations of Motion for the $\phi_{0}$ scenario

This subsection derives the equations of motion and the general conservation
equation for a non-dynamical Brans-Dicke theory, with a decoupled scalar field
$\phi_{0}$. Importantly, given the fixed scalar field in the canonical pair,
this theory shows a totally broken Local Lorentz Invariance, as pointed out in
[22]. The general action in MSS is

$S=V_{c}\int{dt\>\phi_{0}\>\dot{b}a^{2}+\alpha_{3}\dot{m}_{i}\psi_{i}+\phi
Na(b^{2}+kc_{g}^{2})-\alpha_{3}\rho Na^{3}}+S_{U}$ (3.16)

where $S_{U}$ is the unimodular action that will provide the evolution times
and $\dot{m}_{i}\psi_{i}$ is the canonical term from the generic matter
action. The Hamiltonian, obtained via Legendre transform of the Lagrangian in
the action above, is

$H=-\phi Na(b^{2}+kc_{g}^{2})+\alpha_{3}\rho Na^{3}$ (3.17)

Following the standard requirement [93] for the Hamilton’s constraint, we
obtain

$H\overset{!}{=}0\Leftrightarrow b^{2}+kc_{g}^{2}=\frac{\alpha_{3}\rho
a^{2}}{\phi}$ (3.18)

which we use to derive the conservation equation. However, before proceeding,
we need to obtain the equations of motion for the canonical pair $(b,a^{2})$.
In this case, since the scalar field $\phi$ is multiplying the canonical pair,
it is better to express the Hamiltonian in terms of a new canonical variable
$A^{2}=\phi_{0}a^{2}$. Furthermore, when evaluating the Poisson’s brackets, we
will use $A^{2}$ instead of $a^{2}$. This naturally leads

$\dot{(A^{2})}=\\{A^{2},H\\}=\frac{\partial A^{2}}{\partial b}\frac{\partial
H}{\partial A^{2}}-\frac{\partial A^{2}}{\partial A^{2}}\frac{\partial
H}{\partial b}=-\frac{\partial H}{\partial b}$ (3.19)

giving

$2a\dot{a}\phi_{0}=2bNa\phi\Leftrightarrow\dot{a}=\frac{\phi}{\phi_{0}}Nb$
(3.20)

which is the equation of motion for $a$. On the other hand, the same procedure
can be applied to derive the equation of motion for the other conjugate
variable $b$

$\dot{b}=\\{b,H\\}=\frac{\partial b}{\partial b}\frac{\partial H}{\partial
A^{2}}-\frac{\partial b}{\partial A^{2}}\frac{\partial H}{\partial
A^{2}}=\frac{\partial H}{\partial A^{2}}=\frac{1}{2a\phi_{0}}\frac{\partial
H}{\partial a}$ (3.21)

giving

$\displaystyle\dot{b}$ $\displaystyle=\frac{1}{2a\phi_{0}}\frac{\partial
H}{\partial a}$ (3.22)
$\displaystyle=\frac{1}{2a\phi_{0}}\biggl{[}-N\phi(b^{2}+kc_{g}^{2})-3w\frac{\alpha_{3}m}{a^{3w+1}}N\biggl{]}$
(3.23) $\displaystyle=\frac{1}{2a\phi_{0}}\biggl{[}-\alpha_{3}\rho
Na^{2}-3w\frac{\alpha_{3}m}{a^{3w+1}}N\biggl{]}$ (3.24)
$\displaystyle=-\frac{1}{2}\frac{\alpha_{3}\rho}{\phi_{0}}Na-\frac{3}{2}w\frac{\alpha_{3}\rho}{\phi_{0}}Na$
(3.25) $\displaystyle=\frac{\alpha_{3}}{\phi_{0}}\frac{(-\rho-3w\rho)}{2}Na$
(3.26) $\displaystyle=-\frac{\alpha_{3}}{2\phi_{0}}(\rho+3p)Na$ (3.27)

where we have used the Hamilton’s constraint for $(b^{2}+kc_{g}^{2})$ and the
definition of $\rho$ from (2.113). As a side note, we remark that it is
possible to obtain the same results using the Lagrangian approach, as we
showed in subsection (2.3.3). Now we can derive the conservation equation from
the condition that the Hamiltonian must be conserved with time since, if
$H=0$, then it is also true that $\dot{H}=0$. Therefore, taking the time
derivative of the Hamilton’s constraint, we obtain

$2b\dot{b}+k\frac{dc_{g}^{2}}{dt}=\frac{\dot{\alpha_{3}}\rho
a^{2}}{\phi}+\frac{\alpha_{3}\dot{\rho}a^{2}}{\phi}+2\frac{\alpha_{3}\rho
a\dot{a}}{\phi}-\frac{\alpha_{3}\rho a^{2}}{\phi^{2}}\dot{\phi}$ (3.28)

which, rearranged for the standard form of the conservation equation and using
the equations of motion for the canonical pair, gives

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=-\frac{\dot{\alpha_{3}}}{\alpha_{3}}\rho+\frac{\dot{\phi}}{\phi}\rho+\frac{k\phi}{\alpha_{3}a^{2}}\frac{dc_{g}^{2}}{dt}$
(3.29)

Interestingly, we have a $\dot{\phi}$ term behaving like the
$\dot{\alpha_{3}}$ one. This term is new compared to previous theories [22]
and it arises from totally braking Local Lorentz Invariance, because of the
constant scalar field in the canonical term.

####  3.2.2 The Equations of Motion for the full $\phi$ scenario

While the results in the previous subsection are different from the scenario
in subsection (2.3.2) (covered in [22]) because of a varying $\phi$ in the
Hamiltonian but a fixed one in the canonical pair, the following scenario is
fully equivalent to subsection (2.3.3). The only difference between the two is
that, instead of $\alpha_{2}$ which is postulated to vary, we have the scalar
field $\phi$, which is naturally varying by construction of the non-kinetic
Brans-Dicke action. However, the equations of motion and the conservation
equation are the same in both scenarios. We therefore include a brief summary
of the results obtained to show how the scalar field treatment is equivalent
and to provide a few examples of energy conservation or violation. Beginning
from the Brans-Dicke action with $\omega=0$ in MSS

$S=V_{c}\int{dt\>\phi\>\dot{b}a^{2}+\alpha_{3}\dot{m}_{i}\psi_{i}+\phi
Na(b^{2}+kc_{g}^{2})-\alpha_{3}\rho Na^{3}})+S_{U}$ (3.30)

we immediately have the Hamiltonian

$H=-\phi Na(b^{2}+kc_{g}^{2})+\alpha_{3}\rho Na^{3}$ (3.31)

which gives also the Hamilton’s constraint

$b^{2}+kc_{g}^{2}=\frac{\alpha_{3}\rho a^{2}}{\phi}$ (3.32)

Simply following the procedure in subsection (2.3.3), with the substitution
$\alpha_{2}\rightarrow\phi$ and the necessary adjustment for $A^{2}=\phi
a^{2}$, we obtain the equations of motion for $a$ and $b$, which read

$\dot{a}+\frac{\dot{\phi}}{2\phi}=Nb$ (3.33)
$\dot{b}=-\frac{\alpha_{3}}{2\phi}(\rho+3p)Na$ (3.34)

Dotting the Hamilton’s constraint, rearranging and using the equations of
motion we arrive at the general conservation equation

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=-\frac{\dot{\alpha_{3}}}{\alpha_{3}}\rho+\frac{\dot{\phi}}{\phi}\frac{\rho-3p}{2}+\frac{k\phi}{\alpha_{3}a^{2}}\frac{dc_{g}^{2}}{dt}$
(3.35)

From this point onward, it would be just a matter of choice or interest to
consider which constants take part to the $\bm{\alpha}$ and which ones to the
$\bm{\beta}$. As extensively illustrated in [22], we might have scenarios
where $c_{g}^{2}$ depends on the Ricci time $T_{\phi}$, such that
$c_{g}^{2}=c_{g}^{2}(T_{\phi})$. This does not lead to any net energy
violation because we are considering a gravitational parameter depending on a
gravitational clock, following a pattern we will analyse later.
We could, as well, consider the speed of light depending on the unimodular
time, such that $c_{g}^{2}=c_{g}^{2}(T_{\Lambda})$. In this case, we do obtain
a net energy violation, as the variation in $c_{g}^{2}$ is entirely absorbed
by the unimodular clock $T_{\Lambda}$, leading all the energy in the vacuum
energy density $\rho_{\Lambda}$.
Other more exotic scenarios include a matter speed of light $c_{m}^{2}$
depending on both the Ricci and the Newton times. This setup leads, as well,
to net energy violation, because we are considering a matter parameter
depending on a gravitational clock. This situation is the exactly parallel of
the previous one, where we had a gravitational parameter depending on a matter
clock.

####  3.2.3 Fixed $\phi_{0}$ with $c_{g}^{2}=c_{g}^{2}(T_{\phi})$

The first interesting theory to consider is when the speed of light depends on
the Ricci-like time $T_{\phi}$. In this case, we have that $\bm{\alpha}=\phi$
and $\bm{\beta}=c_{g}^{2}$, such that our action reads

$S=V_{c}\int{dt\>\phi_{0}\>\dot{b}a^{2}+\alpha_{3}\dot{m_{i}}\psi_{i}+\phi
Na(b^{2}+kc_{g}^{2})-\alpha_{3}\rho Na^{3}}+\dot{\phi}T_{\phi}$ (3.36)

This leads to the following Hamiltonian:

$H=-\phi Na(b^{2}+kc_{g}^{2})+\alpha_{3}\rho Na^{3}$ (3.37)

The additional equations of motions for $\phi$ and $T_{\phi}$ are obtained by
computing the Poisson’s brackets:

$\dot{\phi}=\\{\phi,H\\}=\frac{\partial H}{\partial T_{\phi}}=-\phi
Nak\frac{dc_{g}^{2}}{dT_{\phi}}$ (3.38)
$\dot{T}_{\phi}=\\{T_{\phi},H\\}=-\frac{\partial
H}{\partial\phi}=Na(b^{2}+kc_{g}^{2})=\frac{\alpha_{3}\rho Na^{3}}{\phi}$
(3.39)

where, in the $T_{\phi}$ equation, we have used Hamilton’s constraint. From
the previous derivation, the most general conservation equation for which
$\phi$ and $c_{g}^{2}$ are varying is

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=\frac{\dot{\phi}}{\phi}\rho+\frac{k\phi}{a^{2}\alpha_{3}}\frac{dc_{g}^{2}}{dt}$
(3.40)

Using the equation of motion for $\phi$ in the first term and the
$\dot{T_{\phi}}$ equation in the second one after applying the chain rule, we
arrive at the standard conservation equation

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=0$ (3.41)

thus not having, as expected, any energy violation. Crucially, in the case of
a scalar field coupled to the canonical pair $\dot{b}a^{2}$ as in the Brans-
Dicke theory, the conjugate variable to $\dot{b}$ is $A^{2}=\phi a^{2}$.
However, this change only applies to theories where the $\phi$ is dynamical in
both the canonical pair and in the Hamiltonian. Unlike these theories, in our
case we have a non-dynamical $\phi_{0}$ coupled to the canonical pair. It is
therefore redundant to express the Hamiltonian in terms of $A^{2}$, so we can
simply use the standard Hamiltonian in terms of $a^{2}$ when taking $\phi$
derivatives.

####  3.2.4 Fixed $\phi_{0}$ with $c_{m}^{2}=c_{m}^{2}(T_{\phi},T_{N})$

In this scenario, we consider the energy violation arising from a matter
parameter depending on both a gravitational and matter clock. This is the case
when $c_{m}^{2}$, the matter speed of light, is part of the $\bm{\beta}$ and
its clocks are given by $\phi$ and $\alpha_{3}$. This is possible because,
from [22], we can specify that $\rho_{i}=\rho_{i}(n_{i},c_{m})$, thus giving
the derivatives of $c_{m}^{2}$ when it is varying. For completeness, we
postulate the same time dependency for $c_{g}^{2}$, as this is a source term
in the conservation equation and doing so we can substitute in it expressions
from the equations of motion of the relational times $T_{\phi}$ and $T_{N}$.
Our starting action is

$S=V_{c}\int{dt\>\phi_{0}\dot{b}a^{2}+\alpha_{3}\dot{m_{i}}\psi_{i}+\phi
Na(b^{2}+kc_{g}^{2})-\alpha_{3}\rho
Na^{3}+\dot{\phi}T_{\phi}+\dot{\alpha_{3}}T_{N}}$ (3.42)

directly giving the Hamiltonian

$H=-\phi Na(b^{2}+kc_{g}^{2})+\alpha_{3}\rho Na^{3}$ (3.43)

From subsection (3.2.1) we have the most general conservation equation, which,
in this case, contains all three source terms:

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=-\frac{\dot{\alpha_{3}}}{\alpha_{3}}\rho+\frac{\dot{\phi}}{\phi}\rho+\frac{\phi
k}{\alpha_{3}a^{2}}\frac{dc_{g}^{2}}{dt}$ (3.44)

We can now obtain the additional Hamilton’s equations. In each pair, the first
equation provides the variation of the constant of motion of the clock (it is
the equation of motion of the constant, if we allow this oxymoron), while the
second one is the equation of motion of the clock itself:

$\dot{\phi}=\\{\phi,H\\}=\frac{\partial H}{\partial T_{\phi}}=-\phi
kNa\frac{\partial c_{g}^{2}}{\partial
T_{\phi}}+\alpha_{3}Na^{3}\frac{\partial\rho}{\partial
c_{m}^{2}}\frac{\partial c_{m}^{2}}{\partial T_{\phi}}$ (3.45)
$\dot{T_{\phi}}=\\{T_{\phi},H\\}=-\frac{\partial
H}{\partial\phi}=\frac{\alpha_{3}\rho}{\phi}Na^{3}$ (3.46)
$\dot{\alpha_{3}}=\\{\alpha_{3},H\\}=\frac{\partial H}{\partial T_{N}}=-\phi
kNa\frac{\partial c_{g}^{2}}{\partial
T_{N}}+\alpha_{3}Na^{3}\frac{\partial\rho}{\partial c_{m}^{2}}\frac{\partial
c_{m}^{2}}{\partial T_{N}}$ (3.47) $\dot{T_{N}}=\\{T_{N},H\\}=-\frac{\partial
H}{\partial\alpha_{3}}=-\rho Na^{3}$ (3.48)

Finally, to substitute the equations of motion for the $\dot{c_{g}}$ term, we
use the chain rule, giving

$\frac{dc_{g}^{2}}{dt}=\frac{\partial c_{g}^{2}}{\partial
T_{\phi}}\dot{T_{\phi}}+\frac{\partial c_{g}^{2}}{\partial T_{N}}\dot{T_{N}}$
(3.49)

Therefore, using this expression for $\dot{c_{g}}^{2}$ and the equations of
motion above we obtain

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=\rho Na^{3}\frac{\partial\rho}{\partial
c_{m}^{2}}\biggl{[}\frac{\alpha_{3}}{\phi}\frac{\partial c_{m}^{2}}{\partial
T_{\phi}}-\frac{\partial c_{m}^{2}}{\partial T_{N}}\biggl{]}$ (3.50)

This result is particularly important in sight of what we will develop in the
next section. In fact, including the scalar field’s kinetic term will enable
us to obtain a similar energy violation equation. However, unlike the one
given here, we will have extra new terms arising from the time ticked by the
Brans-Dicke clock $T_{\omega}$.

###  3.3 Dynamical Brans-Dicke

We now consider the complete case of a dynamical scalar field in the Brans-
Dicke theory. Multiple scenarios will be developed, particularly focusing on
the ones where the Brans-Dicke parameter $\omega$ is either part of the
$\bm{\alpha}$ or of the $\bm{\beta}$. As in the non-dynamical section, we
follow the bifurcation arising from the fully broken Local Lorentz Invariance
due to $\phi_{0}$. Therefore, we derive the equations of motion and the most
general conservation equation for both cases. Interestingly, the equations of
motion are different in the two cases, but lead to the same conservation
equation. This is not a surprise, as the conservation equation derives from
the field equations, which are the same in both cases. As a result, the
differentiation between these two cases is redundant whenever we are not
considering a dependency on the Ricci time $T_{\phi}$ given by the scalar
field $\phi$. So, without further ado, let’s dive into the wild ocean of
Pandora’s Box theories!

####  3.3.1 The Equations of Motion for the $\phi_{0}$ scenario

This subsection will consider the simplified case where the $\phi$ multiplying
the canonical pair $\dot{b}a^{2}$ is kept fixed, but the other terms are left
variable. Importantly, this scenario is not to be confused with the one
explained in subsection (2.3.2). In fact, there the equivalent of $\phi$
(i.e., $\alpha_{2}$) is kept fixed for all the terms appearing in the
Lagrangian. Conversely, in this case, only the $\phi$ multiplying the first
canonical pair is fixed, while the others can vary. This leads to a full
action of the form

$S=V_{c}\int{dt\>\biggl{[}\phi_{0}\dot{b}a^{2}+\dot{\phi}\pi_{\phi}+\alpha_{3}\dot{m}_{i}\psi_{i}+\phi
Na(b^{2}+kc_{g}^{2})-\alpha_{3}\rho
Na^{3}-\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\omega a^{3}}N}\biggl{]}$ (3.51)

which is the usual Brans-Dicke action inclusive of kinetic term, coupled to
matter (for a generic perfect fluid) and reduced to MSS. The Hamiltonian,
obtained in the usual way, is therefore

$H=-\phi Na(b^{2}+kc_{g}^{2})+\alpha_{3}\rho
Na^{3}+\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\omega a^{3}}N$ (3.52)

leading to the Hamilton’s constraint

$H\overset{!}{=}0\Leftrightarrow b^{2}+kc_{g}^{2}=\frac{\alpha_{3}\rho
a^{2}}{\phi}+\frac{1}{4}\frac{\pi_{\phi}^{2}}{\omega a^{4}}$ (3.53)

Following the standard procedure, we derive the $\dot{a}$ equation of motion

$\dot{(A^{2})}=\\{A^{2},H\\}=\frac{\partial A^{2}}{\partial b}\frac{\partial
H}{\partial A^{2}}-\frac{\partial A^{2}}{\partial A^{2}}\frac{\partial
H}{\partial b}=-\frac{\partial H}{\partial b}$ (3.54)

giving

$2a\dot{a}\phi_{0}=2bNa\phi\Leftrightarrow\dot{a}=\frac{\phi}{\phi_{0}}Nb$
(3.55)

On the other hand, the equation of motion for $b$ is obtained via

$\dot{b}=\\{b,H\\}=\frac{\partial b}{\partial b}\frac{\partial H}{\partial
A^{2}}-\frac{\partial b}{\partial A^{2}}\frac{\partial H}{\partial
A^{2}}=\frac{\partial H}{\partial A^{2}}=\frac{1}{2a\phi_{0}}\frac{\partial
H}{\partial a}$ (3.56)

leading to

$\displaystyle\dot{b}$ $\displaystyle=\frac{1}{2a\phi_{0}}\frac{\partial
H}{\partial a}$ (3.57)
$\displaystyle=\frac{1}{2a\phi_{0}}\biggl{[}-N\phi(b^{2}+kc_{g}^{2})-3w\frac{\alpha_{3}m}{a^{3w+1}}N-\frac{3}{4}\frac{\phi\pi_{\phi}^{2}}{\omega
a^{4}}N\biggl{]}$ (3.58)
$\displaystyle=\frac{1}{2a\phi_{0}}\biggl{[}-\alpha_{3}\rho
Na^{2}-\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\omega
a^{4}}N-3w\frac{\alpha_{3}m}{a^{3w+1}}N-\frac{3}{4}\frac{\phi\pi_{\phi}^{2}}{\omega
a^{4}}N\biggl{]}$ (3.59)
$\displaystyle=-\frac{1}{2}\frac{\alpha_{3}\rho}{\phi_{0}}Na-\frac{3}{2}w\frac{\alpha_{3}\rho}{\phi_{0}}Na-\frac{1}{8}\frac{\phi\pi_{\phi}^{2}}{\phi_{0}\omega
a^{5}}N-\frac{3}{8}\frac{\phi\pi_{\phi}^{2}}{\phi_{0}\omega a^{5}}N$ (3.60)
$\displaystyle=\frac{\alpha_{3}}{\phi_{0}}\frac{(-\rho-3w\rho)}{2}Na-\frac{1}{2}\frac{\phi\pi_{\phi}^{2}}{\phi_{0}\omega
a^{5}}N$ (3.61)
$\displaystyle=-\frac{\alpha_{3}}{2\phi_{0}}(\rho+3p)Na-\frac{1}{2}\frac{\phi\pi_{\phi}^{2}}{\phi_{0}\omega
a^{5}}N$ (3.62)

where we have used the Hamilton’s constraint in equation (3.59) and the
definition of $\rho$ from (2.113) in equation (3.62). Remarkably, due to the
$\phi_{0}$ factor in the canonical pair, this equation of motion is different
from the one we will obtain in the next section. Specifically, we lack the
cancellation of $\phi$ in the momentum term.

Finally, we need to obtain an equation of motion for the conjugate momentum
$\pi_{\phi}$. We apply the same procedure, evaluating its related Poisson’s
brackets, giving

$\dot{\pi_{\phi}}=\\{\pi_{\phi},H\\}=\frac{\partial\pi_{\phi}}{\partial\phi}\frac{\partial
H}{\partial\pi_{\phi}}-\frac{\partial\pi_{\phi}}{\partial\pi_{\phi}}\frac{\partial
H}{\partial\phi}=-\frac{\partial H}{\partial\phi}$ (3.63)

and, computing the derivatives, we get

$\displaystyle\dot{\pi_{\phi}}$ $\displaystyle=-\frac{\partial
H}{\partial\phi}$ (3.64)
$\displaystyle=Na(b^{2}+kc_{g}^{2})-\frac{1}{4}\frac{\pi_{\phi}^{2}}{\omega
a^{3}}N$ (3.65) $\displaystyle=Na\frac{\alpha_{3}\rho
a^{2}}{\phi}+\frac{1}{4}\frac{\pi_{\phi}^{2}}{\omega
a^{3}}N-\frac{1}{4}\frac{\pi_{\phi}^{2}}{\omega a^{3}}N$ (3.66)
$\displaystyle=\frac{\alpha_{3}\rho}{\phi}Na^{3}$ (3.67)

where we have used the Hamilton’s constraint in the $(b^{2}+kc_{g})$ term.

The equations of motion above can be equivalently derived using the Lagrangian
approach. We include the derivation to prove that the equations are indeed
correct. Begin by considering action (3.51): integrating by parts the
canonical pair $\dot{b}a^{2}$ and eliminating the boundary term gives

$S=V_{c}\int{dt\>\biggl{[}-2a\dot{a}b\phi_{0}+\dot{\phi}\pi_{\phi}+\alpha_{3}\dot{m}_{i}\psi_{i}+\phi
Na(b^{2}+kc_{g}^{2})-\alpha_{3}\rho
Na^{3}-\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\omega a^{3}}N}\biggl{]}$ (3.68)

Then, using the Euler-Lagrange equation

$\frac{\partial L}{\partial b}-\frac{d}{dt}\frac{\partial
L}{\partial\dot{b}}=0$ (3.69)

we obtain

$\dot{a}=\frac{\phi}{\phi_{0}}Nb$ (3.70)

which is consistent with (3.55). On the other hand, using the Euler-Lagrange
equation for $b$ on action (3.51)

$\frac{\partial L}{\partial a^{2}}-\frac{d}{dt}\frac{\partial
L}{\partial\dot{a^{2}}}=\frac{1}{2a}\frac{\partial L}{\partial
a}-\frac{d}{dt}\biggl{[}\frac{1}{2\dot{a}}\frac{\partial
L}{\partial\dot{a}}\biggl{]}=0$ (3.71)

we arrive at

$\dot{b}=-\frac{\alpha_{3}}{2\phi_{0}}(\rho+3p)Na-\frac{1}{2}\frac{\phi\pi_{\phi}^{2}}{\phi_{0}\omega
a^{5}}N$ (3.72)

which is as well consistent with the equation of motion previously derived
using the Hamiltonian approach.

We now take the time derivative of Hamilton’s constraint and use the equations
of motion to derive a conservation equation for our system. Doing so, we
follow the most general approach were all the constants appearing are assumed
to be varying. This procedure will give the most general energy conservation
equation, which we will simplify in different scenarios where only one
constant is varying. Dotting Hamilton’s constraint we obtain

$2b\dot{b}+k\frac{dc_{g}^{2}}{dt}=\frac{\dot{\alpha_{3}}\rho
a^{2}}{\phi}+\frac{2a\dot{a}\alpha_{3}\rho}{\phi}+\frac{\dot{\rho}\alpha_{3}a^{2}}{\phi}-\frac{\alpha_{3}\rho
a^{2}}{\phi^{2}}\dot{\phi}+\frac{1}{2}\frac{\pi_{\phi}\dot{\pi_{\phi}}}{\omega
a^{4}}-\frac{1}{4}\frac{\pi_{\phi}^{2}}{\omega^{2}a^{4}}\dot{\omega}-\frac{\pi_{\phi}^{2}}{\omega
a^{5}}\dot{a}$ (3.73)

Then, using the equation for $b$ coming from (3.55), the equation for
$\dot{b}$ in the LHS, and multiplying both sides by
$\frac{\phi}{\alpha_{3}a^{2}}$, we obtain

$-(\rho+3p)\frac{\dot{a}}{a}-\frac{\phi\pi_{\phi}^{2}}{\alpha_{3}\omega
a^{7}}\dot{a}+\frac{k\phi}{\alpha_{3}a^{2}}\frac{dc_{g}^{2}}{dt}=\frac{\dot{\alpha_{3}}}{\alpha_{3}}\rho+2\frac{\dot{a}}{a}\rho+\dot{\rho}-\frac{\dot{\phi}}{\phi}\rho\\\
\\\ +\frac{1}{2}\frac{\phi\pi_{\phi}\dot{\pi_{\phi}}}{\alpha_{3}\omega
a^{6}}-\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\alpha_{3}\omega^{2}a^{6}}\dot{\omega}-\frac{\phi\pi_{\phi}^{2}}{\alpha_{3}\omega
a^{7}}\dot{a}$ (3.74)

Rearranging this expression, we obtain the energy conservation equation as

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=-\frac{\dot{\alpha_{3}}}{\alpha_{3}}\rho+\frac{\dot{\phi}}{\phi}\rho-\frac{1}{2}\frac{\phi\pi_{\phi}\dot{\pi_{\phi}}}{\alpha_{3}\omega
a^{6}}+\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\alpha_{3}\omega^{2}a^{6}}\dot{\omega}+\frac{k\phi}{\alpha_{3}a^{2}}\frac{dc_{g}^{2}}{dt}$
(3.75)

We can now use the $\dot{\pi_{\phi}}$ equation to further simplify this
expression, as well as the definition of $\dot{\phi}$ given by the conjugate
momentum, leading to

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=-\frac{\dot{\alpha_{3}}}{\alpha_{3}}\rho+\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\alpha_{3}\omega^{2}a^{6}}\dot{\omega}+\frac{k\phi}{\alpha_{3}a^{2}}\frac{dc_{g}^{2}}{dt}$
(3.76)

####  3.3.2 The Equations of Motion for the full $\phi$ scenario

In this section, we derive the equations of motion for the canonical pair
$(a^{2},b)$ as well as for $\pi_{\phi}$, the conjugate momentum to $\phi$. We
begin by considering the Brans-Dicke action reduced to MSS as done in
subsection (3.1.1). This action, inclusive of the matter part, but exclusive
of any unimodular term, is

$S=V_{c}\int{dt\>\biggl{[}\phi\dot{b}a^{2}+\dot{\phi}\pi_{\phi}+\alpha_{3}\dot{m}_{i}\psi_{i}+\phi
Na(b^{2}+kc_{g}^{2})-\alpha_{3}\rho
Na^{3}-\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\omega a^{3}}N}\biggl{]}$ (3.77)

It is straightforward to extract the usual gravitational Hamiltonian, as well
as the matter and pure kinetic Brans-Dicke Hamiltonian, which read

$H=-\phi Na(b^{2}+kc_{g}^{2})+\alpha_{3}\rho
Na^{3}+\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\omega a^{3}}N$ (3.78)

We can obtain the Hamilton’s constraint by setting the Hamiltonian equal to
zero, such that

$H\overset{!}{=}0\Leftrightarrow b^{2}+kc_{g}^{2}=\frac{\alpha_{3}\rho
a^{2}}{\phi}+\frac{1}{4}\frac{\pi_{\phi}^{2}}{\omega a^{4}}$ (3.79)

Alternatively, we could obtain the Hamilton’s constraint directly from the
action above by considering its variation with respect to the lapse function,
so that

$\frac{\delta S}{\delta N}\overset{!}{=}0\Leftrightarrow
b^{2}+kc_{g}^{2}=\frac{\alpha_{3}\rho
a^{2}}{\phi}+\frac{1}{4}\frac{\pi_{\phi}^{2}}{\omega a^{4}}$ (3.80)

Recalling that the canonical pair is changed by the presence of $\phi$ in the
first term of action (3.77), care must be taken when evaluating the
derivatives leading to the equations of motion. To begin with, we define
$A^{2}=a^{2}\phi$ as the canonical variable to $b$, such that the canonical
pair is now $(A^{2},b)$. Then, we re-express the Hamiltonian in terms of this
new canonical variable, obtaining

$H=-NA\sqrt{\phi}(b^{2}+kc_{g}^{2})+\frac{\alpha_{3}\rho
A^{3}N}{\phi^{\frac{3}{2}}}+\frac{1}{4}\frac{\phi^{\frac{5}{2}}\pi_{\phi}^{2}}{\omega
A^{3}}N$ (3.81)

We continue evaluating the $\dot{a}$ equation in the usual fashion, using
Hamiltonian (3.81)

$\dot{(A^{2})}=\\{A^{2},H\\}=\frac{\partial A^{2}}{\partial b}\frac{\partial
H}{\partial A^{2}}-\frac{\partial A^{2}}{\partial A^{2}}\frac{\partial
H}{\partial b}=-\frac{\partial H}{\partial b}$ (3.82)

leading to

$\dot{a}+\frac{\dot{\phi}}{2\phi}a=Nb$ (3.83)

which we immediately recognise to be equivalent to equation (2.144) in the
case of a non Brans-Dicke theory. This is not a coincidence: Brans-Dicke
theory is indeed a theory with a variable $\alpha_{2}$, with the main
difference given by the addition of a kinetic term for the dynamics of the
scalar field $\phi$ (or $\alpha_{2}$ in the analogy).
Secondly, we compute the $\dot{b}$ equation, re-writing the Hamiltonian in
terms of $a$ and expressing $\rho$ as

$\rho_{i}=\frac{m_{i}}{a^{3(1+w_{i})}}$ (3.84)

as we did before using (2.113) for a generic fluid with different matter
contents. The Hamiltonian is therefore

$H=-Na\phi(b^{2}+kc_{g}^{2})+\frac{\alpha_{3}m}{a^{3w}}N+\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\omega
a^{3}}N$ (3.85)

and the equation for $\dot{b}$ reads

$\dot{b}=\\{b,H\\}=\frac{\partial b}{\partial b}\frac{\partial H}{\partial
A^{2}}-\frac{\partial b}{\partial A^{2}}\frac{\partial H}{\partial
b}=\frac{\partial H}{\partial A^{2}}=\frac{1}{2a\phi}\frac{\partial
H}{\partial a}$ (3.86)

leading to

$\displaystyle\dot{b}$ $\displaystyle=\frac{1}{2\phi a}\frac{\partial
H}{\partial a}$ (3.87) $\displaystyle=\frac{1}{2\phi
a}\biggl{[}-N\phi(b^{2}+kc_{g}^{2})-3w\frac{\alpha_{3}m}{a^{3w+1}}N-\frac{3}{4}\frac{\phi\pi_{\phi}^{2}}{\omega
a^{4}}N\biggl{]}$ (3.88) $\displaystyle=\frac{1}{2\phi
a}\biggl{[}-\alpha_{3}\rho Na^{2}-\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\omega
a^{4}}N-3w\frac{\alpha_{3}m}{a^{3w+1}}N-\frac{3}{4}\frac{\phi\pi_{\phi}^{2}}{\omega
a^{4}}N\biggl{]}$ (3.89)
$\displaystyle=-\frac{1}{2}\frac{\alpha_{3}\rho}{\phi}Na-\frac{3}{2}w\frac{\alpha_{3}\rho}{\phi}Na-\frac{1}{8}\frac{\pi_{\phi}^{2}}{\omega
a^{5}}N-\frac{3}{8}\frac{\pi_{\phi}^{2}}{\omega a^{5}}N$ (3.90)
$\displaystyle=\frac{\alpha_{3}}{\phi}\frac{(-\rho-3w\rho)}{2}Na-\frac{1}{2}\frac{\pi_{\phi}^{2}}{\omega
a^{5}}N$ (3.91)
$\displaystyle=-\frac{\alpha_{3}}{2\phi}(\rho+3p)Na-\frac{1}{2}\frac{\pi_{\phi}^{2}}{\omega
a^{5}}N$ (3.92)

where we have used the Hamilton’s constraint in equation (3.89) and re-
inserted the expression for $\rho$ in equation (3.92).

The final piece of information we need to determine the entire dynamics of the
theory is the equation of motion for $\pi_{\phi}$, the canonical variable of
$\phi$. We begin by considering action (3.77) and the resulting Hamiltonian
(3.78). Expressing the Hamiltonian in terms of the redefined canonical
variable $A^{2}$ and using (2.113) for $\rho$, we obtain

$H=-NA\sqrt{\phi}(b^{2}+kc_{g}^{2})+\frac{\alpha_{3}m}{A^{3w}}\phi^{\frac{3}{2}w}N+\frac{1}{4}\frac{\phi^{\frac{5}{2}}\pi_{\phi}^{2}}{\omega
A^{3}}N$ (3.93)

The Poisson’s brackets for $\dot{\pi_{\phi}}$, given the canonical pair
$(\phi,\pi_{\phi})$ are

$\dot{\pi_{\phi}}=\\{\pi_{\phi},H\\}=\frac{\partial\pi_{\phi}}{\partial\phi}\frac{\partial
H}{\partial\pi_{\phi}}-\frac{\partial\pi_{\phi}}{\partial\pi_{\phi}}\frac{\partial
H}{\partial\phi}=-\frac{\partial H}{\partial\phi}$ (3.94)

which, when evaluated, give

$\displaystyle\dot{\pi_{\phi}}$ $\displaystyle=-\frac{\partial
H}{\partial\phi}$ (3.95)
$\displaystyle=\frac{1}{2}NA\phi^{-\frac{1}{2}}(b^{2}+kc_{g}^{2})-\frac{3}{2}w\frac{\alpha_{3}m}{A^{3w}}\phi^{\frac{3w-2}{2}}N-\frac{5}{8}\frac{\pi_{\phi}^{2}}{\omega
A^{3}}\phi^{\frac{3}{2}}N$ (3.96)
$\displaystyle=\frac{1}{2}Na\biggl{[}\frac{\alpha_{3}\rho
a^{2}}{\phi}+\frac{1}{4}\frac{\pi_{\phi}^{2}}{\omega
a^{4}}\biggl{]}-\frac{3}{2}w\frac{\alpha_{3}m}{a^{3w}\phi^{\frac{3}{2}w}}\phi^{\frac{3-2w}{2}}N-\frac{5}{8}\frac{\pi_{\phi}^{2}}{\omega
a^{3}}N$ (3.97)
$\displaystyle=\frac{\alpha_{3}\rho}{2\phi}Na^{3}-\frac{3}{2}w\frac{\alpha_{3}ma^{3}}{a^{3(1+w)}\phi}N-\frac{1}{2}\frac{\pi_{\phi}^{2}}{\omega
a^{3}}N$ (3.98)
$\displaystyle=\frac{\alpha_{3}\rho}{2\phi}Na^{3}-\frac{3}{2}\frac{w\rho\alpha_{3}a^{3}}{\phi}N-\frac{1}{2}\frac{\pi_{\phi}^{2}}{\omega
a^{3}}N$ (3.99)
$\displaystyle=-Na\phi\biggl{[}\biggl{(}\frac{-1+3w}{2}\biggl{)}\frac{\rho\alpha_{3}a^{2}}{\phi^{2}}+\frac{1}{2}\frac{\pi_{\phi}^{2}}{\omega
a^{4}}\frac{1}{\phi}\biggl{]}$ (3.100)
$\displaystyle=-Na\phi\biggl{[}\biggl{(}\frac{-\rho+3p}{2}\biggl{)}\frac{\alpha_{3}a^{2}}{\phi^{2}}+\frac{1}{2}\frac{\pi_{\phi}^{2}}{\omega
a^{4}}\frac{1}{\phi}\biggl{]}$ (3.101)

As before, it is possible to obtain the same equations of motion following the
Lagrangian approach, but, for simplicity, we will only include the Hamiltonian
derivation. With this equation of motion, we proceed to derive the
conservation equation from the Hamilton’s constraint. In fact, given the
Hamiltonian formalism we are using, it is possible to derive it directly from
the constraint and from the equations of motion for the different variables.
In a normal theory without varying constants we would recover the usual
Friedmann equations, since we are working in a FLRW MSS reduction. However,
assuming varying constants, we will get additional terms arising from all the
$\bm{\beta}$ we wish to use. In this case, trying to get the most general form
of energy conservation, we assume all constants to be varying, but this will
be different in the next sections. In fact, ultimately, we can have different
combinations of constants for different scenarios as we will see. Therefore,
taking the time derivative of (3.166) we obtain

$2b\dot{b}+k\frac{dc_{g}^{2}}{dt}=\frac{\dot{\alpha_{3}}\rho
a^{2}}{\phi}+\frac{2a\dot{a}\alpha_{3}\rho}{\phi}+\frac{\dot{\rho}\alpha_{3}a^{2}}{\phi}-\frac{\alpha_{3}\rho
a^{2}}{\phi^{2}}\dot{\phi}+\frac{1}{2}\frac{\pi_{\phi}\dot{\pi_{\phi}}}{\omega
a^{4}}-\frac{1}{4}\frac{\pi_{\phi}^{2}}{\omega^{2}a^{4}}\dot{\omega}-\frac{\pi_{\phi}^{2}}{\omega
a^{5}}\dot{a}$ (3.102)

Now, considering the LHS of equation (3.102), we can plug in the equation of
motion for $\dot{b}$ and the expression for $b$ coming from the $\dot{a}$
equation, giving, in the $N=1$ gauge,

$\displaystyle 2b\dot{b}+k\frac{dc_{g}^{2}}{dt}$
$\displaystyle=2\biggl{[}\dot{a}+\frac{\dot{\phi}}{2\phi}a\biggl{]}\biggl{[}-\frac{\alpha_{3}}{2\phi}(\rho+3p)a-\frac{1}{2}\frac{\pi_{\phi}^{2}}{\omega
a^{5}}\biggl{]}$ (3.103)
$\displaystyle=-\frac{\alpha_{3}}{\phi}(\rho+3p)a\dot{a}-\frac{\pi_{\phi}^{2}}{\omega
a^{5}}\dot{a}-\frac{\alpha_{3}\dot{\phi}}{2\phi^{2}}(\rho+3p)a^{2}-\frac{\pi_{\phi}^{2}\dot{\phi}}{2\omega\phi
a^{4}}$ (3.104)
$\displaystyle=-(\rho+3p)\frac{\dot{a}}{a}-\frac{\phi\pi_{\phi}^{2}}{\alpha_{3}\omega
a^{7}}\dot{a}-\frac{\dot{\phi}}{2\phi}(\rho+3p)-\frac{\pi_{\phi}^{2}\dot{\phi}}{2\alpha_{3}\omega
a^{6}}$ (3.105)

where in the last line we have multiplied every term by
$\frac{\phi}{\alpha_{3}a^{2}}$. Doing the same on the RHS of equation (3.102)
above gives

$\frac{\dot{\alpha_{3}}}{\alpha_{3}}\rho+2\frac{\dot{a}}{a}\rho+\dot{\rho}-\frac{\dot{\phi}}{\phi}\rho+\frac{1}{2}\frac{\pi_{\phi}\dot{\pi_{\phi}}\phi}{\alpha_{3}\omega
a^{6}}-\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\alpha_{3}\omega^{2}a^{6}}\dot{\omega}-\frac{\phi\pi_{\phi}^{2}}{\alpha_{3}\omega
a^{7}}\dot{a}$ (3.106)

Now, equating the LHS and the RHS and rearranging leads to

$\displaystyle\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=-\frac{\dot{\alpha_{3}}}{\alpha_{3}}\rho+\frac{\dot{\phi}}{\phi}\rho-\frac{\dot{\phi}}{\phi}\frac{\rho+3p}{2}-\frac{\dot{\phi}\pi_{\phi}^{2}}{2\alpha_{3}\omega
a^{6}}-\frac{1}{2}\frac{\pi_{\phi}\dot{\pi_{\phi}}\phi}{\alpha_{3}\omega
a^{6}}+\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\alpha_{3}\omega^{2}a^{6}}\dot{\omega}+\frac{k\phi}{\alpha_{3}a^{2}}\frac{dc_{g}^{2}}{dt}$
(3.107)
$\displaystyle=-\frac{\dot{\alpha_{3}}}{\alpha_{3}}\rho+\frac{\pi_{\phi}}{2\omega
a^{3}}\rho-\frac{\pi_{\phi}}{2\omega
a^{3}}\frac{\rho+3p}{2}-\frac{1}{4}\frac{\phi\pi_{\phi}^{3}}{\alpha_{3}\omega^{2}a^{9}}+\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\alpha_{3}\omega^{2}a^{6}}\dot{\omega}+\frac{k\phi}{\alpha_{3}a^{2}}\frac{dc_{g}^{2}}{dt}-\frac{1}{2}\frac{\pi_{\phi}\dot{\pi_{\phi}}\phi}{\alpha_{3}\omega
a^{6}}$ (3.108)
$\displaystyle=-\frac{\dot{\alpha_{3}}}{\alpha_{3}}\rho+\frac{\pi_{\phi}}{2\omega
a^{3}}\biggl{[}\rho-\frac{\rho}{2}-\frac{3}{2}p\biggl{]}+\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\alpha_{3}\omega^{2}a^{6}}\dot{\omega}-\frac{1}{4}\frac{\pi_{\phi}}{\omega
a^{3}}(\rho-3p)+\frac{k\phi}{\alpha_{3}a^{2}}\frac{dc_{g}^{2}}{dt}$ (3.109)
$\displaystyle=-\frac{\dot{\alpha_{3}}}{\alpha_{3}}\rho+\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\alpha_{3}\omega^{2}a^{6}}\dot{\omega}+\frac{k\phi}{\alpha_{3}a^{2}}\frac{dc_{g}^{2}}{dt}$
(3.110)

where we have used the definition of $\dot{\phi}$,

$\pi_{\phi}=2\omega\frac{\dot{\phi}}{\phi}a^{3}\Leftrightarrow\dot{\phi}=\frac{\phi\pi_{\phi}}{2\omega
a^{3}}$ (3.111)

for (3.108) and we have also used (3.101), the equation of motion for
$\pi_{\phi}$, in line (3.109), always setting the gauge such that $N=1$.

This is the most general form of the energy conservation equation, assuming
that all the constants are varying and using the equations of motion for $a$,
$b$ and $\phi$ to eliminate the extra factors, as well as the definition of
the conjugate momentum $\pi_{\phi}$. From this general equation, it will be
possible to consider specific scenarios where only one of the $\bm{\beta}$ is
varying with respect to one or multiple times. On the other hand, we could
also have multiple $\bm{\beta}$ varying with respect to one single time or
multiple times (the same one or different ones): imagination is really the
limit.

####  3.3.3 Full $\phi$ with $\omega=\omega(T_{\Lambda})$

In this scenario, we assume a Brans-Dicke coupling $\omega$ varying with
respect to the unimodular time $T_{\Lambda}$. This means that, in the choice
of our constants, we have $\bm{\beta}=\omega$ and $\bm{\alpha}=\rho_{\Lambda}$
giving the unimodular time as its canonical variable. In this case, all the
other constants are kept fixed. An important observation to make is that the
case where $\phi=\phi_{0}$ in the canonical pair does not lead to any
different result. This is not, however, the case in the next subsection, as we
will see. Therefore, to consider the dependence $\omega=\omega(T_{\Lambda})$
we start from the full Brans-Dicke action, coupled to matter and inclusive of
the unimodular term

$S=V_{c}\int{dt\>\phi\dot{b}a^{2}+\dot{\phi}\pi_{\phi}+\alpha_{3}\dot{m}_{i}\psi_{i}+\phi
Na(b^{2}+kc_{g}^{2})-\alpha_{3}\rho
Na^{3}-\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{a^{3}\omega}N+\dot{\rho_{\Lambda}}T_{\Lambda}}$
(3.112)

In this scenario, we simply assume that $\alpha_{3}$ does not have any time
dependence. This implies that our general equation for energy conservation
(3.110) becomes

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\alpha_{3}\omega^{2}a^{6}}\dot{\omega}+\frac{k\phi}{\alpha_{3}a^{2}}\frac{dc_{g}^{2}}{dt}$
(3.113)

Furthermore, since we assume that only $\omega$ has a time dependence, the
second term including the varying $c_{g}$ disappears, simplifying equation
(3.113) to

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\alpha_{3}\omega^{2}a^{6}}\dot{\omega}$
(3.114)

Finally, since we are considering the presence of the cosmological constant as
a matter content, the full conservation equation needs to be inclusive of a
cosmological constant density term, therefore giving

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)+\dot{\rho_{\Lambda}}=\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\alpha_{3}\omega^{2}a^{6}}\dot{\omega}$
(3.115)

where $\rho$ and $p$ give the usual matter content density and pressure. Now,
we wish to obtain an equation for $\dot{\rho_{\Lambda}}$ as this will give us
the full conservation equation. We can obtain this from the additional
Hamilton’s equations for the canonical pair $\rho_{\Lambda}T_{\Lambda}$.
Starting from the Hamiltonian

$H=-\phi Na(b^{2}+kc_{g}^{2})+\alpha_{3}\rho
Na^{3}+\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\omega a^{3}}N$ (3.116)

we have the two Hamilton’s equations

$\dot{\rho_{\Lambda}}=\\{\rho_{\Lambda},H\\}=\frac{\partial H}{\partial
T_{\Lambda}}=-N\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\omega^{2}a^{3}}\frac{\partial\omega}{\partial
T_{\Lambda}}$ (3.117) $\dot{T_{\Lambda}}=\\{T_{\Lambda},H\\}=-\frac{\partial
H}{\partial\rho_{\Lambda}}=-\alpha_{3}Na^{3}$ (3.118)

Thus, using the equation for $T_{\Lambda}$ into the one for $\rho_{\Lambda}$,
we obtain

$\dot{\rho_{\Lambda}}=-N\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\omega^{2}a^{3}}\frac{d\omega}{dt}\dot{T_{\Lambda}}^{-1}=\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\alpha_{3}\omega^{2}a^{6}}\dot{\omega}$
(3.119)

We immediately see that $\dot{\rho_{\Lambda}}$ has the same form as the source
term in equation (3.115), therefore we can substitute it in it and confirm
that indeed

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=0$ (3.120)

This result shows that there is no net energy violation solely given by a
unimodular time-dependent $\omega$, confirming that energy is conserved for
all the other matter contents.
Furthermore, can combine the equation of motion for $\rho_{\Lambda}$ with the
definition of $\pi_{\phi}$ from equation (3.12) (where we choose the gauge
$N=1$) to obtain an explicit equation for the evolution of the vacuum energy
density with $\omega$. This substitution leads to

$\dot{\rho}_{\Lambda}=\frac{\dot{\phi}^{2}}{\phi}\dot{\omega}$ (3.121)

where, for simplicity, we have set $\alpha_{3}=1$. This solution is valid for
any $\omega$ depending on $T_{\Lambda}$ and suggests that the vacuum energy
density increases with increasing $\omega$. Further interpretations are
certainly needed, and will be provided in our future research [63].

####  3.3.4 Full $\phi$ with $\omega=\omega(T_{N})$ and
$\alpha_{3}=\alpha_{3}(T_{\omega})$

We can now consider the scenario where $\bm{\alpha}=\alpha_{3}$ and
$\bm{\beta}=\omega$, such that the Brans-Dicke parameter is dependent on the
Newton time $T_{N}$. As it is evident from the title, this scenario can also
be recast as the $\alpha_{3}$ parameter depending on $T_{\omega}$: both these
scenarios are equivalent, leading to the same results. The action we start
from is

$S=V_{c}\int{dt\>\phi\dot{b}a^{2}+\dot{\phi}\pi_{\phi}+\alpha_{3}\dot{m}_{i}\psi_{i}+\phi
Na(b^{2}+kc_{g}^{2})-\alpha_{3}\rho
Na^{3}-\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{a^{3}\omega}N+\dot{\omega}T_{N}}$
(3.122)

If we wish to consider a varying $\omega$ only, the general conservation
equation reduces to

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\alpha_{3}\omega^{2}a^{6}}\dot{\omega}$
(3.123)

where, since $\omega=\omega(T_{N})$ only, we can use the chain rule on
$\dot{\omega}$ thus obtaining

$\frac{d\omega}{dt}=\frac{\partial\omega}{\partial T_{N}}\dot{T_{N}}=-\rho
Na^{3}\frac{\partial\omega}{\partial T_{N}}$ (3.124)

where, in the last step, we have use the $T_{N}$ equation of motion as given
by

$\dot{T_{N}}=\\{T_{N},H\\}=-\frac{\partial H}{\partial\alpha_{3}}=-\rho
Na^{3}$ (3.125)

Equipped with these, we can then plug everything inside the conservation
equation thus arriving at

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=-\frac{\rho}{4}\frac{\phi\pi_{\phi}^{2}N}{\alpha_{3}\omega^{2}a^{3}}\frac{\partial\omega}{\partial
T_{N}}$ (3.126)

which shows that a gravitational parameter depending on a matter clock does
lead to an overall energy violation, given by the source term of the constant
varying with respect to the matter clock. The same situation would be true if
we were to revert the dependency, postulating that
$\alpha_{3}=\alpha_{3}(T_{\omega})$. In this parallel scenario, the
conservation equation would be

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=-\frac{\dot{\alpha_{3}}}{\alpha_{3}}\rho$
(3.127)

since, this time around, $\omega$ is constant. The relevant equation of motion
would be the $\dot{T_{\omega}}$ equation, reading

$\dot{T_{\omega}}=\\{T_{\omega},H\\}=-\frac{\partial
H}{\partial\omega}=\frac{1}{4}\frac{\phi\pi_{\phi}^{2}N}{\omega^{2}a^{3}}$
(3.128)

Using this equation of motion in $\dot{\alpha_{3}}$ leads to

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=-\frac{\rho}{4}\frac{\phi\pi_{\phi}^{2}N}{\alpha_{3}\omega^{2}a^{3}}\frac{\partial\alpha_{3}}{\partial
T_{\omega}}$ (3.129)

which is exactly what we expected: this conservation equation has the same
pre-factor as equation (3.127), with the only difference being the varying
constant. In the first case, $\omega$ varies and $\alpha_{3}$ provides the
clock, while in the second one $\alpha_{3}$ varies and $\omega$ provides the
clock. They both leads to the same net energy violation, but due to different
varying source terms.
To appreciate better this equivalence, we derive a relationship between
$\omega$ and $\alpha_{3}$. We take the equation of motion for $\omega$ (we
could have also picked the one for $\alpha_{3}$),

$\dot{\omega}=\\{\omega,H\\}=\frac{\partial H}{\partial T_{\omega}}=\rho
Na^{3}\frac{\partial\alpha_{3}}{\partial T_{\omega}}$ (3.130)

where $\alpha_{3}$ is assumed to depend on $T_{\omega}$. Then, using the chain
rule on the dependency of $\alpha_{3}$ on $T_{\omega}$ and invoking the
definition of $\pi_{\phi}$ we obtain

$\dot{\omega}=\frac{\phi\rho}{\dot{\phi}^{2}}\dot{\alpha_{3}}$ (3.131)

where we have set $N=1$. This equation could have been equivalently obtained
starting from the $\alpha_{3}$ equation of motion and following the same
procedure. Finally, we note that, when considering both $\alpha_{3}$ and
$\omega$ varying in the same scenario, their source terms contributions to the
energy violation cancel out, giving the usual overall energy conservation
equation

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=0$ (3.132)

####  3.3.5 Fixed $\phi_{0}$ with $\omega=\omega(T_{\phi})$

In this subsection, we explore the main idea of Brans-Dicke theory in the
context of varying constants. In fact, having the additional constant
$\omega$, we can make it dependent on $\phi$, creating a model of Brans-Dicke
parameter dependent on Ricci time. To begin with, we consider the simplified
case where the scalar field factor multiplying the fist canonical pair is
fixed constant, i.e., $\phi=\phi_{0}$. This case is apparently different from
the next one where the equations of motion are different, but, as we will see,
they are actually the same. It happens so because ultimately the conservation
equation is the same one, as it derives directly from the conservation of the
energy-momentum tensor. Furthermore, it is important to notice that, as in the
next subsection, the $\phi$ equation of motion needs to be modified to account
for the extra dependency of $\omega$ on $\pi_{\phi}$. Finally, given the
equivalence of $T_{\phi}$ and $\pi_{\phi}$, we do not need to include the
unimodular term in the action, as the canonical pair from the Brans-Dicke
Hamiltonian suffice. Therefore, the action we will use is

$S=V_{c}\int{dt\>\phi_{0}\dot{b}a^{2}+\dot{\phi}\pi_{\phi}+\alpha_{3}\dot{m}_{i}\psi_{i}+\phi
Na(b^{2}+kc_{g}^{2})-\alpha_{3}Na^{3}\rho-\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\omega
a^{3}}N}$ (3.133)

From subsection (3.3.1) we have the general conservation equation which we
take in the form given by (3.75)

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=\frac{\dot{\phi}}{\phi}\rho-\frac{1}{2}\frac{\phi\pi_{\phi}\dot{\pi_{\phi}}}{\alpha_{3}\omega
a^{6}}+\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\alpha_{3}\omega^{2}a^{6}}\dot{\omega}$
(3.134)

where we have also set $\dot{\alpha_{3}}=\dot{c_{g}^{2}}=0$ because, in this
particular scenario, we are interested only in a varying $\omega$. We can now
compute the $\phi$ equation of motion, which reads

$\dot{\phi}=\\{\phi,H\\}=\frac{\partial
H}{\partial\pi_{\phi}}=\frac{1}{2}\frac{\phi\pi_{\phi}}{\omega
a^{3}}N-\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\omega^{2}a^{3}}\frac{d\omega}{d\pi_{\phi}}N$
(3.135)

where, importantly, we notice the extra term for the $\omega$ dependency. On
the other hand, the $\pi_{\phi}$ equation of motion is the same one as (3.67),
reading

$\dot{\pi_{\phi}}=\\{\pi_{\phi},H\\}=-\frac{\partial
H}{\partial\phi}=\frac{\alpha_{3}\rho}{\phi}Na^{3}$ (3.136)

Finally, expressing $\dot{\omega}$ in terms of $T_{\phi}$ as

$\frac{d\omega}{dt}=\frac{\partial\omega}{\partial
T_{\phi}}\dot{T_{\phi}}=\frac{\partial\omega}{\partial
T_{\phi}}\dot{\pi_{\phi}}=\frac{\alpha_{3}\rho}{\phi}Na^{3}\frac{\partial\omega}{\partial
T_{\phi}}$ (3.137)

we obtain the final form of the conservation equation

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=0$ (3.138)

which shows that energy is indeed conserved. This is an expected result
because, as we will see, gravitational parameters depending on gravitational
clocks do not lead to any net energy violation. As we now see, the full $\phi$
case leads to the same results.

####  3.3.6 Full $\phi$ with $\omega=\omega(T_{\phi})$

In this subsection, we will explore an interesting and unusual scenario, where
the Brans-Dikce parameter $\omega$ depends on the conjugate momentum of the
scalar field, such that $\omega=\omega(\pi_{\phi})$. The first observation we
make is that the $\omega$ dependence on a Ricci-like time as $T_{\phi}$ is
perfectly equivalent to its dependence on $\pi_{\phi}$. In fact, given that
$\pi_{\phi}$ is the conjugate momentum to $\phi$ in the Hamiltonian formalism
and that $T_{\phi}$ is also the conjugate of $\phi$, we see that they are
equivalent, meaning that $\pi_{\phi}$ is indeed the relational Ricci time
given by $\phi$. Secondly, precisely because of this equivalence, the
unimodular term in the action represent an interesting but redundant addition.
In fact, we are not required to add it to include the Ricci time as it is
already provided by the canonical pair $\dot{\phi}\pi_{\phi}$ coming from the
Hamiltonian. This is, however, an interesting occurrence, which we will
potentially explore in our future works [63]. Therefore, the action we will in
our derivations is

$S=V_{c}\int{dt\>\phi\dot{b}a^{2}+\dot{\phi}\pi_{\phi}+\alpha_{3}\dot{m}_{i}\psi_{i}+\phi
Na(b^{2}+kc_{g}^{2})-\alpha_{3}Na^{3}\rho-\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{a^{3}\omega}N}$
(3.139)

Before proceeding, a crucial caveat must be addressed. We might be tempted to
just use the general conservation equation (3.110) with only the
$\dot{\omega}$ term as only $\omega$ is varying, but this would be a mistake.
In fact, when deriving this equation, we have used the equation for
$\dot{\phi}$ obtained by inverting the definition of the conjugate momentum
$\pi_{\phi}$. However, when doing so, we were assuming that only $\phi$
depends on $\pi_{\phi}$ and that no other terms in the Hamiltonian contribute.
This case is different: we have an additional dependency coming from $\omega$,
which implies an extra term in the $\dot{\phi}$ equation of motion. As a
result, we need to re-derive the conservation equation, accounting for this
extra term. We start from its general form (3.107), where, after setting
$\dot{\alpha_{3}}=\dot{c_{g}^{2}}=0$ in this particular case and without using
the standard $\dot{\phi}$ equation, we have

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=\frac{\dot{\phi}}{\phi}\rho-\frac{\dot{\phi}}{\phi}\frac{\rho+3p}{2}-\frac{\dot{\phi}\pi_{\phi}^{2}}{2\alpha_{3}\omega
a^{6}}-\frac{1}{2}\frac{\pi_{\phi}\dot{\pi_{\phi}}\phi}{\alpha_{3}\omega
a^{6}}+\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\alpha_{3}\omega^{2}a^{6}}\dot{\omega}$
(3.140)

It is precisely at this point that we go back to Hamiltonian (3.78) to derive
the $\dot{\phi}$ equation, this time accounting for the fact that, in this
particular case, $\omega=\omega(\pi_{\phi})$

$\dot{\phi}=\\{\phi,H\\}=\frac{\partial
H}{\partial\pi_{\phi}}=\frac{1}{2}\frac{\phi\pi_{\phi}}{\omega
a^{3}}N-\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\omega^{2}a^{3}}\frac{d\omega}{d\pi_{\phi}}N$
(3.141)

while the $\dot{\pi_{\phi}}$ equation of motion takes the usual form as

$\dot{T_{\phi}}=\\{T_{\phi},H\\}=-\frac{\partial
H}{\partial\phi}=-Na\phi\biggl{[}\biggl{(}\frac{-\rho+3p}{2}\biggl{)}\frac{\alpha_{3}a^{2}}{\phi^{2}}+\frac{1}{2}\frac{\pi_{\phi}^{2}}{\omega
a^{4}}\frac{1}{\phi}\biggl{]}=\dot{\pi_{\phi}}$ (3.142)

We may now use equations (3.141) and (3.142) in the conservation equation
above, where we also express $\dot{\omega}$ in terms of $\dot{\pi_{\phi}}$
using the chain rule. This leads to

$\displaystyle\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=-\frac{(-\rho+3p)}{8}\frac{\pi_{\phi}^{2}N}{\omega^{2}a^{3}}\frac{d\omega}{d\pi_{\phi}}-\frac{1}{8}\frac{\phi\pi_{\phi}^{3}N}{\alpha_{3}\omega^{3}a^{9}}\frac{d\omega}{d\pi_{\phi}}+\frac{\rho}{2}\frac{\pi_{\phi}N}{\omega
a^{3}}$ (3.143)
$\displaystyle-\frac{\rho}{4}\frac{\pi_{\phi}^{2}N}{\omega^{2}a^{3}}\frac{d\omega}{d\pi_{\phi}}-\frac{(\rho+3p)}{4}\frac{\pi_{\phi}N}{\omega
a^{3}}+\frac{(\rho+3p)}{8}\frac{\pi_{\phi}^{2}N}{\omega^{2}a^{3}}\frac{d\omega}{d\pi_{\phi}}-\frac{1}{4}\frac{\phi\pi_{\phi}^{3}N}{\alpha_{3}\omega^{2}a^{9}}$
(3.144)
$\displaystyle+\frac{1}{8}\frac{\phi\pi_{\phi}^{3}N}{\alpha_{3}\omega^{3}a^{9}}\frac{d\omega}{d\pi_{\phi}}+\frac{(-\rho+3p)}{4}\frac{\pi_{\phi}N}{\omega
a^{3}}+\frac{1}{4}\frac{\phi\pi_{\phi}^{3}N}{\alpha_{3}\omega^{2}a^{9}}=0$
(3.145)

which simplifies to the standard conservation equation. This scenario shows
two important results. Firstly, once again, we confirm that a gravitational
parameter depending on a gravitational clock does not lead to any net energy
violation. Secondly, we are reminded that any scenario including a dependency
on the Ricci clock should be treated with additional care, as extra terms,
which are normally not included, will arise in the general conservation
equation.

####  3.3.7 Full $\phi$ for $c_{g}=c_{g}(T_{\omega})$

We could, of course, change the pattern by considering $\omega$ as part of the
$\bm{\alpha}$ constants giving times, while a different constant is part of
the $\bm{\beta}$. In this case, we will consider the scenario where
$\bm{\alpha}=\omega$, $T_{\bm{\alpha}}=T_{\omega}$ and $\bm{\beta}=c_{g}$,
while the other constants appearing in the conservation equation are fixed,
i.e., $\dot{\alpha_{3}}=\dot{\omega}=0$. Therefore, our starting point is the
general Brans-Dicke action with the $\omega$ unimodular term, reading

$S=V_{c}\int{dt\>\phi\dot{b}a^{2}+\dot{\phi}\pi_{\phi}+\alpha_{3}\dot{m}_{i}\psi_{i}+\phi
Na(b^{2}+kc_{g}^{2})-\alpha_{3}Na^{3}\rho-\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{a^{3}\omega}N+\omega\dot{T_{\omega}}}$
(3.146)

while the conservation equation for a varying $c_{g}$ only is

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=\frac{k\phi}{\alpha_{3}a^{2}}\frac{dc_{g}^{2}}{dt}$
(3.147)

The Hamiltonian is the same one as (3.116) and we can use it to derive the
Hamilton’s equation for $T_{\omega}$. Also, since we are not computing any
derivative with respect to $\phi$ that would require the use of $A^{2}$, the
$\phi_{0}$ and $\phi$ cases lead to equivalent results. Using the Hamilton’s
equation, we obtain

$\dot{T_{\omega}}=\\{T_{\omega},H\\}=-\frac{\partial
H}{\partial\omega}=\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\omega^{2}a^{3}}N$
(3.148)

The conservation equation, given the dependence of $c_{g}$ on $T_{\omega}$
becomes

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=\frac{k\phi}{\alpha_{3}a^{2}}\frac{\partial
c_{g}^{2}}{\partial T_{\omega}}\dot{T_{\omega}}$ (3.149)

which finally gives, using the $T_{\omega}$ equation of motion and the
definition of $\pi_{\phi}$,

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=\frac{ka\dot{\phi^{2}}}{\alpha_{3}N}\frac{\partial
c_{g}^{2}}{\partial T_{\omega}}$ (3.150)

Furthermore, since $T_{\omega}$ is canonical to $\omega$ and we also have its
equation of motion, we consider the inclusion of the $\dot{\omega}$ term in
the conservation equation, thus obtaining a direct relation between the
evolution of $\omega$ and $c_{g}$. To do so, we consider the more general
conservation equation where also $\omega$ is varying:

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\alpha_{3}\omega^{2}a^{6}}\dot{\omega}+\frac{k\phi}{\alpha_{3}a^{2}}\frac{dc_{g}^{2}}{dt}$
(3.151)

The equation of motion for $T_{\omega}$ is the same one as above, but we
additionally have the $\omega$ Hamilton’s equation, reading

$\dot{\omega}=\\{\omega,H\\}=\frac{\partial H}{\partial T_{\omega}}=-k\phi
Na\frac{\partial c_{g}^{2}}{\partial T_{\omega}}$ (3.152)

Substituting this inside the conservation equation and expanding the time
dependence of $c_{g}$ in terms of $T_{\omega}$, we arrive at

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=-\frac{1}{4}\frac{\phi^{2}k\pi_{\phi}^{2}}{\alpha_{3}\omega^{2}a^{5}}\frac{\partial
c_{g}^{2}}{\partial
T_{\omega}}N+\frac{ka\dot{\phi}^{2}}{\alpha_{3}N}\frac{\partial
c_{g}^{2}}{\partial T_{\omega}}$ (3.153)

which ultimately simplifies to

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=0$ (3.154)

using the definition of $\pi_{\phi}$. Furthermore, we can derive a
relationship between the Brans-Dicke parameter and the speed of light, such
that

$\dot{\omega}=-\frac{k}{a^{2}}\frac{\phi^{2}}{\dot{\phi}^{2}}\dot{c}_{g}^{2}$
(3.155)

####  3.3.8 Full $\phi$ with $\omega=\omega(T_{c_{g}})$

We might, of course, think about reverting the logic used in the scenario
above. In fact, we could consider a fixed speed of light as part of the
$\bm{\alpha}$ giving a relational time for the evolution of $\omega$ which
becomes part of the $\bm{\beta}$. This example just shows how the definitions
of constants and times can be easily interchangeable, crating interesting
mixes of concepts: in some cases a constant could be a time for evolution, in
other cases it varies with respect to a different time. Therefore, it is clear
how the idea of “evolution with respect to what” creates a circular argument
where different quantities can take roles that are dual to each others. Having
established this, we proceed by considering our starting point, the action:

$S=V_{c}\int{dt\>\phi\dot{b}a^{2}+\dot{\phi}\pi_{\phi}+\alpha_{3}\dot{m}_{i}\psi_{i}+\phi
Na(b^{2}+kc_{g}^{2})-\alpha_{3}Na^{3}\rho-\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{a^{3}\omega}N+c_{g}^{2}\dot{T_{c_{g}^{2}}}}$
(3.156)

where we notice that we keep $c_{g}^{2}$ as in the unimodular term as it
appears in the Hamiltonian for dimensional reasons. Given the dependency of
$\omega$ on the gravitational speed of light, the conservation equation
contains the source term given by $\dot{\omega}$ and the one given by
$\dot{c_{g}}^{2}$, i.e.,

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\alpha_{3}\omega^{2}a^{6}}\dot{\omega}+\frac{k\phi}{\alpha_{3}a^{2}}\frac{dc_{g}^{2}}{dt}$
(3.157)

where we have included the $\dot{c_{g}}^{2}$ term because of the equation of
motion for $c_{g}$ coming from the Hamiltonian. The equations of motion for
the canonical pair $(c_{g}^{2},T_{c_{g}^{2}})$ are

$\dot{c_{g}^{2}}=\\{c_{g}^{2},H\\}=\frac{\partial H}{\partial
T_{c_{g}^{2}}}=-\frac{1}{4}\frac{\phi\pi_{\phi}^{2}N}{\omega^{2}a^{3}}\frac{\partial\omega}{\partial
T_{c_{g}^{2}}}$ (3.158)
$\dot{T_{c_{g}^{2}}}=\\{T_{c_{g}^{2}},H\\}=-\frac{\partial H}{\partial
c_{g}^{2}}=\phi Nak$ (3.159)

The time equation can be substituted inside $\dot{\omega}$, giving, while
using also the definition of $\pi_{\phi}$,

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=\frac{ka}{\alpha_{3}N}\dot{\phi}^{2}\frac{\partial\omega}{\partial
T_{c_{g}^{2}}}+\frac{k\phi}{\alpha_{3}a^{2}}\dot{c_{g}^{2}}$ (3.160)

which, when using the $\dot{c_{g}}^{2}$ equation in the second term, finally
simplifies to

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=0$ (3.161)

as expected. This result is not a coincidence: it is, in fact, the same
obtained in (3.154) in the previous section. We might have expected this,
since these two scenarios differ only in whether the $\omega$ and the
$c_{g}^{2}$ are playing the role of a constant or of a time. In both cases,
since they are both gravitational parameters, the change in one is compensated
by the change in the other, thus producing no net energy violation. We can see
how these two parameters are not just related, but they are really dual to
each others, in the sense that they absorb/emit exactly each other’s
variations. To confirm that this is indeed the case, we can express
$\dot{c_{g}^{2}}$ in terms of $\dot{\omega}$, arriving at

$\dot{c_{g}^{2}}=-\frac{a^{2}\dot{\phi}^{2}}{kN^{2}\phi^{2}}\dot{\omega}$
(3.162)

which is exactly the same as equation (3.155) above. The interpretation of
this law is that as the speed of light undergoes the sharp phase transition
from the early Universe until now, $\omega$ increases until the large value
observed today. Of course, the change in order of magnitudes in $c_{g}$ has to
have been very large according to [24], larger then the change in $\omega$.
However, to account for that, we could consider more closely the pre-factor
and $\dot{\phi}$, as we will investigate in [63].

####  3.3.9 Full $\phi$ with $c_{m}^{2}=c_{m}^{2}(T_{\omega})$

In this subsection we enrich the marvellous landscape of varying constants
considering a further dependence on our new parameter, $\omega$. We analyse
the scenario where $\bm{\alpha}=\omega$ and $\bm{\beta}=c_{m}^{2}$.
Furthermore, since the addition of a varying $c_{g}^{2}$ does not affect the
overall conservation equation, we postulate that also
$c_{g}^{2}=c_{g}^{2}(T_{\omega})$ for completeness. The action we begin with
is

$S=V_{c}\int{dt\>\phi\dot{b}a^{2}+\dot{\phi}\pi_{\phi}+\alpha_{3}\dot{m}_{i}\psi_{i}+\phi
Na(b^{2}+kc_{g}^{2})-\alpha_{3}Na^{3}\rho-\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{a^{3}\omega}N+\dot{\omega}{T_{\omega}}}$
(3.163)

which gives us the usual Hamiltonian

$H=-\phi Na(b^{2}+kc_{g}^{2})+\alpha_{3}\rho
Na^{3}+\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\omega a^{3}}N$ (3.164)

The conservation equation for a varying $c_{g}^{2}$ and $c_{m}^{2}$ dependent
on $T_{\omega}$ includes two source terms,

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\alpha_{3}\omega^{2}a^{6}}\dot{\omega}+\frac{k\phi}{\alpha_{3}a^{2}}\frac{dc_{g}^{2}}{dt}$
(3.165)

To find explicit equations for $\dot{\omega}$ and $\dot{c_{g}^{2}}$ above, we
derive the Hamilton’s equations as

$\dot{\omega}=\\{\omega,H\\}=\frac{\partial H}{\partial T_{\omega}}=-\phi
Nak\frac{\partial c_{g}^{2}}{\partial
T_{\omega}}+\alpha_{3}Na^{3}\frac{\partial\rho}{\partial
c_{m}^{2}}\frac{\partial c_{m}^{2}}{\partial T_{\omega}}$ (3.166)
$\dot{T_{\omega}}=\\{T_{\omega},H\\}=-\frac{\partial
H}{\partial\omega}=\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\omega^{2}a^{3}}N$
(3.167)

We can now substitute the $\dot{\omega}$ equation in (3.165) and the
$\dot{T_{\omega}}$ one in the $\dot{c_{g}^{2}}$ source term to obtain

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=\frac{\dot{\phi}^{2}}{\phi}a^{3}\frac{\partial\rho}{\partial
c_{m}^{2}}\frac{\partial c_{m}^{2}}{\partial T_{\omega}}$ (3.168)

where the $\dot{c_{g}^{2}}$ term has cancelled the first contribution to
$\dot{\omega}$ and we have used the definition of $\pi_{\phi}$ and we have set
$N=1$.

We can go beyond this result, exploring the relationship between the Brans-
Dicke parameter and the matter speed of light. Starting form (3.166), we
assume $\dot{c_{g}^{2}}=0$ because we wish to find a relationship only between
$\omega$ and $c_{m}^{2}$, since we already have the one for $c_{g}^{2}$ in
subsection (3.3.8). Therefore, using the $\dot{\omega}$ equation jointly with
the $\dot{T_{\omega}}$ one and the definition of $\pi_{\phi}$ we finally
obtain

$\dot{\omega}=\alpha_{3}\frac{\phi}{\dot{\phi}^{2}}\frac{\partial\rho}{\partial
c_{m}^{2}}\dot{c}_{m}^{2}$ (3.169)

Unlike the previous case where $c_{g}^{2}=c_{g}^{2}(T_{\omega})$, the matter
speed of light increases proportionally with the Brans-Dicke parameter. This
will be further investigated in [63].

####  3.3.10 Full $\phi$ with $c_{m}^{2}=c_{m}^{2}(T_{\phi},T_{\omega})$

In the spirit of considering all the possible extensions of $T_{\omega}$, we
illustrate the scenario where $\bm{\alpha}=(\phi,\omega)$ and
$\bm{\beta}=c_{m}^{2}$. For completeness, we also include a varying
$c_{g}^{2}$ depending, just like $c_{m}^{2}$, on both $T_{\phi}$ and
$T_{\omega}$ as we will see. As in previous sections, the unimodular term for
the Ricci time is redundant as already provided by the canonical pair, so the
only unimodular term is for $T_{\omega}$. The action reads

$S=V_{c}\int{dt\>\phi\>\dot{b}a^{2}+\dot{\phi}\pi_{\phi}+\alpha_{3}\dot{m}_{i}\psi_{i}+\phi
Na(b^{2}+kc_{g}^{2})-\alpha_{3}Na^{3}\rho-\frac{1}{4}\frac{\phi}{\omega}\frac{\pi_{\phi}^{2}}{a^{3}}N+\dot{\omega}T_{\omega}}$
(3.170)

As always in the scenarios were we have some dependency on $T_{\phi}$, we need
to be more careful about which conservation equation to use. In fact, the
general form (3.110) should not be used, as the extra dependency of
$c_{m}^{2}$ on $T_{\phi}$ has not been taken into consideration. Therefore, we
start with a more general form, where the $\dot{\phi}$ terms have not been
substituted for

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=\frac{\dot{\phi}}{\phi}\rho-\frac{\dot{\phi}}{\phi}\frac{\rho+3p}{2}-\frac{\dot{\phi}\pi_{\phi}^{2}}{2\alpha_{3}\omega
a^{6}}-\frac{1}{2}\frac{\pi_{\phi}\dot{\pi_{\phi}}\phi}{\alpha_{3}\omega
a^{6}}+\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\alpha_{3}\omega^{2}a^{6}}\dot{\omega}+\frac{k\phi}{\alpha_{3}a^{3}}\dot{c_{g}^{2}}$
(3.171)

We may now proceed substituting the equations for $\dot{\phi}$ and
$\dot{\pi_{\phi}}$ obtained from the Hamiltonian as

$\dot{\phi}=\\{\phi,H\\}=\frac{\partial
H}{\partial\pi_{\phi}}=\frac{1}{2}\frac{\phi\pi_{\phi}N}{\omega
a^{3}}+\alpha_{3}Na^{3}\frac{\partial\rho}{\partial c_{m}^{2}}\frac{\partial
c_{m}^{2}}{\partial T_{\phi}}-\phi Nak\frac{\partial c_{g}^{2}}{\partial
T_{\phi}}$ (3.172) $\dot{\pi_{\phi}}=\\{\pi_{\phi},H\\}=-\frac{\partial
H}{\partial\phi}=-N\biggl{(}\frac{-\rho+3p}{2}\biggl{)}\frac{\alpha_{3}a^{3}}{\phi}-\frac{1}{2}\frac{\pi_{\phi}^{2}N}{\omega
a^{3}}$ (3.173)

where we notice that the $\dot{\pi_{\phi}}$ equation is the same one we
derived in subsection (3.3.2). Furthermore, since $c_{g}^{2}$ and $c_{m}^{2}$
are also deepening on $T_{\omega}$, we have the additional two Hamilton’s
equations

$\dot{\omega}=\\{\omega,H\\}=\frac{\partial H}{\partial T_{\omega}}=-\phi
Nak\frac{\partial c_{g}^{2}}{\partial
T_{\omega}}+\alpha_{3}Na^{3}\frac{\partial\rho}{\partial
c_{m}^{2}}\frac{\partial c_{m}^{2}}{\partial T_{\omega}}$ (3.174)
$\dot{T_{\omega}}=\\{T_{\omega},H\\}=-\frac{\partial
H}{\partial\omega}=\frac{1}{4}\frac{\phi\pi_{\phi}^{2}N}{\omega^{2}a^{3}}$
(3.175)

Substituting these in equation (3.172) we obtain

$\displaystyle\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=\frac{\rho}{2}\frac{\pi_{\phi}N}{\omega
a^{3}}+\frac{\alpha_{3}\rho Na^{3}}{\phi}\frac{\partial\rho}{\partial
c_{m}^{2}}\frac{\partial c_{m}^{2}}{\partial T_{\phi}}-\rho Nak\frac{\partial
c_{g}^{2}}{\partial T_{\phi}}$ (3.176)
$\displaystyle-\frac{(\rho+3p)}{4}\frac{\pi_{\phi}N}{\omega
a^{3}}-\frac{(\rho+3p)\alpha_{3}Na^{3}}{2\phi}\frac{\partial\rho}{\partial
c_{m}^{2}}\frac{\partial c_{m}^{2}}{\partial
T_{\phi}}+\frac{(\rho+3p)}{2}Nak\frac{\partial c_{g}^{2}}{\partial T_{\phi}}$
(3.177)
$\displaystyle-\frac{1}{4}\frac{\phi\pi_{\phi}^{3}N}{\alpha_{3}\omega^{2}a^{9}}-\frac{1}{2}\frac{\pi_{\phi}^{2}N}{\omega
a^{3}}\frac{\partial\rho}{\partial c_{m}^{2}}\frac{\partial
c_{m}^{2}}{\partial
T_{\phi}}+\frac{1}{2}\frac{\phi\pi_{\phi}^{2}Nk}{\alpha_{3}\omega
a^{5}}\frac{\partial c_{g}^{2}}{\partial
T_{\phi}}+\frac{(-\rho+3p)}{4}\frac{\pi_{\phi}N}{\omega a^{3}}$ (3.178)
$\displaystyle+\frac{1}{4}\frac{\phi\pi_{\phi}^{3}N}{\alpha_{3}\omega^{2}a^{9}}-\frac{1}{4}\frac{\phi^{2}\pi_{\phi}^{2}Nk}{\alpha_{3}\omega^{2}a^{5}}\frac{\partial
c_{g}^{2}}{\partial
T_{\omega}}+\frac{1}{4}\frac{\phi\pi_{\phi}^{2}N}{\omega^{2}a^{3}}\frac{\partial\rho}{\partial
c_{m}^{2}}\frac{\partial c_{m}^{2}}{\partial T_{\omega}}$ (3.179)
$\displaystyle+\frac{1}{4}\frac{\phi^{2}\pi_{\phi}^{2}kN}{\alpha_{3}\omega^{2}a^{5}}\frac{\partial
c_{g}^{2}}{\partial
T_{\omega}}-\biggl{(}\frac{-\rho+3p}{2}\biggl{)}kNa\frac{\partial
c_{g}^{2}}{\partial
T_{\phi}}-\frac{1}{2}\frac{\phi\pi_{\phi}^{2}kN}{\alpha_{3}\omega
a^{5}}\frac{\partial c_{g}^{2}}{\partial T_{\phi}}$ (3.180)

which heavily simplifies to

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=\frac{\partial\rho}{\partial
c_{m}^{2}}\biggl{[}\frac{\alpha_{3}Na^{3}}{\phi}\frac{(\rho-3p)}{2}\frac{\partial
c_{m}^{2}}{\partial T_{\phi}}-\frac{1}{2}\frac{\pi_{\phi}^{2}N}{\omega
a^{3}}\frac{\partial c_{m}^{2}}{\partial
T_{\phi}}+\frac{1}{4}\frac{\phi\pi_{\phi}^{2}N}{\omega^{2}a^{3}}\frac{\partial
c_{m}^{2}}{\partial T_{\omega}}\biggl{]}$ (3.181)

This shows, once again, energy violation in a scenario where a matter
parameter depends on one or more gravitational clocks. Interestingly, the
first source term is equivalent to the one we will find for
$c_{m}^{2}=c_{m}^{2}(T_{\phi},T_{N})$, as well as the second one, arising from
the canonical momentum $\pi_{\phi}$. However, the third term is new and it
provides a source in $c_{m}^{2}$ given by the Brans-Dicke time $T_{\omega}$.

####  3.3.11 Fixed $\phi_{0}$ with $c_{m}^{2}=c_{m}^{2}(T_{\phi},T_{N})$

We would like to derive a scenario for the $\alpha_{3}$-like case. This arises
when we fix $\phi_{0}$ only in the canonical pair. As we have seen in
subsection (3.3.1), doing so leads to different equations of motion, but gives
an overall conservation equation equal to the full $\phi$ (which we analyse in
the next subsection). Therefore, we would like to consider the case where
$\bm{\alpha}=(\phi,\alpha_{3})$ and $\bm{\beta}=c_{m}^{2}$. Once again, for
completeness, also $c_{g}^{2}$ is assumed to vary, such that
$\bm{\beta}=c_{g}^{2}$ as well. The action we begin with is

$S=V_{c}\int{dt\>\phi_{0}\>\dot{b}a^{2}+\dot{\phi}\pi_{\phi}+\alpha_{3}\dot{m}_{i}\psi_{i}+\phi
Na(b^{2}+k)-\alpha_{3}Na^{3}\rho-\frac{1}{4}\frac{\phi}{\omega}\frac{\pi_{\phi}^{2}}{a^{3}}N}$
(3.182)

where the second canonical pair act as a unimodular term for the Ricci clock.
In this case, since the $\phi$ in the canonical pair is fixed, it is not
necessary to express the Hamiltonian in terms of $A^{2}$, so we can directly
consider the general conservation equation (3.75)

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=-\frac{\dot{\alpha_{3}}}{\alpha_{3}}\rho+\frac{\dot{\phi}}{\phi}\rho-\frac{1}{2}\frac{\phi\pi_{\phi}\dot{\pi_{\phi}}}{\alpha_{3}\omega
a^{6}}+\frac{1}{4}\frac{\phi\pi_{\phi}^{2}}{\alpha_{3}\omega^{2}a^{6}}\dot{\omega}+\frac{k\phi}{\alpha_{3}a^{2}}\frac{dc_{g}^{2}}{dt}$
(3.183)

To find equations for the terms with $\dot{\alpha_{3}}$, $\dot{\phi}$ and
$\dot{\pi_{\phi}}$ we use, once again, the Hamiltonian, giving

$\dot{\phi}=\\{\phi,H\\}=\frac{\partial
H}{\partial\pi_{\phi}}=\frac{1}{2}\frac{\phi\pi_{\phi}N}{\omega
a^{3}}+\alpha_{3}Na^{3}\frac{\partial\rho}{\partial c_{m}^{2}}\frac{\partial
c_{m}^{2}}{\partial T_{\phi}}-\phi Nak\frac{\partial c_{g}^{2}}{\partial
T_{\phi}}$ (3.184) $\dot{\pi_{\phi}}=\\{\pi_{\phi},H\\}=-\frac{\partial
H}{\partial\phi}=\frac{\alpha_{3}\rho}{\phi}Na^{3}\equiv\dot{T_{\phi}}$
(3.185)

where the clock equation was already obtained in subsection (3.75) as the
equation of motion of the conjugate momentum of $\phi$. On the other hand,
given the $\alpha_{3}$ dependence, we have

$\dot{\alpha_{3}}=\\{\alpha_{3},H\\}=\frac{\partial H}{\partial T_{N}}=-Na\phi
k\frac{\partial c_{g}^{2}}{\partial
T_{N}}+Na^{3}\alpha_{3}\frac{\partial\rho}{\partial c_{m}^{2}}\frac{\partial
c_{m}^{2}}{\partial T_{N}}$ (3.186)
$\dot{T_{\mathds{N}}}=\\{T_{N},H\\}=-\frac{\partial
H}{\partial\alpha_{3}}=-\rho Na^{3}$ (3.187)

Therefore, substituting these expressions in the conservation equation and
using that

$\frac{dc_{g}^{2}}{dt}=\frac{\partial c_{g}^{2}}{\partial
T_{\phi}}\dot{T_{\phi}}+\frac{\partial c_{g}^{2}}{\partial T_{N}}\dot{T_{N}}$
(3.188)

we arrive at

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=\rho Na^{3}\frac{\partial\rho}{\partial
c_{m}^{2}}\biggl{[}\frac{\alpha_{3}}{\phi}\frac{\partial c_{m}^{2}}{\partial
T_{\phi}}-\frac{\partial c_{m}^{2}}{\partial T_{N}}\biggl{]}$ (3.189)

were we see that both the Ricci and the Newton times create a source term for
the conservation equation. Interestingly, this result closely relates to the
one found in [22], the main difference being the Ricci time source term. In
fact, due to the fixed scalar field in the canonical term, the equation of
motion for $T_{R}$ is different, resulting in a theory with only energy
density.

####  3.3.12 Full $\phi$ with $c_{m}^{2}=c_{m}^{2}(T_{\phi},T_{N})$

We now consider the scenario where $c_{m}^{2}$ depends on the Ricci and the
Newton times, following the same spirit of [22]. As before, we include also a
varying $c_{g}^{2}$ for completeness. The action is

$S=V_{c}\int{dt\>\phi\>\dot{b}a^{2}+\dot{\phi}\pi_{\phi}+\alpha_{3}\dot{m}_{i}\psi_{i}+\phi
Na(b^{2}+k)-\alpha_{3}Na^{3}\rho-\frac{1}{4}\frac{\phi}{\omega}\frac{\pi_{\phi}^{2}}{a^{3}}N}$
(3.190)

giving the usual Hamiltonian as

$H=-\phi
Na(b^{2}+kc_{g}^{2})+\alpha_{3}Na^{3}\rho+\frac{1}{4}\frac{\phi}{\omega}\frac{\pi_{\phi}^{2}}{a^{3}}N$
(3.191)

which may be expressed in terms of $A^{2}=\phi a^{2}$ as

$H=-\phi
N\frac{A}{\phi^{\frac{1}{2}}}(b^{2}+kc_{g}^{2})+\frac{A^{3}}{\phi^{\frac{3}{2}}}\alpha_{3}N\rho+\frac{1}{4}\frac{\phi}{\omega}\pi_{\phi}^{2}N\frac{\phi^{\frac{3}{2}}}{A^{3}}$
(3.192)

This form of the Hamiltonian will prove crucial in deriving the
$\dot{\pi_{\phi}}$ equation, as the correct canonical variable is given by
$\phi a^{2}$ when we include a Ricci time dependence. The general conservation
equation, without using the equation of motion for $\dot{\phi}$ derived by the
definition of $\pi_{\phi}$ is therefore

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=-\frac{\dot{\alpha_{3}}}{\alpha_{3}}\rho+\frac{\dot{\phi}}{\phi}\rho-\frac{\dot{\phi}}{\phi}\frac{\rho+3p}{2}-\frac{\dot{\phi}\pi_{\phi}^{2}}{2\alpha_{3}\omega
a^{6}}-\frac{1}{2}\frac{\pi_{\phi}\dot{\pi_{\phi}}\phi}{\alpha_{3}\omega
a^{6}}+\frac{k\phi}{\alpha_{3}a^{3}}\dot{c_{g}^{2}}$ (3.193)

where we are including the $\dot{\alpha_{3}}$ term because $c_{m}^{2}$ and
$c_{g}^{2}$ depend on the Newton time, while the $\dot{\omega}$ term has been
removed as, in this scenario, $\omega$ is constant. We can now derive the
equations of motion for $\phi$ and $\pi_{\phi}$, obtaining

$\dot{\phi}=\\{\phi,H\\}=\frac{\partial
H}{\partial\pi_{\phi}}=\frac{1}{2}\frac{\phi\pi_{\phi}N}{\omega
a^{3}}+\alpha_{3}Na^{3}\frac{\partial\rho}{\partial c_{m}^{2}}\frac{\partial
c_{m}^{2}}{\partial T_{\phi}}-\phi Nak\frac{\partial c_{g}^{2}}{\partial
T_{\phi}}$ (3.194)
$\dot{\pi_{\phi}}=\\{\pi_{\phi},H\\}=-\frac{\partial
H}{\partial\phi}=-N\biggl{(}\frac{-\rho+3p}{2}\biggl{)}\frac{\alpha_{3}a^{3}}{\phi}-\frac{1}{2}\frac{\pi_{\phi}^{2}N}{\omega
a^{3}}\equiv\dot{T_{\phi}}$ (3.195)

On the other hand, we have the equations of motion for the canonical pair
$(\alpha_{3},T_{N})$

$\dot{\alpha_{3}}=\\{\alpha_{3},H\\}=\frac{\partial H}{\partial T_{N}}=-Na\phi
k\frac{\partial c_{g}^{2}}{\partial
T_{N}}+Na^{3}\alpha_{3}\frac{\partial\rho}{\partial c_{m}^{2}}\frac{\partial
c_{m}^{2}}{\partial T_{N}}$ (3.196)
$\dot{T_{\mathds{N}}}=\\{T_{N},H\\}=-\frac{\partial
H}{\partial\alpha_{3}}=-\rho Na^{3}$ (3.197)

Therefore, combining them inside equation (3.193) and carrying out the
necessary simplifications we arrive at

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=Na^{3}\frac{\partial\rho}{\partial
c_{m}^{2}}\biggl{[}\frac{\alpha_{3}}{\phi}\frac{\partial c_{m}^{2}}{\partial
T_{\phi}}\frac{\rho-3p}{2}-\frac{\partial c_{m}^{2}}{\partial
T_{N}}\rho\biggl{]}-\frac{1}{2}\frac{N\pi_{\phi}^{2}}{\omega
a^{3}}\frac{\partial\rho}{\partial c_{m}^{2}}\frac{\partial
c_{m}^{2}}{\partial T_{\phi}}$ (3.198)

This result, as the one obtained in the previous section, also closely relates
to [22]. In fact, the first term on the RHS of equation (3.198) is exactly the
same with the scalar field giving the Ricci time. However, the second term is
new and it is the result of the scalar field momentum giving an extra source
term for $c_{m}^{2}$.

Having reached the end of our research, we provide a short overview on the
general pattern followed by varying constants. To get net energy violation two
possible cases can be considered: a gravitational parameter depending on a
matter clock or a matter clock depending on a gravitational parameter. In the
first case, we might have $\bm{\beta}=(c_{g}^{2},\omega,\phi)$, all depending
on a matter clock given by the time canonical to an $\bm{\alpha}$ such as
$\alpha_{3}$ or $\rho_{\Lambda}$. These cases lead to net energy violation as
the variation of the parameter can be exchanged with the constant of motion of
the clock, since it is given by a matter parameter. The other scenario occurs
when a matter parameter depends on a gravitational clock. This is the case,
for example, when $\bm{\beta}=c_{m}^{2}$ and $\bm{\alpha}=\omega$. Once again,
the varying matter parameter can exchange energy with the gravitational clock,
thus giving a net energy violation. All the other combinations, as we have
shown above, do not lead to any energy violation, with all the constant’s
variation being absorbed in the clock. As expected, with the exception of
extra terms given by the canonical momentum of $\phi$, our results reproduce
the pattern identified in [22].

##  Chapter 4 Conclusion and Discussion

We have reached the conclusion of our initial work on varying constants in
cosmological Brans-Dicke theories. Our starting point has been a conceptual
analysis of the ideas surrounding physical constants and the possibility they
might be varying with time. We have reviewed the main theoretical intuitions
regarding this topic, ranging from Dirac’s Large Number Hypothesis to Mach’s
Principle, presenting their consequences. Furthermore, we have explored the
experimental evidences supporting these ideas, opening the way to more
extensive applications of varying constants in different fields of physics,
from the Standard Model to Cosmology. These all have been necessary precursors
towards the main objective of this research: the Brans-Dicke theory of
gravity. In fact, motivated by the natural implementation of Mach’s principle
and of a varying $G$ in the Brans-Dicke action, we have used it as the
platform to extend varying constants theories. Doing so, we have produced
several results regarding energy conservation and the Cosmological Constant
problem.

In Chapter 2, we have presented the main open problems is the Standard Big
Bang Cosmology, providing the current solution of inflation. These
cosmological puzzles then lead us to consider an alternative solution, based
on the idea of a time varying speed of light. We have reviewed the main
results of the VSL model in cosmology, as the conceptual basis for our next
steps. Generalising this idea to more constants of Nature, it became necessary
an appropriate mathematical formalism. This was promptly supplied by
unimodular gravity and minisuperspace. The first one allowed to obtain
equations of motion for the constants directly from the action principle,
having demoted them to constants of motion on-shell. Furthermore, unimodular
gravity offered solid definitions of relational physical times, excellent as
evolution parameters for the constants we are studying. On the other hand,
reducing the Einstein-Cartan action to minisuperpace enabled us to fully
exploit the Hamiltonian formalism of General Relativity. Therefore, we
developed the minisuperspace reduction of the Einstein-Cartan and Einstein-
Hilbert actions for a FLRW metric, showing how it naturally produces an
Hamiltonian structure we can use to derive the equations of motion for the
constants and their times.

Continuing, we have applied these formalisms to obtain energy conservation
results in cosmology. Doing so, we were able to characterise different
scenarios where some constants provide clocks while other vary with respect to
these clocks. Interestingly, when considering overall energy violation, a
pattern arose, linking gravitational parameters to matter clocks and vice-
versa. All the other combinations of constants have been found to be sterile,
accordingly to [22], resulting in the standard conservation equation. The
landscape of varying constant resulted to be a vast and manifold one, leading
us to consider multiple combinations in our extensions.

Moreover, in Chapter 3, we developed the core of our original work: energy
conservation due to varying constants in Brans-Dicke cosmologies. Following
the procedure outlined in Chapter 2, we presented the energy conservations and
violations resulting from time dependencies like $T_{\Lambda}$, $T_{\phi}$ and
$T_{N}$ with a dynamical scalar field $\phi$. We also considered the
dependence of several constants on the Brans-Dicke time $T_{\omega}$,
enlarging the set of constants that can be varying. Particularly, this
addition will turn useful in future work on Black Holes applications by
Magueijo [94]. Applying the idea of varying constant to Brans-Dicke theory, we
have obtained several energy conservation and violation scenarios where the
$\omega$ parameter plays the important role of energy source. Specifically,
considering its dependency on the cosmological constant time $T_{\Lambda}$, we
have obtained a relationship between $\rho_{\Lambda}$ and $\omega$ which might
lead to future developments in the context of the vacuum energy sequester
[63]. Furthermore, similar relationships have been obtained for $c_{g}^{2}$
and $\alpha_{3}$, also leading to future extensions connected to VSL and the
early Universe [63]. Finally, the inclusion of the Brans-Dicke parameter
$\omega$ has potentially enabled us to connect our theoretical results in the
field of energy conservation with cosmological observations of the Brans-Dicke
theory.

To conclude, we would like to briefly mention the future developments and
applications of these results, some of which will be included in a paper to
come [63]. Firstly, as Chapter 3 has taught us, the possible combinations of
constants and times are plenty, to the point that only imagination is really
the limit when coupling multiple different actions to gravity. Therefore, in
our future work, we will be selecting the most interesting, relevant and
insightful scenarios and focus on them. As mentioned earlier, these might be
the ones were $\omega$ is linked to $c_{g}^{2}$ and $\rho_{\Lambda}$.
Specifically this ladder case brings us to our next extension of this theory:
the vacuum energy sequester. In fact, a scenario where
$\omega=\omega(T_{\Lambda})$ might provide an explanation to the Cosmological
Constant problem, where the vacuum energy is absorbed by the variation of
$\omega$. This is particularly interesting given the observational constraints
put on $\omega$ [66]. Secondly, when considering the Brans-Dicke kinetic term,
we set the $c$ factor appearing with it to zero. However, this could not be
the case and therefore we will investigate its inclusion, especially in the
$\omega=\omega(T_{\Lambda})$ scenario. Thirdly, the scenarios where $\phi$
provides the Ricci time will be further extended, including the case when both
the unimodular action and the canonical pair provide equivalent times
definitions. Other applications of our theory could be linked to the Standard
Model, where we might have coupling and particle’s parameters depending on
$\omega$, leading to unknown connections between cosmology and particle
physics.

Lastly, among the many extensions of this work, the most fascinating one is a
theory explaining the origin of varying constants beyond the String Theory
formalism. Throughout this dissertation, we have postulated that the constants
of Nature are varying with respect to relational times. To do so, we have used
the unimodular formalism, which provided a solid and rigorous mathematical
platform to implement this idea. However, this formalism does not explain why
the constants of Nature could be varying, omitting the mechanism behind their
dynamics. One simple answer to this would be, as we have done in this work, to
assume that constants do vary, without asking the deeper question of why.
After all, often Physics explains how events happen and not why. But what if
it could be possible to formulate a theory predicting and describing the
evolution of some constants? It would certainly be exciting to discover if
these theory’s predictions agree with the experimental evidences, leading to a
radically different view of the Reality we live in. This speculation, like
many other ideas, is left to the future, when maybe our posterity will finally
consider trivial what we, for so long, chased into darkness.

“Veniet tempus quo posteri nostri tam aperta nos nescisse mirentur”

Lucius Annaeus Seneca

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