1. Introduction
The possibility of the variation of fundamental physical constants has been long explored as a means of solving some issues in the description of nature, especially in astrophysics and cosmology. In spite of being a controversial idea, it has a deep connection with scalar–tensor theories, themselves a popular candidate for viable modified theories of gravity. This paper explored the possible interwoven variation of the couplings G and c (while also accommodating the effects of ) in a Brans–Dicke-like model. Our motivations will be stated and the construction of our setup will begin during this introductory section, which starts off with a brief perspective on the subject of varying fundamental constants.
The two most-studied constants for their potential variation on cosmological time scales are the gravitational constant
G and the fine structure constant
. While
G is a dimensionful constant,
is a dimensionless constant. Uzan [
1,
2] critically reviewed the subject of the variation of fundamental constants and discussed it extensively. Some authors (Ellis and Uzan [
3], Duff [
4,
5]) strongly argue against dimensionful varying fundamental constants. In an attempt to avoid this criticism, we normalized the time derivative of the physical coupling by the value of the associated quantity.
Variable fundamental constant theories can be traced back to at least the late 19th Century, e.g., [
6], and the first half of the last century, e.g., [
7,
8]. However, they gained importance after Dirac [
9,
10] suggested the potential variation of the gravitational constant
G based on his large number hypothesis. Many observational methods were suggested to constrain an eventual variation of
G including neutron star masses and ages [
11]; CMB anisotropies, e.g., [
12]; big bang nucleosynthesis abundances, e.g., [
13]; asteroseismology [
14]; lunar laser ranging, e.g., [
15]; the evolution of planetary orbits, e.g., [
16]; binary pulsars, e.g., [
17]; Supernovae Type-Ia (SNeIa) luminosity evolution, e.g., [
18]; and gravitational wave observations from binary neutron stars [
19]. Almost all of them have resulted in constraints on
well below that predicted by Dirac. There has been significant development on the theoretical side as well. Building on the work of Jordan [
20], Brans and Dicke [
21] developed a scalar–tensor theory of gravitation, wherein
was raised to the status of a scalar field that could vary spatially and temporally. The Brans–Dicke (BD) theory may be seen as one representative of the class of scalar–tensor theories in which gravitation manifests itself by both the metric tensor and a scalar field of a geometrical nature [
22]. A more general example within this class is the scalar–tensor theory in the Lyra manifold [
23,
24]. One of the reasons why scalar–tensor theories of gravity raise so much interest (see [
22,
25] and the references therein) is their well-known equivalence with some modified gravity theories [
26], cf., e.g., [
27,
28]. Among these possible modifications of gravity, we mention the theories
[
29,
30,
31],
[
32],
[
33], and
[
34,
35], which have been proposed as attempts to deal with the GR quantization problem [
36,
37], to realize inflationary models [
38], and to address the dark energy problem [
39]. References [
40,
41,
42] studied the phenomenology of extended Jordan–Brans–Dicke theories and their implication for cosmological datasets. Moreover, significant astrophysical consequences can be found in the literature within the realm of modified theories of gravity including those in [
43,
44,
45,
46].
The
constancy of the speed of light is the foundation of the special and general relativity theories, and arguably, its variation is the most contentious issue in physics. However, even Einstein considered its possible
variation [
47]. There are several theories of the variation of the speed light, e.g., those by Dicke [
48], Petit [
49], and Moffatt [
50,
51]. Some of these proposals break Lorentz invariance, e.g., [
52,
53,
54]; others produce locally invariant theories [
55,
56].
Attempts have also been made to consider the simultaneous variation of two or more constants, e.g., Reference [
57] considered varying
G and
; the variation of
was studied in [
58];
G and
changed concomitantly in [
59]; covarying
c,
G, and
were the subject of [
60,
61,
62]; the set
was allowed to vary in [
63]; finally, References [
64,
65] suggested that all the couplings
G,
c,
,
ħ, and
must covary.
After this contextualization of the subject of varying physical constants, let us concentrate on the topic to be explored here. Our focus in this paper was on the variation of
c and
G and the interrelationship of their variation. We will see that this interrelationship yields naturally an effective cosmological constant. Gupta [
66] introduced the general ansatz:
inspired by the general constraint appearing in the proposal by Costa et al. [
61]. The quantity
is a constant parameter. The value
is strongly favored by several phenomenological applications of Equation (
1) in cosmology [
64,
66,
67,
68] and astrophysics [
65,
69,
70,
71]. This preferred value for
is corroborated by other authors, cf., e.g., [
63]. For this reason, we set off to investigate the fundamental reasons that might be underlying this fact.
We did this by assuming a modification in the standard theory of gravity: General Relativity (GR). This is justified by the fact that GR does not allow for varying physical constants. In fact, the couplings
c, and
showing up in the Einstein field equations:
do not vary in time or space. Recall the definition of the Einstein tensor:
, where
is the Ricci tensor and
R is the curvature scalar. We followed the metric signature and Riemann tensor definition of [
72], i.e., the Minkowski metric in Cartesian coordinates reads
,
with the curvature tensor calculated by
,
, and
stands for the Christoffel symbols built from the metric tensor
and its first-order derivatives. Covariant derivatives were built from the Christoffel, e.g.,
, for an arbitrary contravariant vector
.
Our generalization of GR stems from an action integral that is inspired by the Einstein–Hilbert action:
but with the couplings
being spacetime functions. At this point, we emphasize the natural appearance of the couplings
explicitly in the action integral that is supposed to describe gravity. We feel the urge to underscore this fact for two reasons. The first is: in the ordinary approach of GR, both
G and
c are only multiplicative constants of the kernel
and, as such, can be taken outside the integral with impunity. In fact, people even go further to use units where
, the so-called natural units. These practices are very dangerous here because both
G and
c are space–time-dependent functions; accordingly, they are directly affected by the variation process soon to be carried out. Second, the fact that (
3) displays the set
justifies why we considered only these three couplings, instead of bringing about other fundamental constants as well, such as the Planck constant
ħ and the Boltzmann constant
. Others works considered this enlarged set from a phenomenological perspective—see, e.g., [
64,
65,
66,
67,
68,
71,
73].
As usual, the four-volume in the action
is
. It is worth stressing that the time coordinate
encompasses the speed of light
c, which is allowed to vary in our context. Therefore, it is paramount to work with
throughout. In this way, covariance is guaranteed; caveats such as having to deal with the opened
-differential
are avoided; the speed of light will not appear in the metric components
explicitly. Consequently, the determinant of the metric tensor
will depend explicitly on
, but not on
c. The time component
hides
c in a convenient form: if
c changes, then a time reparameterization
compensates for this fact in each reference frame. This strategy is somewhat different from the procedure of the
c-flation framework developed in [
61]. Here,
has dimensions of length, which explains the power cube for the speed of light in (
3); otherwise, the action would not bear the correct dimension of angular momentum (energy × time).
Gupta’s proposal in Equation (
1) indicates that the couplings
G and
c are intertwined. In fact, it could be recast as
where
is a scalar field comprising the gravitational coupling
G and the speed of light
c. The dot on top of the quantity means a derivative with respect to
, for instance
.
The simplest way to satisfy (
1) is to admit that
due to constant couplings, in which case
and
. This is the way of GR, but it is not the most-general possibility and not the one we adopted here. An earlier attempt to relax the constancy of fundamental couplings in the context of a theory of gravity was the Brans–Dicke (BD) scalar–tensor theory [
21]. Aiming to fully realize the Machian principle in a geometrical theory of gravitation, Brans and Dicke proposed a varying gravitational coupling through the scalar field [
22]:
Our proposal is to extend the scope of the BD field to include the spacetime coupling c as part of its very definition. The presence of c changes the nature of the scalar field , as previously envisioned by Brans and Dicke (in fact, we geometrized not only the gravitational coupling—through G—but also the causality coupling c). This is true even from a dimensional point of view. In the Brans–Dicke picture, is measured in units of . In our scalar–tensor gravity, has dimensions of .
The fact that a varying
c enters the scalar field
in Equation (
5) may be a cause for concern to some. A common criticism upon varying speed of light proposals is that the Maxwell electrodynamics should not be tempered with. This is a wise thought, however one that can be bypassed by arguments such as that by Ellis and Uzan in [
3]. There could be different types of speed of light, viz. the spacetime speed of light
related to Lorentz invariance and causality and the electromagnetic speed of light
built from the couplings in electromagnetism: the (vacuum) electric permittivity
and the magnetic permeability
. Here, we interpreted
c as the causality coupling
and ensured that Lorentz invariance (of Maxwell’s theory) is not broken locally by convenient reparameterizations of
t at each time slice of the space–time manifold. Moreover, our approach deals with cosmology, which makes room for a variation of
c with respect to the cosmic time in such a way that
currently.
The main goal of this paper was to investigate if the dynamics of
can naturally lead to
as an attractor solution. This would give the fundamental reason for why the value
is preferred in several apparently different contexts, such as cosmology [
66], solar astrophysics [
71], and solar system kinematics [
70]. After the eventual relation between
and
is established, a natural question poses itself: “What are the consequences of
for the cosmic evolution?”. Our secondary goal in this paper was to answer this question by investigating cosmological solutions attributable to different epochs in the Universe’s thermal history (e.g., radiation-dominated era, dust matter era, and the recent accelerated regime). We built our scalar–tensor model in a metric-compatible, four-dimensional, torsion-free, Riemannian manifold. Approaches considering a richer geometrical structure [
24], higher-order derivatives of curvature-based objects [
28,
32], or torsion-based invariants [
35] are possible, but these cases shall be explored elsewhere.
The remainder of the paper is organized as follows. In
Section 2, we state the action of our scalar–tensor model and derive the equations of motion for the tensor field
and the scalar field
, which together build the gravitational field. The field equations are specified for the FLRW metric [
74], since our main interest was to study the dynamics of
in the cosmological context. This study is carried out in
Section 3, where it is shown that an attractor solution leading to
is attainable under reasonable hypotheses for the potential
and the dominant matter–energy content during the evolution of
(both the linear stability theory and Lyapunov’s method are briefly reviewed in
Section 3.1 for the sake of completeness of our presentation and convenience for the reader). This finding comes from the linear stability analysis of our dynamical system—performed in
Section 3.2; the result is confirmed and strengthened in
Section 3.3, where we used the general Lyapunov’s method to show that the critical point
leading to
is a globally asymptotically stable fixed point.
Section 3.4 gives some details about the system’s attractor point and emphasizes the consequences of our modified gravity. The cosmic evolution after
reaches equilibrium is analyzed in
Section 4, wherein it is shown that our scalar–tensor model accommodates radiation- and matter-dominated eras evolving naturally to a de Sitter-type accelerated expansion, possibly recognized as dark energy. This is true for the two parameterizations we adopted for the function
, the first assuming a speed of light that decreases as the Universe expands, while the second admits
c increasing with the scale factor. Our final comments appear in
Section 5.
2. Modified Gravity with
The motivations in the previous section led us to describe gravity from a precisely specified form for the action integral. In fact, this form is hinted at by the coefficient of the curvature scalar
R in Equation (
7), namely
, which was defined as the scalar field
. Accordingly, we introduce
where
is a dimensionless constant of order one (it is argued that the Brans–Dicke theory recovers general relativity in the limit
, cf., e.g., [
74]. However, this is not always the case: Reference [
75] presented a counter-example) and
is a scalar field. The constant
appears as the coefficient of the kinetic term for the field
; the denominator of this term includes
for dimensionality consistency. The cosmological coupling
was not written explicitly in (
7) because it can be included in the potential
. In fact, we will show that a constant
can be realized in our framework. The reader can easily check that all terms in (
7) were crafted in order to give
S the correct dimension of
. This includes inserting the factor
explicitly inside the matter term integral containing the matter Lagrangian density
, itself carrying dimensions of energy density.
The action (
7) is formally the same as the Brans–Dicke action. While the Brans–Dicke scalar field is usually interpreted as an effective
(see [
22]), our
includes both
G and
c. Notice, moreover, that (
7) is not the same as the actions in the Varying Speed of Light (VSL) proposals—such as those in [
50,
52,
53]—for reasons ranging from different modeling to the basic fact that variations of the gravitational coupling were disregarded therein. Additionally, it should be pointed out that (
7) does not correspond to the same treatment given in [
61]; indeed, the covariant
c-flation framework considers
G and
c as independent fields, while here, these varying couplings entered the form of a single scalar field; on top of that, the very form of the action integral is different both in its gravitational part and in the kinetic term for the scalar field(s).
The variation of the action in Equation (
7) with respect to the metric
yields the BD-like field equations for
(it is a matter of interpretation to label the model explored in this paper as a Brans–Dicke theory. Our model is not Brans–Dicke if one considers as Brans–Dicke a scalar–tensor theory where the scalar field
is related strictly to
. Our model could be said to be Brans–Dicke if the latter is a scalar–tensor theory for
regardless of its interpretations in terms of the fundamental couplings. We prefer the first interpretation. However, even in the second interpretation, our paper showed that new physics arises from defining
in terms of both
G and
c: this novel perspective allows for the entangled variation of
G and
c in a theoretical structure accommodating a consistent variational principle, the covariant conservation of an effective energy momentum tensor, and a fundamental explanation for recent enticing phenomenological results pointing towards
.):
where
is the stress–energy tensor of ordinary matter. Notice that
in (
9) has the regular dimensions of the energy density. However, it should be underscored that the variation of the matter action:
with respect to
is written as
in the context of our scalar–tensor model. Here, the factor
must participate in the form of
within the integral sign. This is not the case in non-varying speed of light scenarios (including the pure BD theory). The factor
accompanying
(or
) brings about one of the interesting features in our scalar–tensor model: large-valued varying
c setups suppress the ordinary matter contribution to the action, in which case, the dynamics is mostly determined by
through its kinetic and potential terms. In Equation (
9), no dependence of
on the field
is assumed. However, it should be noted that we cannot evade a non-trivial coupling with the (varying) speed of light within the functional derivative entering the definition of
. This feature will be dealt with later—see
Section 4.1; it is related to the definition of an effective stress–energy tensor including
c. According to Noether’s theorem [
76], to every symmetry corresponds a conserved current; in theories of gravity, the conserved current is its own stress–energy tensor, and the symmetry is the invariance under general coordinate transformations (realized in practice by infinitesimal translations involving Killing vectors [
77,
78]). In the paper [
61], this delicate point surrounding the underlying symmetry and the associated energy-momentum tensor appeared in a different form: the covariant
c-flation approach introduces
directly into the action via a Lagrange multiplier.
The variation of
S with respect to the scalar field
leads to the Equation Of Motion (EOM) for the scalar part of our scalar–tensor model:
This EOM says that “ is a dynamical field; it changes in space and time.” As a consequence of the definition of in terms of G and c, the latter conclusion yields the covarying character of the gravitational coupling and the causality coupling. We show below that the simultaneous variation of G and c occurs in such a way that the condition is satisfied after the system evolves to the equilibrium stable point in the phase space.
In the following, we specify our BD-like theory (in the Jordan frame) for cosmology—see, e.g., [
22]. The main hypothesis is that
depends only on the cosmic “time”
due to the requirements of homogeneity and isotropy. These requirements demand the FLRW line element:
In this set of coordinates, the scale factor
bears the dimension of length. Conversely, the comoving coordinates
are dimensionless quantities. The curvature of the space section is determined by the parameter
, as usual [
74]. The Hubble function
H is defined as
and has the dimension of inverse length.
The matter content appearing in both field Equations (
8) and (
12) shall be modeled by the perfect fluid stress–energy tensor:
where
p is the pressure and
is the energy density (both with dimensions of energy per unit volume).
Due to (
15), the 00-component of the
field Equation (
8) reads (additional details on how to calculate Equations (
16)–(
18) are given in
Appendix A):
This is the (first) Friedmann equation of our scalar–tensor cosmology with covarying G and c through the field .
The second Friedmann equation (or acceleration equation) is calculated by taking the trace of Equation (
8), using the curvature scalar built from the metric in (
13), utilizing the trace of the stress energy tensor, and Equations (
12) and (
16). It reads:
Moreover, in the cosmological context, Equation (
12) reduces to
This equation determines the dynamics of
. It depends on
, which means that (
16)–(
18) should be solved simultaneously. The latter also depends on the matter content through the factor
, which demands us to specify the nature of the perfect fluid at play in the cosmological era under scrutiny. Finally, Equation (
18) exhibits a dependence on the potential
and its derivative with respect to the field
. Ultimately, we will have to say what is the form that we expect for potential
, which, in turn, is written to include the physical couplings
G and
c. This challenge will be addressed next: In the following section, we perform the phase space analysis of our model. With regard to dynamical systems approach in cosmology, we refer the reader to the interesting papers [
79,
80,
81,
82,
83,
84,
85,
86,
87].
3. The Dynamics of
The potential
is assumed to be analytical, but otherwise completely general. It was taken as dominant over the contribution of the term involving the matter Lagrangian; more specifically,
This condition may be satisfied either for
or
. We shall consider the first terms of the expansion of
in a power series:
where
(
) are the first three constant coefficients in the truncated series. This type of expansion up to second order in the power series is customary for the potential description around a local minimum of the function. It is usual practice in classical mechanics and quantum mechanics to approximate the potential in the vicinity of a local minimum for that of a harmonic oscillator—e.g., [
88]. The usefulness of a power series expansion of the potential up to its first-order terms is clearly noticeable in the inflationary context—e.g., [
89]—with the quadratic term being a useful example to illustrate the mechanism of a slowly rolling scalar field [
90]. The power series expansion of
V has also applications in classical field theories, such as in [
91]. A power series expansion up to second-order can be found in [
89]; the interpretation of the different terms that participate in a truncated series such as that in Equation (
20) were discussed in [
22,
92], for instance. The form (
20) encodes a few cases of special interest:
Let . This realizes a constant potential that could be rescaled by setting .
For and , one recovers the ordinary cosmological constant () already present in GR. The enticing feature here is that would exhibit all the fundamental couplings in gravity when we substitute . (Nevertheless, recall that would still be a genuine constant here since it defines .)
If and , the potential accounts for a massive term in ’s field equation. In fact, in this case, and , thus corroborating the interpretation. The delicate part is to interpret the role of this mass in . It could be conjectured that the source of ’s mass is either a massive photon or a massive graviton.
In view of the above features, we considered the
in Equation (
20) adequate enough to allow for a sufficiently general dynamical analysis of our scalar–tensor theory of covarying physical couplings in gravity.
Substitution of Equation (
20) into (
18) yields a form for the field equation of
that does not depend on
. This is not the final possible simplification to this equation. The next step is to consider periods in the cosmic history when
or, more specifically,
The above constraints not only guarantee a great simplification of Equation (
18), but they are also very reasonable. Indeed, Equation (
22) is consistent with a radiation-filled Universe—described by the equation of state
. By abiding by Equation (
22), we assumed that the dynamics of the field
takes place in this era.
In standard cosmology, the radiation era spans from the period after inflation and reheating all the way until the matter-dominated phase. It includes major events such as electroweak unification, quark–hadron transition, neutrino decoupling, and nucleosynthesis [
93]. It is a key time period in the cosmic thermal history during which the dynamics of
would have taken place.
In any case, the essential ingredient to the remainder of our analysis is that the condition in Equation (
21) is satisfied during all the dynamics of
. This means that the combination
built from the potential
should dominate the matter–energy content entering into play via
. This configuration could be easily realized within varying
c frameworks during regimes where
, for example in the very early Universe, before the phase transition of the decaying
admitted by some VSL models [
1].
Plugging (
22) into (
18) results in
A solution to this equation leads to
, which constrains the dynamics of
. The physical solution to (
23) must avoid the point
since the notion of a vanishing speed of light or of an infinitely large gravitational coupling is meaningless. In order to realize this physical requirement, we displace the origin of the scalar field
by defining
where we admit
and
finite. In terms of this new field variable, Equation (
23) reads
The solution to this equation is not so easy because
, and this is given by Equation (
16). (Even after we solve Equation (
25) for
using
, we should have to justify a particular ansatz for
, which would lead to
.)
3.1. Elements of Stability Theory for Dynamical Systems
We shall study Equation (
25) from the point of view of dynamical systems. For this reason, it is worth briefly reviewing the elements of standard phase space analysis most-commonly used in modern gravity. This is given in the following subsections, whose discussions are based on the papers [
79,
82] and the references therein. Check also [
94].
3.1.1. Linear Stability Theory
Let us suppose a physical system can be described by
n variables
. Each of these variables is a real-valued function, so that
. Now, consider that the dynamics of this physical system can be described by a set of
n differential equations of first order:
where the dot indicates the derivative with respect to a parameter representing the evolution of this system, say a time coordinate (
). Here,
represents a set of
n analytical functions (which can be nonlinear) of the variables
. Whenever there is no explicit dependence of
with respect to
, the set of equations is called an
autonomous system.
If there exists a point
such that
, then
If there is a time where , then the dynamics of the system ceases and for any value of . This point is called a fixed point, critical point, or equilibrium point.
Whenever a system has a fixed point, it is interesting to analyze the trajectories in a neighborhood of . Roughly speaking, if the trajectories in U diverge from the fixed point, it is considered unstable; if the trajectories converge to , then it is stable.
The simplest stability analysis that can be carried out for such a system consists of considering the equations in a small region
U where
can be expanded in a Taylor series about
and approximated to first order. The advantage of this method is that it allows for an analytical solution for the approximated system (the drawback is that the approximation by a Taylor series is not always a good approximation for a nonlinear system). In
U, we have
A change of variable,
, can always be proposed so that the equilibrium point is the origin of the coordinate system in the new variable:
or in a matrix form:
is called the
Jacobian matrix or
stability matrix. As long as
, we can look for a new transformation of variables
where the transformed Jacobian matrix becomes diagonal: the elements of the diagonal are actually the eigenvalues,
, of the Jacobian matrix. In this case, the set of differential equations will have the following solution:
If the real part of at least one eigenvalue is positive, then the trajectories will diverge from the fixed point and the system is unstable; if the real part of all eigenvalues are negative, then the trajectories converge to the equilibrium point and the system is stable. If the real part of one of the eigenvalues is zero, then the system can be either stable or unstable; in this case, each system has to be considered separately.
The eigenvalues of the Jacobian matrix are called Lyapunov coefficients, and the stability analysis essentially consists of the study of their real parts. Notice that the Jacobian matrix is composed of constant coefficients once is not explicitly -dependent. The stability that is characterized by the Lyapunov coefficients is restricted to a region where the linear approximation for is valid.
The above considerations shall be employed in
Section 3.2.
When the components of the stability matrix are not constant or when the linear approximation of the Taylor series is not a good approximation, other methods have to be applied. For instance, a powerful method to establish the stability of a dynamical system when the linear stability analysis fails is Lyapunov’s method, presented in the next section.
3.1.2. Lyapunov’s Method
Section 3.1.2 follows closely the treatment given in [
79]. This method is more powerful and general than the one in
Section 3.2 in the sense that it does not rely on linear stability: It has the capability of investigating both local and global instability. In addition, Lyapunov’s method applies to non-hyperbolic points as well, something the linear stability theory fails to accomplish. A critical point (fixed point)
of the dynamical system (
26):
is classified as a hyperbolic point if all of the eigenvalues of the stability matrix have a non-zero real part. Otherwise, it is said to be non-hyperbolic.
Lyapunov’s method for checking the stability of a dynamical system consists of finding a Lyapunov function , such that:
- (i)
is differentiable in a neighborhood U of ;
- (ii)
;
- (iii)
, .
There is no known systematic way of deriving the Lyapunov function
. If a Lyapunov function exists, the
Lyapunov stability theorem establishes that the critical point
is a stable fixed point if the requirement
is fulfilled. The critical point
is an asymptotically stable fixed point if there is a Lyapunov function
satisfying the criterion
. In addition, if
,
, then the critical point
is classified as
globally stable or
globally asymptotically stable, respectively. See [
79].
Notice that Requirement (iii) can be cast into the form:
Equation (
33) was used in the last step. The relation in Equation (
34) will be important in
Section 3.3 below.
3.2. Phase Space Analysis with via
the Linear Stability Theory
We avoided the difficulty mentioned below in Equation (
25) by assuming, in this subsection, that the Hubble function is constant during the time interval in which the dynamics of the field
occurs. In other words,
It should be emphasized that this condition is not as restrictive as it may seem. It does not mean that
is always constant, but only during the time interval of
’s evolution. We are not demanding the Universe to be stationary (with
); we are essentially assuming that
H evolves slowly during the time it takes
to reach an eventual attractor point. Again, this is a first-order phase transition scenario common to several models of the VSL [
1].
In order to show that the picture described in the previous paragraph is achievable, let us analyze the dynamics of
in the phase space resorting to the linear stability theory [
79]. Let
be the momentum associated with
. Then, Equations (
25) and (
36) become the following coupled system of equations:
This pair of differential equations has the structure of an autonomous system [
94]. The dynamical system (
37) is already linear; consequently, the linear stability theory is not just an approximation around the critical point(s), but rather an exact description, completely adequate for the case in hand. The result stemming from this analysis should not suffer from any pathologies, such as those mentioned, e.g., in [
83]—see also [
95]. A quick inspection of (
37) revels the equilibrium point:
The next step is to analyze the stability of the fixed point according to the Lyapunov criterion [
94]. The Lyapunov coefficients
are the eigenvalues of the Jacobian matrix (stability matrix) related to the autonomous system [
79]. They are found by solving the characteristic equation associated with this matrix and read:
The autonomous system can be considered stable if the real parts of both eigenvalues satisfy [
79]
The analysis of the direction fields in the phase space allowed us to double check if stability is attained (in an enlarged region surrounding the equilibrium point [
82]) and if it is consistent with the Lyapunov criterion. Here, we assumed that
i.e., our Universe is expanding.
We split our analysis to cover different possible ranges of values for the parameters and .
Case 1: Negative-valued
Here, we took . There are two subcases to be analyzed according to the values of the parameter . These subcases are considered separately next.
Subcase 1.1: Negative-valued with
In the instance
and
, we may write
This condition is important to decide on the behavior of the square root showing up in Equation (
39). As a consequence,
We concluded that
is a positive real number, so that
is always satisfied, and the associated equilibrium point is unstable according to Equation (
40). Since the analysis of
already reveals instability, there is no need to proceed to the study of
. The direction fields for the autonomous system (
37) in the case where
and
are shown in
Figure 1.
Figure 1 clearly shows that the trajectories diverge from the equilibrium point
, thus indicating its instability in the context of the subcase under scrutiny here. This conclusion will guide us towards constraining the interval of values allowed for the parameters
and
if a reasonable physical interpretation for the dynamics of
is to be established. We will decide on that later on, after we conclude our stability analysis.
Subcase 1.2: Values of satisfying
Here, we need both
and
. Under this requirement, the eigenvalues in (
39) can be expressed as
The relevant part for deciding the nature (real or complex) of the eigenvalues
is the second term within the square root. If it is greater than 1, then
and both eigenvalues
become complex numbers with negative real parts. Therefore, the equilibrium point is stable, cf. Equation (
40).
On the other hand, if
the square root in (
43) is a real number smaller than one, and both eigenvalues
are real and negative. This automatically satisfies the criterion
for stability, leading us to conclude, again, that the equilibrium point
is stable.
The direction fields in the plane
are analyzed in
Figure 2. As we can see in the graph, the trajectories in the phase space converge to the equilibrium point
, indicating its stability. The physical interpretation of this key feature will be explored momentarily. Before that, we need to address the case where
is positive.
Case 2: Positive-valued
The stability analysis of the equilibrium point in the case also depends on the range of values assumed by . The two subcases are again and .
Subcase 2.1: Negative-valued with
The case
yields
with
. The analysis here follow exactly the same steps as for Subcase 1.2 above. If
, the eigenvalues are complex numbers with negative real parts, meaning that the equilibrium point is stable. On the other hand, if
, the two eigenvalues are real negative numbers, implying that the system has a stable fixed point.
The direction fields for the dynamics of
when
and
are displayed in
Figure 3. The trajectories converge to
; for this reason, it is classified as a stable equilibrium point.
Subcase 2.2: Values of satisfying
In this situation,
and
, the eigenvalue
is a real positive number. As a consequence, the stability criterion
is violated irrespective of the behavior of the eigenvalue
. Hence, the equilibrium point
is unstable. This conclusion is confirmed by the direction fields in
Figure 4 showing trajectories diverging from the equilibrium point.
3.3. Phase Space Analysis with via Lyapunov’s
Method
Presently, we generalized the autonomous system studied in
Section 3.2 by allowing the Hubble function to be
-dependent. In this case, the linear stability analysis may be insufficient, or even misleading, as per [
83] (and Ref. [
79]). We were, thus, compelled to use other methods of dynamical analysis. In particular, Lyapunov’s method—reviewed in
Section 3.1.2—proved to be convenient in the present case.
The generalization of the system in Equation (
37) that takes into account
is
The equilibrium point is the same as in Equation (
38), namely:
The (vector) function
that determines the dynamics reads
where
as is immediately concluded from the comparison of the system (
45) with the definition (
33).
Now, we propose the following Lyapunov function
:
with
a and
b constants. These constants will be determined in accordance with the Lyapunov stability theorem momentarily.
The function in Equation (
49) immediately satisfies Property (i) in
Section 3.1.2. Moreover, notice that
This fact is relevant for checking if Requirement (ii) is fulfilled .
We can compute the first identity in Equation (
34) for the specific case of our dynamical system:
where we made use of Equation (
45). The constants
a and
b are arbitrary; we used this freedom to choose
Accordingly, Equation (
51) yields
The choice in (
52) led us to satisfy Property (iii) in
Section 3.1.2. This is so because the Hubble function is always positive for an expanding Universe. Actually, we have
since
and
, for
. The equality in (iii) would only be achieved if
, but this choice would lead to
, in which case Property (ii) in
Section 3.1.2 would not be satisfied.
Incidentally, now is the time to attempt to verify Property
(ii). Plugging Equation (
52) into (
49),
Since
and
and
in a neighborhood of the equilibrium point, the decision on the fulfillment of Requirement (ii) depends entirely on the values of parameters
and
. Essentially, the stability will be achieved if
We identify four possibilities:
- (a)
and
, then Condition (
55) is not satisfied, and the system is not stable;
- (b)
and
, then Condition (
55) is satisfied, and the system is asymptotically stable;
- (c)
and , then the system is asymptotically stable;
- (d)
and , then the system is not stable.
These possibilities are consistent with those unveiled by the linear stability analysis performed in
Section 3.2. Here, however, the nature of the equilibrium point is kept even when the Hubble parameter is allowed to change as a function of
.
Figure 5 shows the direction fields for different values of the Hubble function
and fixed values of
and
under Condition (b).
If we restrict our parameter space
to satisfy the conditions for stability, i.e., Conditions (b) or (c) above, then we verify that the equilibrium point
is
globally asymptotically stable, since
cf. the Lyapunov Stability Theorem—enunciated below Point (iii) in
Section 3.1.2.
Finally, we plot the parameter space of the pair
. The stability of the system depends solely on the values of
and
, both when
(
Section 3.2) and when
(this section). The shaded region in
Figure 6 displays the region of the parameter space where the dynamical system is stable.
3.4. Discussion: The Meaning of Stability
The results that have physical interest are those where the equilibrium point is stable, i.e., the case where
and the case with
. These different ranges of values for
lead to two different interpretations. Let us reconsider Equation (
8) and include the potential (
20) explicitly:
As mentioned at the beginning of
Section 3, we see that the parameter
plays the role of a cosmological constant even in the dynamical phase of
. It may be suggestive to rename this parameter
, in such a way that the left-hand side of Equation (
56) is formally the same as the geometrical side of Einstein field equations. Hence, the condition
maps to
, which indicates that the background solution to (
56) is a de Sitter-like spacetime [
74]. Conversely, the condition
would correspond to
, thus leading to an AdS-like background. (Of course, some conditions apply for achieving a strict dS/AdS background: vacuum, derivatives of the field
as null, etc.)
In both cases, as long as we have the stability of the equilibrium point, the system will evolve so that the trajectories of field
in the phase space
converge to
. When this fixed point is reached, we have simultaneously
When the system arrives at the equilibrium point, then
as implied by Equations (
5) and (
57). This means that the condition:
will necessarily be satisfied whenever the system is in equilibrium: Equation (
1) holds true precisely for the parameter value
. This value is exact here: the uncertainty of
is theoretically zero; it is a natural consequence of the definition of
demanded by basic dimensional analysis and out of the dynamics of
in the phase space.
Another consequence of this dynamics of
is that the system can start with arbitrary initial conditions. The condition
may not even be satisfied for this initial condition. It is just a matter of time until the system evolves and converge to Equation (
1).
We also point out the appearance of an effective cosmological constant when the dynamics of
relax to the equilibrium point (regardless of the association
suggested above). In fact, the dynamics of the scalar field ceases at the equilibrium point, i.e.,
with
; in this situation, the gravitational field Equation (
56) approaches
where
is the conserved stress–energy tensor in our scalar–tensor model. The term in the parenthesis above appears as an effective cosmological constant:
and the field
, which converges to
in Equation (
57), can be set as
while Equation (
60) assumes the form of GR’s field equation with the cosmological constant. The quantities
and
are constant values of the couplings, which might as well be considered as their present-day values in cosmological terms. In fact, due to the last two equations, Equation (
60) assumes the following form at the present time:
which is the familiar form of the Einstein field equations, cf. Equation (
2).
It is imperative that the following is crystal clear: the fact that does not mean that G and c will be constants after the equilibrium is reached. All that is required thereafter is . Accordingly, after the equilibrium is attained, it could be and , where is an arbitrary function of the time coordinate: these time-dependent ansatzs for G and c satisfy both the requirements and .
In the face of the comments above, one concludes that, if an astrophysical event takes place after the equilibrium condition is attained, it is essentially impossible to identify the dynamics of
G separately from the dynamics of
c. In order to identify the eventual dynamics of
G and
c with
(by taking into account only gravitational field equations, as we have done here), one would have to consider situations out of the equilibrium for
. If we are currently in the equilibrium condition, then one would have to take into consideration events in the past history of the Universe. How far in that past one should go is still an open question, hinging on the amount of time that the system takes to converge to the equilibrium point. In any case, the condition in Equation (
21) must be preserved for the dynamical analysis in
Section 3 to hold. As an example of what is stated above, we mention the solar system tests constraining the Brans–Dicke parameter
[
96], while as a dimensionless parameter, it would be expected to be of order
. In these tests, the parameterized post-Newtonian approximation is applied assuming that the dynamics of the scalar field plays a role in the solar system evolution. In our case, the dynamics ceased in the radiation era, way before the solar system was formed. From this perspective, it would be meaningless to try and discard our model based on BD constraints.
From the discussion above, the global picture of the dynamics in our scalar–tensor model should be within sight. The system begins with arbitrary initial conditions where
is out of equilibrium. In this phase, the functional form of
G and
c is not constrained to any particular behavior being interwoven. There are two regions in the parameter space of the pair
enabling the system to reach a stable critical point, cf.
Figure 6. By restricting our model parameters to these regions, one guarantees that the stable equilibrium is reached,
, and the couplings
G and
c are forced to evolve respecting
henceforth. The evolution of the Universe from this point onward should be impacted by the fact that
G scales as
. In the next section, we launch ourselves into the task of determining the background cosmological evolution of our scalar–tensor model after the equilibrium. Our final goal is to solve for the scale factor
a as a function of the time coordinate
. Along the way, we build the continuity equation from the conservation of the effective stress–energy tensor mentioned below in Equation (
60)—see
Section 4.1. We will show that this continuity equation differs from the conventionally used in FLRW cosmology as it contains a term depending on
. The ensuing VSL model is studied in
Section 4.2 assuming radiation and dust matter content. In both cases, the evolution of
tends to an accelerated de Sitter-type solution. As it happens, the latter is true for two classes of VSL models: those with a decreasing speed of light and those with an increasing
.
5. Concluding Remarks
In this paper, we studied a scalar–tensor model for gravity in which both the gravitational coupling G and the (causality) speed of light c were included in the scalar sector of the model through the field .
The field equations for
and
were built and specified for the homogeneous and isotropic cosmological background. The dynamics of our field
was then analyzed in the phase space under some working hypotheses. We took the curvature parameter of the space sector
k as null. We used the first three terms in the series expansion of the potential
. We assumed that the dynamics of
happens in a radiation era. We then studied the evolution of
via linear stability theory (reviewed in
Section 3.1) assuming that the dynamical period of
is short enough to ensure that the Hubble function is approximately constant therein (
Section 3.2). Finally, the dynamical system analysis was extended to allow for a time-dependent Hubble parameter; this required the use of Lyapunov’s method (
Section 3.3).
The phase space analysis showed that two sets of conditions upon the free parameters of the model led to trajectories converging to a globally asymptotically stable equilibrium point. These conditions were: (i) and (ii) , where is the constant coefficient of the linear term in the series and is the constant coefficient of the kinetic term of in the model’s action. The physical interpretation of and is commented on momentarily.
We saw in
Section 3.4 that
could be understood as a type of cosmological constant during the dynamical phase of
; therefore, its positivity (or negativity) could impact this version of
. Moreover,
appears in the effective cosmological constant
, which turns up at the end of the dynamical evolution of
, when the trajectories of the phase space converge toward the equilibrium point and stay there for the rest of the cosmic history. The definition of
includes the other coefficients of the potential as well:
is the constant term in the
series, and
is the coefficient of the quadratic term of
V in
. Our treatment demands
, but otherwise unconstrained, while
is totally unconstrained. The equilibrium value of the field,
, also enters the definition of
, which leads to the interesting conclusion that the cosmological constant could depend on
, the (constant) equilibrium values of the gravitational coupling and of the speed of light.
The sign of the parameter
is related to the physical nature of the scalar field
. Indeed, if
, the sign of the kinetic term for
in the action indicates that
is a ghost field. This means that its Hamiltonian is unbounded from below and it would be a source of negative energy. The global sign of the potential
is also important: together with the sign of the kinetic term, it determines if
could also be an unphysical tachyon field breaking causality. In the vicinity of the equilibrium point,
, so that
would characterize the field as a ghost field—see, e.g., page 2 of [
98] and the references therein. This all means that we have to impose
for a physically meaningful behavior of
.
The big picture in our development is the following: After evolving to the equilibrium stable point, the field
reaches it equilibrium value
and stays there. Then, the dynamics of
ceases, and two things happen: (i) the couplings
G and
c that are covarying within
assume the dependence
, where
and
could be interpreted as the numbers that we use as fundamental constants today; (ii) the gravitational field equation of our scalar–tensor model, involving both
and
, degenerates to a field equation that is formally the same as the Einstein field equation of general relativity, but bears a time-dependent factor
alongside the ordinary energy momentum tensor. At the present time
, and the field equation recovers exactly the Einstein equation of GR. That could explain why GR is so successful in describing local and low-redshift data, but should be generalized in a larger scope of application. The relaxation of our model toward a field equation that is formally the same as that of GR is consistent with the works by Damour and Nordtvedt [
99,
100]—see also [
101] by Faraoni and Franconnet.
When the dynamics of
ends,
, and the covarying
G and
c are forced to obey the relation
with
. This number is exact as far as the theoretical prediction is concerned (under the hypotheses we have assumed, cf. the beginning of
Section 3). Our conclusion is that at least some astrophysical phenomena would be unchanged if
G and
c vary concurrently while respecting
. This fact was pointed out in [
65,
69,
70], in relation to orbital timing, the strong gravitational lensing of SNe Ia, and core-collapse supernova events.
Our scalar-field model may be seen as a version of Brans–Dicke allowing for a varying speed of light (alongside the varying
G). Although the field equations in
Section 2 look like those in BD theory, it must be underscored that the varying
c appears explicitly in the term containing the energy momentum tensor. This forces the definition of an effective stress–energy tensor, which is covariantly conserved, thus ensuring the general covariance of the gravity equation. This was discussed also in
Section 4, where we built the Friedmann equations for the cosmology within our scalar–tensor gravity. These equations were solved for two particular ansatzs of
: (i) a speed of light that decreases as
a increases and (ii) a
c that increases as the Universe expands. It was shown that both ansatzs allow for an accelerated expansion consistent with the effect of dark energy. The fact that we were able to fully resolve the field equation for our scalar–tensor gravity with varying coupling constants shows that the generality of our model does not spoil practicality. The following logical step in our research in the future will be to refine the details of the picture by constraining the models with observational data.
Future perspectives include relaxing some of the hypotheses adopted in this manuscript. One possibility is to let the dynamics of evolve in an era dominated by matter, for which and the coupling of with the matter Lagrangian comes into effect. The eventual interrelation between the matter part of the action and brings forth the possibility of c being an independent scalar field. In this context, one would have to take and as scalar fields evolving on their own (not enclosed within the field ). To put it another way, in the stability analysis performed in this paper, we have one scalar degree of freedom, incarnated in ; this is true even as both G and c are allowed to vary. In a future paper, we will consider two independent degrees of freedom, namely G and c, with their own separate kinetic terms and potentials (with or without a mutual interplay).
In this paper, the varying physical couplings were restricted to the set
—with the first two forming the field
. Other works enlarge the set of possible varying fundamental constants to
and perform some phenomenological modeling in astrophysics and cosmology [
65,
102]. Pursuing the fundamentals surrounding an eventual variation of all five couplings from a field theory perspective is something that may be performed in the future.