Homogeneous Cosmological Models in Weyl’s Geometrical Scalar–Tensor Theory

As is well known, scalar–tensor theories of gravity were proposed some years ago by Jordan jor59 , and Brans and Dicke Bra61 . Later, they were extended in a more general framework ber68 ; wag70 ; nor70 . In fact, they represent a generalization of the simplest scalar–tensor theory of gravity which is the Brans–Dicke theory far04 ; wil14 . In general scalar–tensor theories of gravity, the gravitational field is not described only by the usual tensor field $g_{\mu\nu}$ of general relativity. In addition to this, we have one or several long-range scalar fields which also mediate gravitational interaction.

Scalar–tensor theories of gravity have been a subject of renewed interest. Certainly, one motivation for this is the belief that, at least at sufficiently high energy scales, gravity becomes scalar–tensorial in nature dam94 and, therefore, these theories are important in the very early Universe. On the other hand, two important theoretical developments have been achieved such as, for example, unification models based on superstrings, which naturally associate long-range scalar partners to the usual tensor gravity of Einstein theory gre87 . Another motivation for the investigation of scalar–tensor theories is that inflationary cosmology in this framework seems to solve the fine-tuning problem and, in this way, give us a mechanism of terminating inflationary eras la89 . Apart from the solution of this problem, the scalar–tensor theories by themselves have direct implications for cosmology and for experimental tests of the gravitational interaction dam00 and for this reason, they are relevant in the investigation of the early Universe.

Among alternative theories of gravity, scalar–tensor theories are perhaps the most popular ones. As we have pointed out before, in these theories, gravitational effects are described by both a metric field $g_{\mu\nu}$ and a scalar field $\Phi$. A well-known example is the Brans–Dicke theory Bra61 ; tah21 , in which the geometry of the underlying space-time manifold is assumed to be Riemannian, and the scalar field replaces the gravitational constant being interpreted as the inverse of a varying gravitational coupling parameter. In addition to the reasons mentioned above, the scalar–tensor theories are studied because they admit key ingredients of string theories, such as a dilaton-like gravitational scalar field that has a non-minimal coupling to the curvature mor11 . On the other hand, a different approach, in which the scalar field appears as part of the space-time geometry, namely, the Weyl geometrical scalar–tensor theory, has been discussed recently in the literature rom14 . Indeed, in this new approach, one considers the space-time structure as a very special case of the framework adopted in the original Weyl unified field theory wey18 ; wey52 , the geometrical space-time structure being that of a Weyl integrable space-time (WIST) rom08 ; rom09 ; rom10 ; rom11 ; rom12 . It is important to remark that other gravity theories in which a scalar field plays a geometrical role have also been proposed Fonseca ; Fonseca1 ; Fonseca2 .

Recently, some theoretical aspects concerning the Weyl geometrical scalar–tensor theory have been studied, in particular the behaviour of the solutions when $\omega$, the scalar field’s coupling constant, goes to infinity barros . The investigation of cosmological vacuum models for different scalar potentials has also been carried out rom16 . In the present article, we extend this research to include anisotropic models of Kasner type. Here, we take advantage of some similarities between vacuum solutions of the Weyl geometrical scalar–tensor theory and those coming from the Brans–Dicke theory. We also examine cosmological solutions in the presence of matter, a scenario that has not yet been investigated in Weyl geometrical scalar–tensor theory, and at the same time, we compare the results obtained with similar solutions already known from general relativity and the Brans–Dicke theory.

The paper is organized as follows. In Section II, we briefly review Weyl’s original theory, which inspired the geometrical scalar–tensor approach. In Section III, the field equations of the Weyl geometrical scalar–tensor theory are obtained. Then, a Kasner type solution is exhibited in Section IV, while in Section V we work with a homogeneous and isotropic cosmological model having a perfect fluid as a source, such that we find an analytical solution for the stiff matter case and we study the other cases using the qualitative analysis of dynamical systems. Finally, Section VI is devoted to our conclusions.

In the first scalar–tensor theories, the so-called Jordan–Brans–Dicke theories, it is assumed, as in general relativity, that the space-time geometry is purely Riemannian. On the other hand, if the Palatini variational method is applied to deduce the field equations from the action, then in a large class of scalar–tensor theories, a non-Riemannian compatibility condition between the metric and the affine connection appears naturally (for a more general result, see bur98 ). In this way, we have a theory that establishes the space-time geometry from first principles, that is, the space-time manifold is dynamically generated by the choice of the particular coupling of the scalar field in the gravitational sector. In the case where the action is that of the Brans–Dicke theory, this procedure leads to the so-called Weyl integrable space-times, a particular version of the geometry conceived by H. Weyl in his attempt to unify gravity and electromagnetism wey18 . Note, however, that here, it is the scalar field that is being geometrized.

It is true that the Weyl geometry is one of the simplest generalizations of Riemannian geometry, in which the Riemannian compatibility condition between the metric and the affine connection is weakened. This was an ingenious way that Weyl devised to introduce a covariant vector field $\sigma_{\mu}$ in the geometry, which bears a great similarity with the electromagnetic four-potential. Weyl went on and introduced the second-order tensor $F_{\mu\nu}=\partial_{\mu}\sigma_{\nu}-\partial_{\nu}\sigma_{\mu}$, which he interpreted as representing another kind of curvature, namely, the length curvature. As a consequence of this modification in the Riemannian compatibility condition, the covariant derivative of the metric tensor does not vanish, as in Riemannian geometry, and the length of vectors when parallel transported along a curve may change. However, such theory suffered from a severe criticism by Einstein, who objected that the nonintegrability of length implies that the rate at which a clock measures time, i.e., its clock rate, in this case would depend on the past history of the clock. As a consequence of this fact, spectral lines with sharp frequencies would not appear and the spectrum of neighbouring elements of the same kind would be different goe04 . This became known in the literature as the second clock effect (incidentally, the first clock effect refers to the well-known effect corresponding to the “twin paradox”, which is predicted by special and general relativity theories).

Weyl’s new compatibility condition is given by $\nabla_{\alpha}g_{\mu\nu}=\sigma_{\alpha}g_{\mu\nu}$, and is easily verified that this condition is invariant under the conformal transformation $g_{\mu\nu}\rightarrow\bar{g}_{\mu\nu}=e^{f}g_{\mu\nu}$ carried out simultaneously with the gauge transformation $\sigma_{\mu}\rightarrow\bar{\sigma}_{\mu}=\sigma_{\mu}+\partial_{\mu}f$, where $f$ is an arbitrary scalar function. The discovery of this new symmetry is now considered by some authors as the birth of modern gauge theories ale13 . Now, if $F_{\mu\nu}=0$ (null second curvature), which is equivalent to say that the one-form $\sigma$ is closed ($d\sigma=0$), then there is no electromagnetic field. In this case, we know that, from Poincaré’s lemma tu11 , it follows that there exists a scalar field $\phi$, such that $\sigma_{\mu}=\partial_{\mu}\phi$, and, instead of a vector field $\sigma$, we are left with a scalar field $\phi$, which, in addition to the metric, is the fundamental object that characterizes the geometry. A space-time endowed with this particular version of Weyl’s geometry came to be known as aWeyl integrable space-time.

As we have already mentioned, in the Weyl geometrical scalar–tensor theory, the underlying space-time manifold is that of a Weyl integrable space-time rom08 . In this sense, the Weyl nonmetricity condition involves a purely geometrical scalar field $\phi$ and is explicitly given by rom14

Moreover, one can define the Weyl connection, whose coefficients in a local coordinate basis read

with $\{_{\mu\nu}^{\alpha}\}$ representing the usual Christoffel symbols.

In turn, the field equations of the Weyl geometrical scalar–tensor theory can be written as barros

where here, we are using the field variable $\Phi=e^{-\phi}$, $\omega=const$, $V(\phi)$ corresponds to the scalar field potential, and $T_{\mu\nu}$ represents the Weyl invariant energy–momentum tensor of the matter fields. We denote by $G_{\mu\nu}$ and $\square$ the Einstein tensor and the d’Alembertian operator, respectively, defined with respect to the Christoffel symbols. If $V(\Phi)=2\Lambda\Phi$, one can introduce the cosmological constant $\Lambda$. However, let us take $\Lambda=0$, and then the field equations are given by

where $W=\omega-\frac{3}{2}$. Additionally, we can obtain from (5) and (6) that

with $R_{\mu\nu}$ denoting the Ricci tensor and $T=g_{\mu\nu}T^{\mu\nu}$. Equations (6) and (7) constitute the field equations we use in the following.

As is well known, the Kasner metric was obtained by the mathematician E. Kasner in 1921 and represents an exact solution to Einstein’s field equations. It describes an anisotropic universe without matter, that is, it is a vacuum solution. Historically, interest in the Kasner solution came from the fact that, although it may have a singularity (“big bang” or a “big crunch”), an isotropic expansion or contraction of space is not allowed, and this led to the generic singularity studies, the so-called BKL singularities tip79 .

The Kasner type solution in the Brans–Dicke theory of gravity is given by Bra61 ; rub72

with

($i=1,2,3$) and the Brans–Dicke scalar field

where $a$, $b$, $r_{i}$, and $\varphi_{0}$ are constants. The relations ${\textstyle\sum}p_{i}=1$ and

between the constants $p_{i},C$ and the scalar field coupling constant $\omega$ are also satisfied.

The space-time given by (8) corresponds to a homogeneous universe, without matter and rotation, with distinct expansions along the three orthogonal axes, which reflects anisotropy. Note that if $a=1$ and $b=0$, Equations (9) and (10) may be written as

In order to obtain a solution in the Weyl geometrical scalar–tensor theory, let us consider the following result: a vacuum solution of the Weyl geometrical scalar–tensor theory can be found if we make the change $\omega\rightarrow W=\omega-3/2$ in the correspondent vacuum solution of the Brans–Dicke theory. In fact, the two theories are not physically equivalent given that in Weyl’s geometrical scalar–tensor theory test particles follow affine Weyl geodesics (autoparallels) and not metric geodesics as in the case of the Brans–Dicke theory. Nonetheless, there is a formal equality between the vacuum field equations of the two theories rom14 .

Thus, the Kasner type solution in the Weyl geometrical scalar–tensor theory is given by Equation (12) and

where ${\textstyle\sum}p_{i}=1$ and

Now, if we choose $C=\dfrac{2}{W}$, it follows that

Furthermore, (12) and (14) become

In the limit $\omega\rightarrow\infty$, (17) and (18) tend to

where we have taken $r_{i}=1$. On the other hand, from (1) and (2) we find that

by considering the scalar field in the form $\Phi=e^{-\phi}$. Thus, when $\omega\rightarrow\infty$, the space-time geometry becomes Riemannian as we have

Therefore, also taking into account (19) and (20), the Kasner solution of general relativity is recovered in this limit.

The Friedmann–Robertson–Walker metric with a flat spatial section is given by

where $R(t)$ denotes the scale factor. In this cosmological model, the matter content is a perfect fluid represented by the energy–momentum tensor

with $p=\lambda\rho$, $0\leq\lambda\leq 1$, $p$ being the thermodynamic pressure, $\rho$ the energy density, and $u_{\mu}=\left(1,0,0,0\right)$ the four-velocity vector field. Then, field Equations (6) and (7) reduce to

The dot means differentiation with respect to time. Moreover, due to the assumption of spatial homogeneity, the scalar field $\Phi$ is supposed to be a function of $t$ only. Additionally, with the definitions $\theta=\frac{3\dot{R}}{R}$ and $\Psi=\frac{\dot{\Phi}}{\Phi}$, one can express (26)–(28) in the form

By combining (29)–(31), we can derive the equation

After some calculations and by using Equations (5) and (6), it is easy to show that

which reduces to

in the context of the cosmological model considered.

In this paper, we sought to find cosmological solutions in the context of the Weyl geometrical scalar–tensor theory. The vacuum field equations of this theory are formally identical to those of the Brans–Dicke theory, so we were able to obtain a Kasner type solution from the corresponding solution in the Brans–Dicke theory. We also found that, in the limit $\omega\rightarrow\infty$, the Kasner solution of general relativity was recovered. On the other hand, we investigated the existence of solutions for homogeneous and isotropic models sourced by a perfect fluid. In this case, we found an analytic solution for stiff matter and also showed that the corresponding solution of general relativity could be obtained in the limit $W\rightarrow\infty$. For values of the parameter $\lambda\neq 1$, no analytical solution was possible, and we used dynamical systems theory to display the phase diagrams of the solutions in intervals of $W$ (or $\omega$). When $W>0$, we highlighted solutions representing universes with $\rho>0$ and an increasing geometric scalar field, which started with a “big bang” and expanded to a final phase that tended toward Minkowski’s space-time (the curves $C^{\prime}M$).

An interesting fact regarding the phase diagrams examined here is that there was no difference between the cosmological models when different values of the parameter $\lambda$ were considered. In that sense, it can be seen that Equation (62), which determines the invariant rays, did not depend on $\lambda$. Moreover, it should be noted that in the present context, matter was not a source of the geometric scalar field $\Phi$ in Equation (6). By contrast, in the Brans–Dicke theory, the scalar field equation is

where, as is well known, $T$ denotes the trace of the energy–momentum tensor. For the case of a perfect fluid source, $T=T(\lambda)$ and $T=0$ only when $\lambda=\frac{1}{3}$. As a consequence, in the present scalar–tensor theory, cosmological models differed according to the value of the parameter $\lambda$ rom89 .

In this work, we did not consider the presence of the cosmological constant, nor did we take any potential of the scalar field into account. Because of this, we did not find any solution describing the acceleration of the universe. Incidentally, models describing cosmological scenarios in which the acceleration of the cosmos is driven by a scalar field, quintessence models ste99 ; toh23 and Chaplygin gas models fab17 ; ben02 among others sen01 , have been investigated with interest. Two lines of research that we leave for further work are (i) an investigation of the role the geometric scalar field could play in approaching the problem of dark matter and (ii) considering scenarios where the cosmological constant is present with the hope that they can give some light to the problem of dark energy.

Acknowledgements. C. Romero thanks CNPq (Brazil) for financial support.