Complex Scalar fields in Scalar-Tensor and Scalar-Torsion theories

Scalar fields in gravitational theory is a simple mechanism to introduce new degrees of freedom which can play an important role in the description of observable cosmological phenomena ratra . The early acceleration phase of the universe is attributed to a scalar field known as inflaton in0 ; in1 ; in2 ; in3 . In addition, scalar fields have been used to describe also the late-time acceleration phase of the universe attributed to dark energy, or other matter components such as dark matter, see for instance sf1 ; sf2 ; sf3 ; sf4 ; sf5 ; sf6 ; sf7 ; sf8 ; sf9 ; sf10 and references therein.

Multi-scalar fields have been widely studied in the literature. Some well-known two-scalar field models are the quintom qq1 or the Chiral model which leads to hyperbolic inflation hy4 , while other proposed multi-scalar fields theories can be found for instance in cc1 ; mm2 ; mm3 ; mm4 ; mm5 ; mm6 and references therein. A simple mechanism to introduce a multi-scalar field theory is to consider the existence of a complex scalar field, the real and imaginary parts of which give the equivalent of a two scalar-field theory com1 ; com2 .

An inflationary model with a complex scalar field was proposed in com3 . Specifically it was found that, when inflation occurs, the imaginary component of the complex scalar field does not contribute in the cosmological fluid, that is, the phase of the complex scalar field is constant. The cosmological perturbations with a complex scalar field were investigated in com4 . Furthermore, in com5 the authors used cosmological observations to reconstruct the quintessence potential for a complex scalar field. A nonminimally coupled scalar field cosmological model has been studied in com6 , while for some recent studies of complex scalar field cosmological models we refer the reader to com7 ; com8 ; com9 ; com10 ; com11 ; com12 ; com13 ; com14 .

In this study we consider the scalar-tensor and the scalar-torsion theories with a complex scalar field sc1 . In scalar-tensor theory, the scalar field is minimally coupled to gravity. The scalar field interacts with the gravitational Action Integral of Einstein’s General Relativity, that is, the Ricci-scalar of the Levi-Civita connection, the scalar-tensor theory satisfies the Machian Principle Brans and the theory is defined in the so-called Jordan frame Jord . The Brans-Dicke Lagrangian Brans is the most common scalar-tensor theory. On the other hand, the scalar-torsion theory is the equivalent of scalar-tensor model in teleparallelism. In the latter, the fundamental geometric invariant is the torsion scalar determined by the curvatureless Weitzenböck connection Hayashi79 ; Tsamp . There are many important results in the literature on the cosmological studies of the scalar-tensor and scalar-torsion theories, for an extended discussion we refer the reader to omegaBDGR ; st1 ; st2 ; st3 ; st4 ; st5 ; st6 ; st7 ; st8 ; te2 ; te3 ; te4 and references therein.

The purpose of this study is to investigate the effects of a complex scalar field in the evolution of the cosmological dynamics for the background space for these two different gravitational theories and to compare the results. Such analysis provides us with important results in order to understand the differences between the use of the Ricci-scalar and of the torsion scalar in the background geometry. We use dimensionless variables to perform a detailed analysis of the dynamical systems which describe the evolution of the physical variables. Such an approach has been widely studied before with many interesting results about the viability of proposed gravitational theories, see for instance dyn1 ; dyn2 ; dyn3 ; dyn4 ; dyn6 ; dyn5 . The plan of the paper is as follows.

In Section II we consider a spatially flat Friedmann–Lemaître–Robertson–Walker geometry in scalar-tensor gravitational theory with complex scalar field. We derive the field equations and we write the point-like Lagrangian. Moreover, we determined the stationary points for the field equations and we investigate their stability properties. In Section III we perform a similar analysis but now in the context of scalar-tensor theory. Finally, in Section IV we summarize our results and compare the physical results provided by the two theories and draw our conclusions.

Consider the complex scalar field $\psi$ in the case of scalar-tensor theory sc1 , for which the gravitational Action Integral is defined as

where $R$ is the Ricci scalar related to the Levi-Civita connection for the metric tensor $g_{\mu\nu}$; $\left|\psi\right|$ is the norm of the complex field $\psi$, that is, $\left|\psi\right|^{2}=\psi\psi^{\ast}$, $F(\left|\psi\right|)$ is the coupling function between the gravitational and the scalar field $\psi$ and $V\left(\left|\psi\right|\right)$ is the potential function which drives the dynamics. The Action Integral (1) admits the $U\left(1\right)$ symmetry.

The Brans-Dicke theory with a complex field is recovered for $F(\left|\psi\right|)=F_{0}\left|\psi\right|^{2}$, that is, the Action Integral (1) is sc1

In the case of a spatially flat FLRW universe with line element,

we derive for the Ricciscalar

where $H=\frac{1}{N}\frac{\dot{a}}{a}$, $\dot{a}=\frac{da}{dt}$, is the Hubble function.

We substitute (4) into (1) which by integration by parts gives the point-like Lagrangian

or in the case of Brans-Dicke

We focus now in the Brans-Dicke theory in which the field equations are described by the point-like Lagrangian (6). Moreover, the complex scalar field is written with the use of the polar form, $\psi\left(t\right)=\phi\left(t\right)e^{i\theta\left(t\right)}$, such that the Lagrangian (6) becomes

It is obvious that the Lagrangian function (7) describes a multi-scalar field cosmological model, where $\phi$ is the Brans-Dicke field and $\theta$ is a second-scalar field minimally coupled to gravity but coupled to the Brans-Dicke field $\phi$. The $U\left(1\right)$ for the point-like Lagrangian (7) provides the invariant transformation $\theta=\theta+\varepsilon$, and the conservation law $I_{0}=\frac{1}{N}a^{3}\phi^{2}\dot{\theta}$.

Variation with respect to the dynamical variables $\left\{N,a,\phi,\theta\right\}$ of the Lagrangian (7) provides the cosmological field equations which are

and

where without loss of generality we have assumed $N\left(t\right)=1$.

Scalar-torsion theory can be seen as the extension of scalar-tensor theory in teleparallelism. The gravitational Action Integral is te2

where $T$ is the torsion scalar of the antisymmetric Weitzenböck connection, $e=\sqrt{-g}$, $e_{i}$ describes the unholonomic frame, with $g(e_{i},e_{j})=\mathbf{e}_{i}\cdot\mathbf{e}_{i}=\eta_{ij}$ or in terms of coordinates, $e_{i}=h_{i}^{\mu}\left(x^{\kappa}\right)\partial_{i}$, where now the Weitzenböck connection is expressed as revtel

and $T={S_{\beta}}^{\mu\nu}{T^{\beta}}_{\mu\nu}$, with $T_{\mu\nu}^{\beta}$ to be the torsion tensor $T_{\mu\nu}^{\beta}=\hat{\Gamma}_{\nu\mu}^{\beta}-\hat{\Gamma}_{\mu\nu}^{\beta}$ and

where ${K^{\mu\nu}}_{\beta}$ is the contorsion tensor

For the spatially flat FLRW spacetime with line element (3) we consider the diagonal frame

from which we calculate revtel

By replacing in the Action Integral (24) and substituting the complex scalar field as $\psi=\phi e^{i\theta}$, we end up with the point-like Lagrangian function

In an analogue to the Brans-Dicke model we assume now that $F\left(\phi\right)=F_{0}\phi^{2}$. This is not a random choice, the Dilaton field, or the Brans-Dicke field for the specific potential function $V\left(\phi\right)=V_{0}\phi^{2}$ admits a discrete symmetry known as the Gasperini-Veneziano duality transformation. Similarly, the scalar-torsion theory for $F\left(\phi\right)=F_{0}\phi^{2}$ and potential function $V\left(\phi\right)=V_{0}\phi^{2}$ admits a discrete symmetry similar to the Gasperini-Veneziano duality transformation. Thus, the $F\left(\phi\right)=F_{0}\phi^{2}$ can be seen as the teleparallel Brans-Dicke equivalent model.

Variation with respect to the dynamical variables in the Lagrangian function (30) gives the field equations

and

As in the case of the scalar-tensor theory we proceed with the analysis for the dynamics of the latter system of differential equations.

In this piece of work we considered a complex scalar field in the context of scalar-tensor and scalar-torsion theories in a spatially flat FLRW background geometry. For these two gravitational models we derived the field equations and we investigated the dynamical evolution of the physical quantities by using dimensional variables. With the use of the latter variables the field equations for both theories are reduced to a two-dimensional dynamical system of first-order ordinary differential equations. In order to construct the cosmological history as provided by the given models we determined the stationary points and we investigated their stability properties.

For the two different dynamical systems which correspond to the scalar-tensor and scalar-torsion theories respectively, we found that the stationary points for the two dynamical systems are four. They are two points at the finite regime and two points at infinity. The stationary points can describe accelerated asymptotic solutions which can describe the early and late-time acceleration phases of the universe. The stationary points and their stability properties are summarized in Table 1.

We can easily conclude that the two theories, that is, the scalar-torsion theory and the scalar-tensor theory have the same number of stationary points and similar stability properties. The physical variables of the background geometry at the stationary points have similar properties with a different functional dependence on the free variables $F_{0}$ and $\lambda$. The dynamical equivalence of the two theories is an expected result. Torsion is not dynamical and there are no new degrees of freedom in scalar-torsion cosmology with respect to scalar-tensor cosmology.

From the above analysis it is clear that the two theories cannot be distinguished from the evolution of the background geometry and the analysis of the perturbations should be performed. However, that overpasses the scopus of this work and will be published elsewhere.

Finally, as far as the complex scalar field is concerned, we remark that there only at the (families) stationary points $A_{2}$ and $C_{2}$ the imaginary part of the complex scalar field contributes in the physical quantities of the asymptotic solutions. However, these families of points can describe important eras of the cosmological history, for instance, the early inflationary era, the radiation era or the matter dominated era. What will be of special interest is to investigate the existence of a scalar field potential which does not depend only on the norm of the complex scalar field.