Cartan F(R) Gravity and Equivalent Scalar–Tensor Theory

Abstract

We investigate the Cartan formalism in $F\left(R\right)$ gravity. $F\left(R\right)$ gravity has been introduced as a theory to explain cosmologically accelerated expansions by replacing the Ricci scalar R in the Einstein–Hilbert action with a function of R. As is well-known, $F\left(R\right)$ gravity is rewritten as a scalar–tensor theory by using the conformal transformation. Cartan $F\left(R\right)$ gravity is described based on the Riemann–Cartan geometry formulated by the vierbein-associated local Lorenz symmetry. In the Cartan formalism, the Ricci scalar R is divided into two parts: one derived from the Levi–Civita connection and the other from the torsion. Assuming the spin connection-independent matter action, we have successfully rewritten the action of Cartan $F\left(R\right)$ gravity into the Einstein–Hilbert action and a scalar field with canonical kinetic and potential terms without any conformal transformations. red Thus, symmetries in Cartan $F\left(R\right)$ gravity are clearly conserved. The resulting scalar–tensor theory is useful in applications of the usual slow-roll scenario. As a simple case, we employ the Starobinsky model and evaluate fluctuations in cosmological microwave background radiation.

1. Introduction

Cartan formalism is a natural extension of the General Relativity (GR) based on the Riemann–Cartan geometry formulated by the basis called vierbein [1]. Einstein introduced it to unify gravity and electromagnetism in 1928 [2]. Later, the Cartan formalism became known as the Einstein–Cartan–Kibble–Sciama (ECKS) theory in the 1960s [3,4]. The ECSK theory is currently studied for applications in cosmological problems [5,6,7,8,9,10,11,12].

Another extension of GR is to introduce a term that modifies the Einstein–Hilbert action. $F\left(R\right)$ gravity is the theory in which the Ricci scalar, R, in the Einstein–Hilbert action is replaced by an arbitrary function of Ricci scalar, $F\left(R\right)$ [13,14]. The $F\left(R\right)$ gravity can be rewritten as an equivalent scalar–tensor theory by using conformal transformations [14,15]. It should be observed that there is some discussion about the equivalence, such as symmetries before and after the conformal transformation [16,17,18,19].

One of the most famous $F\left(R\right)$ gravity models is the Starobinsky model [20]. After conformal transformation, the model is described by the scalar–tensor theory, for which its scalar field has a potential with a flat plateau at high energy. The potential energy accelerates the expansion of the universe while it dominates the energy density of the universe, As is well-known, the accelerating expansion in the early universe can solve horizon and flatness problems [21,22].

The main purpose of this study is to adapt the Cartan formalism to the $F\left(R\right)$ gravity and apply the result to a Starobinsky-like model. A characteristic property of Cartan $F\left(R\right)$ gravity is that the torsion does not vanish [23]. The Ricci scalar R is then divided into the usual part obtained from the Levi–Civita connection and the additional part obtained from the torsion.

The non-vanishing torsion is also a feature of the Palatini approach to the modified gravity and more general metric-affine geometry. Several studies have been conducted on the basic properties of the metric-affine $F\left(R\right)$ gravity and its application to cosmology [24,25,26]. It is found that the metric-affine $F\left(R\right)$ and Palatini $F\left(R\right)$ gravity are rewritten as a certain class of Brans–Dicke type scalar–tensor theories after a conformal transformation [27,28,29,30,31].

This paper shows that a class of Cartan $F\left(R\right)$ gravity obtains an Einstein–Hilbert action and a scalar field action with canonical kinetics and potentials by the definition of a simple scalar field. The Cartan $F\left(R\right)$ gravity does not require conformal transformations, which is different from conventional $F\left(R\right)$ gravity theories. Therefore, we can evaluate the Cartan $F\left(R\right)$ gravity by using the equivalent scalar–tensor theory in the original frame. Finally, Starobinsky’s potential can be derived from the Cartan $F\left(R\right)$ gravity model. The Cartan $F\left(R\right)$ gravity is useful to consider the usual slow-roll scenario.

2. Cartan $\mathit{F}\left(\mathit{R}\right)$ Gravity

The $F\left(R\right)$ gravity is reformulated on the Riemann-Cartan geometry described by the vierbein and the spin connection. We call it Cartan $F\left(R\right)$ gravity. The fundamental elements of the Cartan $F\left(R\right)$ gravity are the vierbein and the spin connection associated with local Lorenz symmetries. Since it is a natural extension of the $F\left(R\right)$ gravity, we expect that the additional contribution to GR is described as a scalar field theory, similarly to the $F\left(R\right)$ gravity.

Here, we briefly introduce the vierbein and the spin connection. The vierbein, ${e^i}_{\mu }$, is defined to act as a mediator from a flat spacetime metric, ${\eta }_{ij}$, to a curved one, $g_{\mu \nu }$. It contains flat and curved spacetime indices and satisfies the following.

$$\begin{array}{c}\hfill g_{\mu \nu }={\eta }_{ij}{e^i}_{\mu }{e^j}_{\nu }.\end{array}$$

(1)

The affine connection is expressed as follows:

$$\begin{array}{c}\hfill {{\mathsf{\Gamma }}^{\rho }}_{\mu \nu }={e_a}^{\rho }{D}_{\nu }{e^a}_{\mu },\end{array}$$

with the following being the case:

$$\begin{array}{c}\hfill D_{\nu }{e^k}_{\mu }={\partial }_{\nu }{e^a}_{\mu }+{{\omega }^k}_{l\nu }{e^l}_{\mu }.\end{array}$$

where $D_{\nu }$ is the covariant derivative for the local Lorentz transformation, and ${{\omega }^{ij}}_{\nu }$ denotes the spin connection. The Affine connection is not necessarily invariant under the replacement of the lower indices, ${{\mathsf{\Gamma }}^{\rho }}_{\mu \nu }\ne {{\mathsf{\Gamma }}^{\rho }}_{\nu \mu }$. Thus, the geometric tensor, ${T^{\rho }}_{\mu \nu }$, called torsion arises.

$$\begin{array}{c}\hfill {T^{\rho }}_{\mu \nu }\equiv {{\mathsf{\Gamma }}^{\rho }}_{\mu \nu }-{{\mathsf{\Gamma }}^{\rho }}_{\nu \mu }.\end{array}$$

The Riemann tensor is expressed by the spin connection

$$\begin{array}{c}\hfill {R^{ij}}_{\mu \nu }={\partial }_{\mu }{{\omega }^{ij}}_{\nu }-{\partial }_{\nu }{{\omega }^{ij}}_{\mu }+{{\omega }^i}_{k\mu }{{\omega }^{kj}}_{\nu }-{{\omega }^i}_{k\nu }{{\omega }^{kj}}_{\mu },\end{array}$$

and the Ricci scalar by the spin connection and the vierbein

$$\begin{array}{c}\hfill R={e_i}^{\mu }{e_j}^{\nu }{R^{ij}}_{\mu \nu }(\omega ,\partial \omega ).\end{array}$$

The action of the Cartan $F\left(R\right)$ gravity is defined by replacing the Ricci scalar R in the Einstein-–Cartan theory with a function of R:

$$\begin{array}{c}\hfill S=\int d^{4}xe\left(\frac{{M_{\mathrm{Pl}}}^2}{2}F\left(R\right)+{\mathcal{L}}_{\mathrm{m}}\right).\end{array}$$

(2)

where $M_{\mathrm{Pl}}$ indicates the Planck scale and the volume element is given by the determinant of the vierbein, e. ${\mathcal{L}}_{\mathrm{m}}$ shows the Lagrangian density for the matter field. As a simple case, we assume that the matter filed does not depend on the spin connection. The equation of motion can be derived by the variation of the action with respect to the vierbein:

$$\begin{array}{c}\hfill F^{\prime }{R^i}_{\mu }-\frac{1}{2}{e^i}_{\mu }F\left(R\right)={M_{\mathrm{Pl}}}^{-2}{{\Sigma }^i}_{\mu },\end{array}$$

(3)

where $F^{\prime }\left(R\right)$ denotes the derivative of $F\left(R\right)$ with respect to R, and ${{\Sigma }^i}_{\mu }$ is the energy–momentum tensor of the matter filed. From the traces of Equation (3), the Ricci scalar can be represented as a function, $R(\Sigma )$, where $\Sigma$ is the trace of the energy–momentum tensor. In the absence of the matter, the Rich scalar is uniquely determined and constant except for $F\left(R\right)=\alpha R^2$ [26,27,28,29,30,31]. Hereafter, $R(\Sigma )$ is abbreviated as R.

The Cartan equation is derived by the variation of the action with respect to the spin connection. The details of the derivation of the Cartan equation are given in Appendix A. The Cartan equation is described as follows.

$$\begin{array}{c}\hfill ({T^{\mu }}_{kl}-{e_l}^{\mu }{T}_k+{e_k}^{\mu }{T}_l)F^{\prime }\left(R\right)+({e_k}^{\alpha }{e_l}^{\mu }-{e_k}^{\mu }{e_l}^{\alpha }){\partial }_{\alpha }{F}^{\prime }\left(R\right)=0.\end{array}$$

(4)

Since the matter filed is spin-connection independent, the Cartan equation does not include a matter term. From the Cartan equation, the torsion is represented by the derivative of $F\left(R\right)$ and the vierbein.

$$\begin{array}{c}\hfill {T^k}_{ij}=\frac{1}{2}({{\delta }^k}_j{e_i}^{\lambda }-{{\delta }^k}_i{e_j}^{\lambda }){\partial }_{\lambda }ln{F}^{\prime }\left(R\right).\end{array}$$

(5)

It should be noted that the torsion vanishes in the Einstein–Cartan theory, $F\left(R\right)=R$. The contribution from the torsion is induced in the Cartan $F\left(R\right)$-modified gravity. It is extracted from the Ricci scalar:

$$\begin{array}{c}\hfill R=R_E+T-2{{\nabla }_E}_{\mu }{T}^{\mu },\end{array}$$

(6)

where the subscript E in $R_E$ and ${\nabla }_E$ stands for the Ricci scalar and the covariant derivative given by the Levi–Civita connection.

$$\begin{array}{c}\hfill {{\left({\mathsf{\Gamma }}_E\right)}^{\lambda }}_{\mu \nu }=\frac{1}{2}{g}^{\lambda \rho }({\partial }_{\mu }{g}_{\nu \rho }+{\partial }_{\nu }{g}_{\rho \mu }-{\partial }_{\rho }{g}_{\mu \nu }).\end{array}$$

$T_{\mu }$ represents the torsion vector, $T_{\mu }={T^{\lambda }}_{\mu \lambda }$, and T is the torsion scalar defined by the following.

$$\begin{array}{c}\hfill T=\frac{1}{4}{T}^{\rho \mu \nu }{T}_{\rho \mu \nu }-\frac{1}{4}{T}^{\rho \mu \nu }{T}_{\mu \nu \rho }-\frac{1}{4}{T}^{\rho \mu \nu }{T}_{\nu \rho \mu }-T^{\mu }{T}_{\mu }.\end{array}$$

Thus, the Ricci scalar is divided into two parts: $R_E$ and the additional part derived from the torsion. Substituting Equation (5) into Equation (6), the additional part is represented as follows.

$$\begin{array}{c}\hfill R=R_E-\frac{3}{2}{\partial }_{\lambda }ln{F}^{\prime }\left(R\right){\partial }^{\lambda }ln{F}^{\prime }\left(R\right)-3\square ln{F}^{\prime }\left(R\right).\end{array}$$

(7)

Below we consider the class of the Cartan $F\left(R\right)$ gravity expressed as $F\left(R\right)=R+f\left(R\right)$. From Equation (7), the gravity part of the action is rewritten as follows.

$$\begin{array}{cc}\hfill S& =\int d^{4}xe\frac{{M_{\mathrm{Pl}}}^2}{2}\left(R+f\left(R\right)\right)\hfill \\ \hfill & =\int d^{4}xe\frac{{M_{\mathrm{Pl}}}^2}{2}\left(R_E-\frac{3}{2}{\partial }_{\lambda }ln{F}^{\prime }\left(R\right){\partial }^{\lambda }ln{F}^{\prime }\left(R\right)-3\square ln{F}^{\prime }\left(R\right)+f\left(R\right)\right).\hfill \end{array}$$

(8)

The third term in the integrand is in the form of a total derivative. It is assumed that $F^{\prime }\left(R\right)$ vanishes at a distance, and the term proportional to $\square ln{F}^{\prime }\left(R\right)$ is dropped. We introduce a scalar field $\varphi$ by the replacement.

$$\begin{array}{c}\hfill \varphi \equiv -\sqrt{\frac{3}{2}}{M}_{\mathrm{Pl}}ln{F}^{\prime }\left(R\right).\end{array}$$

(9)

Then, the second term in the integrand of the action (8) takes a form of the canonical kinetic term for the scalar field, and the potential term is generated from the last term in the integrand:

$$\begin{array}{c}\hfill S=\int d^{4}xe\left(\frac{{M_{\mathrm{Pl}}}^2}{2}{R}_E-\frac{1}{2}{\partial }_{\lambda }\varphi {\partial }^{\lambda }\varphi -V\left(\varphi \right)\right),\end{array}$$

(10)

where the potential, $V\left(\varphi \right)$, is given by the following.

$$\begin{array}{c}\hfill V\left(\varphi \right)=-\frac{{M_{\mathrm{Pl}}}^2}{2}{\left.f\left(R\right)\right|}_{R=R\left(\varphi \right)}.\end{array}$$

(11)

The potential is expressed as a function of the scalar field $\varphi$ through $R=R\left(\varphi \right)$. The explicit expression for the potential is found in terms of the scalar field by solving Equation (9). It is more convenient to consider the exponential form of the equation.

$$\begin{array}{c}\hfill f^{\prime }\left(R\right)=e^{-\sqrt{\frac{2}{3}}\frac{\varphi }{M_{\mathrm{Pl}}}}-1.\end{array}$$

(12)

Therefore, the action (8) reduces to the scalar–tensor theory (10) without any conformal transformations. It should be observed that potential $V\left(\varphi \right)$ is different from the one in the scalar–tensor theory obtained from the usual $F\left(R\right)$ gravity after the conformal transformation [14].

3. ${\mathit{R}}^{\mathbf{2}}$ Model

We would like to apply the derivation of the scalar–tensor theory to a specific model. We employ a Cartan $F\left(R\right)$ gravity with a $R^2$ correction term as a simple prototype model. The $R^2$ term provides a small correction in weakly curved spacetime. We introduce mass scale M and set $F\left(R\right)=R-R^2/M^2$, $f\left(R\right)=-R^2/M^2$. It is noted that the sign of the modification term, $R^2$, is negative, which is opposite relative to the Starobinsky model.

For $f\left(R\right)=-R^2/M^2$, the action of the equivalent scalar–tensor theory (10) is given by the following:

$$\begin{array}{cc}\hfill S& =\int d^{4}xe\left(\frac{{M_{\mathrm{Pl}}}^2}{2}{R}_E-\frac{1}{2}{\partial }_{\lambda }\varphi {\partial }^{\lambda }\varphi -\frac{{M_{\mathrm{Pl}}}^2}{2}\frac{R^2}{M^2}\right),\hfill \end{array}$$

where we assume that the induced scalar field, $\varphi$, dominates the energy density of the universe and ignores the matter action. From Equation (12) the Ricci scalar, R, is described as follows.

$$\begin{array}{c}\hfill R=\frac{M^2}{2}\left(1-e^{-\sqrt{\frac{2}{3}}\frac{\varphi }{M_{\mathrm{Pl}}}}\right).\end{array}$$

(13)

Thus, the potential, $V\left(\varphi \right)$, is fixed in terms of the scalar field.

$$\begin{array}{c}\hfill V\left(\varphi \right)=\frac{{M_{\mathrm{Pl}}}^2{M}^2}{8}{\left(1-e^{-\sqrt{\frac{2}{3}}\frac{\varphi }{M_{\mathrm{Pl}}}}\right)}^2.\end{array}$$

(14)

Potential (14) reproduces the one induced by the conformal transformation in the usual Starobinsky model with $F\left(R\right)=R+R^2/M^2$ [20]. In other words, the potential in the Cartan $f\left(R\right)=-R^2/M^2$ gravity is identical to that of the Starobinsky model. If we adopt this model to the slow-roll scenario of inflation, we can regard the scalar field as the inflaton field. The slow-roll takes place in the usual Friedmann–Robertson–Walker background. The inflation develops the fluctuations of the cosmic microwave background with spectral index $n_s$ and scalar-tensor ratio r:

$$\begin{array}{cc}\hfill & n_s\simeq 1-\frac{2}{N},\hfill \\ \hfill & r\simeq \frac{12}{N^2},\hfill \end{array}$$

where N is the e-folding number. These results are of course identical to the Starobinsky model and consistent with current observations [32]. It means that the primordial inflation is induced in the $R^2$ model of the Cartan $F\left(R\right)$ gravity.

4. Conclusions

The Cartan $F\left(R\right)$ gravity, an extension of the $F\left(R\right)$ gravity on the Riemann–Cartan geometry, has been studied. We have constructed a derivation of the scalar–tensor theory from Cartan $F\left(R\right)$ gravity. The canonical kinetic term of a scalar field is induced by extracting the torsion from the Ricci scalar. The potential term is derived from the modified gravity action, $f\left(R\right)$. Since the derivation does not require a conformal transformation, it is free from the equivalence problem, such as the symmetry between Jordan and Einstein frames for the scalar–tensor theory derived by the conformal transformation in $F\left(R\right)$ gravity.

In contrast to previous studies, the derived scalar–tensor theory can be applied to usual slow-roll scenarios in the process of inflation.

The derivation has been adapted to a simple model with an $R^2$ correction term. The explicit expression of the scalar–tensor theory has been derived for the model without changing the frame. It is observed that the derived scalar–tensor theory has identical potentials relative to the Starobinsky model. Therefore, we succeeded in finding a Cartan $F\left(R\right)$ gravity model that induces inflation that is consistent with the current observations of cosmic microwave background fluctuations.

In this paper, we have focused on the gravity part of the theory and assumed that the matter action does not depend on spin connections.

Cartan $F\left(R\right)$ gravity is associated with matter fields through the local Lorentz symmetry. It is known that torsion is also affected by matter fields. The original ECKS theory is promoted to introduce a spinor field in a gravity theory [33,34,35]. In the ECKS theory with a spinor field, a non-vanishing torsion introduces a four-fermion interaction called spin–spin interactions or Dirac–Heisenberg–Ivanenko–Hehl–Datta four-body fermi interactions [36,37]. The ECKS theory is also studied with vector fields [38,39] and Rarita–Schwinger spin $3/2$ fields [40].

It is interesting to apply the scalar–tensor theory obtained from Cartan $F\left(R\right)$ gravity to the dark-energy problem by an analogy with quintessence. The origin of dark energy has been investigated in conventional $F\left(R\right)$ gravity theories [41,42]. Geometrically induced interactions between matter fields have also been studied to explain dark energy [43]. This indicates that the Cartan $F\left(R\right)$ gravity can explain early- and late-time accelerating expansion. We hope to extend the derivation to a model with matter fields, and we hope to report the results in the future.