Born-Infeld-$f(R)$ gravity with de Sitter solutions

TL;DR

This paper explores a modified gravity model combining Born-Infeld and $f(R)$ theories in the Palatini formulation, analyzing its solutions and showing it admits de Sitter space as a vacuum solution.

Contribution

It introduces a novel Born-Infeld-$f(R)$ gravity model with explicit $f(R)$ functions and demonstrates the existence of de Sitter solutions within this framework.

Findings

Both models reduce to the same form under conformal transformation.

The model admits de Sitter vacuum solutions.

Explicit analysis for $f(R)$ with positive and negative powers.

Abstract

In this study, we consider the Born--Infeld-$f(R)$ gravity in which the $f(R)$ term enters directly into the square root in the Palatini formulation. We shortly analyzed this model for an explicit $f(R)$ function which includes positive and negative powers of the curvature scalar. We also show that both the ordinary Born--Infeld-$f(R)$ and this modification reduce to the same gravitational action form under the conformal approach. Then we consider the existence of a maximally symmetric vacuum solution for the gravitational field equations and find the de Sitter solution for this modified model.

Equations

$$S$$

$$R_{\mu \beta \nu }^{\alpha }={\partial }_{\beta }{\Gamma }_{\nu \mu }^{\alpha }-{\partial }_{\nu }{\Gamma }_{\mu \beta }^{\alpha }+{\Gamma }_{\beta \lambda }^{\alpha }{\Gamma }_{\nu \mu }^{\lambda }-{\Gamma }_{\nu \lambda }^{\alpha }{\Gamma }_{\mu \beta }^{\lambda },$$

$$\frac{\sqrt{-q}}{\sqrt{-g}}{\left(q^{-1}\right)}^{\mu \nu }-\lambda g^{\mu \nu }+\frac{\kappa }{2}{g}^{\mu \nu }f\left(R\right)-\kappa f_R{R}^{\mu \nu }=-\kappa T^{\mu \nu },$$

$$q_{\mu \nu }=g_{\mu \nu }+\kappa R_{\mu \nu }.$$

$${\nabla }_{\lambda }\left(\sqrt{-q}{q}^{\mu \nu }+\sqrt{-g}{f}_R{g}^{\mu \nu }\right)=0,$$

$${\nabla }_{\mu }\varphi ={\partial }_{\mu }\varphi -{\Gamma }_{\mu \alpha }^{\alpha }\varphi .$$

$$q_{\mu \nu }=k\left(t\right)g_{\mu \nu },$$

$${\nabla }_{\nu }\left[\sqrt{-u}{\left(u^{-1}\right)}^{\mu \nu }\right]=0.$$

$${\Gamma }_{\mu \nu }^{\rho }=\frac{1}{2}{\left(u^{-1}\right)}^{\rho \sigma }\left(u_{\sigma \nu ,\mu }+u_{\mu \sigma ,\nu }-u_{\mu \nu ,\sigma }\right).$$

$$R_{\mu \nu }=\frac{1}{\kappa }\left[k\left(t\right)-1\right]g_{\mu \nu }.$$

$$d{s}^2=-d{t}^2+a^2\left(t\right)\left(d{x}^2+d{y}^2+d{z}^2\right),$$

$$u_{\mu \nu }=u\left(t\right)\text{diag}\left(-1,a^2\left(t\right),a^2\left(t\right),a^2\left(t\right)\right),$$

$$r\left(t\right)=3{\left(H+\frac{\dot{u}}{2u}\right)}^2,$$

$$2\dot{H}=H\frac{\dot{u}}{u}+\frac{3}{2}{\left(\frac{\dot{u}}{u}\right)}^2-\frac{\ddot{u}}{u},$$

$$u\left(t\right)=cr\left(t\right)$$

$$H=\pm \sqrt{\frac{u}{3c}}-\frac{\dot{u}}{2u},$$

$$f\left(R\right)=\Lambda -R+\left(\frac{c-\kappa }{8}\right)R^2,$$

$$S=\frac{2}{\kappa }\int d^{4}x\sqrt{-g}{R}^2.$$

$$S$$

$$F_{\mu \nu }=\alpha R_{\mu \nu }\left(\Gamma \right)+\beta g_{\mu \nu }F\left(R\right),$$

$$p_{\mu \nu }=g_{\mu \nu }+\kappa \left[\alpha R_{\mu \nu }+\beta g_{\mu \nu }F\left(R\right)\right],$$

$$S$$

$$\sqrt{-p}{\left(p^{-1}\right)}^{\mu \nu }\left(1+\kappa \beta F\left(R\right)\right)-\kappa \beta \sqrt{-p}{\left(p^{-1}\right)}^{\sigma \rho }{g}_{\sigma \rho }{F}_R{g}^{\mu \alpha }{g}^{\nu \beta }{R}_{\alpha \beta }-\lambda \sqrt{-g}{g}^{\mu \nu }=-\kappa \sqrt{-g}{T}^{\mu \nu },$$

$${\nabla }_{\lambda }\left[\sqrt{-p}\left(\alpha {\left(p^{-1}\right)}^{\mu \nu }+\beta {\left(p^{-1}\right)}^{\sigma \rho }{g}_{\sigma \rho }{g}^{\mu \nu }{F}_R\right)\right]=0.$$

$$\sqrt{-p}\left(\alpha {\left(p^{-1}\right)}^{\mu \nu }+\beta {\left(p^{-1}\right)}^{\sigma \rho }{g}_{\sigma \rho }{g}^{\mu \nu }{f}_R\right)=\sqrt{-u}{\left(u^{-1}\right)}^{\mu \nu },$$

$$p^{2}det\left(\alpha {\left(p^{-1}\right)}^{\mu \nu }+\beta {\left(p^{-1}\right)}^{\sigma \rho }{g}_{\sigma \rho }{g}^{\mu \nu }{F}_R\right)=u,$$

$${\left(u^{-1}\right)}^{\mu \nu }$$

$${\nabla }_{\lambda }\left[\sqrt{-u}{\left(u^{-1}\right)}^{\mu \nu }\right]=0,$$

$${\Gamma }_{\mu \nu }^{\rho }=\frac{1}{2}{\left(u^{-1}\right)}^{\rho \sigma }\left(u_{\sigma \nu ,\mu }+u_{\mu \sigma ,\nu }-u_{\mu \nu ,\sigma }\right).$$

$$S$$

Full text

de Sitter solution; Modified theory of gravity; Palatini formulation

I Introduction

Einstein’s theory of general relativity (GR) is the most successful theory that explains almost all the gravitational mechanisms for around a century. The validity of this theory has been supported by lots of observation and experiments on a wide range of scales where it has been directly tested. Despite its observational success, there are some reasons to believe that GR is unable to explain some gravitational phenomena that need to be clarified. In particular, GR does not give consistent results at the Planck scale where quantum effects play an important role. In more general words, at extreme conditions such as ultra-high or low energies, GR is expected to be modified.

In this sense, Born-Infeld (BI) like gravitational theories provide a useful framework to modify GR. The original theory proposed by Born and Infeld have a non-linear action for classical electrodynamics [1]. In analogy with this study, Deser and Gibbons established a gravitational model by using Born-Infeld like determinantal structure using the Ricci tensor instead of the electromagnetic field tensor [2]. But this attempt remained unsuccessful because of the presence of ghost fields. To resolve this problem, Vollick succeeded to construct a theory without ghosts by using the Palatini formulation which the connection and the metric tensor are considered independent fields [3, 4]. In 2010, Bañados and Ferreira made this theory in more standard form, especially in the definition of the matter term [5]. After these works, lots of applications and modifications of BI gravity appeared in the literature (for a detailed review see [6]).

One of the important extension is BI-$f\left(R\right)$ [7]. In this model, the BI gravitational action is combined with the well-known $f\left(R\right)$ theories in the Palatini formalism. This model attracted a lot of attention and it is used wide variety of fields especially in cosmology [8, 9, 10, 11, 12, 13, 14]. On the other hand there is different generalization of BI-$f\left(R\right)$ model which $f\left(R\right)$ function directly enters in the determinantal structure [7, 13]. This model has been studied less than the other and it is capable to provide a different framework [13].

From this background, the main purpose of this study is to analyze BI inspired model of gravity coupled to a $f\left(R\right)$ term which is presence in the determinantal structure. In this context, we first give a detailed review of the current situation for this model. Besides we analyze the de Sitter solution which is very important when we consider inflationary period of our universe [15, 16, 17, 18, 19, 20, 21, 22] for this model.

In Section II, we give a brief review of the Born-Infeld-$f\left(R\right)$ gravity. In Section III, a different approximation of the BI-$f\left(R\right)$ gravity which the $f\left(R\right)$ term directly enter into the determinant is given. We also examine three cases for this extension. In the first case, we use an exact $f\left(R\right)$ function which has positive and negative powers of the Ricci scalar. In the second case, we analyze the conformal approximation in the metric tensor. In the last one, we derive the de Sitter solution for this model. In the last section, we conclude the paper with some discussion.

II Born-Infeld-$f\left(R\right)$ theory

Let us briefly review the standard BI-$f\left(R\right)$ theory [7] (see also [9, 8, 10]). In this theory, the original EiBI gravity theory is combined with an additional $f\left(R\right)$ that depends on the Ricci scalar $R=g^{\mu\nu}R_{\mu\nu}\left(\Gamma\right)$. To avoid any ghost instabilities, the theory is formulated within the Palatini formalism (for more detail see [23, 24]), in which the metric $g_{\mu\nu}$ and the connection $\Gamma_{\beta\gamma}^{\alpha}$ are treated as independent variables. The action for this theory is given by

[TABLE]

where the first term represents the standard BI gravitational Lagrangian and the second term is an additional function of the Ricci scalar. $S_{m}$ is the matter action which depends on generically $\psi$ field and the metric tensor $g_{\mu\nu}$.

[TABLE]

is the Riemann tensor of the connection $\Gamma_{\mu\nu}^{\lambda}$. In addition, the connection is also assumed to be torsionless. The parameter $\kappa$ is a constant with inverse dimensions to that of a cosmological constant and $\lambda$ is a dimensionless constant. Note that $g$ is the determinant of the metric. Throughout this paper, we will use Planck units $8\pi G=1$ and set the speed of light to $c=1$.

The variation of this action with respect to the metric tensor leads to a modified metric field equations for the standard BI gravitational model,

[TABLE]

where $f_{R}$ is the derivative of $f\left(R\right)$ with respect to the Ricci scalar, and $T^{\mu\nu}$ is the standard energy-momentum tensor. We have used the notation,

[TABLE]

We denoted the inverse of $q_{\mu\nu}$ by $\left(q^{-1}\right)^{\mu\nu}$ and $q$ represents the determinant of $q_{\mu\nu}$. Similarly, the corresponding equation which follows by variation over the connection $\Gamma$ has the form,

[TABLE]

where the covariant derivative is taken with respect to the connection which is defined for a scalar field $\phi$ as

[TABLE]

If we assume that $q_{\mu\nu}$ is conformally proportional to the metric tensor as

[TABLE]

then Eq.(5) becomes,

[TABLE]

In this case, we have an auxiliary metric defined as $u_{\mu\nu}=\left[k\left(t\right)+f_{R}\right]g_{\mu\nu}$ and $\left(u^{-1}\right)^{\mu\nu}$ represents the inverse representation of $u_{\mu\nu}$. Eq.(8) tells us that the connection can be defined by this auxiliary metric as

[TABLE]

By considering the conformal relation between $q_{\mu\nu}$ and $g_{\mu\nu}$ in Eq.(7), we can easily say that the Ricci tensor must also be proportional to the metric tensor. Thus one can write the relationship between the Ricci tensor and $g_{\mu\nu}$, as

[TABLE]

For a cosmological scenario, let us consider a homogeneous and isotropic the Friedman–Lemaitre–Robertson–Walker (FLRW) universe with metric,

[TABLE]

where $t$ is the cosmic time and $a\left(t\right)$ is the scale factor. Now, we can define the auxiliary metric as

[TABLE]

where $u\left(t\right)=k\left(t\right)+f_{R}$. According to Eq.(10), we can define the Ricci tensor as $R_{\mu\nu}=r\left(t\right)g_{\mu\nu}$. From this background, one can find the following expressions

[TABLE]

[TABLE]

where the upper dot denotes the time derivative and $H=\frac{\dot{a}}{a}$ is the Hubble parameter. Using these equations, one can derive the following relation

[TABLE]

where $c$ is an integration constant. Then combining Eq.(13) and Eq.(15), one can get

[TABLE]

and Eq.(15) lead to find the $f\left(R\right)$ function as

[TABLE]

where the effective cosmological constant is defined by $\Lambda=\frac{\lambda-1}{\kappa}$. If we substitute this function into the action Eq.(1) without matter field, we obtain the following action by setting the constants $\kappa c=16$.

[TABLE]

III $F\left(R\right)$ term in the determinant

In this section, we examine a modified BI gravitational theory with a function of the Ricci scalar being added to the determinantal action. This model has been partly analyzed in [7, 13] (for early development similar to this model in pure metric formalism [25] and in the teleparallel framework[26]). We use the construction method for the BI-like action given in [13] to realize our purpose. To this aim, we use the following action

[TABLE]

here $F_{\mu\nu}$ is defined as follows,

[TABLE]

where $\alpha$ and $\beta$ are dimensionless constants. To find the field equations, we first define a new object as

[TABLE]

here the inverse of $p_{\mu\nu}$ is denoted $\left(p^{-1}\right)^{\mu\nu}$ and these objects satisfy $\left(p^{-1}\right)^{\mu\rho}p_{\rho\nu}=\gamma_{\nu}^{\mu}$. By using Eq.(21), the action Eq.(19) reads

[TABLE]

where $p=det\left(p_{\mu\nu}\right)$. To obtain equations of motion, we look the variation with respect to metric $g_{\mu\nu}$ and the connection $\Gamma_{\beta\mu}^{\alpha}$, respectively,

[TABLE]

[TABLE]

According to the literature, it is well known that the Palatini $F\left(R\right)$ theories can be solved in terms of an auxiliary metric. From this idea and taking account of Eq.(24), we define an auxiliary metric $u_{\mu\nu}$ as follows

[TABLE]

where $u=det\left(u_{\mu\nu}\right)$ and $\left(u^{-1}\right)^{\mu\nu}$ is the inverse of $u_{\mu\nu}$, that is $\left(u^{-1}\right)^{\mu\rho}u_{\rho\nu}=\delta_{\nu}^{\mu}$. By considering the determinant of Eq.(25), we find,

[TABLE]

and this definition leads to

[TABLE]

Now Eq.(24) can be rewritten

[TABLE]

which can be solved with respect to the connection $\Gamma_{\mu\nu}^{\rho}$ as in the Einstein gravity as follows,

[TABLE]

III.1 An explicit $F\left(R\right)$ function

In the low energy limit ($\kappa\rightarrow 0$), the gravitational action takes the following form

[TABLE]

Let us analyze an explicit $F\left(R\right)$ function which include both positive and negative powers. This kind of theories have been use in various kind of application in the context of gravitation [27, 28, 29, 30, 31, 32]. We assume that the function has the following form,

[TABLE]

Firstly, we choose $\gamma=\epsilon=1$ and $\alpha+4\beta=1$ then the action becomes

[TABLE]

here we obtain an action with Einstein’s theory for low curvature limit together with both $R^{2}$ term and inverse power of $R$. Secondly, if we choose $\gamma=1$ and $\epsilon=0$ we reach $F\left(R\right)=R$ then the action reduces to

[TABLE]

In this case, to recover General Relativity, the parameters should satisfy again $\alpha+4\beta=1$. Moreover, if we set $\alpha=0$ and $\beta=\frac{1}{4}$, we derive $R^{2}$ theory of gravity as

[TABLE]

This model has already been studied in [13]. Thirdly, choosing $\gamma=0$ and $\epsilon=1$ we obtain $F\left(R\right)=1/R$ and assuming $\alpha=1$, $\beta=1/4$ then the action takes the following form

[TABLE]

For the last case, the selection $\alpha=1$ and $\beta=0$ goes to standard formulation of the BI gravity.

III.2 Conformal case

The conformal assumption of this model has been already examined in [7]. Here, we wish to analyze this case for our notation. Let us start to give the main definition for this case if we assume the conformal relationship between $p_{\mu\nu}$ and $g_{\mu\nu}$ as

[TABLE]

then Eq.(25) takes the following form,

[TABLE]

From this definition, we can easily write

[TABLE]

In this case, the conditions are satisfied given in Eq.(28) and Eq.(29). The conformal case (36) leads to write a condition in which the Ricci tensor is proportional to the metric tensor $g_{\mu\nu}$ as,

[TABLE]

where $r\left(t\right)$ can be found by using Eq.(21) and Eq.(36) as

[TABLE]

Furthermore, taking trace of Eq.(39), one can also write $r\left(t\right)=\frac{R}{4}$ and Eq.(39) reduces to

[TABLE]

Now if we suppose that for the spatially-flat FLRW universe with metric Eq.(11) and the definition of $u_{\mu\nu}$ in Eq.(12), we find same relationship as given in Eq.(15) and this model also satisfies Eq.(16). By the help of Eq.(15) and Eq.(38), we find the $F\left(R\right)$ function as the following form

[TABLE]

where $c_{1}$ is an integration constant. So substituting Eq.(42) into (1) together with setting $c=16$ and $\lambda=\frac{1}{16c_{1}^{2}\kappa^{2}\beta^{2}}$, the action takes the same form as given before in Eq.(18).

III.3 The de Sitter solution

In this part, our investigation into the existence of the de Sitter vacuum solution. This solution provides a very useful theoretical background for early universe exponential expansion. This model is described by a constant Hubble parameter and from this point, the Riemann tensor can be written

[TABLE]

Taking account of the Bianchi identities, we can write $R=R_{0}$ as a covariantly constant curvature. So the Ricci tensor satisfies,

[TABLE]

Then Eq.(21) becomes,

[TABLE]

[TABLE]

If we apply these results to (25), we get

[TABLE]

where $F_{R_{0}}$ is the derivative with respect to $R_{0}$. Now $\left(u^{-1}\right)^{\mu\nu}$ takes the following form

[TABLE]

which tells, by using (29), we find that the connection $\Gamma_{\mu\nu}^{\rho}$ is nothing but that of the Einstein gravity

[TABLE]

Then by using Eq.(48), Eq.(23) becomes

[TABLE]

where $T_{\mu\nu}=0$ because we are considering the vacuum (anti-)de Sitter space-time. This is an algebraic equation with respect to $R_{0}$. If the solution is positive, $R_{0}>0$, the solutions of Eq. (44) are the de Sitter space-time, the de Sitter-Schwarzschild space-time, and the de Sitter-Kerr space-time and if $R_{0}<0$, they are the anti-de Sitter space-time, the anti-de Sitter-Schwarzschild space-time, and the anti-de Sitter-Kerr space-time.

If we solve (50) with respect to $F\left(R_{0}\right)$ we get

[TABLE]

where $c_{1}$ is the integration constant. Now if we rearrange the action (19), we get

[TABLE]

Inserting Eq.(51) into Eq.(52) and setting $c_{1}=16$, we obtain the following action,

[TABLE]

IV Conclusion

In this work, we examined Born-Infeld-$f\left(R\right)$ gravity in which the $f\left(R\right)$ function is considered in the determinantal structure. We derived the equations of motions by introducing an auxiliary metric without any restrictions in Eq.(23) and Eq.(24).

Then we analyzed this model for three cases. Firstly, we considered the conformal assumption and we showed that the equations of motions reduces to the result given in [7]. According to our results, we note that both actions given in Eq.(1) and Eq.(19) reduce to the same structure which contains only the square of the Ricci scalar

Secondly, we use an exact $f\left(R\right)$ function that includes positive and negative powers of the Ricci scalar with two constants. We analyzed this assumption for three cases $F\left(R\right)=\gamma R+\frac{\epsilon}{R}$, $F\left(R\right)=\gamma R$ and $F\left(R\right)=\frac{\epsilon}{R}$ respectively. The combined case leads to an action that contains Einstein’s theory plus the positive and negative power of the scalar curvature (see Eq.(32)). Then the second condition is reduced to the action (33) which has already been studied in [13]. For the last selection, the action (35) become a similar structure to (32) but the simpler in the constants.

Finally, we looked for the existence of a maximally symmetric vacuum solution for the gravitational field equations, so we assumed that our theory should satisfy the de Sitter condition Eq.(44). Then we found the explicit structure of the $F\left(R_{0}\right)$ function and using this function the action (19) was reduced to Eq.(53) which contains only the square of the constant scalar curvature. Note that the Schwarzschild-de Sitter black hole solution is also constant curvature solution. Such solutions will be discussed elsewhere.

Acknowledgements.

The author wish to thank Sergei D. Odintsov and Shin’ichi Nojiri for useful discussions and comments regarding the results presented in this work. The work of S.K. has been supported by the Scientific and Technological Research Council of Turkey (TUBİTAK) under the grant number 2219.