Modified EddingtoninspiredBornInfeld Gravity with a Trace Term
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Abstract
In this paper, a modified EddingtoninspiredBornInfeld (EiBI) theory with a pure trace term \(g_{\mu \nu }R\) being added to the determinantal action is analysed from a cosmological point of view. It corresponds to the most general action constructed from a rank two tensor that contains up to first order terms in curvature. This term can equally be seen as a conformal factor multiplying the metric \(g_{\mu \nu }\). This very interesting type of amendment has not been considered within the Palatini formalism despite the large amount of works on the BornInfeldinspired theory of gravity. This model can provide smooth bouncing solutions which were not allowed in the EiBI model for the same EiBI coupling. Most interestingly, for a radiation filled universe there are some regions of the parameter space that can naturally lead to a de Sitter inflationary stage without the need of any exotic matter field. Finally, in this model we discover a new type of cosmic “quasisudden” singularity, where the cosmic time derivative of the Hubble rate becomes very large but finite at a finite cosmic time.
Keywords
Hubble Parameter Cosmic Time Cosmological Solution Gravitational Action Hubble Rate1 Introduction
Undeniably, Einstein’s theory of general relativity (GR) has been an extremely successful theory for around a century [1]. However, the theory is expected to break down at some points at very high energies where quantum effects are expected to become crucial, such as in the past expansion of the Universe where GR predicts a big bang singularity [2]. This is one of the motivations for looking for possible modified theories of gravity, which are hoped to not only be able to preserve the huge achievements of GR, but also to shed some light on smoothing the singularities predicted in GR. Such theories could be seen as effective/phenomenological approaches of a more fundamental quantum theory of gravity.
In order to keep the theory free from aforementioned problems, alternative theories formulated within the Palatini formalism and teleparallel representation have been widely studied in Refs. [10, 11, 12, 13, 14, 15, 16, 17, 18]. For example, a theory constructed upon the Palatini approach, which is dubbed EddingtoninspiredBornInfeld theory (EiBI) (see Refs. [17, 18]), has recently attracted a lot of attention and has been studied from both astrophysical and cosmological points of view [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52]. The EiBI theory is shown to be able to cure the big bang singularity for a radiation dominated universe through a loitering effect^{1} or a bounce^{2} in the past, with the coupling constant \(\kappa \) being positive or negative, respectively. The ability of the theory to smooth other cosmological singularities in a phantom dominated universe was also studied in Refs. [21, 22]. Interestingly, in Refs. [39, 40, 41] the authors showed that the bouncing solutions for negative \(\kappa \) are robust against the changes of the Lagrangian through an additional f(R) term or some functional extensions (see as well Ref. [53] for another generalized gravitational theory related to massive gravity and Ref. [54] for the tensorial perturbations of a further generalized gravitational theory within the Palatini formalism.). However, it should be stressed that the amendments through the addition of a pure trace term to the determinantal action have never been considered so far. Besides, the EiBI theory with a negative coupling constant \(\kappa \) was also shown to suffer from instability problems due to the imaginary effective sound speed [34].
On the other hand, a recently proposed determinantal gravity formulated within the teleparallel representation was shown to be able to cure the big bang singularity in the past evolution of the Universe through a de Sitter inflationary phase [16]. Considering the widest generalization, the author added a pure trace term into the Lagrangian of the form of \(g_{\mu \nu }\varvec{T}\), where \(\varvec{T}\) is the Weitzenböck invariant [55]. In our previous work, we exhibited that this theory contains cosmological singularities for some parameters of the model, including the emergence of some cosmological singularities from purely geometrical effects (without the need of exotic matter) [49].
As far as we know, the gravitational actions with a pure trace term added to the BornInfeld determinantal structure have never been considered within the Palatini approach in the literature. Furthermore, we expect the emergence of interesting cosmological solutions with the addition of a pure trace term because it is expected in the teleparallel version. Based on these motivations, in this work we will consider a modified EiBI theory with a pure trace term added to the determinantal action, and analyse its cosmological implications. For simplicity, in this work we will assume a homogeneous and isotropic universe filled with a perfect fluid with a constant equation of state. Because the field equations are complicated, we will follow a method similar to that used in Ref. [39] to demonstrate the results graphically.

A big rip singularity takes place at a finite cosmic time with an infinite scale factor, where the Hubble parameter and its cosmic time derivative diverge [60, 61, 62, 63, 64, 65, 66, 67].

A sudden singularity takes place at a finite cosmic time with a finite scale factor, where the Hubble parameter remains finite but its cosmic time derivative diverges [56, 68, 69].

A big freeze singularity takes place at a finite cosmic time with a finite scale factor, where the Hubble parameter and its cosmic time derivative diverge [56, 70, 71, 72, 73].

A type IV singularity takes place at a finite cosmic time with a finite scale factor, where the Hubble parameter and its cosmic time derivative remain finite, but higher cosmic time derivatives of the Hubble parameter still diverge [56, 70, 72, 73, 74, 75].

A little rip event takes place at an infinite cosmic time with an infinite scale factor, where the Hubble rate and its cosmic time derivative diverge [56, 76, 77, 78, 79, 80, 81].

A little sibling of the big rip takes place at an infinite cosmic time with an infinite scale factor, where the Hubble rate diverges, but its cosmic time derivative remains finite [82, 83].
This table summarizes how the big bang singularity in GR is altered in the modified EiBI theory for a radiation dominated universe. If \(\kappa <0\), the big bang is substituted by a bounce except for the regions of the parameter space \(0<\beta \le 1/4\) where the big bang is still present. If \(\kappa >0\), the big bang singularity can be altered by a loitering effect, a bounce, what we named a quasisudden singularity, or a big freeze singularity in the past. However, for \(1/4\le \beta <1\), the big bang singularity exists. Furthermore, the big bang singularity may be followed by a de Sitter inflationary stage for \(\beta \lesssim 1\)
Our results are clearly shown in Table 1 where we compare them with the original EiBI model [17, 18]. As can be seen the model we are proposing can provide smooth bouncing solutions which were not allowed in the EiBI model for the same EiBI coupling (\(\kappa >0\)). Most interestingly, for a radiation filled universe there are some regions of the parameter space that can naturally lead to a de Sitter inflationary stage without the need of any exotic matter field. Finally, in this model we discover a new type of cosmic “quasisudden” singularity, where the cosmic time derivative of the Hubble rate becomes very large but finite at a finite cosmic time.
This paper is outlined as follows. In section II, we briefly introduce the basis of the modified EiBI theory with the addition of a pure trace term, including its action, field equations, and the low curvature limits of the theory. In section III, we assume a homogeneous and isotropic universe filled with a perfect fluid with a constant equation of state, then follow a similar approach to that used in Ref. [39] to derive a parametric Friedmann equation. In section IV, we exhibit the evolution of the Universe by graphically showing the Hubble rate as a function of the energy density under different assumptions of the parameters characterising the theory. To analyse the evolution of the Universe at the very early time, we then confine ourselves to a radiation dominated universe in our analysis of the modified EiBI theory. We finally present our conclusions in section V.
2 Proposed model: action and field equations
Actually, one can also add the so called zeroth order curvature term; i.e., \(\gamma g_{\mu \nu }\), to the determinant based on the structural completeness. However, This additional term can be rescaled by a conformal transformation \(g_{\mu \nu }\rightarrow (1+\gamma )g_{\mu \nu }\) and then can be absorbed into the cosmological constant term. In this sense this additional term is not expected to affect our results significantly, especially at the high energy regime in which the influence caused by high curvature terms is dominant. In fact, one can easily see from the gravitational action that the higher order curvatures term will dominate over the zeroth order term when curvature gets large. Therefore, we will omit this possible additional term in this work.
3 Modified EiBI gravity: a parametric Friedmann equation
We have now derived the expression of the energy density as a function of x in Eq. (3.9). If we can further express the Hubble rate as a function of x, the graphical relationship between the Hubble rate and the energy density can be completed.
4 The Hubble rate in an expanding universe
In the previous section, we derive the expressions of \(\rho \) [Eq. (3.9)] and \(H^2\) [Eq. (3.23)] as functions of a single variable x. Therefore, we can graphically obtain the representations of \(\kappa H^2\) as a function of \(\kappa \rho \) to exhibit the behaviors of the cosmological solutions of interest. From now on, we will assume a vanishing cosmological constant to simplify the analysis, that is, we assume \(\lambda =1\).
4.1 The original EiBI theory: \(\beta =0\)
As a first glance, we consider the original EiBI theory in which \(\beta =0\) and \(\alpha =1\). The representations of \(\kappa H^2\) as a function of \(\kappa \rho \) are shown in Fig. 1. One can see that the evolution of the energy density terminates at a bounce where \(H^2=0\) and \(\mathrm{d}H/\mathrm{d}\rho \ne 0\) at \(\kappa \rho =1\) for \(\kappa <0\). This bouncing solution is robust against the change of the equation of state w. However, if \(\kappa >0\) it can be seen that the behavior of the Hubble parameter is highly sensitive to the choice of w. There are loitering solutions where \(H^2\rightarrow 0\) and \(\mathrm{d}H/\mathrm{d}\rho \rightarrow 0\) at \(\kappa \rho =1/w\) for \(w>0\), and divergent solutions for \(w\le 0\). Furthermore, it can also be easily seen that the behaviors of the different curves focus around \(H^2=\rho /3\) when \(\rho \approx 0\). This property is not a surprise because of the prior criteria shown in Eq. (2.4), and it can be affirmed in all results shown in the rest of this paper; i.e. we recover GR at low energies. Note that the results summarized in this subsection are compatible with those concluded in the literatures [17, 18, 19].
4.2 Radiation dominated universe
In this subsection, we analyse if the original loitering behaviors and the bouncing solutions within the EiBI theory can be altered with the addition of a pure trace term \(g_{\mu \nu }R\) to the determinantal Lagrangian, i.e., \(\beta \ne 0\), for a radiation dominated universe. The analysis could be easily extended to other equation of state but for simplicity we stick to a radiation dominated universe.
4.2.1 \(\beta \gtrsim 0\)
We first consider the region in which \(\beta \) is slightly larger than zero. One should be reminded that in Refs. [39, 40, 41] the authors concluded that the bouncing solutions in the EiBI theory for negative \(\kappa \) are robust against the amendment to the EiBI action through an additional f(R) term or some functional extents. However, the situations are different in our model. One can see from Fig. 2 that the bouncing solutions for negative \(\kappa \) are quite sensitive to the increase of \(\beta \) from zero by even a small amount. More precisely, the asymptotic behavior of \(H^2\) at large \(\rho \) is \(H^2\propto \rho \). This implies the occurrence of a big bang singularity in the past.
4.2.2 \(0<\beta \le 1/4\)
Furthermore, we have also found that the absolute value of \(\mathrm{d}H^2/\mathrm{d}\rho \), which is proportional to \(\dot{H}\) in this model as the energy momentum tensor is conserved, is a growing function of \(\beta \). As \(\beta \) approaches \(\beta \approx \beta _{\star }=7/50\), \(\mathrm{d}H^2/\mathrm{d}\rho \) gets very large at a finite past cosmic time. Therefore, this singular event can be regarded as a quasisudden singularity in the past on the sense that while H is finite, \(\dot{H}\) almost blows up in a finite past cosmic time.^{4}
As a summary, we find that the original loitering effect for positive \(\kappa \) can be substituted by a point with a minimum scale factor \(a_m\), where a bounce (\(H^2=0\) and \(\mathrm{d}H^2/\mathrm{d}\rho \) remains finite), a past quasisudden singularity (\(H^2=0\) and \(\mathrm{d}H^2/\mathrm{d}\rho \) nearly diverges) or a past big freeze singularity (\(H^2\) and \(\mathrm{d}H^2/\mathrm{d}\rho \) diverge) may emerge.
4.2.3 \(\beta <0\)
In Fig. 4, we show the representations of \(\kappa H^2\) as a function of \(\kappa \rho \) for \(\beta <0\). We find that, unlike what we concluded previously, the loitering effects (\(H^2\propto {\delta \rho }^2\)) and the bouncing solutions (\(H^2\propto {\delta \rho }\)) are robust against the decrease of \(\beta \) below zero. Furthermore, we also find that the smaller the value of \(\beta \), the smaller the value of \(\kappa \rho \) at the loitering event or the bounce.
4.2.4 \(\beta >1/4\)
Interestingly, we also find from the dashed blue and solid green curves in Fig. 6 that for larger values of \(\beta \), there could be a plateau in the \(H^2\) function for positive \(\kappa \), for a radiation dominated universe. This stage may correspond to a de Sitter inflationary expansion phase after the big bang singularity. This inflationary phase is then followed by a classical expansion described well in the context of GR. Furthermore, when \(\beta \ge 1\), the solutions with a loitering effect are again recovered (see the dashed red and dashed green curves in Fig. 6).
Before concluding, we notice that because this theory reduces to GR at the low energy limit, all the radiation dominated universe will be asymptotically flat at that limit.
5 conclusions
Since the proposal of the BornInfeld action for classical electrodynamics, modified theories of gravity inspired on such a proposal and with an elegant determinantal structure in their actions have been widely investigated (see Refs. [5, 6, 7, 8, 9, 10, 11]). Despite the large amount of works in this subject, the very interesting generalization through the addition of a pure trace term into the gravitational Lagrangian in the Palatini formalism has not been considered before. This modification gives rise to the most general action constructed from a rank two tensor that contains up to first order terms in the curvature. Such a theory is expected to not only preserve the great achievement of GR at low energies, but also to generate more drastic deviations from GR than those accomplished within the original BornInfeldinspired theories at high energies. Modified theories with this term have only been investigated in the pure metric formalism [5] and in the teleparallel representation [16]. The former inevitably suffers from troublesome fourth order field equations for the metric or from ghost instabilities [4], which suggests the need of some alternative approach to overcome these problems. The latter, which flees from the ghosts and results in second order field equations, leads in most of the cases to the substitution of the big bang by smoother cosmological singularities [49] or a de Sitter inflationary stage [16].
Inspired by these motivations, in this paper we generalize the EiBI theory, which is formulated within the Palatini formalism, by adding a pure trace term into the determinantal Lagrangian, and analyze the cosmological solutions of this theory by assuming a homogeneous and isotropic universe for its largest scale. As we expect, the early cosmological expansion to be modified as compared with GR or EiBI theory, we assume that the Universe is filled with radiation. Following a similar approach to that proposed in Ref. [39], the behaviors of the cosmological solutions are analyzed using a parametric Friedmann equation.
As a summary, we find that if \(\kappa <0\), the big bang is substituted by a bounce except for the regions of the parameter space \(0<\beta \le 1/4\) where the big bang singularity exists. Note that in Refs. [39, 40, 41] the authors showed that the bouncing solutions in the EiBI theory are robust against the changes of the Lagrangian through an additional f(R) term or some functional extensions. On the other hand, if \(\kappa >0\), we find that the big bang singularity can be altered by a loitering effect (\(\beta \le 0\) or \(\beta \ge 1\)), a bounce (\(0<\beta <\beta _{\star }\)), what we named a quasisudden singularity (\(\beta =\beta _{\star }\)), or a big freeze singularity in the past (\(\beta _{\star }<\beta <1/4\)). However, for \(1/4\le \beta <1\), the big bang singularity remains. Most interestingly, the big bang singularity may be followed by a de Sitter inflationary stage for \(\beta \lesssim 1\). This can be verified by the plateau in the \(H^2\) function as shown in Fig. 6. The inflationary phase is superseded by a standard cosmological expansion. We summarizes our results in Table 1. Moreover, we should emphasize that the cosmological solutions that emerge in this theory are all stemmed from pure geometrical effects. Only a radiation dominated universe is assumed and there is no need of any additional fields or exotic matters to drive these cosmological solutions.
Footnotes
 1.
 2.
A bouncing universe is a universe with a minimum or a maximum scale factor such that after a contracting phase, an expanding phase happens or the other way around. In this kind of model, the big bang is substituted by a bounce.
 3.
 4.
We gave in the introduction the definition of the sudden singularity and the other cosmological singularities related to dark energy. In our case this singular event happens in the finite past of the Universe.
 5.
Please see Sect. 1 for the definition of a big freeze singularity and the classification of the other cosmological singularities related to dark energy. In our case this singular event happens at a finite past of the Universe.
Notes
Acknowledgments
M.B.L. is supported by the Portuguese Agency “Fundação para a Ciência e Tecnologia” through an Investigador FCT Research contract, with reference IF/01442/2013/CP1196/CT0001. She also wishes to acknowledge the hospitality of LeCosPA Center at the National Taiwan University during the completion of part of this work and the support from the Portuguese Grants PTDC/FIS/111032/2009 and UID/MAT/00212/2013 and the partial support from the Basque government Grant No. IT59213 (Spain). C.Y.C. and P.C. are supported by Taiwan National Science Council under Project No. NSC 972112M002026MY3 and by Taiwans National Center for Theoretical Sciences (NCTS). P.C. is in addition supported by US Department of Energy under Contract No. DEAC0376SF00515.
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