# Is Eddington–Born–Infeld theory really free of cosmological singularities?

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## Abstract

The Eddington-inspired-Born–Infeld (EiBI) theory has recently been resurrected. Such a theory is characterized by being equivalent to Einstein theory in vacuum but differing from it in the presence of matter. One of the virtues of the theory is that it avoids the Big Bang singularity for a radiation-filled universe. In this paper, we analyze singularity avoidance in this kind of model. More precisely, we analyze the behavior of a homogeneous and isotropic universe filled with phantom energy in addition to the dark and baryonic matter. Unlike the Big Bang singularity that can be avoided in this kind of model through a bounce or a loitering effect on the physical metric, we find that the Big Rip singularity is unavoidable in the EiBI phantom model even though it can be postponed towards a slightly further future cosmic time as compared with the same singularity in other models based on the standard general relativity and with the same matter content as described above.

### Keywords

Dark Matter Dark Energy Scalar Curvature Hubble Parameter Cosmic TimeThe Einstein theory of general relativity (GR) has been an extremely successful theory for nearly a century [1]. Despite all its advantages, it is expected to break down at some point at very high energies, for example in the past evolution of the Universe where the theory predicts a Big Bang singularity [2] and the laws of physics cease to be valid. This is one of the motivations for looking for a possible extension of GR. In addition, it is hoped that modified theories of general relativity, while preserving the great achievements of GR, would shed some light on the unknown fundamental nature of dark energy or whatever stuff it is that drives the present accelerating expansion of the Universe (see for example Refs. [3, 4, 5, 6, 7]).

There have been many proposals for alternative theories of GR, these being almost as old as the theory itself. One of the oldest was proposed by Eddington [8], where the connection rather than the metric plays the role of the fundamental field of the theory. The gravitational action proposed by Eddington back in 1924 [8] is equivalent to Einstein theory of GR in vacuum. One of the weak points of the theory is that it does not incorporate matter. Recently, an Eddington-inspired-Born–Infeld theory (EiBI) has been proposed in Ref. [9] where matter fields are incorporated into the Lagrangian formulation. More importantly, it turns out that this theory avoids the Big Bang singularity that would face a radiation-dominated universe in standard GR [9]. The *apparent* fulfillment of the energy conditions in EiBI theory was considered in Ref. [10], where the adjective *apparent* refers to quantities defined with respect to a metric compatible with the connection that defines the theory. Their analysis leads to a sufficient condition for singularity avoidance. Besides, the gravitational collapse of noninteracting particles, i.e., dust or equivalently pressureless matter, does not lead to singular states in the nonrelativistic limit (Newtonian regime) [11] (see also [12]). This theory has also been studied as an alternative scenario to the inflationary paradigm [13]. Furthermore, possible constraints on the parameter characterizing the theory have been obtained using solar models [14], neutron stars [15], and nuclear physics [16]. It has also been shown that such an avoidance of the Big Bang singularity is more general and not limited to a radiation-dominated universe [17]. Despite all the virtues of the EiBI theory, a cosmological tensor instability in this model was found in Ref. [18]. In addition, this theory behaves similarly to Palatini \(f(R)\) gravity and shares the same pathologies, such as curvature singularities at the surface of polytropic stars and some unacceptable phenomenology [19].

In this letter, we ask the simple questions: Is EiBI theory really free from cosmological singularities? In particular, is the theory free from dark energy related singularities? In Ref. [10], it was shown that if the null energy condition is fulfilled, then the *apparent* null energy condition is satisfied. It turns out that the null energy conditions are not always fulfilled; a clear example of it is a super-inflationary phase within GR. Moreover, in recent years a new singularity named the Big Rip has been identified where the null energy condition is in fact not fulfilled and the Universe is ripped apart: the Hubble rate and its cosmic derivative approach infinity in a finite cosmic time [20, 21]. Might such a singularity be avoidable in the theory proposed in Ref. [9]? This question is even more pertinent in the aftermath of the release of WMAP9 data, which hints to the possibility of a phantom energy component in the Universe more pronouncedly than that deduced from the WMAP7 data [22]. The analysis of the possible occurrence of a Big Rip in the future of the Universe is therefore timely.

*not*the Levi-Civita connection of the metric \(g_{\mu \nu }\). The parameter \(\kappa \) is a constant with inverse dimensions to that of the cosmological constant (in this letter, we will work with Planck units \(8\pi {G}=1\) and set the speed of light to \(c=1\)), \(\lambda \) is a dimensionless constant and \({\mathcal {S}}_\mathrm{m} (g,\Gamma ,\Psi )\) stands for the matter Lagrangian. This Lagrangian has two well-defined limits: (i) when \(|\kappa R|\) is very large, we recover Eddington’s theory and (ii) when \(|\kappa R|\) is small, we obtain the Hilbert–Einstein action with an effective cosmological constant \(\Lambda =(\lambda -1)/\kappa \) [9]. A solution of the action in Eq. (1) can be characterized by two different Ricci tensors: \(R_{\mu \nu }(\Gamma )\) as presented in Eq. (1) and \(R_{\mu \nu }(g)\) constructed from the metric \(g\). There are in addition three ways of defining the scalar curvature. These are \(g^{\mu \nu }R_{\mu \nu }(g)\), \(g^{\mu \nu }R_{\mu \nu }(\Gamma )\), and \(R(\Gamma )\). The third one is derived from the contraction between \(R_{\mu \nu }(\Gamma )\) and the metric compatible with the connection \(\Gamma \). Therefore whenever one refers to singularity avoidance, one must specify the scalar curvature(s).

The behavior of different possible Ricci tensors and scalar curvatures in the EiBI model at the Big Loitering in a radiation-dominated universe and at the Big Rip singularity in a phantom-dominated universe

Big loitering | Big rip | |
---|---|---|

\(R_{00}(g)\) | \(0\) | \(-\infty \) |

\(R_{ij}(g)\) | \(0\) | \(+\infty \) |

\(g^{\mu \nu }R_{\mu \nu }(g)\) | \(0\) | \(+\infty \) |

\(R_{00}(\Gamma )\) | \(1/\kappa \) | \(-\infty \) |

\(R_{ij}(\Gamma )\) | \(-a^2\delta _{ij}/\kappa \) | \(+\infty \) |

\(g^{\mu \nu }R_{\mu \nu }(\Gamma )\) | \(-4/\kappa \) | \(+\infty \) |

\(R(\Gamma )\) | \(-\infty \) | \(4/\kappa \) |

^{1}:

At small \(\Omega _\kappa , f(\Omega _\kappa , \Omega _m, \Omega _w, w)\) can be written as \(f(\Omega _\kappa ,\) \( \Omega _m, \Omega _w, w)\approx (\Omega _m+\Omega _w)\Omega _\kappa +O({\Omega _\kappa }^2)\), which confirms again that EiBI reduces to GR for vanishing \(\Omega _\kappa \).

In summary, under the above conditions we can conclude the following: (i) \(\Omega _\kappa =0\) whenever \(\Omega _m+\Omega _w\le 1\), where we recover GR. (ii) For \(\Omega _m+\Omega _w>1\), we may consider the second solution in Eq. (10), which is positive in this case. In the latter case, we will assume that \(\Omega _\kappa \) is small, i.e., the deviation of EiBI theory from GR is small, so that \(\Omega _m+\Omega _w\gtrsim 1\) is in agreement with the observational data [22]. (iii) One can always find a suitable value for \(\Omega _\kappa \), or \(\kappa \), to fit a specific set of parameters \(\Omega _m, \Omega _w\), and \(w\).

We now investigate the asymptotic behavior of the Universe within this framework. This amounts to determining the Hubble parameter, \(H\), and its cosmic time derivative, \(\dot{H}\), at large scale factors. From here on we set \(w=-1-\epsilon \), where \(\epsilon \) is positive. As the dark energy corresponds to the phantom matter in our setup and the Universe is expanding, the conservation of such an energy density implies a growth of \(\rho _w\) (see Eq. (2)), unlike the baryonic and dark matter, which would quickly become negligible as compared with \(\rho _w\). We therefore neglect \(\rho _m\) in our estimation of \(H\) and \(\dot{H}\).

We can also prove that a phantom energy-dominated EiBI universe has a well-defined \(H^2\) for any value of \(\rho _w\). In fact, the square of the Hubble parameter in Eq. (6), for \(\bar{\rho }=\kappa \rho _w\), is positive-definite and it vanishes only when \(\rho _w=0\).

The cosmic time elapsed from the present time to the Big Rip singularity time, normalized to the current Hubble parameter; i.e., \(H_0(t_\mathrm{sing}-t_0)\), for different values of \(\epsilon \) in GR and in the EiBI theory. We see that such a cosmic time remains finite in the EiBI theory, meaning that the Big Rip singularity is inevitable. Here we assume \(\Omega _m=0.287\) and \(\Omega _w=0.733\), then we use the constraint Eq. (8) to find the corresponding \(\Omega _\kappa \)

\(\epsilon \) | \(H_0(t_\mathrm{{sing}}-t_0)\) (GR) | \(H_0(t_\mathrm{sing}-t_0)\) (EiBI) |
---|---|---|

0.021 | 37.0149 | 37.1153 |

0.041 | 18.9291 | 19.0294 |

0.061 | 12.7039 | 12.8041 |

0.081 | 9.55371 | 9.65379 |

0.101 | 7.65167 | 7.75166 |

0.121 | 6.37884 | 6.47876 |

0.147 | 5.24246 | 5.34228 |

Our results indicate that the scalar curvature constructed from the physical metric \(g_{\mu \nu }\) will blow up at the Big Rip. It can be shown that \(R_{\mu \nu }(\Gamma )\) and \(g^{\mu \nu }R_{\mu \nu }(\Gamma )\) also blow up at \(t_\mathrm{sing}\) where \(a\) diverges, whereas \(R(\Gamma )\) remains finite. Specifically, \(R_{00}(\Gamma )=(1-U)/\kappa \rightarrow -\infty \) and \(R_{ij}(\Gamma )=[a^2(V-1)\delta _{ij}]/\kappa \rightarrow \infty \), while \(R(\Gamma )=(U-1)/U\kappa +3(V-1)/V\kappa \rightarrow 4/\kappa \), as \(a\rightarrow \infty \).

An interesting model for a modified theory of gravity was suggested in Ref. [9]. It was shown that in this model the Big Bang singularity for a radiation-filled universe can be removed [9], but the scalar curvature constructed from the metric compatible with \(\Gamma \) still blows up as we have shown. On the other hand, it is well known that for the class of dark energy models with \(w< -1\), i.e., the phantom models, the Big Rip singularity is inevitable for a constant \(w\) in the framework of GR. Our main objective of this paper is to see if the Bañados–Ferreira EiBI model can help also to remove the Big Rip singularity. We tackled this issue by investigating the possible occurrence or avoidance of doomsdays in this model. We analyzed an EiBI FLRW universe filled with dark matter and phantom energy with a constant equation of state. It is well known that a universe with such a matter-energy content under GR would face a Big Rip. Our result indicates that the Big Rip singularity remains inevitable in the EiBI theory albeit leading to a minor postponement, as shown in Table 2. The onset of the Big Rip is independent of the amount of dark matter or dark energy; i.e. \(\Omega _m\) and \(\Omega _w\). In fact, the scale factor, the Hubble parameter and its cosmic time derivative all blow up in a finite cosmic time. Consequently, the scalar curvature constructed from the physical metric \(g_{\mu \nu }\) will also blow up. We have shown as well that \(R_{\mu \nu }(\Gamma )\) given in the action in Eq. (1) and \(g^{\mu \nu }R_{\mu \nu }(\Gamma )\) are infinite at the singularity, whereas \(R(\Gamma )\) remains finite. The key message to take home from this letter is that a Big Rip singularity cannot be avoided in the EiBI model but it is smoother than that in GR. This is unlike the Big Loitering^{2} in a radiation-dominant EiBI universe, which is rougher than that in GR, as shown in Table 1.

We will present elsewhere the behavior of other dark energy related singularities/events [23, 24, 25] such as big freeze, sudden singularity, type-IV singularity, little rip, etc., in the EiBI framework [26].

## Footnotes

- 1.
The parameters \(\Omega _m\) and \(\Omega _w\) are defined in the standard way, i.e., \(\Omega _m\equiv \rho _{m_0}/\rho _c\) and \(\Omega _w\equiv \rho _{w_0}/\rho _c\), where \(\rho _c\) is the critical density. In addition, \(\Omega _\kappa \equiv \kappa \rho _c=3{H_0}^2\kappa \), where \(H_0\) is the current Hubble parameter.

- 2.
We refer to this loitering effect as Big Loitering because it takes an infinite cosmic time to occur.

## Notes

### Acknowledgments

M.B.L. is supported by the Basque Foundation for Science IKERBASQUE. She also wishes to acknowledge the hospitality of LeCosPA Center at NTU (Taiwan) during the completion of part of this work and the support of the Portuguese Agency FCT through project No. PTDC/FIS/111032/2009. C.Y.C. and P.C. are supported by Taiwan National Science Council under Project No. NSC 97-2112-M-002-026-MY3 and by Taiwan’s National Center for Theoretical Sciences (NCTS). P.C. is in addition supported by US Department of Energy under Contract No. DE-AC03-76SF00515. This work has been supported by a Spanish-Taiwanese Interchange Program with reference 2011TW0010 (Spain) and NSC 101-2923-M-002-006-MY3 (Taiwan). This work was supported partially by the Basque government Grant No. IT592-13.

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