Eddington–Born–Infeld cosmology: a cosmographic approach, a tale of doomsdays and the fate of bound structures
Abstract
The EddingtoninspiredBorn–Infeld scenario (EiBI) can prevent the big bang singularity for a matter content whose equation of state is constant and positive. In a recent paper [BouhmadiLopez et al. (Eur. Phys. J. C 74:2802, 2014)] we showed that, on the contrary, it is impossible to smooth a big rip in the EiBI setup. In fact the situations are still different for other singularities. In this paper we show that a big freeze singularity in GR can in some cases be smoothed to a sudden or a type IV singularity under the EiBI scenario. Similarly, a sudden or a type IV singularity in GR can be replaced in some regions of the parameter space by a type IV singularity or a loitering behaviour, respectively, in the EiBI framework. Furthermore, we find that the auxiliary metric related to the physical connection usually has a smoother behaviour than that based on the physical metric. In addition, we show that bound structures close to a big rip or a little rip will be destroyed before the advent of the singularity and will remain bound close to a sudden, big freeze or type IV singularity. We then constrain the model following a cosmographic approach, which is well known to be model independent, for a given Friedmann–Lemaître–Robertson–Walker geometry. It turns out that among the various past or present singularities, the cosmographic analysis can pick up the physical region that determines the occurrence of a type IV singularity or a loitering effect in the past. Moreover, to determine which of the future singularities or doomsdays is more probable, observational constraints on the higherorder cosmographic parameters are required.
1 Introduction
With no doubt general relativity (GR) is an extremely successful theory about to become centenary [2]. Nevertheless, it is expected to break down at some point at very high energies where quantum effects can become important, for example in the past evolution of the universe where GR predicts a big bang singularity [3]. This is one of the motivations for looking for possible extension of GR. Moreover, it is hoped that modified theories of GR, while preserving the great achievements of GR, would shed some light over the unknown fundamental nature of dark energy or whatsoever stuff that drives the present accelerating expansion of the universe (see Refs. [4, 5, 6, 7, 8, 9] and references therein), said in other words: What is the hand that started recently to rock the cradle?
Indeed, several observations, ranging from type Ia Supernovae (SNeIa) [10, 11, 12] (which brought the first evidence) to the cosmic microwave background (CMB) [13, 14], the baryon acoustic oscillations (BAO) [15, 16, 17], gamma ray bursts (GRB) [18] and measures of the Hubble parameter at different redshifts [19] among others, showed that the universe has entered in the recent past a state of acceleration if homogeneity and isotropy is assumed on its largest scale. Actually, observations show that such an accelerating state is fuelled by an effective matter whose equation of state is pretty much similar to that of a cosmological constant but which could as well deviate from it by leaving room for quintessence and phantom behaviours, the latter being known to induce future singularities (see Ref. [20] and references therein). Therefore, it is of interest to formulate consistent modified theories of gravity that could appease the cosmological singularities and could shed some light over the latetime acceleration of the universe. Of course, an alternative way to deal with dark energy singularities is to invoke a quantum approach as done in Ref. [21].
A very interesting theory at this regard has been reformulated recently: the EddingtoninspiredBorn–Infeld theory (EiBI) [22, 23, 24], as its name indicates, is based on the gravitational theory proposed by Eddington [25] with an action similar to that of the nonlinear electrodynamics of Born and Infeld [26]. Such an EiBI theory is formulated in the Palatini approach, i.e., the connection that appears in the action is not the LeviCivita connection of the metric in the theory. For a metric approach to the EiBI theory see Ref. [27]. Like Eddington theory [25], EiBI theory is equivalent to GR in vacuum, however, it differs from it in the presence of matter. Indeed while GR cannot avoid the big bang singularity for a universe filled with matter with a constant and a positive equation of state (with flat and hyperbolic spatial section), the EiBI setup does as shown in [24, 28]. The EiBI scenario was as well proposed as an alternative to the inflationary paradigm [29] through a bounce induced by an evolving equation of state fed by a massive scalar field. This model comes with the bonus of overcoming the tensor instability previously found in the EiBI model in Ref. [30] (see also [31] for an analysis of the scalar and vectorial perturbations for a radiation dominated universe and the studies of the large scale structure formation in Ref. [32]). Black hole solutions with charged particles and the strong gravitational lensing within the EiBI theory are studied in Ref. [33]. Besides, the fulfilment of the energy conditions in the EiBI theory was studied in Ref. [34] and a sufficient condition for singularity avoidance under the fulfilment of the null energy condition was obtained. Additionally, it was shown that the gravitational collapse of noninteracting particles does not lead to singular states in the Newtonian limit [35]. Furthermore, the parameter characterising the theory has been constrained using solar models [36], neutron stars [37] and nuclear physics [38]. Very recently, neutron stars and wormhole solutions in the EiBI theory were analysed in [39, 40, 41]. Especially in [41], the authors showed that the universal relations of the fmode oscillation [42], which is the fundamental mode of pulsation in the neutron stars, and the ILoveQ relations [43], which refers to the relation among the moment of inertia, tidal Love numbers (which are parameters measuring the rigidity of a planetary body and the susceptibility of its shape to change in response to a tidal potential) and the quadrupole moment of the neutron stars, found in GR are also valid in the EiBI theory. A theory which combines the EiBI action and the f(R) action is also analysed in Refs. [44, 45] (see also Ref. [46]). A drawback of this theory is that it shares some pathologies with Palatini f(R) gravity such as curvature singularities at the surface of polytropic stars [47] (see also [48]).
We showed recently that despite the big bang avoidance in the EiBI setup, the big rip [49, 50, 51, 52, 53, 54, 55, 56] is unavoidable in the EiBI phantom model [1]. In this paper, we will assume an EiBI model and we will carry a thorough analysis of the possible avoidance of the other dark energy related singularities, known as: (i) sudden, type II, big brake or big démarrage singularity [57, 58, 59, 60], (ii) type III or big breeze singularity [59, 60, 61, 62, 63] and (iii) type IV singularity [59, 60, 64, 65].
Those singularities can show up in GR when a Friedmann–Lemaître–Robertson–Walker (FLRW) universe is filled with a Generalised Chaplygin gas (GCG) [60] (more precisely, a phantom Generalised Chaplygin gas, or pGCG for short) which has a rather chameleonic behaviour despite its simple equation of state [60, 61]. Indeed, the GCG can unify the role of dark matter and dark energy [66, 67] (for a recent update of the subject see Ref. [68]), avoid the big rip singularity [69], describe some primitive epoch of the universe [70] and alleviate the observed low quadruple of the CMB [71]. We will complete our analyses by considering as well the possible avoidance of the little rip event [72] in the above mentioned setup.
In the EiBI theory, there are two metrics, the first one \(g_{\mu \nu }\) appears in the action and couples to matter, the second one is the auxiliary one which is compatible with the connection \(\Gamma \) [24]. The two metrics reduce to the original one in GR when the curvature term is small. Therefore, we will analyse the singularity avoidance with respect to both metrics. Furthermore, we will use the geodesic equations compatible with both metrics to study the behaviour of the physical radius of a Newtonian bounded system near the singularities. For an exhaustive analysis of the geodesics close to the dark energy related singularities in GR see Refs. [73, 74]. As a result, we find that the asymptotic behaviour of \(g_{\mu \nu }\), more precisely the Hubble parameter and its cosmic time derivatives as defined from the metric \(g_{\mu \nu }\), near the singularities is consistent with that of the geodesic behaviour dictated by the same metric \(g_{\mu \nu }\). However, the events corresponding to the singularities with respect to \(g_{\mu \nu }\) are usually well behaved as observed by the connection, and therefore the auxiliary metric, and so do the geodesic equations defined from the physical connection. In addition, we show that bound structures close to a big rip or little rip will be destroyed before the advent of the singularity and will remain bound close to a sudden, big freeze or type IV singularity. This result is independent of the choice of the physical or auxiliary metric.
We will further complete our analyses by getting some observational constraints on the model through the use of a cosmographic approach [75, 76, 77, 78]. This analysis will show that the EiBI model when filled with the matter content mentioned in the previous paragraph on top of the dark and baryonic matter is compatible with the current acceleration of the universe. The cosmographic approach relies on putting constraints on some parameters which quantify the time derivatives of the scale factor and which are called the cosmographic parameters [75, 76, 77, 78]. These parameters depend exclusively on the spacetime geometry, in this case on the geometry of a homogeneous and isotropic spacetime, and not on the gravitational action or the equations of motion that describe the model (see Ref. [76] for a nice review of the subject). Hence, this approach is quite useful because given a set of constraints on the cosmographic parameters [75, 78], it can be applied to a large amount of models in particular to those with relatively messy Friedmann equations like the one we need to deal with [29]. The drawback of this approach is that with the current observational data the errors can be quite large [75, 78, 79, 80, 81, 82, 83]. Nevertheless, we think it is a fair enough approach for the analysis we want to carry out. Essentially, we will show that among the various birth events or past singularities predicted by the theory, the cosmographic analyses pick up the physical region which determines the occurrence of a type IV singularity (or a loitering effect) in the past, which is the most unharmful of all the types of dark energy singularities. Among the various possible future singularities or doomsdays predicted, the use of observational constraints on higherorder cosmographic parameters is necessary to predict which future singularity is more probable.
The paper is outlined as follows. In Sect. 2, we briefly review the idea of the EiBI theory and present a thorough analysis of the avoidance of various singularities in this theory, through deriving the asymptotic behaviours of the Hubble parameter and its cosmic time derivatives near the singularities for both metrics (physical and auxiliary). In Sect. 3, we analyse the effects of the cosmological expansion on local bound systems in the EiBI scenario by analysing the geodesics of test particles for both metrics close to a massive body. In Sect. 4, we use a cosmographic approach to constrain the model and calculate the cosmic time elapsed since now to the possible, past or future, singularities. The conclusions and discussions are presented in Sect. 5.
2 The EiBI model and dark energy related singularities
We will next analyse the possible avoidance of dark energy singularities in the EiBI setup. Those singularities, as we will next review, are characterised by a possible divergence of the Hubble parameter and its cosmic time derivatives at some finite cosmic time. This translates into possible divergences of the scalar curvature and its cosmic time derivatives. The EiBI model we are considering is formulated within the Palatini formalism and therefore there are two ways of defining the Ricci curvature: (i) \(R_{\mu \nu }(\Gamma )\) as presented in the action (2.1) and (ii) \(R_{\mu \nu }(g)\) constructed from the metric \(g_{\mu \nu }\).
There are in addition four ways of defining the scalar curvature: \(g^{\mu \nu }R_{\mu \nu }(\Gamma )\), \(g^{\mu \nu }R_{\mu \nu }(g)\), \(q^{\mu \nu }R_{\mu \nu }(\Gamma )\) and \(q^{\mu \nu }R_{\mu \nu }(g)\). Therefore whenever one refers to singularity avoidance, one must specify the specific curvature one is referring to. For the dark energy singularities the important issue is the behaviour of the Hubble parameter and its cosmic time derivatives and in this case we have two possible quantities for the Hubble parameter: \(H\) related to the physical metric and \(H_q\) related to the physical connection as defined in Eq. (2.8).

The big rip singularity happens at a finite cosmic time with an infinite scale factor where the Hubble parameter and its cosmic time derivative diverge [49, 50, 51, 52, 53, 54, 55, 56].

The 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 [57, 58, 59].

The big freeze singularity happens at a finite cosmic time with a finite scale factor where the Hubble parameter and its cosmic time derivative diverge [59, 60, 61, 62, 63].

Finally the type IV singularity occurs 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 [59, 61, 62, 63, 64, 65].
2.1 The EiBI scenario and the big rip
2.1.1 The physical metric \(g_{\mu \nu }\)
2.1.2 The auxiliary metric \(q_{\mu \nu }\)
2.2 The EiBI scenario and the sudden singularity
2.2.1 The physical metric \(g_{\mu \nu }\)
We seek now the possibility of smoothing the sudden singularity that can appear in GR. We consider a pGCG fulfilling the equation of state (2.13) with \(\alpha >0\) [60]. Note that in GR a universe filled with this fluid hits a past sudden singularity. The presence of matter or radiation cannot remove the occurrence of this cosmic birth in the past of the universe. After integrating the conservation equation (2.4), one can derive the energy density of this kind of pGCG [60] which is shown in Eqs. (2.14) and (2.15).
If, however, \(0<\alpha <2\) and \(\alpha \ne 4/(3n+2)\), we find that as long as \(\alpha \) satisfies \(4/(3p+2)<\alpha <4/(3p1)\) with \(p\) being a positive integer, the \((p+1)\)th derivative of \(\bar{H}\) blows up while the \(1,\ldots ,p\)th derivatives are all finite. This indicates a type IV singularity again.

If \(\alpha >2\), the universe expands from a finite past sudden singularity.

If \(0<\alpha \le 2\), the universe expands from a finite past type IV singularity.
2.2.2 The auxiliary metric \(q_{\mu \nu }\)
2.3 The EiBI scenario and the big freeze
2.3.1 The physical metric \(g_{\mu \nu }\)
We seek now the possibility of smoothing the big freeze singularity that can appear in GR. We consider a pGCG fulfilling the equation of state (2.13) with \(\alpha <1\) [60]. Note that in GR a universe filled with this fluid hits a future big freeze singularity. The presence of matter or radiation cannot remove the occurrence of this cosmic doomsday in the future of the universe. After integrating the conservation equation (2.4), one can derive the energy density of this kind of pGCG [60] which is shown in Eqs. (2.16) and (2.17).

If \(\alpha <3\), the universe will end up into a finite future sudden singularity.

If \(3<\alpha <1\), the universe will end up into a finite future big freeze singularity.

If \(\alpha =3\), the universe will end up into a finite future type IV singularity.
Additionally, there is also a finite future big freeze singularity in GR, which is driven by a very special pGCG whose energy density and pressure are shown in Eq. (2.18), where \(A<0\) and \(1+\alpha =1/(2m)\) with \(m\) being a negative integer [60]. The asymptotic behaviour of \({\bar{H}}^2\) and \(d\bar{H}/d\bar{t}\) in this case are also given by Eq. (2.32). One can easily see that \(3<\alpha <1\), thus the big freeze singularity cannot be avoided in this case.
Actually, the big freeze singularity can also occur in the finite past if the universe is filled with a GCG which fulfils the strong, null and weak energy conditions [60]. However, the universe will not get into an accelerating expansion phase at the present time as implied by astrophysical and cosmological observations, so that the theory will only be worth to analyse from a mathematical point of view. See Ref. [84] for more details of this issue.
2.3.2 The auxiliary metric \(q_{\mu \nu }\)
2.4 The EiBI scenario and the type IV singularity
2.4.1 The physical metric \(g_{\mu \nu }\)
To analyse the possibility of smosothing a type IV singularity within the EiBI theory, we consider the same kind of dark energy pGCG as shown in Eqs. (2.13) and (2.14), with \(1<\alpha <0\). Indeed, this fluid drives a past type IV singularity in GR except for some quantised cases, i.e., if \(\alpha =n/(n+1)\) with \(n\) being natural numbers, the Hubble rate and all of its cosmic time derivatives are all regular in the finite past. Note that this result is different from the one proposed in Ref. [60] because in that case the authors assumed a purely pGCG dominated universe for the analysis, which is not the case in this paper [see Eq. (2.12)]. See Ref. [84] for more details of this issue.
According to Eq. (2.40), one can show that if \(1/2<\alpha <1/3\), the firstorder cosmic time derivative of \(\bar{H}\) goes to infinity and \(\bar{H}\) is finite, implying a finite past sudden singularity.
If, however, \(1/3\le \alpha <0\) and \(\alpha \ne 1/(n+2)\), from Eq. (2.40) and the conservation equation we find that as long as \(\alpha \) satisfies \(1/(p+2)<\alpha <1/(p+3)\) with \(p\) being a positive integer, the \((p+1)\)th derivative of \(\bar{H}\) blows up while the \(1,\ldots ,p\)th derivatives are all finite. This indicates a type IV singularity.
On the other hand, if \(1<\alpha \le 1/2\) and \(\alpha \) cannot be written as \(n/(n+1)\) with \(n\) being a natural number, we find that as long as \(\alpha \) satisfies \((p+1)/(p+2)<\alpha <p/(p+1)\), the \(p\)th derivative of \(\bar{H}\) blows up while the \(1,\ldots ,(p1)\)th derivatives are all finite. This implies that a finite past type IV singularity except for a finite past sudden singularity in which the firstorder cosmic time derivative of the Hubble rate diverges if \(2/3<\alpha <1/2\).

If \(1/2<\alpha <1/3\) or \(2/3<\alpha <1/2\), the universe expands from a finite past sudden singularity.

If \(1/3<\alpha <0\) and \(\alpha \ne 1/(n+2)\), or \(1<\alpha <2/3\) and \(\alpha \ne n/(n+1)\), with \(n\) being a positive integer, the universe expands from a finite past type IV singularity.

If \(\alpha =1/(n+2)\) or \(\alpha =n/(n+1)\), there is no singularity and the universe is born at a finite past.
2.4.2 The auxiliary metric \(q_{\mu \nu }\)
If \(1/2<\alpha <0\) and \(\alpha \) cannot be written as \(1/(n+2)\) where \(n\) is a positive integer, we find that as long as \(\alpha \) satisfies \(1/(p+1)<\alpha <1/(p+2)\), the \((p+1)\)th derivative of \(\bar{H_q}\) blows up while the \(1,\ldots ,p\)th derivatives are all finite. This indicates a type IV singularity of the auxiliary metric.
On the other hand, if \(1<\alpha \le 1/2\) and \(\alpha \) cannot be written as \(n/(n+1)\) with \(n\) being a natural number, we find that as long as \(\alpha \) satisfies \((p+1)/(p+2)<\alpha <p/(p+1)\), the \((p+1)\)th derivative of \(\bar{H_q}\) blows up while the \(1,\ldots ,p\)th derivatives are all finite. This also implies that a past type IV singularity of the auxiliary metric.

If \(\alpha \) cannot be written as \(1/(n+2)\) or \(n/(n+1)\) with \(n\) being a positive integer, the universe expands from a type IV singularity of the auxiliary metric in which \(\bar{H_q}\) and \(\mathrm{d}\bar{H_q}/\mathrm{d}\tilde{t}\) are regular, while higherorder \(\tilde{t}\) derivatives of \(H_q\) blow up at a finite \(\tilde{t}\).

If \(\alpha =1/(n+2)\) or \(\alpha =n/(n+1)\), there is no singularity of the auxiliary metric and the universe is born at a finite past \(\tilde{t}\).
2.5 The EiBI scenario and the little rip
2.5.1 The physical metric \(g_{\mu \nu }\)
We conclude the analysis of this section by considering as well the possibility of smoothing a little rip event within the EiBI formalism. The little rip event is quite similar to the big rip singularity except that the former happens at an infinite future while the latter at a finite cosmic time. Such an event, despite avoiding a future singularity at a finite cosmic time, will still lead to the destruction of all structures in the universe like the big rip. The little rip has been previously analysed under fourdimensional (4D) standard cosmology [72], later on rediscovered in [59, 62, 85]. It can be found in dilatonic braneworld models [86] or other kinds of braneworld models [87, 88]. Forty years after its discovery, the event has been baptised and named the “little rip” [89, 90, 91, 92].
2.5.2 The auxiliary metric \(q_{\mu \nu }\)
A summary of how the asymptotic behaviour of a universe near the singularities in GR is altered in the EiBI theory when the universe is filled with matter, radiation as well as phantom energy. The row labelled by (1) corresponds to \(1/3<\alpha <0\) or \(1<\alpha <2/3\), and where \(\alpha \) cannot be written as \(1/(n+2)\) or \(n/(n+1)\), with \(n\) being a natural number. If \(\alpha =1/(n+2)\) (\(1/3\le \alpha <0\) naturally), which is labelled by (2), there is no singularity, while the universe starts to expand from a finite size at a finite cosmic time. Note that it is possible for the universe to start from a loitering phase of the physical metric instead of a past singularities, as long as the total pressure reaches the value \(\bar{p}=1\) at some particular scale factor \(a_b\) such that \(a_b>a_{\text {min}}\), and it corresponds to a past big bang singularity of the auxiliary metric
Singularity in GR  EiBI physical metric  EiBI auxiliary metric 

Big rip  Big rip  Expanding de Sitter 
Past sudden (\(\alpha >0\))  Past type IV (\(0<\alpha \le 2\))  Contracting de Sitter 
Past sudden (\(\alpha >2\))  
Future big freeze (\(\alpha <1\))  Future big freeze (\(3<\alpha <1\))  Expanding de Sitter 
Future type IV (\(\alpha =3\))  
Future sudden (\(\alpha <3\))  
Past type IV \((1<\alpha <0)\, (\alpha \ne n/(n+1))\)  Past sudden (\(2/3<\alpha <1/3\))  Past type IV 
(1) Past type IV  
(2) Finite past without singularity  Finite past without singularity  
Past loitering effect (\(a_b>a_\text {min}\))  Big bang  
Finite past without singularity \((\alpha =n/(n+1))\, (1<\alpha <0)\)  Finite past without singularity  Finite past without singularity 
Past loitering effect (\(a_b>a_\text {min}\))  Big bang  
Little rip  Little rip  Expanding de Sitter 
3 The geodesic analyses of a Newtonian object within the EiBI setup
In this section, we will consider a spherical Newtonian object with mass \(M\) and a test particle rotating around the mass \(M\) with a physical radius \(r\). We assume that both of them are embedded in a spherically symmetric FLRW background. In Ref. [74], the authors have shown that the bound systems with a strong enough coupling in a de Sitter background will not comove with the accelerating expansion of the universe. However, it is not the case when general accelerating phases are considered, such as the various singularities we have analysed in this paper. Therefore, we will analyse the evolution equations of its physical radius, or the geodesic equations, when the universe approaches those singularities. In the Palatini formalism, there are two metrics, the first one \(g_{\mu \nu }\) couples to matter, the second one \(q_{\mu \nu }\) is the auxiliary one which is compatible with the connection and fixes the curvature of the spacetime. If we regard the first metric as the one used to define the distances, then the geodesic equation is then defined by the LeviCivita connection of \(g_{\mu \nu }\). On the other hand, if we consider the curvature, therefore \(q_{\mu \nu }\), responsible for the geodesic equations then we can define another geodesic equation expressed by the coordinates \(\tilde{t}\) and \(\tilde{a}\) defined in Eq. (2.8).
3.1 Dark energy with a constant equation of state: big rip case
Hence, one can see that \(\phi (t)\rightarrow \phi _0\) as the big rip is approached, which means that the angular motion slows down and freezes near the singularity. These qualitative descriptions of the asymptotic behaviour of the bound system near the singularity confirm the existence of the big rip singularity in the EiBI theory.
3.2 PhantomGCG with \(\alpha >2\): sudden singularity case
3.3 PhantomGCG with \(3<\alpha <1\): big freeze case
3.4 Dark energy driving the little rip event
Furthermore, following a similar approach it can be seen that the bound structures will not be destroyed near the type IV singularity because the singularity is too weak and therefore \(\ddot{a}/a\) will always be finite. For the auxiliary metric we got a similar results as for the type IV singularity within the physical metric.
We would like to stress that the analysis we have performed and which is based on the evolution equation (3.1) is valid for Newtonian objects and under a weak field limit. Actually, there are many choices of metrics one can use to interpolate between a Schwarzschild and a FLRW metric (see Ref. [94]). Our results may be improved if a better interpolating metric is chosen. However, we expect the approximation in Eq. (3.3) is still valid as the universe is close to the cosmological singularities considered above because the matter parts (the terms proportional to \(GM\) in the evolution equation of the bound structure) are small compared with the expansion terms within such situations. Therefore, the use of the evolution equation (3.1) is quite fair in our analysis. Furthermore, we did not focus so much on the kind of gravitating systems we are considering but on the end state of the gravitating system in an expanding FLRW background of the kind we have analysed in the previous section. We have also followed a GR approach in this analysis because (i)] for the strongest singularity like the big rip, the EiBI theory would behave at first order as GR with a different gravitational constant and (ii) for simplicity, we can improve the interpolating metric between a Schwarzschild and a FLRW metric but then we will need to take into account the gravitational theory we are analysing. We think this is far beyond the scope of this paper and we will come back to this issue in the future. Finally, we would like to finish by noticing that even for the strongly gravitating system such as the case of a black hole, the approach we have followed or a more exhaustive one as the one presented in [74] lead to the same result: the black hole event horizon is destroyed. If results are consistent for strongly gravitating system we see no reasons why the results will be modified for other kind of systems. Of course all these hold in GR and we expect it is still valid in the EiBI theory for the reasons stated above.
4 A cosmographic approach of the EiBI scenario
In this section, we will use the cosmographic approach to constrain the parameters of our model, especially in the cases in which the singularities are driven by the pGCG introduced in the previous sections. The cosmographic approach does not assume any particular form of the Friedmann equations and only depends on the assumption that the spacetime is described by a FLRW metric. This makes this approach completely model independent so that we can use it to constrain the parameters of the pGCG model in the EiBI framework [75, 76, 77, 78]. In Sect. 2, one can see that the parameters in this theory have a profound influence on the doomsday or the birth of the universe. Therefore, if the parameters in this theory are somehow constrained, one can further forecast the future evolution of the universe and the possibility of past singularities different from the big bang.
With the above equations and definitions, we can basically use the matter content given in Eq. (2.12), regarding the pGCG as the dark energy component, and rewrite the modified Friedmann equation (2.2) as a function of the redshift \(z\) then taking its \(z\) derivatives. There are six parameters in our model: \(\kappa \), \(\alpha \), \(a_{\text {max}}\) (or \(a_{\text {min}}\)), \(\Omega _m\), \(\Omega _{de}\) and \(\Omega _r\) where the last three are the density parameters of dark and baryonic matter, dark energy and radiation, respectively. For the remainder of this paper, we will assume \(\Omega _r=8.48\times 10^{5}\) according to Ref. [95]. Therefore, we are left with five parameters and we can in principle use Eq. (4.4) and \((H/H_0)^2_{z=0}=1\) to close our system and constrain our model as long as all the cosmographic parameters are given. However, one has to keep in mind that the past evolution of the universe has imposed some physical constraints on the parameters of the model. For example, when one considers the past singularities, i.e., \(\alpha >1\), the minimum scale factor \(a_{\text {min}}\) should be very small to make this model in accordance with the wellknown evolution of the universe. With this assumption, one may expect that these cases should be very close to the \(\Lambda \)CDM version of the EiBI theory as the dark energy density approaches a constant at the present time [see Eq. (2.14)]. On the other hand, when one considers the future big freeze singularities, these physical restrictions are loosened.
In summary, our strategy will be the following. (i) Method A: Even though we have five free parameters (note that \(\Omega _r\) has been fixed), \(\Omega _\kappa \) can be chosen as a small number [37, 38]. Therefore, we are left with only four free parameters which are constrained by the Friedmann equation \(H/H_0\), \(q_0\) and \(j_0\), leaving \(Y\) as the free parameter. (ii) Method B: Again, and even though we have five parameters, \(\Omega _\kappa \) can be chosen as a small number [37, 38] and \(\Omega _m\) can be fixed by Planck data. Therefore, we are left again with only three free parameters which are constrained by the Friedmann equation \(H/H_0\), \(q_0\) and \(j_0\).
Before solving numerically the cosmographic constraints in the EiBI theory, we will provide some qualitative behaviours of \(\alpha \), \(a_s\) and \(\Omega _m\) as functions of \(Y\) in GR [see Eqs. (4.7), (4.8) and (4.9)]. First of all, one can see from Eq. (4.7) that \(\alpha \rightarrow +\infty \) \((\infty )\) for \(Y\rightarrow 0^{+}\) \((0^{})\) if \(1j_0+2\Omega _r\) is positive (negative), and \(\alpha \rightarrow 0\) for \(Y\rightarrow 1/(1j_0+2\Omega _r)\). Note that \(q_0\) is always negative in an accelerating universe as it is in our case. Second, from Eq. (4.8) one can see that \(a_s\rightarrow 1\) for the limits \(Y\rightarrow 0\) and \(Y\rightarrow 1/(1j_0+2\Omega _r)\). Note that the right hand side of Eq. (4.8) is always smaller than \(1\) if \(Y\) and \(\alpha \) are positive, corresponding therefore \(a_s\) to \(a_{\text {min}}\) (See the bottom figure in Fig. 3). For negative \(Y\) and \(\alpha \), the values of \(a_s\) defined in Eq. (4.8) can be divided into \(a_{\text {min}}\) and \(a_{\text {max}}\) by a particular \(Y\) whose absolute value reads \(Y_p\), which corresponds to \(1+\alpha =0\). Besides, we find that \(a_{\text {min}}\) has a local minimum for \(1j_0+2\Omega _r>0\). Furthermore, there is a positive, divergent \(a_{\text {max}}\) at \(Y_p^{}\) corresponding to \(1+\alpha \rightarrow 0^{}\) and a vanishing \(a_{\text {min}}\) at \(Y_p^{+}\) corresponding to \(1+\alpha \rightarrow 0^{+}\) for \(1j_0+2\Omega _r<0\). Finally, one can see from Eq. (4.9) that \(\Omega _m\) is a straight line ranging from \((2+2q_04\Omega _r)/3\) \((Y\rightarrow 0)\), which corresponds exactly to \(\Omega _m\) in the radiation\(+\Lambda \)CDM model, to \(1\Omega _r\) \((Y\rightarrow 1/(1j_0+2\Omega _r))\), which is exactly a pure radiation\(+\)CDM model.
In the following subsections, we will apply the two approaches enumerated previously just after Eq. (4.4).
4.1 The first method: introducing \(Y\)

Fit \((1)\): The analysis using the Taylor approach without priors. (\(H_0=69.90^{+0.438}_{0.433}\), \(q_0=0.528^{+0.092}_{0.088}\), \(j_0=0.506^{+0.489}_{0.428}\))

Fit \((2)\): The analysis using the Padé parametrisation without priors. (\(H_0=70.25^{+0.410}_{0.403}\), \(q_0=0.683^{+0.084}_{0.105}\), \(j_0=2.044^{+1.002}_{0.705}\))

Fit \((3)\): The analysis using the Padé parametrisation with the short redshift range \(z\in [0,0.36]\). (\(H_0=70.090^{+0.460}_{0.450}\), \(q_0=0.658^{+0.098}_{0.098}\), \(j_0=2.412^{+1.065}_{0.978}\))

Fit \((4)\): The analysis presuming priors from Planck’s results on \(H_0\) only. (\(H_0=67.11\), \(q_0=0.069^{+0.051}_{0.055}\), \(j_0=0.955^{+0.228}_{0.175}\))

Fit \((5)\): The analysis presuming priors from Planck’s results on \(q_0\) only. (\(H_0=69.77^{+0.288}_{0.290}\), \(q_0=0.513\), \(j_0=0.785^{+0.220}_{0.208}\))

Fit \((6)\): The analysis presuming priors from Planck’s results on both \(H_0\) and \(q_0\). (\(H_0=67.11\), \(q_0=0.513\), \(j_0=2.227^{+0.245}_{0.237}\))

Fit \((7)\): The analysis presuming priors on \(H_0\) from the firstorder fit of the luminosity distance. (\(H_0=69.96^{+1.12}_{1.16}\), \(q_0=0.561^{+0.055}_{0.042}\), \(j_0=0.999^{+0.346}_{0.468}\))
4.1.1 The analyses for positive \(Y\) in the EiBI theory
From these two figures, one can obtain two simple conclusions: (i) We do not see much difference between using EiBI and GR, the reason of course is that \(\Omega _\kappa \) is very small as predicted in Refs. [37, 38]. (ii) In order to obtain values of \(\Omega _m\) compatible with the \(\Lambda \)CDM model, which we will consider as a guiding line of our analysis, we will stick to small values of \(Y\) which we will consider to be smaller than \(5\).
According to these constraints and the asymptotic behaviour analyses in previous sections (the universe would start from a finite past sudden singularity if \(\alpha >2\) and a finite past type IV singularity if \(0<\alpha \le 2\)), a universe based on the EiBI theory may start its expansion from a past type IV singularity with both \(a_{\text {min}}\) and \(\alpha \) being very small, as long as \(j_0\) is very close to \(1+2\Omega _r\), i.e., the radiation \(+\,\Lambda \)CDM model. It is, however, unlikely that the universe starts from a sudden singularity in this case because this would require a relatively large value of \(\alpha \) which would imply a too large value of \(a_{\text {min}}\) which is incompatible with the history of the universe.
4.1.2 The analyses for negative Y in the EiBI theory
The conclusions are the following: (i) If \(\alpha <1\), which implies the existence of future singularities, the values \(\alpha <3\) (sudden singularities), \(\alpha =3\) (type IV singularities) or \(3<\alpha <1\) (big freeze singularities) are all compatible with the fit \((7)\) in [80]. However, we cannot tell which of these singularities are preferred from an observational point of view and observational constraints on higher cosmographic parameters are necessary. It is worth mentioning that for a fixed \(\alpha \), the closer to \(1+2\Omega _r\) the jerk parameter at present \(j_0\) is, the larger the maximum of the scale factor at the doomsday \(a_{\text {max}}\) would be.
Using the first approach in Sect. 4, here we show the constraints on the parameters derived for various \(\alpha \) and \(\Omega _\kappa \) based on fit \((1)\) of Ref. [80]. Here we assume \(q_0=0.561\), \(\Omega _r=8.48\times 10^{5}\) and \(j_0=0.999\) or \(1.001\), corresponding to positive or negative \(\alpha \). The parameter constraints under the GR framework (\(\Omega _\kappa =0\)) are shown. Note that the cases in which the past singularities are replaced with a loitering effect in an infinite past are also shown. The additional analyses for \(\Omega _\kappa =10^{40}\) and \(10^{43}\) indicate that as long as \(\Omega _\kappa \) is small enough, and \(\alpha \) is close enough to \(1\), it is possible to derive a small enough \(a_s\) to stand for the existence of a past type IV singularity in the EiBI theory. We have to stress that the allowable region of \(\alpha \) in which a small enough \(a_s\) can be obtained also increases as \(j_0\) gets close to \(1+2\Omega _r\), as mentioned in this section
\(\alpha \)  \(\Omega _\kappa \)  \(\Omega _m\)  \(\Omega _{de}\)  \(a_s\) (or \(a_b\))  \(H_0\) (\(t_st_0\))  \(\alpha \)  \(\Omega _\kappa \)  \(\Omega _m\)  \(\Omega _{de}\)  \(a_s\) (or \(a_b\))  \(H_0\) (\(t_st_0\)) 

\(3.5\)  \(10^{6}\)  \(0.292606\)  \(0.707309\)  \(3.5512\)  \(1.39474\)  \(3\)  \(10^{6}\)  \(0.292615\)  \(0.7073\)  \(4.7513\)  \(1.71593\) 
\(10^{7}\)  \(0.292606\)  \(0.707309\)  \(3.55106\)  \(1.39309\)  \(10^{7}\)  \(0.292615\)  \(0.7073\)  \(4.75106\)  \(1.71434 \)  
\(10^{8}\)  \(0.292606\)  \(0.707309\)  \(3.55104\)  \(1.3925\)  \(10^{8}\)  \(0.292615\)  \(0.7073\)  \(4.75103\)  \(1.71377\)  
\(0\) (GR)  \(0.292606\)  \(0.707309\)  \(3.55104\)  \(1.39217\)  \(0\) (GR)  \(0.292615\)  \(0.7073\)  \(4.75103\)  \(1.71345\)  
\(2.5\)  \(10^{6}\)  \(0.292627\)  \(0.707288\)  \(7.67048\)  \(2.23619\)  \(2\)  \(10^{6}\)  \(0.292646\)  \(0.707269\)  \(19.7213\)  \(3.23296 \) 
\(10^{7}\)  \(0.292627\)  \(0.707288\)  \(7.66995\)  \(2.23468\)  \(10^{7}\)  \(0.292646\)  \(0.707269\)  \(19.7192\)  \(3.23158\)  
\(10^{8}\)  \(0.292627\)  \(0.707288\)  \(7.6699\)  \(2.23414\)  \(10^{8}\)  \(0.292646\)  \(0.707269\)  \(19.719\)  \(3.2311\)  
\(0\) (GR)  \(0.292627\)  \(0.707288\)  \(7.66989\)  \(2.23384\)  \(0\) (GR)  \(0.292646\)  \(0.707269\)  \(19.719\)  \(3.23083\)  
\(0.9\)  \(10^{6}\)  \(0.292758\)  \(0.707157\)  \(0.00230578\)  \(\infty \) (Loitering)  \(0.8\)  \(10^{6}\)  \(0.292784\)  \(0.707131\)  \(0.00230578\)  \(\infty \) (Loitering) 
\(10^{7}\)  \(0.292759\)  \(0.707157\)  \(0.00129664\)  \(\infty \) (Loitering)  \(10^{7}\)  \(0.292784\)  \(0.707131\)  \(0.00129664\)  \(\infty \) (Loitering)  
\(10^{8}\)  \(0.292759\)  \(0.707157\)  \(0.0007\)  \(\infty \) (Loitering)  \(10^{8}\)  \(0.292784\)  \(0.707131\)  \(0.000729153\)  \(\infty \) (Loitering)  
\(10^{40}\)  \(0.292759\)  \(0.707157\)  \(7.3\times 10^{12}\)  \(\infty \) (Loitering)  \(10^{40}\)  \(0.292784\)  \(0.707131\)  \(1.54408\times 10^{6}\)  \(0.970345\)  
\(10^{43}\)  \(0.292759\)  \(0.707157\)  \(1.61021\times 10^{12}\)  \(0.970356\)  \(10^{43}\)  \(0.292784\)  \(0.707131\)  \(1.54408\times 10^{6}\)  \(0.970345\)  
\(0\) (GR)  \(0.292759\)  \(0.707157\)  \(1.61021\times 10^{12}\)  \(0.970356\)  \(0\) (GR)  \(0.292784\)  \(0.707131\)  \(1.54408\times 10^{6}\)  \(0.970345\)  
\(0.7\)  \(10^{6}\)  \(0.292817\)  \(0.707098\)  \(0.00230578\)  \(\infty \) (Loitering)  \(0.15\)  \(10^{6}\)  \(0.294282\)  \(0.705633\)  \(0.17496\)  \(0.881185\) 
\(10^{7}\)  \(0.292817\)  \(0.707098\)  \(0.001\)  \(\infty \) (Loitering)  \(10^{7}\)  \(0.294282\)  \(0.705633\)  \(0.174949\)  \(0.881195\)  
\(10^{8}\)  \(0.292817\)  \(0.707098\)  \(0.0007\)  \(\infty \) (Loitering)  \(10^{8}\)  \(0.294282\)  \(0.705633\)  \(0.174947\)  \(0.881196\)  
\(0\) (GR)  \(0.292817\)  \(0.707098\)  \(1.5495\times 10^{4}\)  \(0.970329\)  \(0\) (GR)  \(0.294282\)  \(0.705633\)  \(0.174947\)  \(0.881196\)  
\(0.25\)  \(10^{6}\)  \(0.293592\)  \(0.706323\)  \(0.175573\)  \(0.880798\)  \(0.2\)  \(10^{6}\)  \(0.293851\)  \(0.706064\)  \(0.17372\)  \(0.882178\) 
\(10^{7}\)  \(0.293592\)  \(0.706323\)  \(0.175562\)  \(0.880806\)  \(10^{7}\)  \(0.293851\)  \(0.706064\)  \(0.17371\)  \(0.882187\)  
\(10^{8}\)  \(0.293592\)  \(0.706323\)  \(0.175561\)  \(0.880807\)  \(10^{8}\)  \(0.293851\)  \(0.706064\)  \(0.173709\)  \(0.882188\)  
\(0\) (GR)  \(0.293592\)  \(0.706323\)  \(0.175561\)  \(0.880807\)  \(0\) (GR)  \(0.293851\)  \(0.706064\)  \(0.173709\)  \(0.882188\) 
4.2 The second approach: assuming \(\Omega _m\)
In the previous subsection, we only consider fit \((7)\), which is the closest fit in Ref. [80] to the radiation \(+\,\Lambda \)CDM model. Additionally, we can also use other fits to constrain our model. One can see that other fits from \((1)\) to \((6)\) have a significant difference from fit \((7)\): the jerk parameter \(j_0\) is different from \(1+2\Omega _r\) by a comparable amount. This fact makes these data sets deviate a lot from the radiation \(+\,\Lambda \)CDM model and when applied to the model we are analysing we get a too large \(a_\text {min}\) which is incompatible with the history of the universe, as we mentioned previously. Therefore, we will only analyse the models in which future singularities happen with the data sets from fits \((1)\) to \((6)\).
To analyse the future singularities with these data sets, we use another approach different from the one we followed in the previous subsection: we fix the value of \(\Omega _m\) according to the Planck mission [13, 14] and assume it to be model independent, then we assume that \(\Omega _\kappa \) is roughly within the range \(10^{7}\) to \(10^{4}\). We find that only fits \((2)\) and \((3)\) are compatible with the analyses of the cases in which \(\alpha <1\). The reason is that for the cases in which \(\alpha <1\), the derivative of \(\bar{p}_{de}/\bar{\rho }_{de}\) with respect to the scale factor should be negative. Furthermore, the value of \(\bar{p}_{de}/\bar{\rho }_{de}\) should be smaller than \(1\). On the basis of GR, these criteria are only valid in these two data sets fits \((2)\) and \((3)\). Interestingly, the authors of [80] also claimed that these two fits, in addition to fit \((7)\), are the most reasonable results of their analyses. Hence, we use the data in fits \((2)\) and \((3)\), set the values of \(\Omega _m=0.315\), \(\Omega _r=8.48\times 10^{5}\) and \(\Omega _\kappa \) from \(10^{7}\) to \(10^{4}\), and we numerically solve the resulting \(\alpha \), \(\Omega _{de}\), \(a_{\text {max}}\) as well as the dimensionless cosmic time elapsed from the current time to the doomsday \(H_0(t_\text {max}t_0)\). The results are shown in Tables 3 and 4.
The constraints of the parameters derived according to the data fit (2) in Ref. [80] where \(H_0=70.25\), \(q_0=0.683\), \(j_0=2.044\). Here we use the second approach presented in Sect. 4 in which we assume \(\Omega _m=0.315\) according to the Planck data and \(\Omega _\kappa =10^{4}\), \(10^{5}\), \(10^{6}\), \(10^{7}\). The parameter constraints under the GR framework (\(\Omega _\kappa =0\)) are also shown
\(\Omega _\kappa \)  \(\alpha \)  \(\Omega _{de}\)  \(a_\text {max}\)  \(H_0(t_\text {max}t_0)\) 

\(0\) (GR)  \(1.94103\)  \(0.684915\)  \(2.05115\)  \(0.621529\) 
\(10^{7}\)  \(1.94103\)  \(0.684915\)  \(2.05115\)  \(0.622249\) 
\(10^{6}\)  \(1.94103\)  \(0.684916\)  \(2.05115\)  \(0.6235\) 
\(10^{5}\)  \(1.94102\)  \(0.684919\)  \(2.05113\)  \(0.62681\) 
\(10^{4}\)  \(1.94095\)  \(0.684948\)  \(2.05102\)  \(0.635257\) 
The constraints on the model parameters derived according to the data fit (3) in Ref. [80] where \(H_0=70.09\), \(q_0=0.658\), \(j_0=2.412\). Here we use the second approach presented in Sect. 4 in which we assume \(\Omega _m=0.315\) according to the Planck data and \(\Omega _\kappa =10^{4}\), \(10^{5}\), \(10^{6}\), \(10^{7}\). The parameter constraints under the GR framework (\(\Omega _\kappa =0\)) are also shown
\(\Omega _\kappa \)  \(\alpha \)  \(\Omega _{de}\)  \(a_\text {max}\)  \(H_0(t_\text {max}t_0)\) 

\(0\) (GR)  \(3.19514\)  \(0.684915\)  \(1.39279\)  \(0.317249\) 
\(10^{7}\)  \(3.19514\)  \(0.684915\)  \(1.39279\)  \(0.318152\) 
\(10^{6}\)  \(3.19514\)  \(0.684916\)  \(1.39279\)  \(0.31972\) 
\(10^{5}\)  \(3.19516\)  \(0.68492\)  \(1.39276\)  \(0.323837\) 
\(10^{4}\)  \(3.19532\)  \(0.68496\)  \(1.39248\)  \(0.334114\) 
5 Conclusions
The EddingtoninspiredBorn–Infeld theory (EiBI) proposed recently is characterised by being equivalent to Einstein theory in vacuum but differing from it in the presence of matter. Most importantly, it also features the ability to avoid some singularities such as the big bang singularity in the finite past of the universe, and the singularity formed after the collapse of a star. It is hence interesting to see whether this ability to avoid/smooth other kinds of singularities, especially those driven by the phantom dark energy which could be responsible for the current accelerating expansion of the universe, is efficient enough or not.
In this paper, we give a thorough analysis of the avoidance of all dark energy related singularities by deriving the asymptotic behaviours of the Hubble rate and the cosmic time derivatives of the Hubble rate defined by the physical metric \(g_{\mu \nu }\) coupled to matter, and by the auxiliary metric \(q_{\mu \nu }\) compatible with the physical connection. For the physical metric \(g_{\mu \nu }\) we find that though the big rip singularity and the little rip event driven by phantom dark energy are not cured in the EiBI theory, this theory to some extent smooth the other phantom dark energy related singularities present in GR by leaving some region of the parameter space in which the future big freeze singularity is altered into a future sudden or future type IV singularity. Additionally, the past singularity present in GR is also smoothed in this theory in some parameter space as a past type IV singularity. Note that a past type IV singularity present in GR is, in some parameter space, worsened into a past sudden singularity, while smoothed as a regular birth of the universe at some quantised parameter space or even as a loitering effect in an infinite past. As for the auxiliary metric \(q_{\mu \nu }\) compatible with the physical connection (we remind the reader that the EiBI setup we are dealing with is formulated à la Palatini formalism), all the dark energy related singularities of interest are avoided except for some very specific parameter space in which the past type IV singularities of the auxiliary metric still exist (see Table 1 for a summary).
Furthermore, we analysed the fate of a bound structure near the singularities of the EiBI theory. We find that the bound structure would be destroyed before the universe approaches a big rip singularity and a little rip event, while remains bounded at a sudden, big freeze and type IV singularities.
Besides, we also use the cosmographic approach, which is characterised by its theoretical modelindependence, to constrain the parameters present in our model, so that we in principle can forecast the doomsdays and describe the birth of the universe based on our model. As a result, it turns out that the cosmographic analyses pick up the physical region which determines the occurrence of a type IV singularity in the finite past or the loitering effect in an infinite past. While it is necessary to impose more conditions, such as the use of higherorder cosmographic parameters with more accurate observations or other physical constraints, to forecast the future doomsdays of the universe in this model. According to these results, the EiBI theory is indeed a reliable theory which is able to cure or smooth the singularities predicted originally in GR, thus it makes the theory a convincing alternative to GR as a way to smooth singularities.
Footnotes
 1.
The leading order in the expansion of the scalar curvature with respect to \(\bar{\rho }\) satisfies \(\kappa R\propto \bar{\rho }\) at the low energy density limit, thus we can expand with respect to the energy density when the low curvature assumption is considered.
 2.
We will use the cosmographic results obtained in Ref. [80] but please notice that for the purpose of the current work we could have taken other works from the ones available in the literature.
Notes
Acknowledgments
The work of M.B.L. was supported by the Basque Foundation for Science IKERBASQUE and 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 support from the Portuguese Grants PTDC/FIS/111032/2009 and PEstOE/MAT/UI0212/2014 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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