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Eur. Phys. J. C (2015) 75:90

DOI 10.1140/epjc/s10052-015-3257-4

Regular Article - Theoretical Physics

Eddington–Born–Infeld cosmology: a cosmographic approach,

a tale of doomsdays and the fate of bound structures

Mariam Bouhmadi-López

1,2,3,4,a,b

, Che-Yu Chen

5,7,c

, Pisin Chen

5,6,7,8,d

Departamento de Física, Universidade da Beira Interior, 6200 Covilhã, Portugal

Centro de Matemática e Aplicações da Universidade da Beira Interior (CMA-UBI), 6200 Covilhã, Portugal

Department of Theoretical Physics, University of the Basque Country UPV/EHU, P.O. Box 644, 48080 Bilbao, Spain

IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain

Department of Physics, National Taiwan University, Taipei 10617, Taiwan

LeCosPA, National Taiwan University, Taipei 10617, Taiwan

Graduate Institute of Astrophysics, National Taiwan University, Taipei 10617, Taiwan

SLAC National Accelerator Laboratory, Kavli Institute for Particle Astrophysics and Cosmology,

Stanford University, Stanford, CA 94305, USA

Received: 8 July 2014 / Accepted: 15 December 2014 / Published online: 24 February 2015

Abstract The Eddington-inspired-Born–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 [Bouhmadi-Lopez et al. (Eur. Phys. J. C

74:2802, 2014)] we showed that, on the contrary, it is impos-

sible to smooth a big rip in the EiBI setup. In fact the situa-

tions 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 auxil-

iary 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 singu-

larity 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

e-mail: mbl@ubi.pt

On leave of absence from UPV and IKERBASQUE

e-mail: b97202056@ntu.edu.tw

e-mail: pisinchen@phys.ntu.edu.tw

more probable, observational constraints on the higher-order

cosmographic parameters are required.

1 Introduction

With no doubt general relativity (GR) is an extremely suc-

cessful 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 mod-

iﬁed theories of GR, while preserving the great achievements

of GR, would shed some light over the unknown fundamen-

tal nature of dark energy or whatsoever stuff that drives the

present accelerating expansion of the universe (see Refs. [4–

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 Super-

novae (SNeIa) [10–12] (which brought the ﬁrst evidence)

to the cosmic microwave background (CMB) [13,14], the

baryon acoustic oscillations (BAO) [15–17], gamma ray

bursts (GRB) [18] and measures of the Hubble parameter

at different redshifts [19] among others, showed that the uni-

verse 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

123

90 Page 2 of 23 Eur. Phys. J. C (2015) 75 :90

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 cosmolog-

ical singularities and could shed some light over the late-time

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 refor-

mulated recently: the Eddington-inspired-Born–Infeld the-

ory (EiBI) [22–24], as its name indicates, is based on the grav-

itational theory proposed by Eddington [25] with an action

similar to that of the non-linear 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 Levi-Civita 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 andhyperbolic 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](seealso[31] for an

analysis of the scalar and vectorial perturbations for a radi-

ation 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 fulﬁl-

ment of the energy conditions in the EiBI theory was studied

in Ref. [34] and a sufﬁcient condition for singularity avoid-

ance under the fulﬁlment of the null energy condition was

obtained. Additionally, it was shown that the gravitational

collapse of non-interacting particles does not lead to singular

states in the Newtonian limit [35]. Furthermore, the parame-

ter 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 [39–41]. Especially in [41],

the authors showed that the universal relations of the f-mode

oscillation [42], which is the fundamental mode of pulsation

in the neutron stars, and the I-Love-Q 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](seealsoRef.[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](seealso[48

]).

We showed recently that despite the big bang avoidance in

the EiBI setup, the big rip [49–56] 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 possi-

ble avoidance of the other dark energy related singularities,

known as: (i) sudden, type II, big brake or big démarrage

singularity [57–60], (ii) type III or big breeze singularity

[59–63] 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 alle-

viate 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

μν

appears in the action and couples to matter, the second one is

the auxiliary one which is compatible with the connection 

[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. Further-

more, 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

μν

, more precisely the

Hubble parameter and its cosmic time derivatives as deﬁned

from the metric g

μν

, near the singularities is consistent with

that of the geodesic behaviour dictated by the same metric

μν

. However, the events corresponding to the singularities

with respect to g

μν

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 singu-

larity 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 [75–78]. This analysis will show that

the EiBI model when ﬁlled with the matter content mentioned

in the previousparagraph on top of the dark and baryonic mat-

ter is compatible with the current acceleration of the universe.

123

Eur. Phys. J. C (2015) 75 :90 Page 3 of 23 90

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 param-

eters [75–78]. These parameters depend exclusively on the

space-time geometry, in this case on the geometry of a homo-

geneous and isotropic space-time, and not on the gravita-

tional 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 usefulbecause 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 [75,78–83].

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 phys-

ical 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 dooms-

days predicted, the use of observationalconstraints on higher-

order 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 the-

ory, through deriving the asymptotic behaviours of the Hub-

ble parameter and its cosmic time derivatives near the sin-

gularities 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 mas-

sive 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 start reviewing the EiBI model whose gravitational action

in terms of the metric g

μν

and the connection 

μν

reads [24]

EiBI

(g,,) =







μν

+ κ R

μν

()|−λ



|g|



(g,). (2.1)

The theory is formulated within the Palatini approach and

therefore the Ricci tensor is purely constructed from the con-

nection . In addition, R

μν

() in the action (2.1) is chosen

to be the symmetric part of the Ricci tensor and the con-

nection is also assumed to be torsionless. Within the Pala-

tini formalism we are assuming here, the connection 

μν

and the metric g

μν

are treated as independent variables. The

parameter κ is a constant with inverse dimensions to that of

a cosmological constant (in this paper, we will work with

Planck units 8π G = 1 and set the speed of light to c = 1),

λ is a dimensionless constant and S

(g,) stands for the

matter Lagrangian in which matter is assumed to be cou-

pled covariantly to the metric g only. Therefore, the energy

momentum tensor derived from Eq. (2.1) is conserved like in

GR [24]. One can also note that the action (2.1) will recover

the Einstein–Hilbert action as |κ R| gets very small with an

effective cosmological constant  = (λ − 1)/κ [24]. From

now on we will assume a vanishing effective cosmological

constant, i.e., λ = 1. In addition, we will restrict our analy-

sis to a positive κ, in order to avoid the imaginary effective

sound speed instabilities usually present in the EiBI theory

with negative κ [37].

Fora FLRW universe ﬁlled with a perfect ﬂuid with energy

density ρ and pressure p, the Friedmann equation reads [29]



¯ρ + 3 ¯p − 2 + 2



(1 +¯ρ)(1 −¯p)



(1 +¯ρ)(1 −¯p)

[(1 −¯p)(4 +¯ρ − 3 ¯p) + 3

d ¯p

d ¯ρ

(1 +¯ρ)( ¯ρ +¯p)]

(2.2)

where

H ≡

√

κ H, H is the Hubble parameter as deﬁned

from the physical metric, ¯ρ = κρ, ¯p = κp and d ¯p/d ¯ρ ≡

denotes the derivative of the pressure with respect to the

energy density. For simplicity, we will also use the following

dimensionless cosmic time:

t ≡ t/

√

κ where t corresponds

to the cosmic time as deﬁned from the physical metric g

μν

When the curvature gets very small, i.e., |κ R||g|,the

Friedmann equation (2.2) becomes

≈

¯ρ

−

+ 2w − 15

( ¯ρ)

+ higher order of ¯ρ,

(2.3)

where a constant equation of state ¯p = w ¯ρ is considered.

Recall that the EiBI theory recovers GR when |κ R| is very

small as shown in [24]. On the other hand, the conservation

equation, as mentioned previously, takes the standard form

d ¯ρ

+ 3

H( ¯ρ +¯p) = 0. (2.4)

It can easily be veriﬁed that the big bang singularity can

be avoided in this theory for a radiation dominated universe

[24]; i.e. ¯p =¯ρ/3, and in general a universe ﬁlled with a

perfect ﬂuid with a constant and positive equation of state w;

The leading order in the expansion of the scalar curvature with respect

to ¯ρ satisﬁes κ R ∝¯ρ at the low energy density limit, thus we can expand

with respect to the energy density when the low curvature assumption

is considered.

123

90 Page 4 of 23 Eur. Phys. J. C (2015) 75 :90

i.e., fulﬁlling the null energy conditions [3], bounces in the

past for κ<0 or has a loitering behaviour in the inﬁnite past

for κ>0[28].

Aside, we can deﬁne an auxiliary metric q

μν

which is

compatible with the connection  [24]:

μν

=−U(t)dt

+ a

(t)V (t)(dx

+ dy

+ dz

(2.5)

where

U =



(1 −¯p)

1 +¯ρ

, (2.6)

V =



(1 +¯ρ)(1 −¯p), (2.7)

and a is the scale factor of the physical metric g

μν

.From

the auxiliary metric q

μν

we can deﬁne as well an auxiliary

Hubble parameter H

whose rescaled dimensionless value

can be expressed as

≡

√

κ H

and reads

√

˜a

d˜a

√

ln(a

√

V ), (2.8)

where ˜a ≡

√

Va and d

t ≡

√

Udt. Besides, we ﬁnd that

satisﬁes

κq

μν

() = 12

+ 6

√

= 4 −

−

, (2.9)

where

¯ρ + 3 ¯p − 2



(1 +¯ρ)(1 −¯p)

. (2.10)

Notice that

does not depend on c

, unlike

in Eq. (2.2).

One can see that this auxiliary Hubble parameter also recov-

ers the Hubble parameter in standard GR as the curvature

gets small:

≈

¯ρ

+ 6w − 5

( ¯ρ)

+ higher order of ¯ρ,

(2.11)

where w is also a constant equation of state parameter. This

auxiliary metric which is compatible with the physical con-

nection cannot avoid the big bang singularity in the past

because both H

and dH

t diverge at a vanishing ˜a and at a

ﬁnite past

t, and so does the Ricci scalar deﬁned in Eq. (2.9).

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 Pala-

tini formalism and therefore there are two ways of deﬁning

the Ricci curvature: (i) R

μν

() as presented in the action

(2.1) and (ii) R

μν

(g) constructed from the metric g

μν

There are in addition four ways of deﬁning the scalar cur-

vature: g

μν

(), g

μν

(g),q

μν

() andq

μν

(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

to the physical connection as deﬁned in Eq. (2.8).

In general, the universe is ﬁlled with radiation, dark and

baryonic matter and dark energy:

¯ρ =¯ρ

+¯ρ

¯p =

¯ρ

+¯p

( ¯ρ

), (2.12)

where ¯ρ

= κρ

, ¯ρ

= κρ

, ¯ρ

= κρ

and ¯p

= κ p

are the energy density of radiation, matter, dark energy and

the pressure of dark energy, respectively. Note that p

( ¯ρ

)

means that the equation of state of dark energy is purely a

function of the dark energy density. For the sake of complete-

ness, we will assume a universe ﬁlled with a matter contents

as shown in Eq. (2.12) throughout the analysis in this paper.

Note that even though dark matter and radiation are unim-

portant for the analysis of future singularities, they are not

for the analysis of past singularities.

Before starting our analysis, we will review the deﬁnition

of these dark energy related singularities:

• 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 [49–56].

• 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–

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 [59–63].

• Finally the type IV singularity occurs at a ﬁnite cosmic

time with a ﬁnite scale factor where the Hubble parame-

ter and its cosmic time derivative remain ﬁnite, but higher

cosmic time derivatives of the Hubble parameter still

diverge [59,61–65].

To analyse the big freeze, sudden and type IV singularities,

we regard the phantom generalised Chaplygin gas (pGCG) as

the dark energy component in this model [60,69].Its equation

of state takes the form:

¯p

=−

( ¯ρ

)

, (2.13)

123

Eur. Phys. J. C (2015) 75 :90 Page 5 of 23 90

where α and A > 0 are two dimensionless constants. In

GR, this kind of phantom energy will drive a past sudden

singularity for α>0, a future big freeze singularity for α<

−1, and a past type IV singularity for −1 <α<0 except for

some quantised values of α in which the Hubble rate and its

higher-order derivatives are all regular in the ﬁnite past [60].

Note that the last case is different from the results shown

in Ref. [60] because in that reference the authors assumed

a universe ﬁlled only with a pGCG instead of the matter

content given in Eq. (2.12) to which we will stick in this paper.

Actually, the addition of radiation and matter contributions

does not make any comparable difference from the cases in

which past sudden and future big freeze occur in GR, i.e.,

α>0 and α<−1, respectively. However, the conclusion is

different when −1 <α<0. See Ref. [84] for more details

of this issue.

After integrating the conservation equation (2.4) and

assuming α>−1, one can derive the energy density of this

kind of pGCG which drives the ﬁnite past sudden or type IV

singularity in GR [60]:

¯ρ

= A

1+α



1 −



min



−3(1+α)



1+α

, (2.14)

where a

min

is the scale factor corresponding to the singularity.

For later convenience, we also rewrite the energy density

in terms of the scale factor as

¯ρ

=¯ρ

de0



1 −



min



3(1+α)

1 − a

min

3(1+α)



1+α

. (2.15)

Note here that we have set the scale factor at present, a

as a

= 1 and we will use this convention in the rest of this

paper. A subscript 0 stands for quantities evaluated today. On

the other hand, if α<−1 and A > 0, the energy density of

this pGCG which drives the ﬁnite future big freeze singularity

in GR reads [60]:

¯ρ

= A

1+α



1 −



max



−3(1+α)



1+α

, (2.16)

where a

max

is the scale factor corresponding to the future

singularity.

We also rewrite the energy density in terms of the scale

factor as follows for the sake of later convenience:

¯ρ

=¯ρ

de0



1 −



max



3(1+α)

1 − a

max

3(1+α)



1+α

. (2.17)

Additionally, there are some special case in which the

phantom character shares the same equation of state (2.13)

while does not imply A > 0, as shown in Refs. [60,69]. This

special pGCG will drive a ﬁnite future big freeze singularity

in GR and its energy density and pressure are

¯ρ

=|A|

1+α





max



−3(1+α)

− 1



1+α

¯p

=−

( ¯ρ

)

=|A|

1+α





max

− 1



−3(1+α)

− 1



1+α

−1

, (2.18)

where A < 0 and 1 + α = 1/(2m) with m being a negative

integer [60]. We will also discuss this special case within the

EiBI scenario in the upcoming subsection.

2.1 The EiBI scenario and the big rip

2.1.1 The physical metric g

μν

We showed recently that despite the big bang avoidance in

the EiBI setup, the big rip singularity [49,50] is unavoid-

able in the EiBI phantom model [1]. Indeed, we have shown

analytically and numerically that in the EiBI theory, a uni-

verse ﬁlled with matter and phantom energy with a constant

equation of state w<−1 will still hit a big rip singularity;

i.e. the Hubble parameter

H and d

H/d

t blow up in a ﬁnite

future cosmic time and at an inﬁnite scale factor. Essentially,

the square of the dimensionless Hubble parameter

H and its

cosmic time derivative near the singularity are almost linear

functions of the energy density:

≈



|w|

3(3w + 1)

¯ρ →∞,

≈



|w|

(3w + 1)

|1 + w|¯ρ →∞.

(2.19)

Therefore, at very large scale factor and energy density

(which grows as ¯ρ ∝ a

−3(1+w)

for w<−1 and constant),

and d

H/d

t get equally large. This happens at a ﬁnite future

cosmic time [1].

2.1.2 The auxiliary metric q

μν

As for the quantities deﬁned by the auxiliary metric, it can

be shown that

≈

1 + 3w



|w|

¯ρ

→

√

≈

|1 + w|



|w|

¯ρ

→ 0,

(2.20)

and second- and higher-order derivatives of

with respect

t vanish when ¯ρ →∞because their leading order in the

expansion in ¯ρ is inversely proportional to ¯ρ. Furthermore,

we also ﬁnd that the energy density blows up and

123

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