Eur. Phys. J. C manuscript No.(will be inserted by the editor)

A unique equation of state for the universe evolution fromAdS5 space-time

M.M. Lapolaa,1, P.H.R.S. Moraesb,1,2, W. de Paulac,1, J. F. Jesus d,3, R.

Valentim e,4, M. Malheiro f,1

1ITA - Instituto Tecnologico de Aeronautica - Departamento de Fısica,12228-900, Sao Jose dos Campos, SP, Brazil2UNINA - Universita degli Studi di Napoli Federico II - Dipartamento di FisicaNapoli I-80126, Italy3Universidade Estadual Paulista (Unesp), Campus Experimental de Itapeva - R. Geraldo Alckmin 519, 18409-010, Itapeva,SP, Brazil4Departamento de Fısica, Instituto de Ciencias Ambientais, Quımicas e Farmaceuticas (ICAQF), Universidade Federal deSao Paulo - UNIFESP, Rua Sao Nicolau no.210, Centro, 09913-030, Diadema - SP, Brazil

Received: date / Accepted: date

Abstract We apply the Induced Matter Model to a

five-dimensional metric. For the case with null cosmo-

logical constant, we obtain a solution able to describe

the radiation-dominated era of the universe. The posi-

tive Λ case yields a bounce cosmological model. In the

negative five-dimensional cosmological constant case,

the scale factor is obtained as a(t) ∼√

sinh t, which is

able to describe not only the late-time cosmic acceler-

ation but also the non-accelerated stages of the cosmic

expansion in a continuous form. This solution together

with the extra-dimensional scale factor solution yields

the material content of the model to be remarkably re-

lated through an equation of state analogous to the

renowned MIT bag model equation of state for quark

matter p = (ρ− 4B)/3. In our case, ρ = ρm + B, with

ρm being the energy density of relativistic and non-

relativistic matter and B = |Λ|/16π represents the bag

energy constant, which plays the role of the dark energy

in the four-dimensional universe, with Λ being the cos-

mological constant of the AdS5 space-time. Our model

satisfactorily fits the observational data for the low red-

shift sample of the experimental measurement of the

Hubble parameter, which resulted in H0 = 72.2+5.3−5.5km

s−1 Mpc−1.

Keywords Cosmology · Induced matter model · dark

energy · observational data

ae-mail: marcelo.lapola@gmail.combe-mail: moraes.phrs@gmail.comce-mail: wayne@ita.brde-mail: jfjesus@unesp.bree-mail: valentim.rodolfo@unifesp.brfe-mail: malheiro@ita.br

1 Introduction

Since the late 90’s of the last century, a lot of efforts

have been made to describe the observable universe as

a brane embedded in a higher dimensional space [1]-[6].

Some results obtained from such a set up for the uni-

verse are remarkable. Braneworld models of dark energy

were recently presented in References [7]-[9]. In Ref-

erence [10], the possibility of the ΛCDM cosmological

model be a braneworld model in disguise was investi-

gated. In the astrophysics of compact objects context,

braneworld models are able to predict some deviations

from standard General Relativity (GR) outcomes and

get in touch with some peculiar observations [11]-[15].To be in touch with recent literature on braneworld

models applications, we suggest References [16]-[22].

The braneworld scenario was originally proposed as

an alternative to the hierarchy problem, as it can be

checked, for instance, in References [23]-[25]. The con-

cept of extra dimensions has also been used in attempts

to unify the four fundamental forces of nature [26]-[28].

Not only on braneworld scenarios lie the extra di-

mensional universe configurations. There are also the

renowned Kaluza-Klein (KK) models [29]-[36]. The relic

density of KK dark matter in universal extra dimen-

sions was calculated [37]. The virtual effects of KK

states on Higgs physics in universal extra dimensional

models were examined [38]. F. Darabi and P.S. Wesson

introduced a generalized gravitational conformal invari-

ance in the context of non-compactified five-dimensional

(5D) KK theory [39]. In astrophysics, stability of strange

stars in extra dimensions has been investigated recently

[40].

arX

iv:1

909.

0869

8v1

[gr

-qc]

18

Sep

2019

2

P.S. Wesson, has contributed very significantly to

KK cosmology as well as for the interpretation of the

extra dimension [41]-[44]. Together with collaborators,

Wesson has also investigated the effective properties of

matter in KK theory [45], outlined a Machian interpre-

tation of KK gravity [46], applied some classical tests

to the theory [47] and derived the referred equation of

motion [48].

The outcomes of some more recent articles by Wes-

son et al. on five-dimensional (5D) universe can be ap-

preciated in the following. J.M. Overduin et al. have

used measurements of geodesic precession from Grav-

ity Probe B experiment and constrained possible de-

partures from Einstein’s GR for a spinning test body in

KK theory [49]. C. Zhang and collaborators have used

Wetterich’s parameterization equation of state (EoS)

to obtain cosmological solutions in a 5D Ricci-flat Uni-

verse [50]. In [51], some relations for the embedding

of spatially flat Friedmann-Lemaıtre-Robertson-Walker

(FLRW) cosmological models in flat KK manifolds were

presented.

The cosmological constant problem, namely, the huge

discrepancy between theoretical and observed values of

the cosmological constant in standard ΛCDM cosmol-

ogy, was investigated in KK gravity by P.S. Wesson

and H. Liu [52]. F. Darabi et al. have derived a quan-

tum cosmology from KK theory with non-compactified

extra dimension [53]. In [54], Wesson et al. have ob-

tained an exact solution of the 5D field equations that

describes a shock wave moving in time and extra KK

coordinate. Such a solution suggested that the four-

dimensional (4D) big bang was a 5D shock wave.

Particularly regarding the interpretation of the ex-

tra dimension in the 4D observable universe, Wesson

has proposed the so-called Induced Matter Model (IMM),

which can be appreciated in [45],[55]-[59]. It consists of

the following concept. The KK field equations read

GAB = 0, (1)

with GAB being the Einstein tensor and the indices

A,B run from 0 to 4. From Eq.(1), it can be seen

that the KK field equations depend only on the 5D

metric gAB . The Wesson’s idea consists of collecting in

Eq.(1) the terms that depend on the extra coordinate

and make them play the role of an induced energy-

momentum tensor in 4D. Further applications of the

IMM can be appreciated in Refs.[60]-[68].

In the present article we intend to develop - by

meaning of Wesson’s model - and investigate the Friedmann-

like equations derived from a 5D metric. The field equa-

tions will be taken as Eq.(1) in the presence of a 5D

cosmological constant Λ, that is [69]

GAB + ΛgAB = 0. (2)

We will be particularly concerned with the role of

the extra-dimension scale factor in the metric [70]-[72]

ds2 = dt2−a(t)2[dr2+r2(dθ2+sin2 θdφ2)]−ξ(t)2dl2.(3)

In Eq.(3), a(t) is the scale factor of the observable uni-

verse and ξ(t) is the extra-dimension scale factor. More-

over, we are assuming the spatial curvature of the uni-

verse to be null, in accordance with recent observational

data on the fluctuations of temperature of the cosmic

microwave background radiation [73]. Still in (3), t is

the time coordinate, r, θ and φ are the polar spheri-

cal coordinates and l is the extra spatial coordinate.

Throughout the article, natural units will be assumed,

unless otherwise advised.

In the present article, we shall investigate the cos-

mological solutions obtained from the substitution of

(3) in (1) and in (2). We will search for being in touch

with recent cosmological observational data, which shall

naturally constrain the extra dimensional features of

the model.

2 4D dynamics from 5D empty space

In the present section we will substitute Eq.(3) in Eqs.(1)

and (2). For all cases we will consider that matter in

the 4D observable universe is a manifestation of a 5D

universe devoid of matter, through the application of

the IMM. That is to say that the terms on the 5D Ein-

stein tensor for (3) which somehow depend on the extra

coordinate will “be moved” to the rhs of Eqs.(1) and

(2) to play the role of an induced energy-momentum

tensor.

Throughout the whole article, the energy-momentum

tensor of a perfect fluid will be assumed, that is, TBA =

diag(ρ,−p,−p,−p, 0), with ρ being the matter-energy

density and p the pressure of the universe. Note that

T 44 = 0 since we will consider, such as in braneworld

models, that matter is restricted to the 4D universe.

2.1 Field equations without cosmological constant

The non-null components of the Einstein tensor ob-

tained when substituting metric (3) in field equations

(1) read:

G00 = 3

[(a

a

)2

+

(a

a

)(ξ

ξ

)], (4)

G11 = G2

2 = G33 =

(a

a

)2

+ 2

(a

a

)+ 2

(a

a

)+ξ

ξ, (5)

G44 = 3

[(a

a

)2

+

(a

a

)], (6)

3

where dots indicate time derivatives.

By applying the IMM in such components, one has

ρ = − 3

8π

a

a

ξ

ξ, (7)

p =1

4π

(a

a

ξ

ξ+

1

2

ξ

ξ

), (8)

and from G44 = 0 we also obtain the constrain equation(a

a

)2

+a

a= 0. (9)

By solving Eq.(9), we have

a(t) = c1√t. (10)

Throughout the article, ci, with i = 1, 2, 3, ..., are con-

stants.

It is worth to remark that Eq.(10) is in agreement

with a radiation-dominated universe, since in standard

cosmology, a ∼ t1/2 occurs exactly for such a stage of

the universe evolution [74]. Remarkably, when Kaluza

developed his extradimensional theory of gravity, to-

day called KK gravity, his intent was to describe from

GAB = 0 uniquely, both 4D Einstein’s field equations

with matter and Maxwell’s equations for electromag-

netism, as it can be checked in [75].

We can substitute (10) in the G00 component of

Eq.(1) for metric (3) and derive the solution for ξ(t) in

a

a+ξ

ξ= 0. (11)

The result is

ξ(t) =c2√t. (12)

It can be verified that Eqs. (10) and (12) are solutions

of Eq.(1).

It is interesting to remark that the solution obtained

for ξ(t) may indicate a compactification of the extra

coordinate as time passes by. This can be clearly verified

by deriving the referred Hubble parameter Hl = ξ/ξ,

which reads

Hl(t) = − 1

2 t, (13)

and a negative Hubble parameter would indicate com-

pactification rather than expansion of the referred space.

Solutions (10) and (12) when substituted in (7) and

(8) yield, respectively,

ρ(t) =3

32π t2, (14)

p(t) =1

32π t2. (15)

We can see from Eqs.(14) and (15) that ρ and p, in this

model, have a quadratic term on t. Such a behaviour

can also be seen in braneworld models [69,76].

We can also note that, remarkably, by dividing (15)

by (14), one has ω = p/ρ = 1/3, which is the EoS

parameter of a radiation-dominated universe [74]. This

result can also be verified in the Friedmann-like equa-

tions (7)-(8).

2.2 Field equations with cosmological constant

2.2.1 Case I: Λ > 0

Let us now work with Eq.(2). By substituting metric

(3) in (2), we can write, through the IMM application,

the following Friedmann-like equations:

ρ = − 3

8π

(a

a

ξ

ξ+Λ

3

), (16)

p =1

8π

(2a

a

ξ

ξ+ξ

ξ+ Λ

), (17)

(a

a

)2

+a

a= −Λ

3. (18)

Eq.(18) can be solved for the scale factor, yielding

a(t) = c3

√√√√∣∣∣ sin(√2

3Λ t

)∣∣∣. (19)

The evolution of the scale factor (19) in time can be

appreciated in Fig.1.

By analysing Fig.1 we are led to conclude that a pos-

itive 5D cosmological constant yields a cyclic or bounc-

ing universe [77]-[79].

In possession of Eq.(19), we can use the non-null

components of Eq.(2) to write

ξ(t) = c4

∣∣ cos(√

23Λ t

) ∣∣√∣∣ sin(√ 23Λ t

) ∣∣ . (20)

From Fig.2, we can see that ξ completes each cy-

cle in the same time scale as a does. We can also see

that, by keeping in mind that a = 1 at present, the

length scale of the extra dimension is minimum today,

which could justify the absence of evidences of extra

dimensions in the Large Hadron Collider [80]-[83].

From (19) and (20), we can write the explicit solu-

tions for ρ(t) and p(t) as

ρ(t) =Λ

16πcot2

(√2

3Λ t

), (21)

4

0 2 4 6 8 10 12 14t

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

a(t)

Fig. 1 Evolution of the scale factor as a function of time in natural units, for c3 = Λ = 1.

0 2 4 6 8 10 12 14t

0

0.5

1

1.5

2

2.5

3

3.5

4

ξ (t)

Fig. 2 Evolution of the extra-dimension scale factor as a function of time in natural units, for c4 = Λ = 1.

p(t) =Λ

48π

[cot2

(√2

3Λ t

)+ 4

]. (22)

Although bouncing models have their importance

specially because they evade the Big-Bang singularity,

we should discard the present model due to the impossi-

bility of predicting the late-time accelerated expansion

regime of the universe [84,85] from Eq.(19).

2.2.2 Case II: Λ < 0

Following the same approach of the previous section

now for Λ < 0, we obtain the scale factors as

a(t) = c5

√√√√sinh

(√2

3|Λ| t

), (23)

5

ξ(t) = c6cosh

(√23 |Λ| t

)√

sinh(√

23 |Λ| t

) . (24)

The evolution of those scale factors can be appreci-

ated in Figures 3-4 below.

We can see from Fig.3 that a(t) assumes an expo-

nential behaviour as time grows, which may be an in-

dication of the recent cosmic acceleration [84,85]. This

will be clarified in Fig.5 below.

From Fig.4, we can see that the extra dimension

apparently is large for the primordial stages of the uni-

verse. Then, it naturally suffers a process of compacti-

fication, assuming its minimum value for t ∼ 1. Then,

it maximizes its length scale once again.

It is possible to derive a relation between the scale

factors a(t) and ξ(t). Starting from (4) and (6) for Λ < 0

we obtain the system of equations

a

a

ξ

ξ+

(a

a

)2

=Λ

3(25)(

a

a

)+

(a

a

)2

=Λ

3(26)

Subtracting (27) from (28), leads to

ξ

ξ=a

a. (27)

Thus, we obtain a relation between the extra-dimension

scale factor ξ and the time derivative of a as

ξ = Ka, (28)

with constant K. Therefore,

ξ

a= KH =

c6c5

√6

|Λ|H, (29)

where H = H(t) = aa is the Hubble parameter.

The solutions for the induced matter content read

ρ(t) =|Λ|16π

coth2

(√2

3|Λ| t

), (30)

p(t) =|Λ|48π

[coth2

(√2

3|Λ| t

)− 4

], (31)

and the induced density can be rewritten as

ρ =|Λ|16π

(c5c6

ξ

a

)2

(32)

or

ρ =3H2

8π, (33)

where

H(t) =

√|Λ|6

coth

(√2

3|Λ| t

). (34)

The Hubble parameter has its evolution in time

shown in Figure 5.

We see from Fig.5 that the predicted Hubble param-

eter starts evolving as ∼ 1/t, which is, indeed, expected

from standard model predictions [74]. After a period of

time, H(t) ∼ constant. It is known that an exponen-

tial scale factor describes the cosmic acceleration. From

the definition of the Hubble parameter, an exponential

scale factor yields a constant Hubble parameter. In this

way, the constant behaviour that H(t) assumes for high

values of time is an indication of the recent cosmic ac-

celeration.

3 Bag model-like equation of state for the

universe evolution and the deceleration

parameter

In this section we will deeper investigate the solutions

obtained in the previous section for the negative 5D

cosmological constant case. We will show that the in-

duced matter-energy density and pressure can be re-

lated through a bag model-like EoS. We will also derive,

from the scale factor solution, the deceleration param-

eter of the model.

3.1 Unified equation of state for the universe evolution

Considering the cases in which Λ 6= 0, the expressions

for the density and pressure can be remarkably written

in a unified form as

p =(ρ± 4B)

3, (35)

where the constant B = |Λ|16π and the positive sign

stands for Λ > 0 while the negative sign for Λ < 0.

Eq.(35) remarkably resembles the MIT bag model equa-

tion of state (check, for instance [86,87,88]), for which

B is the so-called bag constant. Naturally we are not

claiming that the universe is made of quarks confined

inside a bag, but that the EoS for the universe has the

same mathematical form. The bag constant here is in

fact the bulk energy density necessary to create the vac-

uum in the flat 5D space, and in this sense, it plays the

same role of the bag constant in the MIT model, that

is, the energy density necessary to create a bag in the

QCD vacuum. As we will see, for the universe evolution

given by this EoS, the constant B will be identified with

the dark energy in the 4D universe.

6

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5t

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

a(t)

Fig. 3 Evolution of the scale factor as a function of time in natural units, for c5 = 1 and Λ = −1.

0 1 2 3 4 5t

0

2

4

6

8

10

ξ (t)

Fig. 4 Evolution of the extra-dimension scale factor as a function of time in natural units, for c6 = 1 and Λ = −1.

For the case of cosmological interest, namely the

case Λ < 0, we can separate the density in matter-

radiation and dark energy components, so that

ρ = ρm + ρΛ , (36)

ρm =|Λ|16π

cosech2

(√2

3|Λ|t

), (37)

ρΛ =|Λ|16π

. (38)

We can see from (35) and (38) that the bag energy

constant B in this bag model-like unified EoS for the

universe plays the role of dark energy in the 4D uni-

verse.

One can write the EoS parameter as

ω = −1 +4

3sech2

(√2

3|Λ|t

)(39)

whose evolution in time can be appreciated in Fig.6.

The model prediction for the evolution of the EoS

parameter in time, according to Fig.6, is remarkable.

7

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5t

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

H(t)

Fig. 5 Evolution of the Hubble parameter as a function of time in natural units, for Λ = −1.

0 1 2 3 4 5t

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

ω (t

)

Fig. 6 Evolution of the equation of state parameter as a function of time in natural units, for Λ = −1.

One can note that for small values of time, ω ∼ 1/3. Ac-

cording to standard model (as it was mentioned above),

the primordial value of ω is indeed 1/3, as the primor-

dial universe dynamics is dominated by radiation, such

that p = ρ/3 [74]. As the universe expands and cools

down, it allows pressureless matter to be formed. This

stage represents the matter-dominated stage of the uni-

verse, for which ω ∼ 0, that is also depicted in Fig.6.

Last, but definitely not least, Fig.6 indicates that for

high values of time, ω ∼ −1. According to recent ob-

servations on the cosmic microwave background radia-

tion temperature fluctuations, ω = −1.073+0.090−0.089 [73].

This negative pressure fluid is the responsible for the

cosmic acceleration in standard model. Therefore, our

present approach has also revealed a dominant negative

pressure fluid for high values of time, but remarkably,

it has also predicted other stages of the universe evolu-

tion, named radiation and matter-dominated eras, in a

continuous and analytical form.

8

It is important to show that the expression (36)

for the density satisfies the continuity equation in 4D.

Starting from the continuity equation

ρ+ 3a

a(ρ+ p) = 0, (40)

substituting (35) and integrating on both sides leads to

ρ(t) =c7a(t)4

+B. (41)

Comparing the last expression with Eqs. (36) and (38)

and using the expression (23) for a(t), we obtain c7 =

c54 |Λ|16π , that proves that the continuity equation in 4D

is satisfied in our model.

3.2 The deceleration parameter

The deceleration parameter is defined as

q(t) = − a aa2, (42)

so that q > 0 indicates a decelerated expansion and

q < 0 indicates an accelerated expansion.

In the present model, it can be show that

q =ξ

ξ

a

a= −Hl

H. (43)

Therefore, remarkably the deceleration factor in our

model is the negative ratio between the Hubble param-

eter of the extra-dimension scale factor and the Hubble

parameter in 4D.

Explicitly, the deceleration parameter for Λ < 0

reads

q(t) = 1− 2 tanh2

(√2

3|Λ| t

). (44)

4 Cosmological parameters in terms of redshift

and the observational analysis

With the purpose of confronting our solutions with ob-

servational data, we will study the behavior of the Hub-

ble parameter and other cosmological parameters in

terms of the redshift rather than of time. We will con-

centrate our attention in the case Λ < 0.

Taking into account the scale factor obtained in

(23), the redshift can be written as

z(t) = −1 +1

c5

√sinh

(√23 |Λ| t

) . (45)

The Hubble parameter is then expressed in terms of

redshift as follows

H(z) =

√|Λ|6

coth

[arcsinh

[1

c5(1 + z)

]2}. (46)

The above equation gives a relation between the

Hubble constant, the cosmological constant and the in-

tegration constant c5 as

H0 =

√|Λ|6

coth

[arcsinh

(1

c5

)2]. (47)

4.1 Observational constraints

Hubble parameter data as a function of redshift yields

one of the most straightforward cosmological tests to-

day. It consists on constraining the cosmological models

with values of the expansion rate as a function of red-

shift. It is even more interesting when the Hubble pa-

rameter data come from estimates of differential ages

of objects at high redshifts, because it is inferred from

astrophysical observations alone, not depending on any

background cosmological models (check References [89,

90]).

The data we use here comes from the 51 H(z) data

compilation from Magana et al. [93]. This compilation

consists of 20 clustering (from Baryon Acoustic Oscil-

lations and Luminous Red Galaxies) and 31 differential

age H(z) data.

We choose to work here only with the 31 differen-

tial age H(z) data1, because it does not depend on any

background cosmological model. The age estimates de-

pend only on models of chemical evolution of objects

at high redshifts. H(z) estimates from clustering like

Baryon Acoustic Oscillations usually assume a stan-

dard cosmological model in order to obtain the data

from surveys.

In all analyses here, we have written a χ2 func-

tion for parameters, with the likelihood given by L ∝e−χ

2/2. The χ2 function for H(z) data is given by the

following:

χ2H =

31∑i=1

[Hobs,i −H(zi, s)]2

σ2Hi,obs

, (48)

where s is the parameter vector, which we choose to

be s = (c5, H0). Λ can be related to these parameters

through Eq.(47).

In Figure 7 below, we can see the 31 H(z) data used

here and the best fit H(z) we have found by minimizing

χ2H .

In order to find the constraints over the free parame-

ters, we have assumed flat priors for c5 and H0 and have

sampled the posteriors with the so called Affine Invari-

ant Monte Carlo Markov Chain Ensemble Sampler by

[94], which was implemented in Python language with

1Marked as “DA” in Table 1 of Ref. [93].

9

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

z

50

100

150

200

250

300

350

H(z

)

H0 = 72.2, c5 = 0.60

H0 = 67.4, c5 = 0.58

31 H(z) data

Fig. 7 Hubble parameter as a function of redshift for the best fit parameters from the 31 H(z) data (H0 = 72.2 km/s/Mpc,c5 = 0.60). We also show a curve with H0 = 67.4 km/s/Mpc, in agreement with Planck data [91,92] for the Hubble parameterand for a universe age of 13.8 Gyr, that corresponds to c5 = 0.58. The blue region corresponds to a 2σ (95.4%) c.l. around thebest fit.

the emcee software by [95]. In order to plot all the con-

straints on each model, we have used the freely available

software getdist2, in its Python version.

The results of this analysis can be seen in Fig.8 and

Table 1.

Parameter 95% limits

c5 0.600+0.061−0.058

H0 72.2+5.3−5.5

t0 12.59+0.69−0.62

Table 1 Mean value and 95% limits of the model parame-ters. In bold face are the free parameters and t0 is a derivedparameter. H0 is in units of km/s/Mpc and t0 in Gyr.

In Fig. 9 we present the scale factor of the extra di-

mension as function of the redshift, which can be writ-

ten as

ξ(z) = c5 c6 (1 + z)

√1 +

1

c45 (1 + z)4. (49)

It is interesting to note that the extra-dimension

scale factor has a free constant c6, which is not fixed

by the cosmological analysis. This happens because the

extra dimensional dependence of the cosmological pa-

rameters only appears through the fraction ξ/ξ. This

means that, in this model, even if the scale of the extra

dimension is very small, the cosmological effects would

still be measurable. As a consequence, the extra dimen-

sional lenght scale can be arbitrarly small.

2getdist is part of the great Affine Invariant Monte CarloMarkov Chain Ensemble Sampler, COSMOMC [96].

The deceleration parameter as function of the red-

shift is given by

q(z) = 1− 2

1 + c45 (1 + z)4, (50)

whose behavior can be seen in Fig.10. We can see that

the model gives an accelerated expansion of the uni-

verse (q < 0) for the present epoch. Also, the solution

obtained within the IMM prescription gives a transi-

tion from a decelerated to an accelerating universe ex-

pansion, as expected from supernova observations, in

particular the SN 1997f data photometric observations

by the Hubble Space Telescope [97]. In the analysis of

[98], the transition is expected to occur for z ∼ 0.5,

which is qualitatively compatible to our model predic-

tion (z ∼ 0.66).

The analytical expression for the EoS parameter as

function of redshift is

ω(z) =1

3

[1− 4

1 + c45(1 + z)4

], (51)

whose pattern is shown in Fig.11. It presents the EoS

parameter evolution for different epochs of the universe.

As expected by the standard cosmological model [99],

for recent redshifts the parameter is < −1/3 (dark en-

ergy era) and for past times the EoS parameter presents

a null value, which is compatible with the matter-dominated

phase.

5 Discussion and Conclusion

In the present article we have applied the IMM to a

general 5D metric with scale factors acting in the usual

10

12.0 12.6 13.2 13.8

t0

65

70

75

80

H0

0.50 0.55 0.60 0.65 0.70

c5

12

12.6

13.2

13.8

t 0

65 70 75 80

H0

Fig. 8 Confidence contours from 31 H(z) data analysis of the free parameters of the model, c5 and H0. We also show theconstraints over the total age, t0, which is a derived parameter (H0 in km/s/Mpc, t0 in Gyr). The contours correspond to 68%and 95% c.l..

three space coordinates and in the extra spatial coordi-

nate. We have considered 5D field equations with null

and non-null (namely, positive and negative) cosmolog-

ical constant.

The IMM is a purely geometrical approach in the

sense that matter in the 4D universe appears as a man-

ifestation of a 5D empty space. The mechanism which

describes that is the collection of the extradimensional

dependent terms in the 5D Einstein tensor, which are

“moved” to the rhs of the field equations, playing the

role of an induced energy-momentum tensor. In this

sense, what we have is a realization of the Mach’s prin-

ciple [100]-[101], which was desired by Einstein for a

theory of gravity.

From a quite general approach we have obtained

some cosmological features particularly interesting. We

have shown in Section 2.1 that general KK models with

null cosmological constant are restrict to a radiation-

dominated universe - which evolves as a ∼ t1/2. Since

the matter content obtained from the IMM applica-

tion has a traceless energy-momentum tensor, this is in

agreement with Kaluza’s original idea of unifying gravi-

tation and electromagnetism. Also, we have shown that

the extra-dimension scale factor yields a negative Hub-

ble parameter for the extra coordinate, i.e., the extra

coordinate length compactifies.

In Section 2.2.1, we have inserted a positive 5D cos-

mological constant in the field equations. The approach

has led to a cyclic or bouncing universe, i.e., a universe

that goes from a collapsing era to an expanding era

without displaying the singularity that standard model

carries. Bouncing cosmological models are well-known

alternatives to inflation and also provide the cosmo-

logical perturbations we see today. For a deeper un-

11

0 0.5 1 1.5z

1.4

1.6

1.8 ξ

(z)

Fig. 9 Evolution of the extra-dimension scale factor as a function of the redshift in natural units, for c5 = 0.60 and c6 = 1.

0 0.5 1 1.5z

-1

-0.5

0

0.5

1

q(z)

Fig. 10 Evolution of the universe deceleration parameter as a function of the redshift, for c5 = 0.60.

derstanding of bouncing cosmological models, besides

[77]-[79], we refer the reader to [102].

In Section 2.2.2, we considered Λ < 0. It is inter-

esting to remark here that usually braneworld models

contain negative bulk cosmological constant as a con-

sequence of the appearance of terms ∼√−Λ in their

Friedmann-like equations [103,104].

Our negative cosmological constant model has shown

to be able to uniquely describe the radiation, matter

and dark energy eras of the universe evolution in a con-

tinuous and analytical form, what can be clearly seen,

for instance in Fig. 6.

This is a quite non-trivial result. Cosmological mod-

els able to describe from a single analytical equation of

state the whole history of the universe evolution are

12

0 0.5 1 1.5z

-0.8

-0.4

0

ω (z

)

Fig. 11 Evolution of the EoS parameter as a function of the redshift in natural units, for c5 = 0.60.

rarely obtained in the literature [105,106]. References

[105,106] show cosmological scenarios obtained from

f(R, Tφ) gravity, with R being the Ricci scalar and Tφ

the trace of the energy-momentum tensor of a scalar

field φ, and decaying vacuum models, respectively.

This interesting feature is a consequence of the re-

markable hyperbolic solution obtained for the scale fac-

tor. While we have obtained such a feature from the

model, some other approaches use this solution as a

prior ansatz [107,108,109,110,111,112]. This kind of

hyperbolic solution also is found in the flat ΛCDM con-

cordance model, by neglecting radiation. However, in

this case, the time dependence of the scale factor is like

a(t) ∼ [sinh(t)]2/3

, and not a(t) ∼ [sinh(t)]1/2

, as we

have found here [113,114,115,116].

Furthermore, from our solution for the matter-energy

density (25), it is clear that ρ→ constant for high val-

ues of time, which is also in agreement with standard

model. Here, this constant reads |Λ|/16π, while in stan-

dard model it is Λ/8π. The factor 2 between these en-

ergy densities may be due to the fact that the former

refers to a 5D space-time, and therefore should be more

diluted than a 4D cosmological constant.

Our model also satisfactorily fits the observational

data for the experimental measurement of the Hub-

ble parameter, as shown in Section 4.1. The adopted

method resulted in H0 = 72.2+5.3−5.5 km/s/Mpc, which

is in agreement with the most recent estimate from lo-

cal observations, H0 = 74.03 ± 1.42 km/s/Mpc [117],

and, in the limit, also in agreement with the Planck

collaboration estimate, H0 = 67.4±0.5 km/s/Mpc [92],

in the context of flat ΛCDM cosmology. As a derived

parameter, we have obtained the total age of the Uni-

verse as t0 = 12.59+0.69−0.62 Gyr, which is in agreement

with most of age estimates of objects today. Jimenez

et al. [118] have obtained, with 22 globular clusters

(GC) age estimates from [119], an weighted average of

tGC = 13.0 ± 0.4 Gyr, which is in agreement with our

superior limit (t0 = 13.28 Gyr at 95% c.l.). Our result

is also in agreement with estimates of absolute ages

of very-low-metallicity stars, estimated in the range of

13.0 – 13.535 Gyr, as explained on [118] and references

therein.

Finally, since our unified EoS for the universe was

obtained from the AdS5 space-time and |Λ| is of the

same order of Λ4, the very small value observed for

the cosmological constant has its origin in the energy

to create the vacuum in the 5D space-time and is not

necessarily related to the vacuum energy of quantum

fields in 4D [113].

Acknowledgements M.M.Lapola thanks CAPES, for finan-cial support. PHRSM would like to thank Sao Paulo ResearchFoundation (FAPESP), grants 2015/08476-0 and 2018/20689-7, for financial support. WP thanks CAPES, grant 88881.309870/2018-01 and CNPQ, grants 313236/2018-6 and 438562/2018-6. M.Malheirothanks CAPES, CNPq and the FAPESP thematic project2013/26258-5 for financial support. R. Valentim would like tothank by Fundacao de Amparo a Pesquisa do Estado de SaoPaulo - FAPESP who was supported by thematic project pro-cess no. 2013/26258-2 and regular project process no. 2016/09831-

13

0. JFJ is supported by Fundacao de Amparo a Pesquisa doEstado de Sao Paulo - FAPESP (Process number 2017/05859-0).

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