We consider a single field governed expansion of the universe from a five
dimensional (5D) vacuum state. Under an appropiate change of variables the
universe can be viewed in a effective manner as expanding in 4D with an
effective equation of state which describes different epochs of its evolution.
In the example here worked the universe fistly describes an inflationary phase,
followed by a decelerated expansion. Thereafter, the universe is accelerated
and describes a quintessential expansion to finally, in the future, be vacuum
dominated.Comment: version accepted in Phys. Lett. 

Abolghasem

Balbi

Bellini

Bertolami

Campbell

Carroll

Chang

Davidson

Katqourakis

Katzourakis

Ledesma

Liko

Madriz Aguilar

Mauricio Bellini

McManus

Netterfield

Overduin

Permutter

Ponce de Leon

Romero

Wesson

Physics Letters B

English

arXiv

AbstractWe consider a single field governed expansion of the universe from a five dimensional (5D) vacuum state. Under an appropiate change of variables the universe can be viewed in a effective manner as expanding in 4D with an effective equation of state which describes different epochs of its evolution. In the example here worked the universe firstly describes an inflationary phase, followed by a decelerated expansion. Thereafter, the universe is accelerated and describes a quintessential expansion to finally, in the future, be vacuum dominated

Bellini, Mauricio

Elsevier - Publisher Connector 

Cosmological expansion governed by a scalar field from a 5D vacuum 

Crossref

Cosmological expansion governed by a scalar field from a 5D vacuum

Physics Letters B 637 (2006) 16–20
www.elsevier.com/locate/physletb
Cosmological expansion governed by a scalar field from a 5D vacuum
Mauricio Bellini a,b
a Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, Funes 3350, (7600) Mar del Plata, Argentina
b Consejo Nacional de Ciencia y Tecnología (CONICET), Argentina
Received 6 February 2006; received in revised form 6 April 2006; accepted 19 April 2006
Available online 2 May 2006
Editor: N. Glover
Abstract
We consider a single field governed expansion of the universe from a five dimensional (5D) vacuum state. Under an appropiate change of
variables the universe can be viewed in a effective manner as expanding in 4D with an effective equation of state which describes different epochs
of its evolution. In the example here worked the universe firstly describes an inflationary phase, followed by a decelerated expansion. Thereafter,
the universe is accelerated and describes a quintessential expansion to finally, in the future, be vacuum dominated.
© 2006 Elsevier B.V.
PACS: 04.20.Jb; 11.10.kk; 98.80.Cq
Open access under CC BY license. 1. Introduction and overview of the 5D formalism
In the last years has been an uprising interest in finding ex-
act solutions of the Kaluza–Klein field equations in 5D, where
the fifth coordinate is considered as non-compact [1]. Unlike
the usual Kaluza–Klein theory in which a cyclic symmetry as-
sociated with the extra dimension is assumed, the new approach
removes the cyclic condition on the extra dimension and deriv-
atives of the metric with respect to the extra coordinate are
retained. This induces non-trivial matter on the hypersurfaces
with ψ = const and other non-trivial frames. This theory re-
produces and extends known solutions of the Einstein field
equations in 4D. Particular interest revolves around solutions
which are not only Ricci flat [2], but also Riemann flat [3]:
RABCD = 0.1 This is because it is possible to have a flat 5D
manifold which contains a curved 4D submanifold, as implied
by the Campbell theorem [4]. So, the universe may be “empty”
and simple in 5D, but contain matter of complicated forms in
4D [5].
E-mail address: mbellini@mdp.edu.ar (M. Bellini).
1 In our conventions, capital Latin indices run from 0 to 4 and Greek indices
run from 0 to 3.0370-2693 © 2006 Elsevier B.V.
doi:10.1016/j.physletb.2006.04.033
Open access under CC BY license. One of the greatest challenges of modern cosmology is un-
derstanding the nature of the observed late-time acceleration
of the universe. Recent measurements of type Ia Supernovae
(SNIa) [6] at redshifts z ∼ 1 and also the observational results
coming from the Cosmic Microwave Background Radiation
(CMBR) along with the Maxima [7] and Boomerang data [8]
indicate that the expansion of the present universe is acceler-
ated. In fact the present day results show that supernovae are
moving faster than expected from the luminosity redshift rela-
tionship in a decelerating universe. A possible explanation is
that in the universe there exists an important matter component
which, in its most simple description, has the characteristic of
a cosmological constant as vacuum energy density which con-
tributes to a large component of negative pressure [9,10]. The
idea that the expansion of the universe could be governed by
a scalar field has been developed in the quintessential models,
where the dynamics of the scalar field is governed by an ap-
propiate potential V (ϕ) [11]. This paper is devoted to study the
evolution of the universe which firstly suffers an inflationary
expansion followed by a decelerated expansion (matter and ra-
diation dominated) to finally be monotonically accelerated until
a quasi-de Sitter expansion from a 5D vacuum state, where the
(space-like) fifth dimension is considered as non-compact. We
shall suppose that the universe is governed by a scalar field,
M. Bellini / Physics Letters B 637 (2006) 16–20 17which is minimally coupled to gravity. For the system, we shall
consider that the action is
(1)I = −
∫
d4x dψ
√∣∣∣∣ (5)g(5)g0
∣∣∣∣
[ (5)R
16πG
+ (5)L(ϕ,ϕ,A)
]
,
where ϕ is a scalar (neutral) quantum field, G = M−2p is the
gravitational constant [being Mp = 1.2×1019 GeV the Planck-
ian mass], and (5)R is the Ricci scalar. Furtheremore (5)g is the
determinant of the covariant tensor metric gAB (A,B can take
the values 0,1,2,3,4). We are interested to describe a mani-
fold in apparent vacuum, so that the Lagrangian density L in
(1) should be only kinetic
(2)(5)L(ϕ,ϕ,A) = 12g
ABϕ,Aϕ,B.
To describe a 3D spatially isotropic and homogeneous universe,
which is Ricci flat: RABCD = 0 and describes a 5D vacuum:
GAB = 0, we shall consider the background metric [12]
(3)dS2 = ψ2 dN2 − ψ2e2N dr2 − dψ2,
where ψ is the space-like fifth coordinate, r(x, y, z) =√
x2 + y2 + z2 and N , x, y, z are dimensionless coordinates.
Furthermore, the metric (3) is 3D spatially isotropic, homoge-
neous and flat. For the metric (3), the determinant of the covari-
ant metric tensor gAB is |(5)g| = ψ8e6N and |(5)g0| = ψ80 e6N0
is a constant of dimensionalization, for the constants ψ = ψ0
and N = N0. On the other hand, the energy–momentum tensor
is given by
(4)TAB = ϕ,Aϕ,B − 12gABϕ,Cϕ
,C ,
which is symmetric because the symmetry of gAB . The dynam-
ics for ϕ is described by the Lagrange equation
(5)∂
2ϕ
∂N2
+ 3 ∂ϕ
∂N
− e−2N∇2r ϕ − ψ
(
ψ
∂2ϕ
∂ψ2
+ 4 ∂ϕ
∂ψ
)
= 0.
In absence of 3D spatially isotropic field fluctuations, the back-
ground field ϕb(N,ψ) corresponding to the vacuum equation
(
∂ϕb
∂N
)2 + ψ2( ∂ϕb
∂ψ
)2 = TAB |b = 0 on the background metric (3),
is a constant of N and ψ : ϕb(N,ψ) = const. The commutator
between ϕ and

ϕ will be
(6)
[
ϕ(N, r,ψ), ϕ (N, r ′,ψ ′)]= i∣∣∣∣ (5)g0(5)g
∣∣∣∣δ(3)(r − r ′)δ(ψ − ψ ′).
In order to simplify the structure of Eq. (5) we can make the
transformation ϕ = χe−3N/2(ψ0
ψ
)2. The dynamics for χ(N,
r,ψ) being described by the equation of motion
(7)χ −
[
e−2N∇2r +
(
ψ2
∂2
∂ψ2
+ 1
4
)]
χ = 0,
and the commutator between χ and

χ is
(8)[χ(N, r,ψ), χ (N, r ′,ψ ′)]= iδ(3)(r − r ′)δ(ψ − ψ ′).We can make a Fourier expansion for χ
χ(N, r,ψ)
= 1
(2π)3/2
∫
d3kr
∫
dkψ
[
akrkψ e
i( kr ·r+kψ ·ψ)ξkrkψ (N,ψ)
(9)+ a†kr kψ e−i(
kr ·r+kψ ·ψ)ξ∗kr kψ (N,ψ)
]
,
where the asterisk denotes the complex conjugate and (akrkψ ,
a
†
kr kψ
) are respectively the annihilation and creation operators.
They satisfy the following commutation expressions
(10)[akrkψ , a†kr kψ ]= δ(3)(kr − k′r )δ(kψ − k′ψ),
(11)[a†kr kψ , a†k′r k′ψ ]= [akrkψ , ak′r k′ψ ] = 0.
The expression (8) complies if the modes are normalized by the
following condition:
(12)ξkrkψ (

ξkrkψ )
∗ − (ξkr kψ )∗

ξkr kψ = i.
This equation provides the normalization for the complete set
of solutions on all the spectrum (kr , kψ ). As was demonstrated
in a previous work [13], ξkrkψ (N,ψ) = e−ikψ . ψ ξ¯kr (N), where
ξ¯kr (N) is a solution of
(13)

ξ¯ kr +
(
k2r e
−2N − 1
4
)
ξ¯kr = 0,
such that the normalization condition for ξ¯kr (N) becomes
(14)ξ¯kr
(
ξ¯ kr
)∗ − (ξ¯kr )∗ ξ¯ kr= i,
where the overstar denotes the derivative with respect to N .
Hence, the field χ in Eq. (9) can be rewritten as
χ(N, r) = 1
(2π)3/2
∫
d3kr
∫
dkψ
[
akrkψ e
ikr .r ξ¯kr (N)
(15)+ a†kr kψ e−i
kr .r ξ¯∗kr (N)
]
.
Furthermore, the functions ξ¯kr (N) are given by [13]
(16)ξ¯kr (N) =
i
√
π
2
H(2)1/2
[
kre
−N ].
Finally, the field ϕ is given by
(17)ϕ(N, r,ψ) = e− 3N2
(
ψ0
ψ
)2
χ(N, r),
with χ(N, r) given by Eq. (15). It is very important to notate
that exponentials e±ikψ . ψ disappear in χ(N, r) and there is not
dependence on the fifth coordinate ψ in this field. This fact is
an evidence of that the field ϕ(N, r,ψ) propagates only on the
3D spatially isotropic space r(x, y, z), but not on the additional
space-like coordinate ψ .
18 M. Bellini / Physics Letters B 637 (2006) 16–202. An effective 4D model of expansion for the universe
In order to develope an effective 4D model for an universe
which is governed by the scalar field ϕ, we shall use the follow-
ing change of variables on the metric (3)
(18)dt = ψ dN, dR = ψ dr, ψ = ψ
so that the resulting metric becomes
(19)dS2 = dt2 − e2
∫
dt
ψ dR2 − dψ2.
On hypersurfaces ψ = 1/H(t) the metric (19) becomes
(20)dS2 = dt2 − e2
∫
H dtdR2 − [d(H−1)]2,
which also can be rewritten as an effective 4D metric
(21)dS2 → ds2 =
[
1 −
(
H˙
H 2
)2]
dt2 − e2
∫
H dt dR2,
which is well defined when 1 − ( H˙
H 2
)2 	= 0. Notice that, from
the mathematical point of view, the change of variables (18)
do not describes a transformation of coordinates. This change
of variables describes a map from the particular frame Uψ = 0
[of the metric (3)], to the particular frame uR = 0 of the effec-
tive 4D metric (21). Here, UA = dxA
dS
are the penta velocities
of the original metric (3) and uμ = dXμ
ds
are the tetra velocities
of the effective 4D metric (21), such that gABUAUB = 1 and
gμνuμuν = 1, respectively.
Furthermore, the density Lagrangian L of Eq. (2) can be ex-
panded as
(22)L(ϕ,ϕ,A) = 12
[
gμνϕ,μϕ,ν + gψψϕ,ψϕ,ψ
]
,
so that we can make the following identification for the scalar
potential on the effective 4D metric (21):
(23)V (ϕ) = 1
2
gψψϕ,ψϕ,ψ |ψ=1/H = 2H 2ϕ2(t, R).
Using the change of variables (18), the equation (5) can be
rewritten on the 4D effective submanifold (21), as
(24)ϕ¨ +
(
3H − H˙
H
)
ϕ˙ − e2
∫
H dt∇2Rϕ + V ′(ϕ) = 0,
with
(25)V ′(ϕ) = −H 2[ψ2ϕ,ψψ + 4ψϕ,ψ]∣∣ψ=H−1 = 2H 2ϕ.
2.1. The effective 4D equation of state
Using the effective 4D metric (21), we obtain the following
Einstein’s equations
(26)3H
6
H˙ 2 − H 4 = 8πGρ,
H 6
(H˙ 2 − H 4)2
[
3H 4 − 3H˙ 2 + 2H˙H 2 − 6 H˙
3
H 2
+ 2H¨ H˙
H
]
(27)= 8πGp,where ρ and p are, respectively, the energy density and the
pressure. The effective 4D equation of state for the universe
becomes
(28)p
ρ
= ωeff(t)
with
(29)ωeff(t) = −
[
1 + (2H˙H
2 − 6 H˙ 3
H 2
+ 2 H˙ H¨
H
)
3(H 4 − H˙ 2)
]
,
which, for a Hubble parameter H = p/t (with constant p)
agrees exactly with that of a spatially flat 4D Friedmann–
Robertson–Walker (FRW) metric ds2 = dt2 − e2
∫
H dt dR2:
ωeff(t)|H=p/t = 2−3p3p = ωFRW(t). Some particular cases p →∞, p = 2/3 and p = 1/2, give us respectively ωeff → −1,
ωeff = 0 and ωeff = 1/3, which describes respectively expan-
sions dominated by vacuum, matter and radiation. Note that for
p > 1 the effective 4D metric (21) is always Lorentzian in na-
ture. However for p = 1 this metric is not well defined. On the
other hand, for p(t) < 1 the metric (21) is Euclidean and hence
losses its relativistic nature. Hence, a well defined model for the
expansion of the universe must be developed using p > 1 in the
metric (21). For a more realistic model with a time dependent
p(t), one obtains ωeff(t)|H=p(t)/t 	= ωFRW(t). In Section 3 we
shall study with more detail this last case.
2.2. Comoving frame
In order to describe the evolution of the universe on a co-
moving frame, which is the relevant for cosmological models,
we can make use of the hyperbolic condition gμνuμuν = 1 on
the 4D effective metric (21). Here, uμ = dxμ
dS(t)
, are the tetrave-
locities being uR = 0 in the comoving frame. In this frame the
velocity ut is
(30)ut = 1√
1 − (H˙ /H 2)2
.
Note that when
(31)−H˙ /H 2  1,
we obtain ut  1 and the metric (21) describes an asymptotic
4D FRW metric, which usually is used to describe the universe
in cosmological models.
2.3. Evolution of the background field ϕb
The effective 4D evolution of the background field ϕb(t) is
given by Eq. (24) with ∇2Rϕ = 0
(32)ϕ¨b +
(
3H − H˙
H
)
ϕ˙b + 2H 2ϕb = 0,
which has the general solution
(33)ϕb(t) = ϕb(0)2 e
− ∫ H(t) dt[1 + e− ∫ H(t) dt],
where ϕb(0) = ϕb(t = 0). It is evident that ϕb(t) decreases
monotonically with the time and rolls down the minimum of
the potential, so that ϕb(t → ∞) → 0.
M. Bellini / Physics Letters B 637 (2006) 16–20 19Fig. 1. Evolution of p[x(t)] as a function of x(t) = log10(t/t0).
3. Model of expansion of the universe
In order to describe all the evolution of the universe we
can propose the following expression for the Hubble: H(t) =
p(t)/t , such that
(34)p(t) = 1.8at−n +
(
b2
4a
+ 0.62
)
+ ct,
where a = 1/6 × 1030nGn/2, b = 8/7 × 1015nGn/4, c = 2.0 ×
10−61G−1/2 and n = 0.352. The general solution (33) in the
example we are worked assume the explicit expression
ϕb(t) = ϕb(0)e− 3ct2
(
t
t0
)−31/25{(
t
t0
)−b2/(2a)
e−[
36at−n−5cnt
10n ]
(35)+
(
t
t0
) 62a−25b2
100a
e
18at−n+5cnt
10n
}
.
In Fig. 1 is ploted the parameter p[x(t)] [where x(t) =
log10(t)], which always remains with values p > 1. In the
Fig. 2 is shown the evolution for the cosmological parameter
weff[x(t)]. Note that for x(t) < 20 the universe is governed
by vacuum and describes an inflationary expansion, but later
weff[x(t)] increases to thereafter describe a phase with pos-
itive pressure: ωeff > 0 (in the range 30 < x(t) < 60). Af-
ter it, the pressure decreases and for x(t) > 60 the universe
expands with negative pressure until the present day, when
ωeff[x  60.652]  −0.7. This result agrees with the experi-
mental data [14]. In the Fig. 3 is ploted (ωeff − ωFRW)[x(t)].
It is very clear that the discrepance between ωeff and ωFRW be-
comes more notorious in epochs where the universe expands
with positive pressure. However, in both epochs (in the very
early universe and the present day universe), the equation of
state for the universe agrees with that predicted by a 4D spa-
tially flat FRW metric: ds2 = dt2 − e2
∫
H dt dR2. More exactly,
the condition (31) holds for
(36)1
p(t)
− t p˙(t)
p(t)2
 1.Fig. 2. Evolution of ωeff as a function of x(t) = log10(t/t0).
Fig. 3. Evolution of ωeff − ωFRW as a function of x(t) = log10(t/t0).
4. Final comments
In this Letter we have studied the evolution of the universe
which is considered as governed by a single scalar field from
a 5D vacuum state, where the fifth dimension is considered as
non-compact [see the metric (3)]. However when we make the
change of variables (18) the universe describes an effective 4D
evolution which can be described by the metric (21). This met-
ric is well defined for (H˙ /H 2)2 	= 1, which is the case we are
considered in this Letter. If we consider a Hubble parameter
H(t) = p(t)/t , this metric remains Lorentzian for p > 1. In the
example here considered, with a power law expansion (34), the
universe initially describes an equation of state with ωeff  −1,
which increases until take values of the order of ωeff  0.4.
Thereafter, this cosmological parameter begins to decrease to
take values which agree very good with the present day ex-
perimental data ωeff  −0.7. This result is consistent with a
quintessential expansion. The model predicts that, in the future,
the universe will be more and more accelerated to finally de-
scribe an effective 4D vacuum dominated expansion ωeff  −1.
Acknowledgements
M.B. acknowledges CONICET (Argentina) and UNMdP for
financial support.
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Cosmological expansion governed by a scalar field from a 5D vacuum

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