Higher-dimensional models in gravitational theories of quarticLagrangians

Abstract

Ten-dimensional models, arising from a gravitational action which includes terms up to the fourth order in curvature tensor, are discussed. The spacetime consists of one timelike dimension and two maximally symmetric subspaces, filled with matter in the form of an anisotropic fluid. Numerical integration of the cosmological field equations indicates that exponential, as well as power law, solutions are possible. We carry out a dynamical study of the results in the H_{ext} - H_{int} plane and confirm the existence of "attractors" in the evolution of the Universe. Those attracting points correspond to "extended De Sitter" spacetimes, in which the external space exhibits inflationary expansion, while the internal one contracts. Comment: LaTeXfile, 22 pages

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arXiv:gr-qc/9905042v1 13 May 1999
Higher-dimensional models in gravitational theories
of quartic Lagrangians
K. Kleidis, A. Kuiroukidis, D. B. Papadopoulos and H. Varvoglis
Department of Physics
Section of Astrophysics, Astronomy and Mechanics
Aristotle University of Thessaloniki
54006 Thessaloniki, GREECE
February 7, 2008
Abstract
Ten-dimensional models, arising from a gravitational action which includes
terms up to the fourth order in curvature tensor, are discussed. The spacetime
consists of one time direction and two maximally symmetric subspaces, filled
with matter in the form of an anisotropic fluid. Numerical integration of the
cosmological field equations indicates that exponential, as well as power-law,
solutions are possible. We carry out a dynamical study of the results in the
Hext−Hintplane and confirm the existence of attractors in the evolution of the
Universe. Those attracting points correspond to ”extended” De Sitter space-
times in which the external space exhibits inflationary expansion, while the
internal one contracts.
PACS Codes: 98.80.Hw , 11.10.Kk
I. Introduction
The mathematical background for a non-linear Lagrangian theory of gravity was
first formulated by Lovelock1, who proposed that the most general gravitational La-
grangian is
L =√−g
m=0
where λ(m) are coupling constants, n denotes the manifold’s dimensions, g is the
determinant of the metric tensor and L(m)are functions of the Riemann curvature
n/2
?
λ(m)L(m)
(1.1)
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tensor, of the form
L(m)=
1
2mδβ1...β2m
α1...a2mRα1α2
β1β2... Rα2m−1α2m
β2m−1β2m
(1.2)
where δα
cosmological constant, L(1)=1
the quadratic Gauss-Bonnett (GB) combination2. Euler variation of the gravitational
action corresponding to Eq.(1.1) yields the most general symmetric and divergenceless
tensor, which describes the propagation of the gravitational field and depends only
on the metric and its first and second order derivatives1.
While quadratic Lagrangians have been widely studied (e.g. see Refs. [3,4] and ref-
erences therein), cubic and/or quartic Lagrangians only recently have been introduced
in the discussion of cosmological models in the framework of superstring theories5−10.
The reason is that, it is very hard to derive and (even harder) to solve the cor-
responding field equations. In this case, solutions may be obtained only through
certain numerical techniques11,12, where the idea of ”attractor” plays a central role13:
If some special spacetime is the attractor for a wide range of initial conditions, such a
spacetime is naturally realized asymptotically. Since the ten-dimensional superstring
theory is a candidate for a realistic unified theory, it is very important to investigate
whether a similar attractor exists in this theory.
In the present paper we integrate numerically the field equations, resulting from a
quartic gravitational Lagrangian, to obtain anisotropic, ten-dimensional cosmological
models. The spacetime consists of one time direction and two maximally symmet-
ric subspaces, FRW⊗FRW: The external space, representing the ordinary Universe
and the internal one, constituted by the extra dimensions. The internal space is a
compact manifold of very small ”physical size” with respect to that of the ”visible”
space at the present epoch14,15. Since, on the other hand, at the origin the two sub-
spaces were of comparable physical size, the internal one must have somehow been
contracted towards a static value of the order of Planck length, lPl ∼ 10−33cm, to
achieve ”spontaneous compactification”16. Compactification is a topological process
of quantum origin, which leads to the separation of the extra dimensions from the or-
dinary ones17. In what follows we consider models of an already compactified internal
space, i.e. we study only its contraction.
In Section II we derive the explicit form of the field equations for a quartic theory
in ten dimensions, in which both subspaces are filled with an anisotropic fluid. In
Section III we solve numerically the field equations, for a wide range of initial condi-
tions and for several values of the ”free” parameters involved, as regards (1) vacuum
models of flat subspaces and (2) perfect fluid models of positively curved subspaces.
Next, we carry out a dynamical study in the Hext−Hintplane, where each Hjrepre-
sents the Hubble parameter of the corresponding subspace. Accordingly, we confirm
the existence of attracting points and investigate their evolution with respect to the
variation of the coupling constants λ(m). The explicit time-dependence of the un-
known scale functions may be subsequently determined by solving the linearized field
equations around these attracting points. The corresponding analysis is presented in
βis the Kronecker symbol, L(0)is the volume n-form which gives rise to the
2R is the Einstein-Hilbert (EH) Lagrangian and L(2)is
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Section IV.
II. The field equations in a quartic gravity theory
We consider a ten-dimensional line element, representing cosmological models which
consist of two homogeneous and isotropic factor spaces, of the form
ds2= − dt2+ R2(t)
?3
i=1(dxi)2
4kext
1 +
1
?3
i=1(xi)2+ S2(t)
?9
j=4(dxj)2
?9
1 +
1
4kint
j=4(xj)2
(2.1)
where ¯ h = 1 = c, R(t) and S(t) are the cosmic scale functions of the external and
the internal space respectively, kext = −1,0,+1 is the curvature parameter of the
”ordinary” space and kint = 0,+1 is the corresponding parameter of the internal
one. Therefore, the extra dimensions may be compactified either in a six-dimensional
sphere, for kint= +1, or in a six-dimensional torus, for kint= 0. The spatial section
of the metric (2.1) can be viewed as the direct product of two FRW models with three
and six dimensions respectively6. These models may be obtained through Hamilton’s
principle, from a ten-dimensional action in which the gravitational part is of the form
I =
1
Vint
?√−g
?
λ(0)L(0)+ λ(1)L(1)+ λ(2)L(2)+ λ(3)L(3)+ λ(4)L(4)
?
d10x(2.2)
where each of L(m)is given by Eq.(1.2), λ(m)are the corresponding coupling constants
and Vintis a normalization constant18, corresponding to the ”physical size volume” of
the internal space, once it may be considered static3. The field equations read
Lµν = − 8πG10Tµν
(2.3)
where Lµνis the Lovelock tensor up to the fourth order in curvature (Greek indices
refer to the ten-dimensional spacetime)1and G10 = G Vint is the ten-dimensional
gravitational constant19. Tµνis the energy-momentum tensor of an anisotropic per-
fect fluid source, of the form Tµν= diag (ρ,−pext,...,−pint,...), where ρ is the total
mass-energy density, while pextand pintare the pressures associated to each factor
space, separately. For the metric (2.1) Eq.(2.3) is decomposed into three independent
equations of the form (cf. Ref. [9])
16πG10ρ = λ(0)+ 6λ(1)
?
P + 5Q + 6(
˙R
R)(
˙R
R)2(
˙S
S)
˙S
S)2+ 2P(
?
+ 72λ(2)
?
5Q2+ 5PQ + 10(
˙R
R)(
˙R
R)(
˙S
S) + 20Q(
˙S
S) + 12PQ(
˙R
R)(
˙S
S)
˙R
R)(
?
+ 720λ(3)
?
Q3+ 8(
˙R
R)3(
˙S
S)3+ 9PQ2+ 18Q2(
˙S
S)
+36Q(
˙R
R)2(
˙S
S)2
?
+ 17280λ(4)
?
PQ3
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+ 6Q2(
˙R
R)2(
˙S
S)2+ 6PQ2(
˙R
R)(
˙S
S) + 8Q(
˙R
R)3(
˙S
S)3
?
(2.4a)
− 16πG10pext = λ(0)+ 2λ(1)
?
P + 15Q + 12(
˙R
R)(
˙S
S) + 2(
¨R
R) + 6(
˙R
R)(
¨R
R)(
¨S
S)
˙S
S) + 20Q(
˙S
S)
?
+ 24λ(2)
?
15Q2+ 10(
˙R
R)2(
˙S
S)2+ 5PQ + 40Q(
¨S
S)(
˙R
R)(
¨S
S)(
?
¨R
R)(
¨S
S)
+2P(
¨S
S) + 10Q(
¨R
R) + 20(
˙R
R)(
˙S
S) + 4(
˙S
S) + 6Q2(
˙R
R)2(
¨R
R)
˙S
S) + 6PQ2(
˙R
R)2(
˙R
R)(
¨S
S) + 6Q2(
?
+ 720λ(3)
?
Q3+ 3Q2P + 12Q2(
¨R
R)
˙R
R)(
+ 12Q(
˙R
R)2(
¨R
R)(
¨S
S)(
¨S
S)(
˙S
S) + 4PQ(
˙S
S)
˙R
R)(
˙R
R)2(
¨S
S) + 8(
˙S
S)2+ 24Q(
¨S
S)(
˙S
S)
+8Q(
˙R
R)(
?
+ 5760λ(4)
2Q3(
+ 12Q2(
˙S
S) + 12Q2(
˙S
S)2+ PQ3+ 6Q2(
˙R
R)(
¨S
S)
+ 24Q(
˙S
S)2
?
(2.4b)
− 16πG10pint = λ(0)+ 2λ(1)
?
3P + 10Q + 15(
˙R
R)(
˙S
S) + 3(
¨R
R) + 5(
˙R
R)(
¨S
S) + 10(
¨S
S)
?
+ 24λ(2)
?
5Q2+ 20(
˙R
R)2(
˙S
S)2+ 10PQ + 5P(
¨S
S) + 10Q(
˙S
S) + P(
¨R
R)(
¨R
R)
˙R
R)(
+30Q(
˙R
R)(
˙S
S) + 10Q(
˙S
S)
˙R
R)3(
¨R
R) + 5P(
˙S
S)
+20(
¨S
S)(
˙R
R)(
?
+ 720λ(3)
?
4(
˙S
S)3+ 3PQ2+ 3Q2(
˙R
R)(
˙R
R)2(
¨S
S)(
¨R
R)(
˙R
R)(
˙S
S)2+ 2PQ(
˙R
R)2(
?
¨S
S) + 12Q(
˙S
S) + 3Q2(
¨R
R)
+2Q2(
¨S
S) + 6PQ(
¨S
S) + 4(
¨R
R)(
˙S
S) + 12Q(
˙S
S)2+ 12(
˙R
R)2(
¨S
S)(
˙S
S)
¨R
R)
+6PQ(
¨R
R)(
˙S
S) + 12Q(
˙R
R)(
˙S
S)2+ 4P(
¨S
S)(
˙R
R)(
˙S
S)
+ 12Q(
˙R
R)(
˙R
R)(
+ 5760λ(4)
?
6Q2(
˙S
S) + 3PQ2(
¨S
S)(
˙R
R)2(
˙S
S)2
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+ 12PQ(
¨S
S)(
˙R
R)(
˙S
S) + 3PQ2(
˙R
R)(
˙S
S) + 4Q(
˙R
R)3(
˙S
S)3
+ 3PQ2(
¨R
R) + 12Q(
¨R
R)(
˙R
R)2(
˙S
S)2+ 8(
¨S
S)(
˙R
R)3(
˙S
S)3
?
(2.4c)
where an overdot denotes derivative with respect to time and we have set
P = (
˙R
R)2+kext
R2
, Q = (
˙S
S)2+kint
S2
(2.5)
Since the Lovelock tensor is divergenceless, Lµν
Tµν
;ν= 0, we obtain the conservation law
;ν= 0, which gives
˙ ρ + 3(ρ + pext)
˙R
R
+ 6(ρ + pint)
˙S
S= 0 (2.6)
Further inspection of the system of Eqs. (2.4) and (2.6) shows that only three of
them are truly independent. Thus, the problem is completely determined by those,
plus the two equations of state for the matter content, one for each subspace17. In the
present article we consider two cases with regard to the energy-momentum tensor:
(a) Vacuum models, ρ = 0, in connection to flat spatial sections (kext= 0 = kint) and
(b) Models of an heterotic superstring gas20, pext=1
possitively curved spatial sections (kext= 1 = kint). In the later case, the conservation
law (2.6) gives
3ρ and pint= 0, in connection to
ρ =
M
R4S6
(2.7)
where M is an integration constant. Thus, the external space is radiation dominated9,11,20.
In principle, we may integrate the system of Eqs. (2.4) and (2.6) to obtain the
form of the unknown scale functions. However this is not an easy task, even in the
most simple and symmetric cases9. Nevertheless, we may get a good estimation of
their dynamic behaviour through numerical integration11.
Once the two equations of state are determined, Eq.(2.6) may be readily solved to
give the unknown energy density and pressures, as functions of R(t) and S(t). These
expressions are subsequently introduced in the r.h.s. of Eqs.(2.4). Now, only two of
these equations are truly independent. The third one corresponds to a constraint,
to be satisfied by the solutions of the system. As such, we choose Eq.(2.4a). The
remaining independent field equations (2.4b) and (2.4c) may be recast in the form of
a first order system (see also [11]), as follows
˙Hext= G1(Hext, Hint, X , Y )(2.8a)
˙Hint= G2(Hext, Hint, X , Y )
˙X = − X Hext
˙Y = − Y Hint
(2.8b)
(2.8c)
(2.8d)
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where we have set
Hext=
˙R
R, Hint=
˙S
S, X2=kext
R2, Y2=kint
S2
(2.9)
and the explicit forms of the functions G1and G2are given in the Appendix A.
Finally, it is convenient to make a parameter rescaling in the field equations, of
the form
κm =
λ(1)
λ(m)
, m = 0, 1, 2, 3, 4(2.10)
where λ(1) = (16πG)−1is the coupling constant in the four-dimensional General
Relativity (GR). The value of the normalized coupling constants, κm (κm ≤ 1), is
directly proportional to the contribution in the field equations of the corresponding
m−th order non-linear term, with respect to the results obtained in the EH cosmology.
Clearly, κ1= 1.
III. Numerical Results
We integrate numerically the system of Eqs.(2.8). The constraint (2.4a) is checked
to be satisfied with an accuracy of 10−10along integration. The initial conditions
Hext
spaces are separated, but of the same ”physical size”16. (b) Hext
ordinary space expands, in accordance to what we observe at the present epoch12,21.
(c) Hint
0
< 0, i.e. at the origin, the internal space contracts, in correspondance to
”spontaneous compactification”12,16,17,21. The cases where either Hext
are not permitted, since the constraint equation is not satisfied. Nevertheless, the case
where both conditions Hext
0
< 0 and Hint
0
analysis. Actually, it corresponds to the time-reversed solution of the system (2.8).
The time coordinate is measured in dimensionless units, being normalized with
respect to the Planck time, τ = t/tPl(tPl=√G ∼ 10−43sec). The limits of numerical
integration range from τ = 0 to τ = 105. The upper limit coincides with the origin
of the GUT epoch21, tGUT= 105tPl, corresponding to the end of the string regime22.
However we have to point out that, although the origin of the time coordinate is set
at τ = 0, the equations (2.8) may not be valid in the region 0 < τ ≤ 1 since, in the
absence of a quantum gravity theory, there is always a region of ambiguity around
t = 0, of the order of Planck time23−25.
The solution of the system (2.8) may be represented as curves in the Hext− Hint
plane. Any point located on these curves always satisfies the constraint condition
(2.4a). Thus, the curves actually represent ”orbits” of the dynamical system under
study. Each curve, corresponding to a different set of initial conditions, is bounded by
fixed points (or infinities) and represents a different type of evolution for the Universe.
In what follows, we focus attention on the existence and the evolution of attracting
points in the Hext− Hint plane. The reason rests in the physical meaning of the
0, Hint
0, X0, Y0are chosen so that: (a) X0= Y0, i.e. at the origin, the two factor
0
> 0, i.e. initially the
0
< 0 or Hint
0
> 0
> 0 are valid is acceptable by numerical
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attractor: No matter what the behaviour of a cosmological model at the origin might
be, it will always end up to evolve as indicated by the location of the attracting point
in the Hext− Hintplane.
(1) Vacuum models with spatially flat subspaces
We study the evolution of vacuum ten-dimensional cosmological models, with metric
of the form (2.1), in which both subspaces are spatially flat, i.e. kext = 0 = kint.
Thus, X = 0 = Y .
The first case to study are the GB models (see also Refs. [11,12]). In this case,
κ2= 1 and κ0= 0 = κ3= κ4. The non-linear curvature contributions to the field
equations come out from the quadratic terms alone. The time evolution of the Hubble
parameters is presented in Fig. 1a. We see that both parameters evolve to approach
constant values in the later stages. This situation verifies the existence of attracting
points in the Hext− Hintplane during the evolution of the Universe. Therefore, for
a wide range of initial conditions, both subspaces will end up to evolve as De Sitter
spaces, in complete correspondence to the results of Ishihara12.
We also observe that Hext> 0 and Hint< 0. Therefore, while the internal space
contracts exponentially to achieve spontaneous compactification, the external one
expands, a fact that corresponds to an inflationary phase. This result indicates that
the introduction of the non-linear curvature terms into the gravitational action may
play an important role as far as the inflation is conserned26−30. The explicit location
of the attracting point is shown in Fig. 1b. The attractor corresponds to the fixed
point D2recognized by Ishihara12in the evolution of the extended De Sitter models
in GB theory.
The next step is to introduce into the problem a ”bare” cosmological constant,
Λ, corresponding to the expectation value of the vacuum energy density25. Now, in
addition to κ2, we also have κ0?= 0, while κ3= 0 = κ4. When κ0 ∈ [0 , 1] the value
of the cosmological constant in physical units is Λ = 2κ0×10−48cm−2, which is quite
small.
The behaviour of the model is qualitatively similar to the previous case. Again we
verify the existence of an ”attractor”. Both subspaces correspond to De Sitter models.
The external space exhibits inflationary expansion, while the internal one contracts.
However, in this case, the location of the attracting point D2has changed to higher
absolute values in the evolution of Hext and Hint (Fig. 2a). We may determine
explicitly the law of the attractor’s displacement in the Hext− Hintplane, caused by
variations of the cosmological constant.
In general, to determine the exact location of the attracting points in an Hext−Hint
plane, requires to set
G1(Hext,Hint,X,Y ) = 0(3.1.1a)
G2(Hext,Hint,X,Y ) = 0(3.1.1b)
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In the case of flat and vacuum subspaces (X = Y = pext= pint= 0), Eqs.(3.1.1) read
f1(Hext,Hint,κm) = [G12G20− G22G10]X=Y =0= 0
f2(Hext,Hint,κm) = [G21G10− G11G20]X=Y =0= 0
where m = 0,2,3,4 and the quantities Gij are presented in the Appendix A. We
differentiate the functions f1and f2with respect to Hext, Hintand κm, to obtain a
system of first order differential equations (”variational equations”)
(3.1.2a)
(3.1.2b)
df1= (∂f1
∂Hext)PdHext+ (∂f1
∂Hint)PdHint+
?
j
(∂f1
∂κm)Pdκm= 0 (3.1.3a)
df2= (∂f2
∂Hext)PdHext+ (∂f2
∂Hint)PdHint+
?
j
(∂f2
∂κm)Pdκm= 0 (3.1.3b)
The system (3.1.3) may be used, to determine the evolution of the attracting point
D2 (Hext,Hint), under the variation of the normalized coupling constants κm. For
κ2= 1, in the case of vanishing κ3and κ4, the evolution of the attractor D2(Hext,Hint)
with respect to the variation of the cosmological constant κ0, is given by
(dHext
dκ0
) = (QQ1
PP) (3.1.4a)
(dHint
dκ0
) = (QQ2
PP) (3.1.4b)
where the functions PP, QQ1and QQ2are given in the Appendix B. Subsequently,
the system (3.1.4) is evaluated by numerical integration. The corresponding results
are shown in Fig. 2b. Using least square fitting, we see that the displacement of D2
takes place along the straight line
Hint= −0.075Hext− 0.071(3.1.5)
The investigation of the behaviour of the models under consideration by including
a third order curvature term, corresponds to study them at earlier epochs in the
history of the Universe. Indeed, if we are interested in the behaviour of the model
very close to the initial singularity, the leading terms to consider in the field equations
are those with the highest power in
?1
terms in the gravitational action9.
The time-evolution of the model is quite similar to the previous cases. In the later
stages it corresponds to an extended De Sitter model, in which both subspaces evolve
exponentially. The external space expands, while the internal one contracts (Fig. 3a).
Again, we verify the existence of an attracting point P in the evolution of the
Hubble parameters and we investigate its behaviour as κ3increases, from 0 to 1, i.e.
until it becomes as important as the quadratic term. The evolution of the attractor in
the L(3)-theory, with respect to the variation of κ3, may be obtained in a similar way
t
?
, i.e. those obtained from the highest order
8

Page 9

as in the κ0case. We differentiate the functions f1and f2with respect to Hext, Hint
and κ3to obtain a first order system of differential equations which, for κ2= 1 and
for vanishing κ0and κ4, will determine the displacement of P in the Hext−Hintplane,
under the variation of κ3.
The corresponding results are presented in Fig. 3b. We observe that the attractor
moves to higher absolute values of Hintas κ3increases. This result has a clear physical
meaning. Since increasing κ3corresponds to study the earlier stages in the evolution
of the Universe, we see that at these epochs the internal space contracts at higher
rates than those of the GB theory. Then Fig. 3b verifies that at the late stages, where
the GB theory holds alone, the value of the internal Hubble parameter decreases in
order to achieve stabilization.
Again, the law of displacement of P in the Hext− Hintplane may be estimated
using best-fit methods. In this context, we find that it may be represented by a sixth-
order polynomial Hext= p6(Hint), with coefficients: a0= 0.7373,a1= −25.594, a2=
−457.324 , a3= −3323.9 , a4= −12156.9 , a5= −22135.2 and a6= −15905.6.
Finally, to solve the cosmological field equations when all terms in the action (2.2)
are included (i.e. κ4?= 0), corresponds to study the dynamic behaviour of the model
under consideration at even earlier epochs. The results are slightly different from
those of the previous case (Fig. 4a). Again, in the later stages, the model consists
of two De Sitter subspaces and there exists an attracting point. The attractor’s
displacement in the Hext− Hintplane is obtained in a way similar to the κ0and κ3
cases and may be represented by a third-order polynomial, Hint = p3(Hext), with
coefficients: b0= 7.47, b1= −34.27, b2= 51.33 and b3= −25.74. The corresponding
result is shown in Fig. 4b.
Hence, we may conclude that in every case where non linear terms are included,
the ”extended” De Sitter solution (i.e. an exponentially expanding external space in
connection to an exponentially contracting internal one) corresponds to an ”attractor”
of the dynamical system under consideration. Accordingly, (in our model) no matter
how the Universe may originate, there is at least one period during its time-evolution
in which it exhibits inflation of the ordinary space, accompanied by spontaneous
compactification of the internal one12,30.
(2) Perfect fluid models of curved subspaces
We consider a ten-dimensional metric of the form (2.1), which now represents a class
of cosmological models with positively curved subspaces (kext = 1 = kint). Then,
X = R−1(t) and Y = S−1(t) and we study the time-evolution of the cosmological
models as results from the solution of the system (2.8).
The numerical analysis is carried out in the same fashion as in the previous case
of vacuum models. We consider that at the origin both subspaces are of the same
”physical size”, i.e. X0= Y0, but they have different expansion rates, Hext
As such we choose the corresponding range used in the vacuum case. We normalize
both scale functions R(t) and S(t) to unity, with respect to their value at the Planck
0
and Hint
0.
9

Page 10

epoch. That is
R(t) →R(t)
RPl
, S(t) →S(t)
SPl
(3.2.1)
where RPl = SPl. As initial conditions we choose R0= 100 = S0.
We represent the matter filling the Universe by a closed or heterotic superstring
perfect gas, with the following equation of state, deduced by Matsuo20
pext =
1
3ρ ,pint = 0 (3.2.2)
Thus, the external space is radiation-dominated, while the internal one is pressure-
less. It has been recently shown that, in this case, the two subspaces are completely
disjoint4,31. The time-evolution of the total mass-energy density ρ is accordingly given
by Eq.(2.7).
As regards the GB models (κ3= 0 = κ4), we have performed a number of com-
putational runs, varying the initial values of the Hubble parameters and the coupling
constant κ2as well, from κ2= 0.1 to κ2= 1. Numerical results in this case indicate
that there is a considerable difference with respect to the vacuum-flat models. It rests
in the fact that the range of values of the coupling constant κ2may be splitted into
two parts. Each one of these parts leads to a different time-evolution of both the
external and the internal scale functions.
The first part consists of values of κ2in the interval 0.1 ≤ κ2≤ 0.65, i.e. when
the contribution of the quadratic curvature terms is relatively small. In this case we
expect that the time-evolution of the Universe will be only slightly different from the
corresponding EH one. Indeed, the numerical results indicate that the system (2.8)
admits solutions with a power law dependence of the scale functions upon time, of
the form
R(t) ∝ tm1
S(t) ∝ t−m2
where the values of the indices m1and m2are continuously increasing in the ranges
0.25 ≤ m1 ≤ 0.55 and 0.01 ≤ m2 ≤ 0.11, as κ2 increases from 0.1 to 0.65. In
this case, there are no attracting points in the evolution of the Universe. The last
values in those ranges (0.55 and 0.11, respectively), both corresponding to the value
κ2 = 0.65, represent a Kasner-type regime12,32−34of the GB models. Indeed, the
analytic approach in this case suggests that the two subspaces evolve as
(3.2.3a)
(3.2.3b)
R(t) ∼ tp1, S(t) ∼ t−p2
(3.2.4)
where both p1and p2are possitive and in a ten-dimesional spacetime they satisfy the
conditions
3p1− 6p2= 1
3p2
(3.2.5a)
1+ 6p2
2= 1 (3.2.5b)
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The only physically acceptable solution of the system (3.2.5), compatible with the
condition p1, p2 > 0, is
p1=5
9= 0.555 , p2=1
9= 0.111
Therefore, when κ2 = 0.65, although the spatial sections are curved, the time-
evolution of the Universe admits a Kasner-type solution. This solution actually lies
on the interface between two different types of cosmological behaviour (Figs. 5a and
5b).
The second type of time-evolution arises when 0.65 < κ2≤ 1. Then the Universe
behaves, again, as an extended De Sitter spacetime (where the external space expands
while the internal one contracts, both exponentially). In this case there exists an
attracting point as in the vacuum-flat models (Fig. 6a).
In conclusion, for a curved ten-dimensional GB cosmological model, filled with
matter in the form of a superstring perfect gas, we may obtain three different types
of cosmological behaviour, depending on the exact value of the normalized coupling
constant κ2:
(a) Power-law solutions, with no attracting points, when 0.1 ≤ κ2< 0.65.
(b) A Kasner-type model, when κ2= 0.65.
(c) Extended De Sitter models, with an attracting point, when 0.65 < κ2≤ 1.
In all cases, the external space expands, while the internal one contracts. The
inclusion of the contribution of the third and/or the fourth order terms in the field
equations simply amounts to a modulation of those results (Fig. 6b).
IV. Analytic Results
Analytic expressions, for the time-evolution of the model Universe considered, may
be obtained by solving the cosmological field equations (2.8) around the attracting
points. Accordingly, we investigate the cosmological behaviour of a vacuum, ten-
dimensional model with spatially flat subspaces (X = 0 = Y ) within the context
of the quartic Lagrangian theory under consideration. Clearly, setting some of the
coupling constants λ(m)equal to zero corresponds to reducing the general theory to
its lower case counterparts (EH-cosmology, GB-theory etc.).
Since we are interested in the behaviour of the model around the attracting points,
we consider the linearized equations
Hext= A1+ H1(t)(4.1a)
Hint= A2+ H2(t) (4.1b)
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where A1and A2are the coordinates of the attractor, while H1(t) and H2(t) represent
small perturbations around those values (|H1| , |H2| ≪ 1). Therefore, to obtain the
time-evolution of Hextand Hint, we only have to solve the system (2.8) linearized with
respect to H1and H2.
The system of the cosmological field equations (2.8), linearized with respect to H1
and H2, may be written in the form
˙H1(t) =β1H1+ β2H2+ β3
α1H1+ α2H2+ α3
˙H2(t) =γ1H1+ γ2H2+ γ3
α1H1+ α2H2+ α3
where αj,βjand γj(j = 1,2,3) are constants, calculated directly from the lineariza-
tion of the original equations, which depend on A1, A2and λ(m)(m = 0,1,2,3,4).
From Eqs.(4.2) we obtain
dH1
dH2
γ1H1+ γ2H2+ γ3
The solution of Eq.(4.3), in connection to Eqs.(4.1), will give, in the linear approxi-
mation, the analytic expression of Hextin terms of Hint. To solve Eq.(4.3), we need
to have the solution (h1, h2) of the algebraic system
(4.2a)
(4.2b)
=β1H1+ β2H2+ β3
(4.3)
β1H1+ β2H2+ β3= 0(4.4a)
γ1H1+ γ2H2+ γ3= 0 (4.4b)
We choose
γ1?= 0 , β2γ1− β1γ2?= 0 (4.5)
and furthermore, we set
w = H1− h1
z = H2− h2
(4.6a)
(4.6b)
We verify that the solution of Eq.(4.3) depends on several algebraic combinations of
the constants αj, βjand γj, something that leads to several conditions between the
coupling constants λ(m). Therefore, we consider the following cases:
(a) β1+ γ2?= 0: This combination corresponds to the most general case. Setting
4β2γ1+ (β1− γ2)2?
the solution of Eq.(4.3) reads35
∆ = −
?
(4.7)
ln1
c
?
β2z2+ (β1− γ2)zw − γ1w2?
=



















β1+γ2
√−∆ln
β1−γ2−√−∆−2γ1w
β1−γ2+√−∆−2γ1w
z
z
for ∆ < 0
−
2(β1+γ2)
(β1−γ2)−2γ1w
z
for ∆ = 0
2β1+γ2
√∆arctan
(β1−γ2)−2γ1w
√∆
z
for ∆ > 0
(4.8)
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where c is an arbitrary integration constant.
(b) β1+ γ2= 0 and ∆ = 0 with β2γ1< 0: In this case we may proceed to derive
the explicit time-dependence of the Hubble parameters and the corresponding scale
functions for both subspaces. From Eq.(4.8) we obtain
H2= c1H1+ c2
(4.9)
where
c1=
?
|γ1
β2| ,c2= 1 + h2− c1h1
(4.10)
Now, Eq.(4.9) is inserted into Eq.(4.2a) to give
˙H1=δH1+ ǫ
ζH1+ η
(4.11)
where the constants δ , ǫ , ζ and η stand for the combinations
δ = β1c1+ β2 , ǫ = β3+ β1c2
ζ = α1c1+ α2 , η = α3+ α1c2
(4.12)
We consider the following cases:
(i) δ , ζ ?= 0: In this case, Eq.(4.11) results in
ζ
δH1+ζ
δ
?η
ζ−ǫ
δ
?
ln
?
H1+ǫ
δ
?
= t − t0
(4.13)
where t0 is an integration constant. Now, Eq.(4.1a) in connection with Eq.(4.13),
may be easily integrated to give the form of R(t), when the condition
η
ζ−ǫ
δ= 0 (4.14)
holds. Then, we obtain
lnR(t) ∼ A1(t − t0) +δ
2ζ(t − t0)2
(4.15)
which introduces a quadratic correction to the expected De Sitter solution.
(ii) δ , η ?= 0 and ζ = 0: In this case we rediscover the solutions of Ishihara12,
obtained in the GB theory, as a particular case of the general solution. Indeed, from
Eq.(4.11) we obtain
H1= Ce
δ
ηt−ǫ
δ
(4.16)
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where C is an arbitrary integration constant. Therefore the corresponding external
scale function is of the form
lnR(t) ∼
?
A1−ǫ
δ
?
(t − t0) + Cη
δe
δ
η(t−t0)
(4.17)
Since the external space expands, we must have A1 >
ǫ
δ. For C = 0, Eq.(4.17) reads
R(t) ∼ e(A1−ǫ
δ)(t−t0)
(4.18a)
corresponding again to a De Sitter phase, while forηC
δ
≪ 1 it yields
R(t) ∼ e(A1−ǫ
δ)(t−t0)
?
1 + C(η
δ)2e
δ
η(t−t0)?
(4.18b)
For ǫ = 0 Eq.(4.18b) corresponds to the solution of Ishihara (Eq.(15) of Ref. [12])
obtained in the framework of the GB theory.
(iii) δ = 0 , ζ ?= 0: Finally, in this case, Eqs. (4.1a) and (4.11) result in
?
ζ
lnR(t) ∼A1−η
?
(t − t0) ±
1
2ǫζ2
?
η2+ 2ǫζ(t − t0)
?3/2
(4.19)
where, in connection with the numerical results we must have A1>η
In concluding, we see that the coupling constants λ(m)may not be arbitrary. In
every case, they should satisfy certain algebraic relations, depending on the form of
the corresponding solution around the attracting points.
Since both Eqs.(4.2) are almost of the same functional form, in all of the preceding
cases, similar functional results may be obtained for the internal space, through the
solution of Eq.(4.2b). In this case, however, we must take into account the fact that
the numerical results indicate that the extra dimensions contract (A2 < 0). This
argument may lead to additional constraints on the coupling constants λ(m).
ζ.
V. Discussion and Conclusions
In the present paper we have studied the time evolution of anisotropic, ten-dimensional
cosmological models in the framework of a quartic Lovelock-Lagrangian theory of
gravity1,9−11. The cosmological models under consideration consist of one time di-
rection and two homogeneous and isotropic subspaces: A three-dimensional external
space, which represents the ordinary Universe, and a compact internal space, which is
constituted by the extra dimensions. The evolution of the Universe depends on four
free parameters. These are the coefficients λ(m)which introduce the extra curvature
terms in the gravitational Lagrangian (m = 0, 2, 3, 4). They are to be regarded as the
coupling constants3,9. Since we have considered models of an already compactified
internal space17, we accordingly examine the process of its contraction15−17,19.
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The Universe is filled with matter in the form of an anisotropic perfect fluid. Given
an equation of state for the matter content of each subspace, the time evolution of
the Universe is completely determined by a system of three second-order, non-linear
differential equations, consisting of the field equations (2.4b) and (2.4c) together with
the conservation law (2.6). The initial value field equation (2.4a) corresponds to a
constraint which should be satisfied by the cosmological solutions. As regards the
energy momentum tensor, we have considered two cases: (a) Vacuum models, ρ = 0,
in connection with spatially flat subspaces and (b) Models of an heterotic superstring
gas, with pext=1
The three independent equations may be subsequently expressed in the form of
a first order system, Eqs.(2.8), involving the Hubble parameters Hext, Hintand the
corresponding scale functions R(t) , S(t) of the two factor spaces. This system is
evaluated numerically, for a wide range of initial conditions of the form Hext
Hint
0
< 0. Its solutions may be represented by curves in the Hext−Hintplane. Those
curves correspond to the ”orbits” of the dynamical system under study and each one
of them, associated with a different set of initial conditions, represents a different
type of evolution for the Universe.
In the case of vacuum models with flat subspaces (kext= 0 = kint), the numerical
results indicate that for all values of the coupling constants involved and also for a
wide range of initial conditions, the Universe will always end up to evolve according
to an extended De Sitter solution, i.e. an exponentially expanding external space,
accompanied by an exponentially contracting internal one. Indeed, in this case the
Hubble parameters of both subspaces approach constant values in the later stages.
We have confirmed that those values actually represent the attracting points of the
dynamical system under consideration11−13. The appearence of attractors in the so-
lution of the cosmological field equations is very important, since, if a spacetime is an
attractor for a wide range of initial conditions, then it may be realized asymptotically
in the later stages11,13. Those results indicate that the existence of the non-linear
curvature terms in the gravitational action may lead to inflation without the use of
any phase transition19,27−30,36.
Furthermore, we have investigated the evolution of the attractors under the varia-
tion of the normalized coupling constants κm= λ(m)/λ(1)(m = 0, 3, 4). In all cases,
the attracting points are displaced at higher absolute values of Hintas κmincreases
from 0 to 1. As regards the variation of κ3and κ4, this result has a clear physical
meaning.
In determining the cosmological behaviour of the model very close to the initial
singularity, the leading terms to consider are those with the highest power in (1
those obtained from the highest order terms in the gravitational action5,6. Therefore,
the increase of κm(m = 3 , 4) corresponds to a more accurate study of the earlier
stages in the evolution of the Universe9. Then, from Figs. 3b and 4b, we see that at
those epochs the internal space contracts at higher rates than those of the GB theory.
This means that in the later stages of the time evolution, where the GB theory holds
alone, the absolute value of the internal Hubble parameter decreases, in order for the
3ρ and pint= 0, in connection with positively curved subspaces.
0
> 0 and
t), i.e.
15