Integration of D-dimensional 2-factor spaces cosmological models by reducing to the generalized Emden-Fowler equation

14 Pages·0.15 MB·English

by  V. R. Gavrilov

INTEGRATION OF D-DIMENSIONAL 2-FACTOR SPACES
COSMOLOGICAL MODELS BY REDUCING TO THE GENERALIZED
EMDEN-FOWLER EQUATION
V. R. Gavrilov,V.N. Melnikov
Center for Gravitation and Fundamental Metrology, VNIIMS,
3-1 Ulyanova St., Moscow, 117313, Russia
e-mail: [email protected]
Abstract
8
9
9 The D-dimensional cosmological model on the manifold M = R×M1×M2 describing the evolution of 2
1 Einsteinian factor spaces, M1 and M2, in the presence of multicomponent perfect fluid source is considered.
The barotropic equation of state for mass-energy densities and the pressures of the components is assumed
n
a in each space. When the number of the non Ricci-flat factor spaces and the number of the perfect fluid
J componentsarebothequalto2,theEinsteinequationsforthemodelarereducedtothegeneralizedEmden-

3 Fowler (second-order ordinary differential) equation, which has been recently investigated by Zaitsev and
1 Polyanin within discrete-group analysis. Using the integrable classes of this equation one generates the

integrable cosmological models. The corresponding metrics are presented. The method is demonstrated for
1
the special model with Ricci-flat spaces M ,M and the 2-component perfect fluid source.
v 1 2
2 PACS numbers: 04.20.J,04.60.+n, 03.65.Ge
4
0
1 1 Introduction
0
8
Following the purpose to study the early universe we develop the multidimensional generalization [8]-[13],
9
/ [16]-[21],[25],[26],[28] of the standard Friedman-Robertson-Walker world model. If the extra dimensions of
c
the space-time manifold really exist, the unique conceivable site, where they might become dynamically
q
- important, seems to be possible. This is some early stage of the evolution. Usually within multidimensional
r
cosmology (see, for instance, [1]-[2], [5]-[13],[15]-[26],[28],[31]-[36] and references therein) it is assumed the
g
: occurrence of the topological partition for the multidimensional space-time on the external 3-dimensional
v
space and additional so called internal space (or spaces) due to the quantum processes at the beginning
i
X of this stage. In correspondence with such partition the space-time acquires the topology M = R×M ×
1
r ...×Mn, where R is the time axis, one part of the manifolds M1,...,Mn is interpreted as 3-dimensional
a
external space and the other part stands for internal spaces. Usually the internal spaces are compact,
however the models with noncompact internal spaces are also discussed [14],[20], [29] [30]. The subsequent
evolution of the multidimensional Universe is considered as classical admitting the description by means of
the multidimensional Einstein equations. Achieving the integrability of these equations is the main goal of
ourinvestigation. As the presentworldseems to be 4-dimensional,there is the assumptionthat the internal
space(s) had contracted to extremely small sizes, which are inaccessible for experiment. This contraction
accompanied by the expansion of the external space is described by some models (the first model of such
type has been found in [6]) within multidimensional cosmology and is called dynamical compactification.
We consider a mixture of several perfect fluid components as a source for the multidimensional Einstein
equations. Such multicomponent systems are usually employed in 4-dimensional cosmology and are quite
adequate type of matter for description some early epochs in the history of the universe [4].
The paperis organizedasfollows. Insection2 wedescribethe multidimensionalcosmologicalmodeland
obtain the Einstein equations in the form of the Lagrange-Eulerequations following from some Lagrangian.
Herewedevelopthen-dimensionalvectorformalismfortheintegratingoftheequationsofmotion. Conclud-
ing Section 2 we present the review of the all known integrable models. In Section 3 we suggestthe method
forobtainingthenewclassoftheintegrablemodelsonthemanifoldM =R×M ×M . Themethodisbased
1 2
on the reducing of the Einstein equations to the generalized Emden-Fowler (second-order ordinary differen-
tial) equation. The method is useful for any 2-componentmodel on the manifold M =R×M ×M except
1 2
for the cases admitting the integration by more simple way. The total number of the model components is
equaltothesumofthenumberofthenonRicci-flatspaceswiththenumberoftheperfectfluidcomponents.
1
The integrable classes recently derived by Zaitsev and Polyanin of the generalized Emden-Fowler equation
allow to generate the new integrable cosmological models. Their metrics are presented. In Section 3 the
method is applied for the models with Ricci-flat spaces M ,M and 2-component perfect fluid.
1 2
2 The model and the equations of motion
Withinn-factorspacescosmologicalmodelD-dimensionalspace-timemanifoldM isconsideredasaproduct
of the time axis R and n manifolds M ,...,M , i.e.
1 n
M =R×M ×...×M , (2.1)
1 n
The product of one part of the manifolds gives the external 3-dimensional space and the remaining part
stands for so called internal spaces. The internal spaces are supposed to be compact. Further, for sake of
generality, we admit that dimensions N =dimM for i=1,...,n are arbitrary.
i i
The manifold M is equipped with the metric
n
g =−e2γ(t)dt⊗dt+ exp[2xi(t)]g(i), (2.2)
i=1
X
where γ(t) is an arbitrary function determining the time t and g(i) is the metric on the manifold M . We
i
suppose that the manifolds M ,...,M are the Einstein spaces, i.e.
1 n
R [g(i)]=λ g(i) , k ,l =1,...,N , i=1,...,n, (2.3)
kili i kili i i i
whereλ isconstant. Inthe specialcase,whenM isaspaceofconstantRiemanncurvatureK theconstant
i i i
λ reads: λ =K (N −1) (here N >1).
i i i i i
Using the assumptions (2.3) we obtain the following non-zero components of the Ricci tensor for the
metric (2.2) [19]
n
R0 =e−2γ N (x˙i)2+γ¨ −γ˙γ˙ (2.4)
0 i 0 0
!
i=1
X
Rmi = λ exp[−2xi]+ x¨i+x˙i(γ˙ −γ˙) e−2γ δmi (2.5)
ni i 0 ni
where we denoted (cid:8) (cid:2) (cid:3) (cid:9)
n
γ = N xi. (2.6)
0 i
i=1
X
Indices m and n in (2.4),(2.5) for i = 1,...,n run over from (D − n N ) to (D − n N + N )
i i j=i j j=i j i
(D =1+ n N =dimM).
i=1 i P P
We consider a source of gravitational field in the form of multicomponent perfect fluid. The energy-
P
momentum tensor of such source under the comoving observer condition reads
m¯
TM = TM(µ), (2.7)
N N
µ=1
X
TM(µ) =diag −ρ(µ)(t),p(µ)(t)δk1,...,p(µ)(t)δkn , (2.8)
N 1 l1 n ln
(cid:16) (cid:17) (cid:16) (cid:17)
Furthermore we suppose that for any µ-th component of the perfect fluid the barotropic equation of state
holds
p(µ)(t)= 1−h(µ) ρ(µ)(t), µ=1,...,m¯, (2.9)
i i
(cid:16) (cid:17)
where h(µ) = const. It should be noted that each µ-th component admits different barotropic equations of
i
state in the different spaces M ,...,M . From the physical viewpoint this follows from the separation of
1 n
the internal spaces with respect to the external one and with respect to each others.
2
One easily shows that the equation of motion ▽ TM(µ) =0 for the µ-th component of the perfect fluid
M 0
described by the tensor (2.8) reads
n
ρ˙(µ)+ N x˙i(ρ(µ)+p(µ))=0. (2.10)
i i
i=1
X
Using the equations of state (2.9), we obtain from (2.10) the following integrals of motion
n
A(µ) =ρ(µ)exp 2γ − N h(µ)xi =const. (2.11)
0 i i
" #
i=1
X
The Einstein equations RM − RδM/2 = κ2TM (κ2 is the gravitational constant), can be written as
N N N
RM = κ2[TM −TδM/(D−2)]. Further we employ the equations R0−R/2 = κ2T0 and Rmi = κ2[Tmi −
N N N 0 0 ni ni
Tδmi/(D−2)]. Using (2.4)-(2.9), we obtain for them
ni
n
1
G x˙ix˙j +V =0, (2.12)
ij
2
i,j=1
X
m¯ n N h(µ)
λie−2xi +[x¨i+x˙i(γ˙ −γ˙)]e−2γ = −κ2 A(µ) h(µ)− k=1 k k
0 i D−2
µ=1 P !
X
n
× exp N h(µ)xi−2γ . (2.13)
i i 0
" #
i=1
X
Here
G =N δ −N N (2.14)
ij i ij i j
are the components of the minisuperspace metric,
n m¯ n
1
V =e2γ − λiN e−2xi +κ2 A(µ)exp N h(µ)xi−2γ . (2.15)
2 i i i 0
" #!
i=1 µ=1 i=1
X X X
The dependence on the densities ρ(µ) in (2.12),(2.13) has been canceled according to the relations (2.11).
It is not difficult to verify that after the gauge fixing γ = F(x1,...,xn) the equations of motion (2.13)
may be considered as the Lagrange-Eulerequations obtained from the Lagrangian
n
1
L=eγ0−γ G x˙ix˙j −V (2.16)
ij
2 
i,j=1
X
 
under the zero-energy constraint (2.12).
Now we introduce n-dimensional real vector space Rn. By e ,...e we denote the canonicalbasis in Rn
1 n
(e =(1,0,...,0) etc.). Hereafter we use the following vectors:
1
the vector we need to obtain
x=x1(t)e +...+xn(t)e , (2.17)
1 n
the vector induced by the curvature of the space M
k
n
2 −2
v =− e = δie , (2.18)
k N k N k i
k k
i=1
X
the vector induced by µ-th component of the perfect fluid
n n N h(µ)
u = h(µ)− k=1 k k e . (2.19)
µ i D−2 i
i=1 P !
X
3
Let <.,.> be a symmetrical bilinear form defined on Rn such that
<e ,e >=G . (2.20)
i j ij
The form is nongenerated and the inverse matrix to (G ) has the components
ij
δij 1
Gij = + . (2.21)
N 2−D
i
The form < .,. > endows the space Rn with the metric, which signature is (−,+,...,+) [17],[18]. By the
usual way we may introduce the covariantcomponents of vectors. For the vectors v and u we have
k µ
δi n
vi =−2 k , v(k) = G vj =2(N −δk), (2.22)
(k) N i ij (k) i i
k
i=1
X
n N h(µ) n
ui =h(µ)− k=1 k k , u(µ) = G uj =N h(µ). (2.23)
(µ) i D−2 i ij (µ) i i
P i=1
X
The values of <v ,v >, <v ,u > and <u ,u > are presented in Table 1.
k i k µ µ ν
<.,.> v u
j ν
v 4(δij −1) −2h(ν)
i Ni i
u −2h(µ) n h(µ)h(ν)N + 1 [ n h(µ)N ][ n h(ν)N ]
µ j i=1 i i i 2−D i=1 i i j=1 j j
P P P
TABLEI.Valuesofthebilinearform<.,.>forthevectorsv andu ,inducedbycurvatureofthespace
i µ
M and µ-th component of the perfect fluid correspondingly.
i
A vector y ∈ Rn is called time-like, space-like or isotropic, if < y,y > has negative, positive or null
values correspondingly. Vectors y and z are called orthogonal if < y,z >= 0. It should be noted that the
curvature induced vector v is always time-like, while the perfect fluid induced vector u admits any value
i µ
of <u ,u > (see Table 1).
µ µ
Using the notation <.,.> and the vectors (2.17)-(2.19),we may write the zero-energyconstraint (2.12)
and the Lagrangian(2.16) in the form
1
E = <x˙,x˙ >+V =0, (2.24)
2
1
L=eγ0−γ <x˙,x˙ >−V , (2.25)
2
(cid:18) (cid:19)
where
n m¯
1
V =e2(γ−γ0) − λiN e<vi,x>+κ2 A(µ)e<uµ,x> . (2.26)
i
2
" #
i=1 µ=1
X X
It is obviously from (2.26) that the term induced in the potential by the non-Ricci flat space M is similar
i
to the term induced by µ-component of the perfect fluid. Due to this fact the non-zero curvature of the
manifoldM maybealsocalledacomponentandnowweusethenotionofthecomponentinsuchnewsense.
i
Further we employ the so called harmonic time gauge, which implies
n
γ(t)=γ = N xi. (2.27)
0 i
i=1
X
4
¿Fromthe mathematical viewpointthe problemconsistin integrabilityof the systemwith n≥2 degrees
of freedom, described by the Lagrangianof the form
m
1
L= <x˙,x˙ >− a(µ)e<bµ,x>, (2.28)
2
µ=1
X
wherex,b ∈Rn. In(2.28)mdenotes the totalnumber ofthe componentsincluding the curvaturesandthe
µ
perfectfluidcomponents. Itshouldbe notedthatthe kineticterm<x˙,x˙ >isnotpositivelydefinite bilinear
form as it usually takes place in classical mechanics. Due to the pseudo-Euclideansignature (−,+,...,+) of
the form <.,.> such systems may be called pseudo-Euclidean Toda-like systems as the potential like that
given in (2.28) defines well known in classical mechanics Toda lattices [34]. In the papers [8],[19],[20] the
following classes of the integrable pseudo-Euclidean Toda-like systems have been found
1. m = 0. This case corresponds to the vacuum multidimensional cosmological model on the manifold
M =R×M ×...×M withallRicci-flatspacesM . Thecorrespondingmetric isamultidimensional
1 n i
generalization of the well-known Kasner solution [19].
2. m=1, the vector b is arbitrary. The metrics for this 1-component case were obtained in [20]. This
1
integrableclass may be enlargedby the addition ofthe new components inducing the vectorscollinear
to the vector b .
1
3. m ≥ 2, n = 2, b = b+C b , where b is an arbitrary vector and b is an arbitrary isotropic vector,
µ µ 0 0
C =const. This class was integrated in [20] only under the zero energy constraint.
µ
4. m ≥ 2, the vectors b ,...,b are linear independent and satisfy the conditions
1 m
< b ,b >= 0 for µ 6= ν. This integrable class may be enlarged by the addition of the new com-
µ ν
ponents inducing the vectors collinear to one from the orthogonal set b ,...,b . The corresponding
1 m
cosmologicalmodels are studied in [8].
5. m ≥ 2, the vectors b ,...,b are space-like and may be interpreted as a set of admissible roots [3]
1 m
of a simple complex Lie’s algebra G. In this case the pseudo-Euclidean Toda-like system is trivially
reducible to the Toda lattice associatedwith the Lie algebraG [34]. For G=A ≡sl(3,C)the metric
2
of the corresponding cosmologicalmodel was explicitly written in [8].
In the present paper we consider only 2-component (m=2) pseudo-Euclidean Toda-like systems with 2
degrees of freedom (n = 2) under the zero energy constraint. The corresponding multidimensional cosmo-
logical models are 2-factor spaces, i.e.
M =R×M ×M (2.29)
1 2
and admit the following combinations of the components: curvature of M and curvature of M (vacuum
1 2
models); curvature of M or M and 1-component perfect fluid; 2-component perfect fluid in Ricci-flat
1 2
spaces M and M . In our recent paper [10] we have integrated the vacuum model of the type (2.29) with
1 2
2 curvatures for the dimensions (dimM ,dimM )=(6,3),(8,2),(5,5). Now we develop more generalprocedure
1 2
useful for any combination of the 2 components.
3 Reducing to the generalized Emden-Fowler equation
Let us consider the equations of motion following from the Lagrangian (2.28) with n = m = 2 under the
zero energy constraint. If the vectors b and b satisfy one of the following conditions
1 2
1. b and b are linearly dependent,
1 2
2. <b ,b >=0, i.e. b and b are orthogonal,
1 2 1 2
3. <b −b ,b −b >=0, i.e vector b −b is isotropic,
1 2 1 2 1 2
5
the equations of motion are easily integrable and the corresponding exact solutions have been obtained in
the papers [8],[20]. Now we aim to develop the integration procedure just for all remaining cases. Then,
further we suppose that the vectors b and b do not satisfy any condition from 1-3.
1 2
Let us introduce in R2 an orthogonal basis forming by the following two vectors
f =(u −u )b +(u −u )b , f =b −b , (3.1)
1 22 12 1 11 12 2 2 2 1
where we denoted
u =<b ,b >, µ,ν =1,2. (3.2)
µν µ ν
According to the admission accepted f is not isotropic vector, i.e.
2
<f ,f >=u +u −2u 6=0. (3.3)
2 2 11 22 12
One may easily check that u2 −u u ≥ 0 for any vectors b ,b ∈ R2 and u2 −u u =0 if and only if
12 11 22 1 2 12 11 22
b and b are linearly dependent. Then in the case under consideration
1 2
<f ,f >=−(u2 −u u )(u +u −2u )6=0, (3.4)
1 1 12 11 22 11 22 12
<f ,f >/<f ,f >=−(u2 −u u )<0, (3.5)
1 1 2 2 12 11 22
i.e. one from the orthogonalvectors f and f is space-like and the other is time-like.
1 2
The vector x(t) we have to find decomposes as follows
<x,f > <x,f >
1 2
x= f + f . (3.6)
1 2
<f ,f > <f ,f >
1 1 2 2
For the new configuration variables
<x,f > a(2)
2
z(t)= +ln , (3.7)
2 s a(1)
(cid:12) (cid:12)
(cid:12) (cid:12)
1 <f ,f > (cid:12) (cid:12)
2 2 (cid:12) (cid:12)
y(t)= − <x,f > (3.8)
1
2s <f1,f1 >
the Lagrangian(2.28) and the corresponding zero-energy constraint look as follows
L=2β z˙2−y˙2 −V(z,y), (3.9)
E =2β z˙2−(cid:0) y˙2 +(cid:1)V(z,y)=0, (3.10)
where the potential V(z,y) has the form (cid:0) (cid:1)
V(z,y)=V e2αβy sgn a(1) e2β1z +sgn a(2) e2β2z . (3.11)
0
(cid:16) h i h i (cid:17)
In formulas (3.9)-(3.11) the following constants are used
α= u2 −u u , β =(u +u −2u )−1, (3.12)
12 11 22 11 22 12
β =−(u −u )β, β =β +1=(u −u )β, (3.13)
1 p 11 12 2 1 22 12
V0 =|a(1)|β2|a(2)|−β1. (3.14)
It should be mentioned that using of a basis in the form (3.1) provides the factorization of the potential
(3.11) with respect to the coordinates of the vector x(t) (the additional linear transformation (3.7),(3.8)
does not matter in this situation). Such factorization of the potential is essential under the developing of
the following procedure proposed in [1]. Using the equation of motion following from the Lagrangian(3.9)
1
z¨ = − V e2αβy β sgn a(1) e2β1z +β sgn a(2) e2β2z , (3.15)
0 1 2
2β
α (cid:16) h i h i (cid:17)
y¨ = V(z,y), (3.16)
4
6
the zero-energy condition (3.10) written in the form
1 V(z,y) 1 V(z,y)
z˙2 = = (3.17)
2β(y˙/z˙)2−1 2β(dy/dz)2−1
and the relation
d2y y¨−z¨dy
= dz (3.18)
dz2 z˙2
we obtain the following second-order ordinary differential equation
d2y dy 2 1 e2z −ε dy
= −1 β +β + +αβ , (3.19)
dz2 dz 2 1 2 e2z +ε dz
"(cid:18) (cid:19) #(cid:26) (cid:18) (cid:19) (cid:27)
where
ε=sgn a(1)a(2) . (3.20)
h i
Wenoticethatduetothe factorizationofthe potentialtherightsideofthe equation(3.19)doesnotcontain
y, so, in fact, the equation is the first-order one with respect to dy/dz.
This procedure is valid for the solutions such that z˙ 6≡0. Under the zero energy constraint the solutions
of (3.15),(3.16) with z˙ ≡0 gives the following vector x(t)
x(t)=pln|t−t |+q, (3.21)
0
where the constant vectors p,q∈R2 are such that
2
p= f , (3.22)
α2 1
β β
e<q,b1> = 2 >0, e<q,b2> =− 1 >0. (3.23)
a(1)α2β a(2)α2β
We note that the exceptional solution (3.21) exists only if the inequalities in (3.23) are satisfied. It should
be mentioned, that the set of the equations (3.10),(3.15),(3.16) does not admit static solutions z˙ = y˙ ≡ 0
due to the condition (3.3). The solutions with z˙ = ±y˙ are also impossible, so using the relation (3.17) we
do not lose any solutions of the set (3.10),(3.15),(3.16) except, possibly, the solution (3.21).
Let us suppose that one is able to obtain the general solution of the equation (3.19) in the parametrical
form z = z(τ), y = y(τ), where τ is a parameter. Then using (3.6)-(3.8) we obtain the vector x as the
function of the parameter τ
2y(τ) a(2)
x(τ)= (−β b +β b )+2β z(τ)−ln (b −b ). (3.24)
α 2 1 1 2 " s(cid:12)a(1)(cid:12)# 2 1
(cid:12) (cid:12)
(cid:12) (cid:12)
Werecallthatcoordinatesofthe vectorx(τ) inthe canonicalbasisare(cid:12) the(cid:12)logarithmsofthe scalefactorsfor
the spaces M ,M . The relation between the harmonic time t and the parameter τ may be always derived
1 2
by integration of the zero-energy constraint written in the form of the separable equation
dy 2− dz 2
dt2 =2β dτ dτ dτ2. (3.25)
(cid:16)V(z(cid:17)(τ),y(cid:0)(τ)(cid:1))
Thustheproblemoftheintegrabilitybyquadratureofthepseudo-EuclideanToda-likesystemswith2degrees
of freedom under the zero-energy constraint is reduced to the integrability of the equation (3.19).
Fordy/dz theequation(3.19)representsthefirst-ordernonlinearordinarydifferentialequation. Itsright
side is third-order polynom (with the coefficients depending on z) with respect to the dy/dz. An equation
of such type is called Abel’s equation (see, for instance [27],[37]). There are no methods to integrate
arbitraryAbel’s equation, howeverthe equation(3.19) maybe integratedfor some values of the parameters
αβ and β +β . First of all let us notice that the equation (3.19) has the partial integrals y±z =const,
1 2
7
which make the relation (3.17) singular and as was already mentioned are not partial integrals of the set
(3.10),(3.15),(3.16). Existenceofthis partialsolutionofthe Abelequation(3.19)allowstofindthefollowing
nontrivial transformation
X dY
e2z = −ε , (3.26)
Y dX
Y
y = δ z+ln +lnC , δ =±1, C >0, (3.27)
X
(cid:20) (cid:12) (cid:12) (cid:21)
(cid:12) (cid:12)
(cid:12) (cid:12)
which reduces the Abel equation (3.19) to the s(cid:12)o c(cid:12)alled generalized Emden-Fowler equation
d2Y dY l
=XnYm , (3.28)
dX2 dX
(cid:18) (cid:19)
where the constant parameters n,m and l read
1 −2u −u +3u −δ u2 −u u
n = (β +β −2δαβ−3)= 11 22 12 12 11 22, (3.29)
1 2
2 u +u −2u
11 22 p12
1 −u −2u +3u +δ u2 −u u
m =− (β +β −2δαβ+3)= 11 22 12 12 11 22, (3.30)
1 2
2 u +u −2u
11 22 p12
1 2u +u −3u −δ u2 −u u
l =− (β +β +2δαβ−3)= 11 22 12 12 11 22. (3.31)
1 2
2 u +u −2u
11 22 p 12
For our models the parameters in the generalized Emden-Fawler equation are not independent. It follows
from (3.29),(3.30) that
n+m=−3. (3.32)
In the special case l =0 the equation (3.28) is known as the Emden-Fowler equation.
If the parameters l and m given by (3.31),(3.30) are such that l = 0, m 6= 1 there exists one more
transformation
2 X dY
1+εe2z =− , (3.33)
m−1Y dX
1
y =δ z− lnY2+C , δ =±1, C ∈R, (3.34)
m−1
(cid:20) (cid:21)
which reduces the Abel equation (3.19) to the following integrable Emden-Fowler equation
d2Y m+3
=Y m−1. (3.35)
dX2
There are no methods for integratingof the generalizedEmden-Fowler equationwith arbitraryindependent
parameters n,m and l. However, the discrete-group methods recently developed by Zaitsev and Polyanin
[37] allows to integrate by quadrature 3 two-parametricalclasses, 11 one-parametrical classes and about 90
separatedpoints intheparametricalspace(n,m,l)ofthegeneralizedEmden-Fowlerequation. Forinstance,
the two-parametrical integrable classes arise when m and l are arbitrary and n = 0 or when n and l are
arbitraryandm=0. Theone-parametricalclasswithl =0andn+m=−3isalsointegrablebyquadrature.
Let us suppose that the two components of the 2-factor spaces cosmological model under consideration
induce such vectors b and b that the corresponding to the model generalized Emden-Fowler equation
1 2
(3.28) with the parameters defined by (3.2),(3.29)-(3.31) is integrable in the parametrical form X = X(τ),
Y = Y(τ), where τ is a parameter. Then, using the parameter τ as the new time coordinate we obtain by
the formulas (3.26),(3.27),(3.24),(3.25)the following final result for the metric (2.2)
g =−f2(τ)[a (τ)]2N1[a (τ)]2N2dτ ⊗dτ +[a (τ)]2g(1)+[a (τ)]2g(2), (3.36)
1 2 1 2
8
where we denoted
2|β| Y′(τ) l [X′(τ)]2
f(τ)= Cn+l , (3.37)
s V0 (cid:20)X′(τ)(cid:21) X(τ)Y(τ)Y′(τ)
2β
a (τ)≡exi(τ) =eγi Y′(τ) (2−l)bi(1)−(1−l)bi(2) Y(τ) (2+n)bi(1)−(1+n)bi(2) l+n . (3.38)
i X′(τ) X(τ)
((cid:12) (cid:12) (cid:12) (cid:12) )
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
By γi for i=1,2 we denoted the follow(cid:12)ing con(cid:12)stants (cid:12) (cid:12)
lnC a(2)
γi =2β (n−l+4)bi −(n−l+2)bi −ln bi −bi . (3.39)
(n+l (1) (2) s(cid:12)a(1)(cid:12) (2) (1) )
h i (cid:12) (cid:12)h i
(cid:12) (cid:12)
We recall that bi are coordinates of the vector b in the canonical b(cid:12)asis.(cid:12)In the special case l = 0 one
(µ) µ
may also use by the similar manner the transformation (3.33),(3.34) and the result of integration of the
equation (3.35) to write the metric. This transformationwas used in [10] for integrating of the models with
two curvatures.
Thus the method described allows to integrate the cosmologicalmodels if the corresponding generalized
Emden-Fowler equation is integrable. Note that if the model with some vectors b and b is integrable by
1 2
such manner then any model with the vectors αb and αb (α is an arbitrary non-zero constant) is also
1 2
integrableasthe parametersn,mandl donotchangeundersuchtransformationofthevectors. Takinginto
account the classes 1-4 (the class 5 does not arise for n = 2) and the additional to them class, which may
be integrated by the method described, we obtain the quite large variety of the integrable 2-factor spaces
cosmologicalmodels with 2 components.
4 Examples of the integrable models
Now we apply the method proposed in Section 3 to the cosmological models on the manifold (2.29) with
both Ricci-flat spaces M ,M and the 2-component perfect fluid source. Let us represent such model by
1 2
Table 2
manifold/source external space MN1 internal space MN2
1 2
1-st component of
the perfect fluid h(1) h(1)
1 2
2-nd component of
the perfect fluid h(2) h(2)
1 2
TABLE2. RepresentationofthemodelonthemanifoldM =R×M ×M withRicci-flatspacesM ,M
1 2 1 2
for the 2-component perfect fluid.
We recall that N =dimM and h(µ) are the constant parameters in the barotropic equation of state
i i i
(2.9). The model is entirely defined by these 6 parameters. One easily shows [12] that the dominant energy
(µ)
condition applied to the stress-energy tensor (2.8) implies 0 ≤ h ≤ 2. Usually rational values of the
i
parameter h(µ) are employed in cosmology, for instance, h(µ) = (N −1)/N - radiation, h(µ) = 1 - dust,
i i i i i
h(µ) =0- Zeldovich(stiff) matter, h(µ) =2- false vacuum (Λ-term),h(µ) =(D−1)/D - superradiationetc.
i i i
Onthe other handthe most knowncases,when the generalizedEmden-Fowlerequation(3.28)is integrable,
arise for the rationalparameters n,m and l. So if one demands the rationality of the parametersn,m and l
9
in the equation (3.28) corresponding to the model under the condition of the rationality for the parameters
h(µ), then due to the following relation
i
N N 2
α2 =u2 −u u = 1 2 h(1)h(2)−h(1)h(2) , (4.1)
12 11 22 N +N −1 1 2 2 1
1 2
(cid:16) (cid:17)
thedimensionsN ,N withintegervalueoftheexpressionR≡ N N (N +N −1)aresingledout. Forin-
1 2 1 2 1 2
stance,theexpressionRisintegerforthefollowingdimensions: (N ,N )=(3,6),(2,8),(5,5),(7,8),(3,25),(N ,1).
p1 2 1
From the physical viewpoint the following cases may be of interest: (2,1),(3,1),(3,6),(3,25).
Let us consider the models of the type represented in Table 2 leading to the generalized Emden-Fowler
equation (3.28) with
l=0. (4.2)
Duetotherelation(3.32)arisingforourmodelstheequationisintegrableforarbitraryparametermandits
exactsolution has been written by Zaitsev and Polyanin[27]. It is worthto mention that other 2 integrable
classes of the generalized Emden-Fowler equation (3.28), arising when n = 0 or m = 0, describe the same
cosmological models. It follows easily from (3.29),(3.31) that if the model is such that l = 0 for δ = 1 (or
δ = −1) then n = 0 for δ = −1 (correspondingly, δ = 1). It is easy to see also from (3.30),(3.31) that
the condition l = 0 transforms to the condition m = 0 under the inverse numbering of the components.
Thus, from 3 integrable classes, arising for n = 0, m = 0 and l = 0, correspondingly, of the generalized
Emden-Fowler equation (3.28) for our models it is enough to study any one from them, let it be the class
with l=0. In this case the equation (3.28) has the form
d2Y
=X−m−3Ym, m=−β −β . (4.3)
dX2 1 2
By the following transformation
τ 1
Y = , X = (4.4)
ξ ξ
it reduces to the equation
d2τ
=τm, (4.5)
dξ2
which is easily integrable. Then the general solution of the equation (4.3) has the form
τ 1
Y =± , X =± , (4.6)
F(τ) F(τ)
where
2 −1/2
F(τ) = τm+1+C dτ +C , m6=−1, (4.7)
1 2
m+1
Z (cid:20) (cid:21)
= [2ln|τ|+C ]−1/2dτ +C , m=−1. (4.8)
1 2
Z
We suppose that the both components of the perfect fluid have the positive mass-energy densities given by
(2.11). It means a(µ) = κ2A(µ) > 0 for µ = 1,2, so ε = sgn a(1)a(2) = 1. Then taking into account the
formula (3.26), we must consider the general solution (4.6) on such interval of the variable τ where
(cid:2) (cid:3)
′
X(τ)Y (τ) F(τ)
G(τ)≡− = −1>0. (4.9)
Y(τ)X′(τ) τF′(τ)
Finally using the results of Section 3 we obtain the following exact solution for the cosmological model
represented by Table 2 in the special case l=0:
the metric is given by the formula (3.36), where
2|β|C−2δαβ F′(τ) 2
f2(τ)= , (4.10)
A(1) β2 A(2) −β1 G(τ)τ2
(cid:20) (cid:21)
(cid:2) (cid:3) (cid:2) (cid:3)
10