Stable exponential cosmological solutions with two factor spaces in the Einstein-Gauss-Bonnet model with a $Λ$-term
V. D. Ivashchuk A. A. Kobtsev
gr-qc



 


 Stable  exponential cosmological solutions with 
 two factor spaces in the Einstein-Gauss-Bonnet  model  with a Λ-term
  
  


 





 V. D. Ivashchuk^1,2 and A. A. Kobtsev^3 




 ^1 Institute of Gravitation and Cosmology,
   Peoples' Friendship University of Russia (RUDN University),
   6 Miklukho-Maklaya St.,  Moscow 117198, Russian Federation 
 

 ^2 Center for Gravitation and Fundamental Metrology,
 VNIIMS, 46 Ozyornaya ul., Moscow 119361,  Russian Federation
 
 ^3 Institute for Nuclear Research of the Russian Academy of Sciences,
            Moscow, Troitsk, 142190,  Russian Federation 
 
 



 





We study   D-dimensional Einstein-Gauss-Bonnet  gravitational model including  the Gauss-Bonnet term and the cosmological term Λ. We find a class of solutions with  exponential time dependence of two scale factors, governed by two Hubble-like parameters H >0 and h, corresponding to factor spaces of dimensions m >2 and l > 2, respectively. These solutions contain
a fine-tuned Λ = Λ (x, m, l, α), which depends upon the ratio h/H = x, dimensions 
of factor spaces m and l, and the ratio  α = α_2/α_1  
of two constants (α_2 and α_1) of the model.  
The master equation Λ(x, m, l,α) = Λ
is equivalent to a polynomial equation of either  fourth or third order and may be solved
in radicals. The explicit solution for m = l is presented in Appendix. Imposing certain restrictions on x,  we  prove  the stability of the solutions  in a class of cosmological solutions with diagonal  metrics. 
We also consider a subclass of solutions with small enough variation of the effective gravitational constant G and show the stability of all solutions from this subclass. 












§ INTRODUCTION


In this paper  we deal with the so-called  Einstein-Gauss-Bonnet (EGB) gravitational model 
in D dimensions, which contains  Gauss-Bonnet term and cosmological term Λ.  
The so-called Gauss-Bonnet term appeared in (super)string theory  as a correction 
(in appropriate frame) to the  effective (super)string action 
(e.g. heterotic one)  <cit.>-<cit.>. 

It should be noted that at present EGB gravitational model, e.g. with 
cosmological term Λ, and  its modifications, 
see   <cit.>-<cit.> and refs. therein,  are under intensive studies in  cosmology,
aimed at  possible explanation  of  accelerating  expansion of the Universe,
which follows from supernovae (type Ia) observational data <cit.>.

In this paper  we  study the cosmological model with diagonal metric  governed by n >3 scale factors
(D = n+1) depending upon one variable , which is the synchronous time variable. 
As it is well-known,  the multidimensional gravitational model  (D > 4) with quadratic in curvatures 
terms governed by the Lagrangian 
R - 2 Λ + α R^2 + β  R_mn R^mn + γ R_mnpq R^mnpq
leads us for generic values of coupling constants  α, β and  γ
  to the fourth-order differential
equations for the components of the metric, while for the unique choice of the constants α = γ = -4 β 
(yielding  the Gauss-Bonnet term) the equations of motion are of the second order <cit.>.  
In cosmological model with the Gauss-Bonnet term the equations of motion are governed by an effective 
Lagrangian which contains 2-metric (or minisupermetric) G_ij and finslerian metric G_ijkl, see
<cit.>  for Λ = 0 and   <cit.> for Λ≠ 0.  

 Here we consider the cosmological solutions  with exponential dependence of scale factors and obtain 
a class of  solutions with  two scale factors, governed by two Hubble-like parameters H >0 and h, which correspond to factor spaces of dimensions m > 2 and l > 2 , respectively, (D = 1 + m + l)
and obey relations: m H + lh ≠ 0 and  H ≠ h. The solutions depend upon Λ and
 α = α_2/α_1. The latter is the ratio of two constants of the model: α_2 and α_1.
 (In string inspired models α corresponds to Regge slope parameter α' which is inverse proportional 
 to the  tension of the string.)
Any of  these solutions  describes   an exponential (e.g. accelerated) expansion  of 3-dimensional subspace with Hubble parameter H > 0 <cit.>.  Here, as in our previous paper <cit.>, we use the Chirkov-Pavluchenko-Toporensky scheme of reduction  (of the set of polynomial equations)  <cit.>, which gives us a drastical simplification of  the search and analysis of the exact solutions under consideration.  

The special m =3 case   was studied recently in ref. <cit.>.
The analysis of the number of solutions for given Λ and  α 
(denoted as n(Λ,α)), carried out in  ref. <cit.>, 
tells us  that  exponential solutions exist if and only the following bounds
 on  Λ and α   are valid:

    Λα≤3l^2 -7l + 6/8l(l-1) = 3D^2-31D+82/8(D - 4)(D - 5) = λ_c

 for α > 0 and 

    Λ |α| > (l+2)(l+3)/8l(l +1) = (D -2)(D-1)/8(D - 4)(D - 3) =|λ_a|

for α < 0. Here D = 4 + l.

Relations (<ref>) and (<ref>) were obtained (independently) in ref. 
<cit.> [1][1] The second  relation (<ref>) was extended 
in ref. <cit.> to Λ |α| ≥  |λ_a| by adding into 
consideration the case H=h  <cit.>. In ref. <cit.> the cosmological constant Λ_P  is related to our one as  Λ_P = 2 Λ and the internal space dimension l is denoted as D., 
 where they were compared with some other relations obtained in physical literature, e.g. in refs.
<cit.> (Λ < 0, AdS/CFT correspondence), <cit.> (Λ < 0, black holes) etc.
In this paper we generalize these bounds to m ≥ 3 case. 

We  study   the stability of the  solutions in a class of cosmological solutions with diagonal metrics by using results of  refs. <cit.> (see also approach of ref. <cit.>) and show  that the solutions, considered here, are stable if certain restrictions on ratio h/H are imposed.  

Here we also study solutions with a small enough  variation  of the effective gravitational constant 
G  in Jordan  frame <cit.> (see also  <cit.> and refs. therein)
which obey the most severe restrictions on  variation of G from ref. <cit.>. These
solutions are shown to be stable.

We note that a class of stable  solutions with zero variation  of the effective gravitational constant G (either in Jordan or Einstein frames) was considered recently in <cit.> 
for (m,l) ≠  (6,6), (7,4), (9,3). Two  special solutions  for D = 22, 28 and Λ = 0
were found earlier in ref.  <cit.>; in ref. <cit.> it was proved that they are stable.
Another six stable  exponential solutions (five in dimensions D = 7, 8, 9, 13 and two - for D = 14) 
were found recently in <cit.>. 

It should be stressed here that  the main physical motivation for this paper and numerous other articles   devoted to higher dimensional cosmological solutions in EGB gravity is based on (i) a possible solution of “dark energy” problem, i.e.   explaining  the  accelerated expansion of the Universe. Here, as in some  previous papers, we try to strengthen the physical counterpart of the investigations (e.g.  item (i)) by studying also: (ii) the stability of the obtained solutions and  (iii) by considering the solutions with small enough variation of the effective gravitational constant G (in Jordan  frame) which satisfy the observational restrictions.  At the moment the item (ii) is rarely  considered  in physical publications on multidimensional EGB cosmology while the item (iii) is mostly studied (to our knowledge) in our papers. 



§ THE COSMOLOGICAL MODEL


The action of the model reads as follows

    S =  ∫_M d^Dz √(|g|){α_1 (R[g] - 2 Λ) +
                  α_2  L_2[g] },

where g = g_MN dz^M⊗ dz^N is the metric defined on
the manifold M, M = D, |g| = | (g_MN)|, Λ is
the cosmological term, R[g] is scalar curvature,

    L_2[g] = R_MNPQ R^MNPQ - 4 R_MN R^MN +R^2

is the  Gauss-Bonnet term and  α_1, α_2 are
nonzero constants.

We consider the following (product) manifold

    M = ×   M_1 ×…× M_n

with the cosmological metric

    g= - d t ⊗ d t  +
          ∑_i=1^n B_i e^2v^i t dy^i ⊗ dy^i,

where   B_i > 0 are arbitrary constants, i = 1, …, n. In (<ref>) 
M_1, …,  M_n  are one-dimensional manifolds, either   or S^1, 
and n > 3.

The equations of motion for the action (<ref>) 
leads us the to the set of following polynomial equations <cit.>

    G_ij v^i v^j + 2 Λ
      - α   G_ijkl v^i v^j v^k v^l = 0,  
    [ 2   G_ij v^j
        - 4/3α  G_ijkl  v^j v^k v^l ] ∑_k =1^n v^k 
        - 2/3   G_sj v^s v^j  +  8/3Λ = 0,

i = 1,…, n, where  α = α_2/α_1. 
Here 

    G_ij = δ_ij -1, 
           G_ijkl  = G_ij G_ik G_il G_jk G_jl G_kl

are, respectively, the components of two  metrics on  ^n <cit.>: 
2-metric and  Finslerian 4-metric, respectively.
For the case n > 3  (or D > 4) we have a set of forth-order polynomial  equations.

For Λ =0  and n > 3 the set of equations (<ref>)
 and (<ref>) has  an isotropic
 solution v^1 = … = v^n = H only if α  < 0 <cit.>.
In <cit.> an isotropic solution was obtained for Λ≠ 0.

It was proved  in ref. <cit.> that there are no more than
three different  numbers among  v^1,… ,v^n, when Λ =0. 
This is also valid for the case Λ≠ 0 when ∑_i = 1^n v^i ≠ 0  <cit.>.



§ SOLUTIONS WITH TWO HUBBLE-LIKE PARAMETERS


In this section we find a class of solutions to the set of equations (<ref>), 
(<ref>) with the following set of Hubble-like parameters

    v =(H,H,H_“our"  space,H, …, H^m-3, 
       h, …, h^l_internal  space).

where H is the Hubble-like parameter corresponding  
to an m-dimensional factor space with m > 2, while  h is the Hubble-like parameter 
corresponding to the l-dimensional factor space, l > 2. For future 
cosmological applications we split the m-dimensional  
factor space into the  product of two subspaces of dimensions 3 and m-3, respectively. 
The first one is considered as “our" 3d space while the second one is identified with 
a subspace of (m-3 +l)-dimensional internal space.
 
For a description of an accelerated expansion of a
3-dimensional subspace (which may describe our Universe) we put 

    H > 0.


It is  well-known that the (four-dimensional) effective gravitational constant G = G_eff in the Brans-Dicke-Jordan (or simply Jordan) frame <cit.>
is proportional to the inverse volume scale factor
of the internal space, see  <cit.> and references therein.

 We note, that due to  ansatz (<ref>) “our” 3d space expands (isotropically) with 
Hubble parameter H and (m -3)-dimensional part of internal space expands (isotropically) with the same Hubble parameter H too. Here  we consider for physical applications (in our epoch) the internal space to be compact one, i.e. we put in (<ref>) M_4 = … = M_n = S^1). We also put the internal scale factors corresponding to present time  to be small enough as compared to the scale factor of “our” space.                    

Due to  ansatz (<ref>),  the m-dimensional factor space is expanding with the Hubble parameter H >0, while  the evolution of the l-dimensional factor space is described  by the Hubble-like  parameter h.

It was shown earlier in refs. <cit.> 
 that the ansatz  (<ref>) with two  restrictions on parameters H and h  
  imposed 
   
    m H + lh ≠ 0,       H ≠ h,

 leads us to a reduction of the relations (<ref>) and (<ref>)  to the  following set of two polynomial equations
  
    E =  m H^2 + l h^2 - (mH + lh)^2  + 2 Λ 
            - α [m (m-1) (m-2) (m - 3) H^4
              
    
           + 4 m (m-1) (m-2) l H^3 h   
           + 6 m (m-1) l (l - 1) H^2 h^2
             
    
           + 4 m l (l - 1) (l - 2) H h^3  
           + l (l - 1) (l - 2) (l - 3) h^4] = 0,   
    
      Q =  (m - 1)(m - 2)H^2 + 2 (m - 1)(l - 1) H h 
            
     
           + (l - 1)(l - 2)h^2 = - 1/2 α.


Using equation  (<ref>) we get for m > 2 and l > 2 

    H   =     (- 2 α P)^-1/2,

where we denote

    P  =   P(x,m,l)  =   (m - 1)(m - 2) 
      
      
      + 2 (m - 1)(l - 1) x  + (l - 1)(l - 2)x^2, 
     
    
        x  = h/H,

and 
 
    α P < 0.

We note that the following “duality” identity
is obeyed 

    P(x,m,l) = x^2  P(1/x,l,m),

for x ≠ 0.

Due to restrictions (<ref>)  we have 

    x ≠ x_d = x_d(m,l) ≡  - m/l,       x ≠ x_a ≡ 1,

where 
 
    x_d(m,l) =  1/x_d(l,m).

 
 The relation  (<ref>) is valid only if        

    P(x,m,l) ≠ 0.

We note, that for P(x,m,l) = 0 the relation (<ref>) is not obeyed.

The substitution of relation (<ref>) into (<ref>)
leads us to following formulas

    Λα = λ  =  λ(x,m,l) ≡1/4 ( P(x,m,l))^-1 M(x,m,l)
             
    
          + 1/8 (  P(x,m,l))^-2 R(x,m,l),  
                      
     M(x,m,l) ≡  m  + l x^2 -(m  + l x)^2, 
                      
     R(x,m,l) ≡  m (m-1) (m-2) (m - 3)
                     + 4 m (m-1) (m-2) l x 
                        
        
           + 6 m (m-1) l (l - 1) x^2        
           + 4 m l (l - 1) (l - 2)  x^3
                
    
           + l (l - 1) (l - 2)(l - 3) x^4.


In what follows we use the following duality identities

    M(x,m,l) = x^2 M(1/x,l,m), 
                      
     R(x,m,l) =  x^4 R(1/x,l,m)

 for x ≠ 0.  
 The identities  (<ref>), (<ref>)  and (<ref>) imply
 
    λ(x,m,l) = λ (1/x,l,m)

for x ≠ 0.

Using (<ref>) we obtain
 
    x ≠ x_± = x_±(m,l) 
      ≡-(m - 1)(l - 1) ±√(Δ(m,l))/(l - 1)(l - 2), 
         
    Δ(m,l) ≡  (m - 1)(l - 1)(m + l - 3) = Δ(l,m).

Here x_±(m,l) are roots of the quadratic equation P(x,m,l) =0.
These roots obey the following relations

    x_+(m,l) x_-(m,l) =  (m - 1)(m - 2)/(l - 1)(l - 2), 
                 
    
       x_+(m,l) + x_-(m,l) =  - 2 (m - 1)/l - 2,

which lead us to the  inequalities

    x_-(m,l) < x_+(m,l) < 0.

These relations and the duality identity (<ref>) imply 

    x_∓(m,l) = 1/x_±(l,m)

 for all m > 2 and l > 2.

Using (<ref>) and (<ref>) we get 
 
    Λ   = α^-1λ(x,m,l),

where 

    x_-(m,l) < x <  x_+(m,l)   for α > 0
    
  and    
   
    x <  x_-(m,l),   or x > x_+(m,l)   for α < 0.


For α < 0 we have the following limit 
  
    lim_x →±∞λ(x,m,l) = 
       λ_∞(l) ≡  - l(l + 1)/8 (l - 1)(l - 2) < 0.

  Hence 
 
    lim_x →±∞Λ = 
       Λ_∞≡  -  l(l + 1)/8 α (l - 1)(l - 2) > 0,

l > 2. We note that Λ_∞ does not depend upon m.
For x = 0 we get in agreement with the duality identity (<ref>) 

    Λ = Λ_0 = α^-1λ(0,m,l) = 
        -  m(m + 1)/8 α (m - 1)(m - 2) > 0,

m > 2. We see that Λ_0 does not depend upon l. For x = 0
the Hubble-like parameters read 

    H  = H_0 = (- 2 α (m - 1)(m - 2))^-1/2,      h = 0
  
and due to relations  (<ref>),  (<ref>) we get the product
of (a part of) (m+1)-dimensional de-Sitter space and l-dimensional 
Euclidean space.

“Master” equation. We rewrite eq. (<ref>)  
in the following form

    2  P(x,m,l)  M(x,m,l)
           +   R(x,m,l) - 8 λ (  P(x,m,l))^2 = 0.
                
This equation may be called as a master equation, since the 
solutions under consideration are governed by it.
The master equation is of fourth order in x  
for λ≠λ_∞(l) or less (of third order for
λ = λ_∞(l)). 
For any m > 2 and l > 2 the  equation (<ref>) may be solved  in radicals, 
though the general  solution has a rather cumbersome form and will not be presented here.
It is worth for any given m and l to find the solution just 
by using Maple or Mathematica. An example of explicit (generic) solution for m = l > 2 
is presented in Appendix A (for  m = l = 3 see ref. <cit.>). 
Several special solutions with m=3 and l =4  were given in ref.  <cit.>. 

Now we consider the behaviour of the function  λ(x,m,l)   in the vicinity of 
 the points x_-(m,l) and  x_+(m,l). Here the following proposition takes place.

Proposition 1. For m > 2, l > 2 
 
    λ(x,m,l) ∼ B_±(m,l) (x - x_±(m,l))^-2,

 as x → x_± = x_±(m,l), where B_±(m,l) < 0 and hence
 
    lim_x → x_±λ(x,m,l) = - ∞.

 

The  Proposition 1 can be proved by using  the following lemma.

Lemma. For all m > 2, l > 2 
 
    R_±(m,l) ≡ R(x_±(m,l),m,l) < 0.



 The proof of the Lemma is presented in the Appendix B. 

Proof of the Proposition 1. Relation (<ref>)
follows from  (<ref>), P(x,m,l) = (l-1)(l-2) (x - x_+)(x - x_-) and Lemma.
Here 
 
    B_±(m,l) =  R_±(m,l)/8 (l-1)^2 (l-2)^2 (x_+ - x_-)^2
      
      =  R_±(m,l)/32 (m-1) (l-1)(m + l - 3)   < 0,
 
where  m > 2 and l > 2. Relation (<ref>)  follows from (<ref>) and (<ref>).
Thus, the Proposition 1 is proved. 

Now we analyze the behaviour of the function  λ(x,m,l), for fixed 
m,l and x ≠ x_±(m,l).
We find the  extremum  points  obeying ∂/∂ xλ(x,m,l) = 0.
By straightforward calculations we obtain 

    ∂/∂ xλ(x,m,l) = - f(x,m,l) ( P(x,m,l))^-3,
                        
    
            f(x,m,l) = (l-1)(m-1)(x-1)(lx+m) ×
    × [(l-2)x + m-1][(l-1)x + m-2],

x ≠ x_±(m,l). Using these relations we are led to the following
extremum points
 
    x_a = 1,
                    
    
            x_b = x_b(m,l) ≡ - m-1/l-2 < 0,
                     
    
            x_c = x_c(m,l) ≡ - m-2/l-1 < 0,
                             
    
            x_d = x_d(m,l) ≡ - m/l < 0.


These points obey  the following duality identities
 
    x_d(m,l) = 1/x_d(l,m),
                    
    
         x_b(m,l) = 1/x_c(l,m).

We also obtain the inequality

    x_b(m,l) < x_c(m,l),

which is valid since 

    x_c(m,l) - x_b(m,l) = m + l -3/(l-1)(l-2) > 0

for all m >2, l>2.
   
   The points x_b, x_c, x_d obey the following 
   inclusion
  
    x_i(m,l) ∈  (x_-(m,l),x_+(m,l)),

  i = b,c,d for  m >2, l>2. 
  This inclusion just follows from relations  P_i(m,l) =  P(x_i(m,l),m,l) < 0,
  i = b,c,d, since
  
    P_b(m,l) = - (m - 1)(m + l -3)/(l - 2) < 0,
                      
     P_c(m,l) = - (m - 2)(m + l -3)/(l - 1) < 0,
                       
     P_d(m,l) = - (l - 2) m^2 + 2 l m + (m - 2) l^2/l^2 < 0,

 for all m > 2 and l > 2.

By using relations

    x_d - x_c    = m-2l/l(l-1), 
    
    
    x_d - x_b    = 2m-l/l(l-2),

we get 

    (1)  x_b < x_c < x_d,    for  l < m/2,  
    
    (2)  x_b < x_d < x_c,    for   m/2 < l < 2m, 
    
    (3)  x_d < x_b < x_c,    for  l > 2m, ,
	
and 

    (1_0)  x_b < x_c = x_d,    if  l = m/2, 
    
    (3_0)  x_d = x_b < x_c,    for  l = 2m.
	
We note that m/2 < l < 2m may be rewritten as l/2 < m < 2l. 

Let us calculate λ_i=λ(x_i,m,l), i=a,b,c,d.
We obtain

    λ_a=-(m+l-1)(m+l)/8(m+l-3)(m+l-2)<0,



    λ_b=lm^2+(l^2-8l+8)m+l^2-l/8(l-2)(m-1)(l+m-3)>0,



    λ_c=ml^2+(m^2-8m+8)l+m^2-m/8(m-2)(l-1)(l+m-3)>0,

and 

    λ_d=ml(m+l)/8(lm^2+ml^2-2m^2-2l^2+2lm)>0,

In the proof of inequalities  (<ref>), (<ref>) the following relations were
used: 

    u(m,l) = l m^2+(l^2-8l+8)m+l^2-l  > 0,       
    
       v(m,l) = m l^2+(m^2-8m+8)l+m^2-m  > 0        
    
       w(m,l) = m^2(l-2) + l^2(m -2) + 2lm >0
 
for  m > 2, l > 2. Indeed,  for m ≥ 4, l ≥ 4  we get 
u(m,l) = ml(m + l- 8) + 8m +l^2-l > 0 and u(4,3) = 26, u(3,4) = 24, u(3,3) = 12.   
Thus, the first relation (<ref>) is valid. The second one (<ref>)
just follows from the first one and v(m,l)= u(l,m).

We also get 

    λ_b-λ_c=(m-l)(m+l-3)/4(l-2)(l-1)(m-2)(m-1)
     >0, if  m>l, 
    
     =0, if  m=l, 
    
     <0, if  m<l.

and

    λ_d-λ_c=(m - 1)(m-2l)^3/4(l-1)(m-2)(m+l-3)w(m,l), 
                 
    λ_d-λ_b=(l-1)(l-2m)^3/4(m-1)(l-2)(m+l-3)w(m,l),

  for  m > 2, l > 2. ( w(m,l) is defined in (<ref>).)
By using these relations we obtain 

    λ_d-λ_c 
     >0, if  m > 2l, 
    
     =0, if  m = 2l, 
    
     <0, if  m < 2l.

and 

    λ_d - λ_b 
     >0, if  l > 2m, 
    
     =0, if  l = 2m, 
    
     <0, if  l < 2m.


Now we study the behaviour of the function λ(x,m,l) with respect to variable x for fixed  integer numbers m >2, l>2. We denote by n(Λ, α) the number of solutions 
(in x) of the equation  Λα = λ(x,m,l). We calculate n(Λ, α)
by using unequalities for points of extremum  x_i and λ_i (i = b,c,d) presented above and 
relations (<ref>) and  (<ref>).	

First, we start with the case α > 0 and x_- < x < x_+. 
 
(1) m > 2l. We get x_b < x_c < x_d and  λ_c < λ_d < λ_b.
Here x_b and x_d are points of local maximum 
(x_b is a point of maximum on interval (x_-, x_+))
  and   x_c is a point of local minimum.  We obtain          
 
    n(Λ, α) = 
       0,  Λα > λ_b, 
    
       1,  Λα= λ_b, 
    
       2,  λ_d < Λα < λ_b, 
    
       2,  Λα = λ_d, 
    
       4,  λ_c < Λα < λ_d, 
    
       3,  Λα = λ_c, 
    
       2,  Λα < λ_c.

Here and in what follows we use x ≠ x_d.

We present an example of the function λ(x) = Λα for α > 0,  m=12 and l=3  at
Figure 1. At this and other figures  the point (x_i, λ_i) is marked by i, where i = a,b,c,d.



 (1_0) m = 2l. We have x_b < x_c = x_d and λ_c = λ_d < λ_b.
Here x_b is the point of maximum on the interval (x_-, x_+) and   x_c = x_d
 is the point of inflection.
     We obtain          
  
    n(Λ, α) = 
        0,  Λα > λ_b, 
    
        1,  Λα= λ_b, 
    
        2,  λ_d < Λα < λ_b, 
    
         1,  Λα = λ_d, 
    
          2,  Λα < λ_d.


An example of the function λ(x) = Λ(x) α for α > 0,  m=12 and l=6 is depicted at
Figure 2.





 (2) l/2 < m < 2l. 
 We obtain x_b < x_d < x_c and λ_d < λ_c,  λ_d  < λ_b.
 The points x_b,  x_c are points of local maximum and x_d is point of local minimum.
  
  Now we split this case on three subcases: 
  (2_+) l< m < 2l, (2_0) m = l and (2_-) l/2 < m < l.
 
 (2_+) l < m < 2l. In this subcase we have λ_d < λ_c < λ_b and 
 hence 

 
    n(Λ, α) = 
       0,  Λα > λ_b, 
    
       1,  Λα= λ_b, 
    
       2,  λ_c < Λα < λ_b, 
    
       3,  Λα = λ_c, 
    
       4,  λ_d < Λα < λ_c, 
    
       2,  Λα = λ_d, 
    
       2,  Λα < λ_d.

 
 An example of the function λ(x) = Λ(x) α for α > 0,  m=9 and l=7 is depicted at
 Figure 3.
 
 
 

(2_0) m = l. In this subcase we find  λ_d < λ_c = λ_b.  
 Hence 

 
    n(Λ, α) = 
       0,  Λα > λ_b = λ_c, 
    
       2,  Λα= λ_b = λ_c, 
    
       4,  λ_d < Λα < λ_b = λ_c, 
    
       2,  Λα = λ_d, 
    
       2,  Λα < λ_d.



At  Figure 4 an example of the function λ(x) = Λ(x) α for α > 0 and  m=l=4 is presented. 
 
 


 (2_-) l/2 < m < l. In this subcase one gets  λ_d < λ_b < λ_c and 
  
    n(Λ, α) = 
        0,  Λα > λ_c, 
    
        1,  Λα= λ_c, 
    
        2,  λ_b < Λα < λ_c, 
    
        3,  Λα = λ_b, 
    
        4,  λ_d < Λα < λ_b, 
    
        2,  Λα = λ_d, 
    
        2,  Λα < λ_d.


An example of the function λ(x) = Λ(x) α for α > 0,  m=4 and l=6 is depicted at
 Figure 5.
 
 



(3) m < l/2. We have x_d < x_b < x_c and  λ_b < λ_d < λ_c. Here
x_c and x_d are points of local maximum (x_c is the point of maximum on interval (x_-, x_+))
  and   x_b is a point of local minimum.  We find           
 
    n(Λ, α) = 
       0,  Λα > λ_c, 
    
       1,  Λα= λ_c, 
    
       2,  λ_d < Λα < λ_c, 
    
       2,  Λα = λ_d, 
    
       4,  λ_b < Λα < λ_d, 
    
       3,  Λα = λ_b, 
    
       2,  Λα < λ_b.

 
 An example of the function λ(x) = Λ(x) α for α > 0,  m=4 and l=12 is depicted at
  Figure 6.
  
  

(3_0) m = l/2. We get relations x_d = x_b < x_c and  λ_b = λ_d < λ_c.
Here x_c is the point of maximum on interval (x_-, x_+) and   x_b = x_d
 is the point of inflection.
     We obtain          
  
    n(Λ, α) = 
        0,  Λα > λ_c, 
    
        1,  Λα= λ_c, 
    
        2,  λ_d < Λα < λ_c, 
    
        1,  Λα = λ_d, 
    
        2,  Λα < λ_d.


Bounds on  Λα for α > 0.
Summarizing all cases presented above we find that for α > 0
exact solutions under consideration exist if and only if
 
    Λα≤ λ_b,  for m ≥ l, 
     λ_c,   for m < l,

where λ_b = λ_b (m,l) and  λ_c = λ_c (m,l) 
are defined in  (<ref>) and  (<ref>), respectively.
For m = 3 and l ≥ 3 we are led to relation  (<ref>).

 An example of the function λ(x) = Λ(x) α for α > 0,  m=4 and l=8 is presented at
  Figure 7. By using our analysis  we find  that for α > 0 and small enough value of Λ
   there exist at least a pair of solutions x_1, x_2, obeying x_- < x_1 < x_2 < x_+ < 0.
  
  


Let us consider the case  α < 0. 
We have Λ |α| = - λ(x), where x < x_- or x > x_+.  Due to to 
the relations (<ref>), (<ref>)  and Proposition 1 
the function - λ(x)  is monotonically increasing in two intervals: i) in the
interval (-∞,x_-) from - λ_∞ to + ∞ and ii) in the
interval (x_a = 1, +∞) from - λ_a to - λ_∞. 
The function - λ(x)  is monotonically decreasing in the  interval  
(x_+, x_a) from + ∞ to - λ_a.
Here x_a is a point of local minimum of the function Λ(x) |α| = - λ(x), which
is excluded from the solution and   - λ_a < - λ_∞.  
This inequality may be readily verified: 
 due to relations (<ref>), (<ref>) we obtain
  
 
    λ_a - λ_∞ = 2(m-1)[(2l-1)m + 2l(l-2)]/8(l - 1)(l - 2)(m+l-3)(m+l-2) > 0,

 for m > 2 and l > 2. The  functions 
 Λ(x)  = λ(x)/α   for α = +1, -1, respectively, and m = l =4 are 
 presented at Figure 8.

   For the number of solutions (for α < 0) we obtain          
   
    n(Λ, α) = 
         2,  Λ |α| > |λ_∞|, 
    
         1,  Λ |α| = |λ_∞|, 
    
         2,   |λ_a| < Λ |α| < |λ_∞|. 
    
         0,  Λ |α| ≤ |λ_a|.

Here  x ≠ x_a =1. Hence, for α < 0 and big enough values of Λ
there exist  two solutions x_1, x_2: x_1 < x_- < 0  and  x_2 > x_+.

Bounds on  Λ |α| for α < 0.
It follows from (<ref>) that for α < 0
exact solutions  under consideration exist if and only if
 
    Λ |α| >     |λ_a| = (D -2)(D-1)/8(D - 4)(D - 3),

where λ_a  is defined in  (<ref>).
This relation is valid for all m > 2, l > 2 (D = m + l +1), 
e.g. for m = 3 (see (<ref>)).


 




§ STABILITY ANALYSIS


Here,  using results of refs. <cit.>, we study the stability of exponential solutions (<ref>)
with non-static total volume factor, i.e. we put

    S_1(v) = ∑_i = 1^n v^i ≠ 0.


Here, as in  <cit.>, we  impose the following restriction 

    ( R )    (L_ij(v)) ≠ 0

on the symmetric matrix 

    L =(L_ij(v)) = (2 G_ij - 4 α G_ijks v^k v^s).


For general cosmological ansatz with the (diagonal) metric 

    g= - dt ⊗ dt + ∑_i=1^n e^2β^i(t)  dy^i ⊗ dy^i,

we have  the  set of (algebraic and differential) equations <cit.>

    E = G_ij h^i h^j + 2 Λ  - α G_ijkl h^i h^j h^k h^l = 0,
             
    
             Y_i =  d L_i/dt  +  (∑_j=1^n h^j) L_i -
                     2/3 (G_sj h^s h^j -  4 Λ) = 0,

where h^i = β̇^i,           
 
    L_i = L_i(h) = 2  G_ij h^j
           - 4/3α  G_ijkl  h^j h^k h^l  
           ,

 i = 1,…, n.

Earlier, it was proved  <cit.> that a constant solution
(h^i(t)) = (v^i) (i = 1, …, n; n >3) to eqs. (<ref>), (<ref>)
obeying restrictions (<ref>), (<ref>) is  stable under perturbations

    h^i(t) = v^i +  δ h^i(t),

 i = 1,…, n,  (as t → + ∞)  in the following case

    S_1(v) = ∑_k = 1^n v^k > 0

and  it is unstable (as t → + ∞) when 

    S_1(v) = ∑_k = 1^n v^k < 0.


For our considertion we have S_1(v) = m H + l h and hence due to H >0  the restriction
(<ref>) may be written in the following form

    x > - m/l = x_d,

while the  restriction (<ref>) may be written as

    x < - m/l = x_d.


The  perturbations δ h^i 
obey (in the linear approximation) the following set of linear equations
<cit.>
 
    C_i(v) δ h^i = 0, 
    
       L_ij(v) δḣ^j =  B_ij(v) δ h^j.

  Here
 
    C_i(v)  =  2 v_i - 4 α G_ijks  v^j v^k v^s, 
    
     L_ij(v) = 2 G_ij - 4 α G_ijks v^k v^s,
        
    
     B_ij(v) = - (∑_k = 1^n v^k)  L_ij(v) - L_i(v) + 4/3 v_j,

 where v_i = G_ij v^j,  L_i(v) =  2 v_i - 4/3α  G_ijks  v^j v^k v^s
 and i,j,k,s = 1, …, n.

In case when restrictions (<ref>), (<ref>) are imposed,  the set of  equations on  perturbations (<ref>), (<ref>)  has the following solution <cit.>
   
    δ h^i = A^i exp( -  S_1(v) t ),
           
    ∑_i =1^n C_i(v)  A^i =0,

    (A^i are constants) i = 1, …, n.    

It was shown in  <cit.> that  for  the vector v from  (<ref>), obeying
 relations (<ref>), the matrix L is a block-diagonal one

    (L_ij) =  diag(L_μν, L_αβ ),

where

    L_μν =  G_μν (2 + 4 α S_HH),
     
    
      L_αβ = G_αβ (2 + 4 α S_hh)

and

    S_HH =  (m-2)(m -3) H^2  + 2(m-2)lHh + l(l - 1)h^2 ,
      
    
      S_hh = m(m-1)H^2  + 2m(l - 2)Hh+ (l- 2)(l- 3)h^2.


The matrix (<ref>) is invertible  only if  m > 1,  l > 1 and

    S_HH≠ - 1/2 α, 
    
     S_hh≠ - 1/2 α.

We remind (the reader) that  the matrices  (G_μν) = (δ_μν -1 ) and
(G_αβ) = (δ_αβ - 1) are invertible only if  m > 1 and l > 1.


Now, we prove that inequalities (<ref>), (<ref>) are obeyed
if   
 
    x ≠  -  m-2/l -1 = x_c

and
 
    x ≠  - m -1/l - 2 = x_b

for l > 2.

Let us suppose that  (<ref>) does not take place, i.e.  
S_HH = - 1/2 α.  Then using (<ref>) we obtain  

    S_HH - Q = - 2 (H - h) ((m-2) H + (l -1) h ) = 0,

which implies  due to H - h ≠ 0 (see  (<ref>))         

    (m-2) H + (l -1) h = 0.

 This relation contradicts to the  restriction (<ref>). 
 The obtained contradiction proves the inequality (<ref>).     
      
 Now let us  suppose that  (<ref>) is not valid,
 i.e. S_hh = - 1/2 α. Then using (<ref>) we find  
 
    S_hh - Q =  - 2 (h - H)  ((l - 2) h + (m -1) H ) = 0.

 Due to H - h ≠ 0 this implies           
 
    (l - 2) h + (m -1) H = 0,

  which is in the contradiction with the restrictions (<ref>)
  and H > 0. This contradiction lead  us to the proof of the inequality
 (<ref>).         

Thus, we have proved that relations (<ref>) and (<ref>) are valid and hence the
 restriction (<ref>) is satisfied for our solutions. 

Thus we have proved the following proposition. 

Proposition 2. The cosmological solutions under consideration, which obey  x = h/H ≠ x_i, i = a,b,c,d, where x_a =1, x_b = - m -1/l - 2, x_c = - m-2/l -1,  x_d = - m/l, 
are  stable if i) x > x_d and unstable if ii) x < x_d. 

Here it should be noted that our anisotropic solutions with non-static volume factor
are not defined for x= x_a and x= x_d. Meanwhile, they
are defined when x = x_b or x = x_c, if x ≠ x_d. The stability 
analysis of these special solutions can not be covered by the equations for perturbations 
(<ref>), (<ref>) in the linear approximation. As it was pointed 
 out in ref. <cit.> this analysis needs a special consideration.  

Now we  consider  the number of non-special stable solutions which are
given by Proposition 2 (see item i)). We denote this number as n_+(Λ, α).
By using the results  from the previous section (e.g. illustrated by figures) we 
obtain for α > 0:   

  (1), (1_0) m ≥ 2l          
  
    n_+(Λ, α) = 
        0,  Λα≥λ_d, 
    
        1,  Λα < λ_d;

 
  (2) l/2 < m < 2l 
     
    n_+(Λ, α) = 
        0,  Λα≥λ_c, 
    
        2,  λ_d < Λα < λ_c, 
    
        1,  Λα≤λ_d;

  
  (3) m < l/2         
  
    n_+(Λ, α) = 
        0,  Λα≥λ_c, 
    
        2,  λ_d ≤Λα < λ_c, 
    
        3,  λ_b < Λα < λ_d, 
    
        1,  Λα≤λ_b;

  
   (3_0) m = l/2          
   
    n_+(Λ, α) = 
         0,  Λα≥λ_c, 
    
         2,  λ_d < Λα < λ_c, 
    
         1,  Λα≤λ_d.

 
 We see, that for α > 0 and small enough value of Λ there exists at least one stable solution 
 with x ∈ (x_-,x_+). 
 
 Bounds on  Λα for stable solutions with α > 0.
 Summarizing all cases presented above we find that for α > 0
 stable exact solutions under consideration exist if and only if
  
    Λα <  λ_d,  for m ≥ 2l, 
     λ_c,   for m < 2l,

 where λ_c = λ_c (m,l) and  λ_d = λ_d (m,l) 
 are defined in  (<ref>) and  (<ref>), respectively.
 For m = 3 and l > 2 we are led to relation  
  Λα <  λ_c instead of  (<ref>).
 
 
 In the case  α < 0 we obtain          
    
    n_+(Λ, α) = 
          1,  Λ |α| ≥ |λ_∞|, 
    
         
          2,   |λ_a| < Λ |α| < |λ_∞|, 
    
          0,  Λ |α| ≤ |λ_a|.

 Here the inequality  x ≠ x_a =1 was used. Our analysis tells us that 
  for α < 0 and big enough value of Λ
 there exists at least one stable solution governed by x which obey x > x_+. 
 We also find that the solution with x < x_- is  unstable. 
 
 Bounds on  Λ |α| for stable solutions with α < 0.
 It follows from (<ref>) that for α < 0
 stable exact solutions  under consideration exist if and only if
 the relation (<ref>) (Λ |α| >   |λ_a|)  is obeyed.
 
  
 


§ SOLUTIONS DESCRIBING A SMALL ENOUGH VARIATION OF G



  Here we analyze the solutions  by using the restriction on variation of the effective
  gravitational constant G, which is inversely proportional (in the Jordan frame) 
  to the  volume scale factor  of the (anisotropic) internal space <cit.> 
  (see also references therein), i.e.

  
    G =  const exp[- (m - 3) H t - l h t].

 
By using (<ref>) one can get the following formula 
for a dimensionless parameter of temporal variation of G (G-dot):

    δ≡Ġ/GH  =  - (m - 3 + l x),      x = h/H.

Here  H >0 is the Hubble parameter.

Due to observational data, the variation of the gravitational constant is 
on the level of 10^-13 per year and less.
For example, one can use, as it was done in ref. <cit.>,  the following bounds on the 
value of the dimensionless variation of the  effective gravitational constant:

 
    - 0,65 · 10^-3 < δ < 1,12 · 10^-3.

They come from the most stringent limitation
on G-dot obtained by the set of ephemerides <cit.>

 



and  value of the Hubble parameter (at present) <cit.>

 
 
 
 when both are written with 95% confidence level <cit.>. 
      
 
  When the value  δ is fixed we get from (<ref>)
    
    x =  x_0(δ) = x_0(δ,m,l) ≡  - (m - 3 + δ)/l.


We remind (the reader) that our solutions are defined if 
 
    P(x_0(δ,m,l),m,l) ≠ 0,

 or, if
 
    x_0(δ,m,l) ≠  x_±(m,l).

   

The substitution of  x = x_0(δ,m,l) into quadratic polynomial (<ref>) 
gives us 

    P(x_0(δ,m,l),m,l)  =  P(x_0(0,m,l),m,l)
        
    
       -  4  (l-1)(m + l -3) /l^2δ  +  (l-1)(l-2) /l^2δ^2,

where (see  <cit.>)

    P(x_0(0,m,l),m,l) ≡ P_0(m,l) =   1/l^2 (m + l - 3)[(5-m)l +2m -6].

 
 We note that equation P_0(m,l) = 0 implies relation 
 l = l_0(m) = 2m - 6/m - 5 = 2 + 4/m - 5, m ≠ 5. 
 For m > 9 we get 2 < l_0(m) < 3, that
 means that integer solutions are absent in this interval. 
 For 3 ≤ m ≤ 9 and  m ≠ 5, the only integer 
 values of l_0(m) > 2  takes place for m = 6,7,9 and we get a special set of pairs (m,l):
   
    A)    (m,l) = (6,6), (7,4), (9,3),

 which was obtained in <cit.>. In the case A) the restriction (<ref>)
 gives us (see (<ref>)) δ≠ 0 and - 4 (l +m -3) δ + (l-2) δ^2 ≠ 0 
 for l > 2 which lead
 us to two restrictions: δ≠ 0 and δ≠  4l + m -3/l-2 = 9, 16,36 for 
 (m,l) = (6,6), (7,4), (9,3), respectively.
 But the second one may be omitted due to bounds (<ref>). 

    Let us consider the second case
   
    B)    m =5,    l > 2.

     In this case the restriction (<ref>) reads
     
    4 (l + 2) - 4 (l-1) (l+ 2) δ  +  (l-1)(l-2)  δ^2 ≠ 0,

   l > 2.  It may be rewritten
   as  
   
    δ≠δ_±(5,l) ≡  2 (l+2)/(l-2)(1 ±√(l^2/(l-1)(l + 2))).

  The first restriction  δ≠δ_+(5,l), l >2, may be omitted due to the bounds
  (<ref>) since δ_+(5,l) > 2 (l + 2)/(l-2) >  2 for l > 2. So,
  the only  second restriction δ≠δ_-(5,l), l >2, should be imposed.
  Since 
  
    δ_-(5,l) =  2/l-1(1 + √(l^2/(l-1)(l + 2)))^-1∼ 1/(l-1)

  as l → + ∞, this restriction forbids one 
  value  of δ
  obeying the bounds  (<ref>) for big anough value of l (e.g., for l > 1000). 
      
Now we consider the last case
  
    C)  (m,l)   do  not  belong  to  cases  A  and  B .


In the case C) the restriction (<ref>) reads
     
    (l + m - 3) [(5-m)l +2m -6] - 4 (l-1) (l + m - 3) δ  +  (l-1)(l-2)  δ^2 ≠ 0,

   l > 2.  It may be rewritten
   as  
   
    δ≠δ_±(m,l) ≡  2 (l + m - 3)/(l-2)(1 ±√(l^2 (m-1)/4(l-1)(l + m - 3))).

  The first restriction  δ≠δ_+(m,l) (l >2) may be omitted due to the bounds
  (<ref>) since δ_+(m,l) > 2 (l + m -3)/(l-2) >  2 for m >2, l > 2. So,
  the only  second restriction δ≠δ_-(m,l), should be imposed.
  Here another equivalent relation may be used
  
    δ_-(m,l) =  (5-m)l +2m -6/2(l-1)(1 + √(l^2 (m-1) /4(l-1)(l + m - 3)))^-1.


Thus, for our special values of δ obeying the bounds  (<ref>) the
only restriction on δ coming from (<ref>) or (<ref>) are the following
ones 
 
    δ≠ 0,  in  the  case  A,  
    δ≠δ_-(5,l),   in  the  case  B,
    δ≠δ_-(m,l),  in  the  case  C,

where δ_-(5,l) is defined in (<ref>) and δ_-(m,l) is defined in (<ref>).

Now we analyse the stability of these special solutions. The main condition
for stability x_0(δ) > x_d is satisfied  since
 
    x_0(δ) -  x_d = 3 - δ/l > 0

 due to our bounds  (<ref>).
 
 Other three conditions (see Proposition 2) : 
 x_0(δ) ≠  x_a, x_0(δ) ≠  x_b and x_0(δ) ≠  x_c read  
   
    δ≠δ_a =  - (m + l - 3),    
    δ≠δ_b = 2(m + l - 3)/l - 2,  
    δ≠δ_c = 2(m + l - 3)/l - 1.

They are satisfied due to bounds (<ref>) and inequalities:  
δ_a ≤  -3, δ_b > 2 and δ_c > 1.  

Thus, we have shown that all well-defined solutions under consideration, which  obey restrictions 
(<ref>), (<ref>), (<ref>) and the physical bounds (<ref>), are stable.   



§ CONCLUSIONS


We have considered the  D-dimensional  Einstein-Gauss-Bonnet (EGB) model
with the Λ-term and two non-zero constants α_1 and α_2.  
By dealing with diagonal  cosmological  metrics, we have found 
for certain  (fine-tuned) Λ = Λ(x,m,l,α) with  
α = α_2 / α_1 a class of solutions with  exponential 
time dependence of two scale factors.
This exponential dependence is governed by two Hubble-like parameters H >0 and h, 
corresponding to submanifolds of dimensions m > 2 and l > 2, respectively, with  D = 1 + m + l. 
Here m > 2 is the dimension  of the expanding subspace and l > 2
is the dimension of another  one, the dimensionless parameter x = h/H satisfies the 
following restrictions:  x ≠ 1,  x ≠ x_d = - m/l and 
 (m - 1)(m - 2) + 2 (m - 1)(l - 1) x   + (l - 1)(l - 2)x^2 ≠ 0.

Any obtained solution describes an exponential expansion of  3-dimensional subspace
 (which may be identified with “our” space) with
the Hubble parameter H > 0 and anisotropic behaviour of (m-3+ l)-dimensional internal space:
expanding in (m-3) dimensions (with Hubble-like parameter H) and  either contracting, 
or expanding (with Hubble-like parameter h) or stable in l dimensions.
The solutions are governed by master equation Λ(x,m,l,α) = Λ, which 
may be solved in radicals for all values of Λ, since it is equivalent to a polynomial equation of either  
fourth or third order (depending upon Λ). The analytical solution for m = l <cit.> 
 is presented in  Appendix A.

Here we have obtained the bounds on  Λ which guarantee the existence
of the exponential cosmological solutions under consideration:
 
    Λα≤λ_*

 for α > 0 and
 
    Λ |α| >    (D -2)(D-1)/8(D - 4)(D - 3)

for α < 0. In (<ref>) we denote: λ_* = λ_b for m ≥ l 
and  λ_* = λ_c   for m < l, where  λ_b  and  λ_c 
are defined in (<ref>)  and  (<ref>), respectively. These bounds 
generalize the bounds (<ref>) and (<ref>) for m =3 <cit.>.
It should be noted that the bounds (<ref>) and (<ref>) were obtained here  without solving 
the equations of motion (e.g. the master equation). They were obtained by 
analyzing the function λ = λ(x,m,l)      from (<ref>) (λ =  Λα ), 
e.g. by using the “duality” identity  λ(x,m,l) = λ (1/x,l,m). 
(The “duality” transformation (x,m,l) ↦ (1/x,l,m) describes  just a trivial 
 interchange of factor spaces which corresponds to the replacement (H,h, m,l) ↦ (h, H,l,m).)
  
Using the scheme, which was developed in ref. <cit.>,  
we  have proved that any of these solutions obeying the additional restrictions: 
x ≠  - m -2/l - 1 and  x ≠  - m -1/l - 2,
 is stable (as  t → + ∞) if  x > x_d = - m/l and unstable if  x < x_d.

We have also found that for α > 0  stable exact solutions
exist if and only if: 

    Λα < λ_d

 for m ≥ 2l    and  
 
    Λα <  λ_c,

 for m < 2l,     
 where   λ_d is defined in   (<ref>).
 For α < 0  stable exact solutions exist only if
 the relation (<ref>)   is obeyed.

It was also shown that all (well-defined) solutions with small enough varation of 
the effective gravitational constant G (in the Jordan frame)
are stable.

Here an open problem is to extend  the cosmological solutions from this paper 
to the cosmological type solutions in the Lovelock gravitational model <cit.>,
 e.g. to solutions  describing cosmological and static configurations. Another problem
is related to  search and analysis of  the solutions with three factor spaces.
These and some orther topics may be addressed in our separate publications. 

It should be noted here that the results obtained in this paper and its possible  extensions to static and other
 cases may  be used in  other areas of physics (e.g. chromodynamics, condensed matter etc) by  applying powerful holographic methods based on  AdS/CFT, dS/CFT approaches and its generalizations. 











§ APPENDIX




 §.§ The analytical solution for m=l


For any m = l > 2 the master equation (<ref>)  reads
 
    A x^4+B x^3 + C x^2 + B x + A = 0,
 
where

    A= 8  λ (m-2)^2(m-1) + m(m+1)(m-2), 
    
    B= 32 λ (m-2)(m-1)^2 + 4 m (m-1)^2, 
    
    C= 16 λ (m-1)(3m^2-8m+6) + 2m(m-1)(3m -4).
 

It may be readily solved  in radicals, by using the substitution y = x + 1/x <cit.>.
For A ≠ 0 we obtain 


    x= 1/4A( - B + ν_1 √(E - 2 B ν_2 √(d)) + ν_2 √(d)),

where ν_1 = ± 1, ν_2 = ± 1 and 

    d = 8A^2 - 4 C A + B^2,       E = - 8A^2 - 4CA + 2 B^2.

We get 

    d  = 16 m^2(2m^2-7m+7)  
    - 128 m(m-1)(m-2) (2m-3) λ, 
    
     E = 1024 λ ^2(m-2)^2(m-1)^2(2m-3)   
    
            + 128 λ (m-2)(m-1)m(4m-7)  
    
            -16 m^2 (2 m^3- 11 m^2+15 m-4).


 For A = 0,   the  solution reads 
 
    x= 1/2B [ - C ±√(C^2 -4 B^2 ) ],   or x=0,

where 

    B = - 8 m (m-1),      C = - 4m/m-2 (4 m^2-10m+7).


The special solution for m=3 was considered recently in ref. <cit.>.





 §.§ The proof of the Lemma


Here we give the proof of the Lemma from Section 2.
The calculations (by using Mathematica) lead us to following relations

    R_±(m,l)= R(x_±(m,l),m,l)=A(m,l) ± B(m,l)√(Δ(m,l))/C(l)

where

    A(m,l)   =-2(m-1)(l+m-3)A_*(m,l),
    
    A_*(m,l)   =l^2m^2+4lm^2-4m^2+l^3m-4l^2m-8lm+8m-2l^3+8l^2-4l,
    
    B(m,l)   =8l(m-1)^2(l+m-3)>0,
    Δ(m,l)   =(m-1)(l-1)(l+m-3)>0,
    
    C(l)   =  (l - 2)^3 (l -1) > 0.


In order to prove R_-(m,l) <0  it is sufficient to prove that A_*(m,l) > 0 
for m>2 and l>2.

Let m ≥ 4. Then we group A_*(m,l) as the sum of the non-negative terms:


    A_*(m,l)= (l^2m^2-4l^2m)_1+(4lm^2-4m^2-8lm)_2 
       +(l^3m-2l^3)_3+(8m)_4+(8l^2-4l)_5;

where

    (.)_1   =l^2m^2-4l^2m=l^2m(m-4)≥0, 
    
    (.)_2   =4lm^2-4m^2-8lm=2(l-2)m^2+2lm(m-4)>0, 
    
    (.)_3   =l^3(m-2)>0,  
    
    (.)_4   >0, 
    
    (.)_5   =4l(2l-1)>0.



Thus, we get  A_*(m,l) > 0 for m ≥ 4 and l > 2.
For m=3 we have A_*(3,l)=l^3+5l^2+8l-12 ≥ 84 (as l≥3).
Thus,  R_-(m,l) < 0  (m>2, l>2) is proved.

Now we prove R_+(m,l) < 0 (m>2, l>2)).
By using the identities  (<ref>), (<ref>) 
and definitions of R_±(m,l) we obtain

    R_+(m,l)=  R(x_+(m,l),m,l) 
    = (x_+(m,l))^4  R(1/x_+(m,l),l,m) 
    
    = (x_+(m,l))^4  R(x_-(l,m),l,m) = 
       (x_+(m,l))^4  R_-(l,m) < 0.

By this we complete the proof of the Lemma. 





 Acknowledgments


The publication has been prepared with the support of the “RUDN University Program 5-100”.
It was also partially supported by the  Russian Foundation for Basic Research,  grant  Nr. 16-02-00602.








99

Zwiebach
 B. Zwiebach, Curvature squared terms and string theories,
  Phys. Lett.  B 156, 315 (1985).

 FrTs2
 E.S. Fradkin and A.A. Tseytlin, Effective action approach to superstring
 theory,  Phys. Lett. B 160, 69-76 (1985).

 GW
 D. Gross and E. Witten,
 Superstrings modifications of  Einstein's equations,
 Nucl. Phys. B 277,  1 (1986).

  MTs
  R.R. Metsaev and A.A. Tseytlin,
  Two loop beta function for the generalized bosonic sigma model,
   Phys. Lett.  B  191, 354  (1987).

Ishihara
 H. Ishihara, Cosmological solutions of the extended Einstein
  gravity with the Gauss-Bonnet term,
  Phys. Lett. B 179, 217 (1986).

 Deruelle
 N. Deruelle, On the approach to the cosmological
 singularity in quadratic theories of gravity: the Kasner
 regimes,   Nucl. Phys. B  327,  253-266 (1989).

NojOd0
S. Nojiri and S.D. Odintsov, Introduction to modified gravity and
gravitational alternative for Dark Energy,
 Int. J. Geom. Meth. Mod. Phys. 4, 115-146 (2007);
   hep-th/0601213.


ElMakObOsFil
 E. Elizalde, A.N. Makarenko, V.V. Obukhov, K.E. Osetrin  and
 A.E. Filippov,  Stationary vs. singular points in an accelerating
 FRW cosmology derived from six-dimensional Einstein-Gauss-Bonnet
 gravity,  Phys. Lett. B 644, 1-6 (2007);  hep-th/0611213.

  BambaGuoOhta
 K. Bamba, Z.-K. Guo and N. Ohta, Accelerating Cosmologies in the
 Einstein-Gauss-Bonnet theory with dilaton, Prog. Theor. Phys.
 118,  879-892 (2007); arXiv: 0707.4334.

KirMak
I.V. Kirnos and A.N. Makarenko,
Accelerating cosmologies in Lovelock gravity with dilaton,
 Open Astron. J. 3, 37-48 (2010);  arXiv: 0903.0083.

Pavl
 S.A. Pavluchenko,
 On the general features of Bianchi-I cosmological models in
 Lovelock gravity,  Phys. Rev. D 80, 107501 (2009);
 arXiv: 0906.0141.

 KirMPTop
 I.V. Kirnos, A.N. Makarenko, S.A. Pavluchenko and A.V. Toporensky,
 The nature of singularity in multidimensional anisotropic
 Gauss-Bonnet cosmology with a perfect fluid,
  Gen. Relativ. Gravit. 42, 2633-2641 (2010);
  arXiv: 0906.0140.

Iv-09
  V.D. Ivashchuk,
  On anisotropic Gauss-Bonnet cosmologies in (n + 1) dimensions,
  governed by an n-dimensional Finslerian 4-metric,  Grav. Cosmol.
  16(2), 118-125 (2010); arXiv: 0909.5462.

 Iv-10
 V.D. Ivashchuk,  On cosmological-type solutions in
 multidimensional model with  Gauss-Bonnet term,
 Int. J. Geom. Meth. Mod. Phys.
 7(5), 797-819 (2010);  arXiv: 0910.3426.

MaOh
 K.-i. Maeda and N. Ohta,
 Cosmic acceleration with a negative cosmological constant in
 higher dimensions,  JHEP 1406: 095 (2014); arXiv:1404.0561.

ChPavTop
D. Chirkov, S. Pavluchenko and A. Toporensky, Exact exponential solutions
in Einstein-Gauss-Bonnet flat anisotropic cosmology,
 Mod. Phys. Lett. A 29,  1450093 (11 pages) (2014);  arXiv:1401.2962.

ChPavTop1
D. Chirkov, S.A. Pavluchenko and A. Toporensky,
Non-constant volume exponential solutions in higher-dimensional
Lovelock cosmologies,  Gen. Relativ. Gravit.  47: 137 (33 pages) (2015); arXiv: 1501.04360.

IvKob
 V.D. Ivashchuk and A.A. Kobtsev,
 On exponential cosmological type solutions in the model
 with Gauss-Bonnet term and variation of gravitational constant,
 Eur. Phys. J.  C  75: 177 (12 pages) (2015);
 Erratum, Eur. Phys. J. C (2016) 76: 584; arXiv:1503.00860.

 Pavl-15
 S.A. Pavluchenko, Stability analysis of exponential solutions in Lovelock cosmologies,
 Phys. Rev. D 92, 104017 (2015); arXiv: 1507.01871.

Pavl-16
 S.A. Pavluchenko, Cosmological dynamics of spatially flat Einstein-Gauss-Bonnet models in various dimensions: Low-dimensional 
 Λ-term case, Phys. Rev. D 94, 084019 (2016); arXiv: 1607.07347.  
    
ErIvKob-16
 K.K. Ernazarov, V.D. Ivashchuk and A.A. Kobtsev,
 On exponential solutions in the Einstein-Gauss-Bonnet cosmology,
 stability and variation of G,  
   Grav.  Cosmol., 22 (3), 245-250 (2016).

Ivas-16
V.D. Ivashchuk, On stability of exponential cosmological solutions
with non-static volume factor in the Einstein-Gauss-Bonnet model,
 Eur. Phys. J.  C 76 431 (2016) (10 pages); arXiv: 1607.01244v2.

Ivas-16-2
V.D. Ivashchuk,  On Stable Exponential Solutions in Einstein-Gauss-Bonnet 
Cosmology with Zero Variation of G,  Grav. Cosmol. 22 (4), 329-332 (2016); 
Erratum, Grav.  Cosmol. 23 (4), 401 (2017); arXiv: 1612.07178.

ErIv-17-1
K.K. Ernazarov and V.D. Ivashchuk, Stable exponential cosmological solutions with zero variation of G in the Einstein-Gauss-Bonnet model with a Λ-term, Eur. Phys. J.  C  77: 89 (2017) (6 pages); arXiv:1612.08451.

ErIv-17-2
K.K. Ernazarov and V.D. Ivashchuk, Stable exponential cosmological solutions with zero variation of G and three different Hubble-like parameters in the Einstein-Gauss-Bonnet model with a Λ-term, 
Eur. Phys. J. C 77: 402  (2017) (7 pages);  arXiv:1705.05456.

ChTop-17
D. Chirkov, and A. Toporensky, On stable exponential cosmological
solutions in the EGB model with a cosmological constant in
dimensions D = 5, 6, 7, 8, Grav. Cosmol. 23, 359-366 (2017);
arXiv: 1706.08889.

IvKob-m3-18
 V.D. Ivashchuk and A.A. Kobtsev, 
 Stable  exponential cosmological solutions with 3- 
 and l-dimensional  factor spaces in the Einstein-Gauss-Bonnet  model  with a Λ-term, 
  Eur. Phys. J.  C  78:100   (11 pages) (2018).

Pavl-p-18
S. Pabluchenko, 
Realistic Compactification Models in Einstein-Gauss-Bonnet Gravity,
Particles, 1 (1), 4 (21 pages) (2018); arXiv: 1803.01887



Riess
 A.G. Riess  et al. Observational evidence from supernovae
 for an accelerating  universe and a cosmological constant,
  Astron. J. 116, 1009-1038 (1998).

Perl
 S. Perlmutter  et al., Measurements of Omega and Lambda from 42 High-Redshift  Supernovae,
  Astrophys.  J. 517,  565-586 (1999).

 Kowalski
 M. Kowalski, D. Rubin et al., Improved cosmological constraints from new,
 old and combined supernova datasets,
  Astrophys.  J. 686 (2), 749-778 (2008); arXiv: 0804.4142.

GG
C. Garraffo and  G. Giribet, The Lovelock Black Holes,
Mod. Phys. Lett. A 23: 1801-1818 (2008); arXiv:0805.3575v4 [gr-qc].

IvKob-18mm
V.D. Ivashchuk and A.A. Kobtsev, Exact  exponential cosmological solutions with two factor spaces 
 of dimension m in EGB  model  with a Λ-term, unpublished. 


Ade
P.A.R. Ade  et al. [Planck Collaboration], Planck 2013 results.
I. Overview of products and scientific results,
 Astron. Astrophys. 571, A1 (2014); arXiv: 1303.5076.
 
 BEMPSS-10
 A. Buchel, J. Escobedo, R.C. Myers, M.F. Paulos, A. Sinha and M. Smolkin, 
 Holographic GB gravity in arbitrary dimensions, 
 JHEP 1003, 111 (2010), arXiv:0911.4257 [hep-th].
 
 CG-10
 R.G.-Cai and Q. Quo, Gauss-Bonnet black holes in dS spaces, 
 Physical Review D, 69, no. 10, 104025 (9 pages) (2004);
 arXiv: hep-th/0311020.  
    
 
RZ-98
M. Rainer and A. Zhuk,
Einstein and Brans-Dicke frames in multidimensional cosmology,
 Gen. Relativ. Gravit. 32, 79-104 (2000); gr-qc/9808073.

I-96
V.D. Ivashchuk and V.N. Melnikov,
Multidimensional Gravity with Einstein Internal Spaces,
 Grav. Cosmol. 2 (3),  211-220 (1996);
hep-th/9612054.

 BIM
 K.A. Bronnikov, V.D. Ivashchuk  and V.N. Melnikov,
  Time Variation of Gravitational Constant in Multidimensional Cosmology,
 Nuovo Cimento  B 102, 209-215 (1998).

 Mel
 V.N. Melnikov,  Models of G time variations in diverse dimensions,
  Front. Phys. China 4, 75-93 (2009).

 Pitjeva
    E.V. Pitjeva,
    Updated IAA RAS Planetary Ephemerides-EPM2011
    and Their Use in Scientific Research, Astron. Vestnik
    47 (5),  419-435 (2013), arXiv: 1308.6416.

 
Lov
 D. Lovelock, The four-dimensionality of space and the Einstein tensor, J. Math. Phys. 13, 874 (1972).




