Stable exponential cosmological solutions with zero variation of G and three different Hubblelike parameters in the Einstein–Gauss–Bonnet model with a \(\Lambda \) term
Eur. Phys. J. C
Stable exponential cosmological solutions with zero variation of G and three different Hubblelike parameters in the EinsteinGaussBonnet model with a term
K. K. Ernazarov 1
V. D. Ivashchuk 0 1
0 Center for Gravitation and Fundamental Metrology, VNIIMS , 46 Ozyornaya ul., Moscow 119361 , Russia
1 Institute of Gravitation and Cosmology, RUDN University , 6 MiklukhoMaklaya ul., Moscow 117198 , Russia
We consider a Ddimensional gravitational model with a GaussBonnet term and the cosmological term . We restrict the metrics to diagonal cosmological ones and find for certain a class of solutions with exponential time dependence of three scale factors, governed by three noncoinciding Hubblelike parameters H > 0, h1 and h2, corresponding to factor spaces of dimensions m > 2, k1 > 1 and k2 > 1, respectively, with k1 = k2 and D = 1 + m + k1 + k2. Any of these solutions describes an exponential expansion of 3d subspace with Hubble parameter H and zero variation of the effective gravitational constant G. We prove the stability of these solutions in a class of cosmological solutions with diagonal metrics.

In this paper we consider a Ddimensional gravitational
model with Gauss–Bonnet term and cosmological term .
The socalled Gauss–Bonnet term appeared in string
theory as a first order correction (in α ) to the effective action
[1–4].
We note that at present the Einstein–Gauss–Bonnet (EGB)
gravitational model and its modifications, see [5–28] and the
references therein, are intensively studied in cosmology, e.g.
for possible explanation of accelerating expansion of the
Universe which follow from supernova (type Ia) observational
data [29–31].
In Ref. [28] we were dealing with the cosmological
solutions with diagonal metrics governed by n > 3 scale
factors depending upon one variable, which is the synchronous
time variable. We have restricted ourselves by the
solutions with exponential dependence of scale factors and have
presented a class of such solutions with two scale
factors, governed by two Hubblelike parameters H > 0 and
h < 0, which correspond to factor spaces of dimensions
m > 3 and l > 1, respectively, with D = 1 + m + l
and (m, l) = (6, 6), (7, 4), (9, 3). Any of these solutions
describes an exponential expansion of 3d subspace with
Hubble parameters H > 0 [32] and has a constant
volume factor of (m − 3 + l)dimensional internal space, which
implies zero variation of the effective gravitational constant
G either in a Jordan or in an Einstein frame [33,34]; see
also [35–37] and the references therein. These solutions
satisfy the most severe restrictions on variation of G [38].
We have studied the stability of these solutions in a class
of cosmological solutions with diagonal metrics by using
results of Refs. [24,26] (see also approach of Ref. [22])
and have shown that all solutions, presented in Ref. [28],
are stable. It should be noted that two special solutions for
D = 22, 28 and = 0 were found earlier in Ref. [21];
in Ref. [24] it was proved that these solutions are stable.
Another set of six stable exponential solutions, five in
dimensions D = 7, 8, 9, 13 and two for D = 14, were considered
earlier in [27].
In this paper we extend the results of Ref. [28] to the case
of solutions with three noncoinciding Hubblelike
parameters. The structure of the paper is as follows. In Sect. 2
we present a setup. A class of exact cosmological solutions
with diagonal metrics is found for certain in Sect. 3. Any
of these solutions describes an exponential expansion of
3dimensional subspace with Hubble parameter H and zero
variation of the effective gravitational constant G. In Sect. 4
we prove the stability of the solutions in a class of
cosmological solutions with diagonal metrics. Certain examples are
presented in Sect. 5.
2 The cosmological model
The action of the model reads
S =
dD z g{α1(R[g] − 2 ) + α2L2[g]},
where g = gM N dz M ⊗ d z N is the metric defined on the
manifold M , dim M = D, g =  det(gM N ), is the
cosmological term, R[g] is scalar curvature,
L2[g] = RM N P Q R M N P Q − 4RM N R M N + R2
is the standard Gauss–Bonnet term and α1, α2 are nonzero
constants.
We consider the manifold
M = R × M1 × · · · × Mn
with the metric
g = −dt ⊗ dt +
i=1
Bi e2vi t dyi ⊗ dyi ,
= 0,
where Bi > 0 are arbitrary constants, i = 1, . . . , n, and
M1, . . . , Mn are 1dimensional manifolds (either R or S1)
and n > 3.
The equations of motion for the action (2.1) give us the
set of polynomial equations [24]
E = Gi j vi v j + 2
Yi =
2Gi j v j − 43 αGi jkl v j vk vl
i=1
i = 1, . . . , n, where α = α2/α1. Here
Gi j = δi j − 1, Gi jkl = Gi j Gik Gil G jk G jl Gkl ,
are, respectively, the components of two metrics on Rn
[16,17]. The first one is a 2metric and the second one is
a Finslerian 4metric. For n > 3 we get a set of fourthorder
polynomial equations.
We note that for = 0 and n > 3 the set of Eqs. (2.4)
and (2.5) has an isotropic solution v1 = · · · = vn = H only
if α < 0 [16,17]. This solution was generalized in [19] to the
case = 0.
It was shown in [16,17] that there are no more than three
different numbers among v1, . . . , vn when = 0. This is
valid also for = 0 if in=1 vi = 0 [26].
3 Solutions with constant G
In this section we present a class of solutions to the set of
equations (2.4), (2.5) of the following form:
m−3
“our space
internal space
where H is the Hubblelike parameter corresponding to an
mdimensional factor space with m > 2, h1 is the Hubblelike
parameter corresponding to an k1dimensional factor space
with k1 > 1 and h2 (h2 = h1) is the Hubblelike parameter
corresponding to an k2dimensional factor space with k2 > 1.
We split the mdimensional factor space into the product
of two subspaces of dimensions 3 and m − 3, respectively.
The first one is identified with “our” 3d space, while the
second one is considered as a subspace of (m − 3 + k1 +
k2)dimensional internal space.
We put
for a description of an accelerated expansion of a
3dimensional subspace (which may describe our Universe)
and also put
(m − 3)H + k1h1 + k2h2 = 0
for a description of a zero variation of the effective
gravitational constant G.
We remind the reader that the effective gravitational
constant G = Ge f f in the Brans–Dicke–Jordan (or simply
Jordan) frame [33] (see also [34]) is proportional to the inverse
volume scale factor of the internal space; see [35–37] and
references therein.
Due to (3.1) “our” 3d space expands isotropically with
Hubble parameter H , while the (m − 3)dimensional part of
the internal space expands isotropically with the same Hubble
parameter H too. Here, like in Ref. [28], we consider for
cosmological applications (in our epoch) the internal space to
be a compact one, i.e. we put in (2.2) M4 = · · · = Mn = S1.
We put the internal scale factors corresponding to present
time t0: a j (t0) = B 1j/2 exp(v j t0), j = 4, . . . , n, (see (2.3))
to be small enough in comparison with the scale factor of
“our” space for t = t0: a(t0) = B1/2 exp(H t0), where B1 =
B2 = B3 = B.
According to the ansatz (3.1), the mdimensional
factor space is expanding with the Hubble parameter H > 0,
while the ki dimensional factor space is contracting with the
Hubblelike parameter hi < 0, where i is either 1 or 2.
Now we consider the ansatz (3.1) with three Hubble
parameters H , h1 and h2 which obey the following
restrictions:
S1 = m H + k1h1 + k2h2 = 0,
H = h1,
H = h2, h1 = h2.
The first inequality in (3.4) is valid since S1 = 3H > 0 due
to (3.2) and (3.3).
In this case the set of n + 1 equations (2.4), (2.5) is
equivalent to the set of three equations
E = 0, YH = 0, Yh1 = 0, Yh2 = 0,
YH = Yμ, Yh1 = Yα, Yh2 = Ya ,
for all μ = 1, . . . , m; α = m + 1, . . . , m + k1 and a =
m + k1 + 1, . . . , n. These relations follow from the definition
of Yi in (2.5) and the identities [16,17]
vi = Gi j v j = vi − S1,
Ai = Gi jkl v j vk vl = S13 + 2S3 − 3S1 S2
+ 3(S2 − S12)vi + 6S1(vi )2 − 6(vi )3,
i = 1, . . . , n, where here and in what follows
Due to (2.5), (3.7), (3.8) we obtain
Yhi − Yh j = (hi − h j )S1[2 + 4α Qhi ,h j ],
Qhi h j = S1 − S2 −2S1(hi + h j )+2(hi2 + hi h j + h2j ), (3.11)
2
i = j ; i, j = 0, 1, 2 and h0 = H . Equations (3.4), (3.5) and
(3.10) imply
Sk =
i=1
i = j and i, j = 0, 1, 2.
Due to S1 = m H + k1h1 + k2h2 = 0 the set of equations
(3.5) is equivalent to the following set of equations:
E = 0, YH − Yh1 = 0, Yh1 − Yh2 = 0,
m H YH + k1h1Yh1 + k2h2Yh2 = 0.
The last relation in (3.13) may be omitted since E = 0
implies Yi hi = m H YH + k1h1Yh1 + k2h2Yh2 = 0 [26].
Using this fact and Eqs. (3.4) and (3.10) we reduce the
system (3.13) to the following one:
E = 0,
Using the identity
we reduce the set of equations (3.14) to the equivalent set
E = 0,
H + h1 + h2 − S1 = 0.
Here we put Q = Qh1h2 , though other choices, Q = Q H h1
or Q = Q H h2 , give us equivalent sets of equations. Thus the
set of (n + 1) polynomial equations (2.4), (2.5) under ansatz
(3.1) and restrictions (3.4) imposed is reduced to a set (3.16)
of three polynomial equations (of fourth, second and first
orders). This reduction is a special case of the more general
prescription from Ref. [20].
Using the condition (3.3) of zero variation of G and the
linear equation from (3.16) we obtain for k1 = k2,
h1 =
m + 2k2 − 3
k2 − k1
H, h2 =
m + 2k1 − 3 H.
k1 − k2
For k1 = k2 we get H = 0, which is not appropriate for our
consideration.
1
The substitution of (3.17) into relation Qh1h2 = − 2α
gives us the following relation:
(k2 − k1)2
for k1 = k2, where
+ k1(2k2 − 5) + k2(2k1 − 5) + 6) = 0,
H = k1 − k2(−2α P)−1/2, α P < 0.
It may be readily verified that
P = P(m, k1, k2) < 0
= −F1 H 2 − F2 H 4
for all m > 2, k1 > 1, k2 > 1, k1 = k2 and hence our
solutions take place for α > 0.
The substitution of (3.17) into (3.5) gives us
1 2
F1 = (k2 − k1)2 [(k1 +k2)m2 +(k12 +6k1k2 +k2 − 6k1 −6k2)m
− 9(k12 + k2 − k1 − k2) + 2(2k1 + 2k2 − 3)k1k2]
2
F2 = −
[(k1 + k2)(k1 + k2 − 2)m3
− 42(k1 + k2 + 16k1k2 + 63)k1k2))m
+ 27(k13 + k2 ) − 81(k12 + k2 ) + 54(k1 + k2)
3 2
− (40(k12 + k2 ) − 16(k1 + k2 − 6)k1k2 + 162
2
− 153(k1 + k2))k1k2].
Using Eqs. (3.20), (3.22), (3.23), (3.24) we obtain
× [(k1 + k2)(k1 + k2 − 2)m3
+ (k13 + k2 + 11(k12k2 + k1k22) − 19(k12 + k2 )
3 2
− 8k22(k2 − 11)k1) − 32k12k22 + 54(k1 + k2))m
− (9(k13 + k2 ) + 45(k12 + k2 ) − 54(k1 + k2)
3 2
− 16(k1 + k2 −10)k12k22 −9(21k1 + 21k2 − 26)k1k2)],
(3.25)
where P = P(m, k1, k2) is defined in (3.19).
The function (m, k1, k2) in (3.25) is symmetric with
respect to k1 and k2, i.e.
(m, k1, k2) =
(m, k2, k1).
For k2 = 0 we get a function (m, k1, 0) = (m, k1), where
(m, k1) was obtained in Ref. [28] for the case of two
different Hubblelike parameters.
It may be readily verified that for k1(k) = n1k + q1 and
k2(k) = n2k + q2, where k, n1 > 0, q1, n2 > 0, q2 are
integer numbers, we get
1
(m, k1(k), k2(k)) → 8α ,
as k → +∞ for any fixed m ≥ 3. We note that the limit
(3.27) is positive and does not depend upon m. For fixed
integer m > 2 and k2 ≥ 1 we are led to the following limit:
1
(m, k1, k2) → 8α(m + 4k2 − 5)2
m2 − 8(1 − k2)m − 9
−8k2 + 16k22 =
(m, ∞, k2),
as k1 → +∞ and there is an analogous relation (due to
(3.26)) for fixed m > 2, k1 ≥ 1 and k2 → +∞. It can
4 The proof of stability
Here, as in [28], we have due to (3.3)
K = K (v) =
vi = 3H > 0.
i=1
Let us put the restriction
det(Li j (v)) = 0
on the matrix
L = (Li j (v)) = (2Gi j − 4αGi jks vk vs ).
We recall that, for a general cosmological setup with the
metric
i=1
g = −dt ⊗ dt +
we have the set of equations [24]
E = Gi j hi h j + 2
Yi =
j=1
be easily verified that, for these values of m, k1 we get
(m, ∞, k2) > 0.
Equations (3.27) and (3.28) may be used in a context of
(1/D)expansion for large D in the model under
consideration; see [25] and the references therein.
4
Li = Li (h) = 2Gi j h j − 3 αGi jkl h j hk hl ,
i = 1, . . . , n.
Due to the results of Ref. [26] a fixed point solution
(hi (t )) = (vi ) (i = 1, . . . , n; n > 3) to Eqs. (4.5), (4.6)
obeying restrictions (4.1), (4.2) is stable under perturbations,
i = 1, . . . , n, as t → +∞.
In order to prove the stability of solutions we should prove
Eq. (4.2). First, we show that for the vector v from (3.1),
obeying Eqs. (3.4) the matrix L has a blockdiagonal form,
Here we use the notation Sk = in=1(vi )k and the identity
∂∂v j Sk = k(v j )k−1. It follows from (3.11) and (4.10) that
SH h1 ≡ Sμα = Sαμ = Q H h1 ,
SH h2 ≡ Sμa = Saμ = Q H h2 ,
Sh1h2 ≡ Sαa = Saα = Qh1h2
and hence Lμα = Lαμ = 0, Lμa = Laμ = 0 and Lαa =
Laα = 0 due to Eq. (3.12). Thus, the matrix (Li j ) is
blockdiagonal.
For the other three blocks we have
Shi hi = S1 − S2 + 6hi2 − 4S1hi ,
2
i = 0, 1, 2 and h0 = H . Here we denote SH H = Sμν , μ = ν;
Sh1h1 = Sαβ , α = β and Sh1h1 = Sab, a = b.
Due to Eqs. (4.9), (4.14), (4.15), (4.16) the matrix (4.9) is
invertible if and only if m > 1, k1 > 1, k2 > 1 and
where here and in what follows: μ, ν = 1, . . . , m; α, β =
m + 1, . . . , m + k1 and a, b = m + k1 + 1, . . . , n.
Indeed, denoting Si j = Gi jkl vk vl we get from (3.8)
But due to (3.16)
hi0 + hi1 + hi2 − S1 = 0,
where i2 ∈ {0, 1, 2} and i2 = i0, i2 = i1. Subtracting (4.22)
from (4.21) we obtain hi0 − hi2 = 0, i.e. hi0 = hi2 . But due
to restrictions (3.4) we have hi0 = hi2 . We are led to a
contradiction, which proves the inequalities (4.18) and hence the
matrix L from (4.9) is invertible (m > 2, k1 > 1, k2 > 1), i.e.
Eq. (4.2) is obeyed. Thus, the solutions under consideration
are stable.
5 Examples
Here we present several examples of stable solutions under
consideration.
5.1 The case m = 3
Let us consider the case m = 3. From (3.25) we get
− 8k1k2)k1k2).
= − 4α (k1 − 2k1k2 + k2)2(k1 + k1)
× (3(k13 + k2 ) − (2(k12 + k2 ) + (k1 + k2)(3 + 2k1k2)
3 2
(5.1)
For (m, k1, k2) = (3, 3, 2) we have P = −70,
i = 0, 1, 2.
Now, we prove that inequalities (4.18) are satisfied for the
solutions under consideration. Let us suppose that (4.18) is
not satisfied for some i0 ∈ {0, 1, 2}, i.e.
Shi0 hi0 = S12 − S2 + 6hi20 − 4S1hi0 = − 21α .
Let i1 ∈ {0, 1, 2} and i1 = i0. Then using Eqs. (3.11) and
(3.12) we get
2hi0 + hi1 − S1 = 0.
Now we present other examples of stable solutions for m = 4
and m = 5.
Now we put (m, k1, k2) = (3, 4, 2). We obtain P =
−120,
According to our analysis from the previous section both
solutions are stable.
5.2 Examples for m = 4 and m = 5
In this case we obtain
H = √
H = √
6 Conclusions
h1 = −5H,
h2 = 7H.
Now we enlarge the value of m by putting (m, k1, k2) =
(5, 3, 2). We find P = −140,
h1 = −6H,
h2 = 8H.
We note that in all examples above
We have considered the Ddimensional Einstein–Gauss–
Bonnet (EGB) model with the term and two constants
α1 and α2. By using the ansatz with diagonal cosmological
metrics, we have found, for certain = (m, k1.k2) and
α = α2/α1 < 0, a class of solutions with exponential time
dependence of three scale factors, governed by three different
Hubblelike parameters H > 0, h1 and h2, corresponding to
submanifolds of dimensions m > 2, k1 > 1, k2 > 1,
respectively, with k1 = k2 and D = 1 + m + k1 + k2. Here m > 2
is the dimension of the expanding subspace.
Any of these solutions describes an exponential expansion
of “our” 3dimensional subspace with the Hubble parameter
H > 0 and anisotropic behaviour of (m − 3 + k1 +
k2)dimensional internal space: expanding in (m −3) dimensions
(with Hubblelike parameter H ) and either contracting in k1
dimensions (with Hubblelike parameter h1) and expanding
in k2 dimensions (with Hubblelike parameter h2) for k1 > k2
or expanding in k1 dimensions and contracting in k2
dimensions for k1 < k2. Each solution has a constant volume factor
of internal space and hence it describes zero variation of the
effective gravitational constant G. By using the results of
Ref. [26] we have proved that all these solutions are stable
as t → +∞. We have presented several examples of stable
solutions for m = 3, 4, 5.
Acknowledgements This paper was funded by the Ministry of
Education and Science of the Russian Federation in the Program to increase
the competitiveness of Peoples Friendship University (RUDN
University) among the world’s leading research and education centers in the
2016–2020 and by the Russian Foundation for Basic Research, Grant
No.160200602.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
1. B. Zwiebach , Curvature squared terms and string theories . Phys. Lett. B 156 , 315 ( 1985 )
2. E.S. Fradkin , A.A. Tseytlin , Effective field theory from quantized strings . Phys. Lett. B 158 , 316  322 ( 1985 )
3. E.S. Fradkin , A.A. Tseytlin , Effective action approach to superstring theory . Phys. Lett. B 160 , 69  76 ( 1985 )
4. D. Gross , E. Witten , Superstrings modifications of Einstein's equations . Nucl. Phys . B 277 , 1 ( 1986 )
5. H. Ishihara , Cosmological solutions of the extended Einstein gravity with the GaussBonnet term . Phys. Lett. B 179 , 217 ( 1986 )
6. N. Deruelle , On the approach to the cosmological singularity in quadratic theories of gravity: the Kasner regimes . Nucl. Phys. B 327 , 253  266 ( 1989 )
7. S. Nojiri , S.D. Odintsov , Introduction to modified gravity and gravitational alternative for Dark Energy . Int. J. Geom. Methods Mod. Phys . 4 , 115  146 ( 2007 ). arXiv:hepth/0601213
8. G. Cognola , E. Elizalde , S. Nojiri , S.D. Odintsov , S. Zerbini , Oneloop effective action for nonlocal modified GaussBonnet gravity in de Sitter space . Eur. Phys. J. C 64 ( 3 ), 483  494 ( 2009 ). arXiv:0905.0543
9. E. Elizalde , A.N. Makarenko , V.V. Obukhov , K.E. Osetrin , A.E. Filippov , Stationary vs. singular points in an accelerating FRW cosmology derived from sixdimensional EinsteinGaussBonnet gravity . Phys. Lett. B 644 , 1  6 ( 2007 ). arXiv:hepth/0611213
10. K. Bamba , Z.K. Guo , N. Ohta , Accelerating cosmologies in the EinsteinGaussBonnet theory with dilaton . Progr. Theor. Phys . 118 , 879  892 ( 2007 ). arXiv:0707.4334
11. A. Toporensky , P. Tretyakov , Powerlaw anisotropic cosmological solution in 5+1 dimensional GaussBonnet gravity . Gravit. Cosmol . 13 , 207  210 ( 2007 ). arXiv:0705.1346
12. S.A. Pavluchenko , A.V. Toporensky , A note on differences between (4 + 1) and (5 + 1 )dimensional anisotropic cosmology in the presence of the GaussBonnet term . Mod. Phys. Lett. A 24 , 513  521 ( 2009 )
13. I.V. Kirnos , A.N. Makarenko , Accelerating cosmologies in Lovelock gravity with dilaton . Open Astron. J . 3 , 37  48 ( 2010 ). arXiv:0903.0083
14. S.A. Pavluchenko , On the general features of BianchiI cosmological models in Lovelock gravity . Phys. Rev. D 80 , 107501 ( 2009 ). arXiv:0906.0141
15. I.V. Kirnos , A.N. Makarenko , S.A. Pavluchenko , A.V. Toporensky , The nature of singularity in multidimensional anisotropic GaussBonnet cosmology with a perfect fluid . Gen. Relat. Gravit . 42 , 2633  2641 ( 2010 ). arXiv:0906.0140
16. V.D. Ivashchuk , On anisotropic GaussBonnet cosmologies in (n + 1) dimensions, governed by an ndimensional Finslerian 4metric . Gravit. Cosmol. 16 ( 2 ), 118  125 ( 2010 ). arXiv:0909.5462
17. V.D. Ivashchuk , On cosmologicaltype solutions in multidimensional model with GaussBonnet term . Int. J. Geom. Methods Mod. Phys . 7 ( 5 ), 797  819 ( 2010 ). arXiv:0910.3426
18. K.I. Maeda , N. Ohta , Cosmic acceleration with a negative cosmological constant in higher dimensions . JHEP 1406 , 095 ( 2014 ). arXiv:1404.0561
19. D. Chirkov , S. Pavluchenko , A. Toporensky , Exact exponential solutions in EinsteinGaussBonnet flat anisotropic cosmology . Mod. Phys. Lett. A 29 , 1450093 ( 11 pages) ( 2014 ). arXiv:1401.2962
20. D. Chirkov , S.A. Pavluchenko , A. Toporensky , Nonconstant volume exponential solutions in higherdimensional Lovelock cosmologies . Gen. Relat. Gravit . 47 , 137 ( 33 pages) ( 2015 ). arXiv:1501.04360
21. V.D. Ivashchuk , A.A. Kobtsev , On exponential cosmological type solutions in the model with GaussBonnet term and variation of gravitational constant . Eur. Phys. J. C 75 , 177 ( 12 pages) ( 2015 ). arXiv:1503.00860
22. S.A. Pavluchenko , Stability analysis of exponential solutions in Lovelock cosmologies . Phys. Rev. D 92 , 104017 ( 2015 ). arXiv:1507.01871
23. S.A. Pavluchenko , Cosmological dynamics of spatially flat EinsteinGaussBonnet models in various dimensions: lowdimensional term case . Phys. Rev. D 94 , 084019 ( 2016 ). arXiv:1607.07347
24. K.K. Ernazarov , V.D. Ivashchuk , A.A. Kobtsev , On exponential solutions in the EinsteinGaussBonnet cosmology, stability and variation of G . Gravit. Cosmol. 22 ( 3 ), 245  250 ( 2016 )
25. F. Canfora , A. Giacomini , S.A. Pavluchenko , A. Toporensky , Friedmann dynamics recovered from compactified EinsteinGaussBonnet cosmology . arXiv:1605.00041
26. V.D. Ivashchuk , On stability of exponential cosmological solutions with nonstatic volume factor in the EinsteinGaussBonnet model . Eur. Phys. J. C 76 , 431 ( 2016 ). arXiv: 1607 .01244v2
27. V.D. Ivashchuk , On stable exponential solutions in EinsteinGaussBonnet cosmology with zero variation of G . Gravit. Cosmol. 22 ( 4 ), 329  332 ( 2016 ). See corrected version in arXiv:1612.07178
28. K.K. Ernazarov , V.D. Ivashchuk , Stable exponential cosmological solutions with zero variation of G in the EinsteinGaussBonnet model with a term . Eur. Phys. J. C 77 , 89 ( 2017 ). arXiv:1612.08451
29. A.G. Riess et al., Observational evidence from supernovae for an accelerating universe and a cosmological constant . Astron. J . 116 , 1009  1038 ( 1998 )
30. S. Perlmutter et al., Measurements of Omega and Lambda from 42 highredshift supernovae . Astrophys. J . 517 , 565  586 ( 1999 )
31. 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
32. 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
33. M. Rainer , A. Zhuk , Einstein and BransDicke frames in multidimensional cosmology . Gen. Relat. Gravit . 32 , 79  104 ( 2000 ). arXiv:grqc/9808073
34. V.D. Ivashchuk , V.N. Melnikov , Multidimensional gravity with Einstein internal spaces . Gravit. Cosmol . 2 ( 3 ), 211  220 ( 1996 ). arXiv:hepth/9612054
35. K.A. Bronnikov , V.D. Ivashchuk , V.N. Melnikov , Time variation of gravitational constant in multidimensional cosmology . Nuovo Cimento B 102 , 209  215 ( 1998 )
36. V.N. Melnikov , Models of G time variations in diverse dimensions . Front. Phys. Chin . 4 , 75  93 ( 2009 )
37. V.D. Ivashchuk , V.N. Melnikov , On time variations of gravitational and YangMills constants in a cosmological model of superstring origin . Gravit. Cosmol . 20 ( 1 ), 26  29 ( 2014 ). arXiv:1401.5491
38. E.V. Pitjeva , Updated IAA RAS planetary ephemeridesEPM2011 and their use in scientific research . Astron. Vestnik 47 ( 5 ), 419  435 ( 2013 ). arXiv:1308.6416