Stable exponential cosmological solutions with zero variation of G and three different Hubble-like parameters in the Einstein–Gauss–Bonnet model with a \(\Lambda \) -term

The European Physical Journal C, Jun 2017

We consider a D-dimensional gravitational model with a Gauss–Bonnet term and the cosmological term \(\Lambda \). We restrict the metrics to diagonal cosmological ones and find for certain \(\Lambda \) a class of solutions with exponential time dependence of three scale factors, governed by three non-coinciding Hubble-like parameters \(H >0\), \(h_1\) and \(h_2\), corresponding to factor spaces of dimensions \(m > 2\), \(k_1 > 1\) and \(k_2 > 1\), respectively, with \(k_1 \ne k_2\) and \(D = 1 + m + k_1 + k_2\). 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.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

https://link.springer.com/content/pdf/10.1140%2Fepjc%2Fs10052-017-4974-7.pdf

Stable exponential cosmological solutions with zero variation of G and three different Hubble-like 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 Hubble-like parameters in the Einstein-Gauss-Bonnet 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 Miklukho-Maklaya ul., Moscow 117198 , Russia We consider a D-dimensional gravitational model with a Gauss-Bonnet 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 non-coinciding Hubble-like 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 D-dimensional gravitational model with Gauss–Bonnet term and cosmological term . The so-called 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 Hubble-like 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 non-coinciding Hubble-like 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 1-dimensional 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 2-metric and the second one is a Finslerian 4-metric. For n > 3 we get a set of fourth-order 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 Hubble-like parameter corresponding to an mdimensional factor space with m > 2, h1 is the Hubble-like parameter corresponding to an k1-dimensional factor space with k1 > 1 and h2 (h2 = h1) is the Hubble-like parameter corresponding to an k2-dimensional factor space with k2 > 1. We split the m-dimensional 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 m-dimensional factor space is expanding with the Hubble parameter H > 0, while the ki -dimensional factor space is contracting with the Hubble-like 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 Hubble-like 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 block-diagonal 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 D-dimensional 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 Hubble-like 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” 3-dimensional subspace with the Hubble parameter H > 0 and anisotropic behaviour of (m − 3 + k1 + k2)dimensional internal space: expanding in (m −3) dimensions (with Hubble-like parameter H ) and either contracting in k1 dimensions (with Hubble-like parameter h1) and expanding in k2 dimensions (with Hubble-like 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.16-02-00602. 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 Gauss-Bonnet 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:hep-th/0601213 8. G. Cognola , E. Elizalde , S. Nojiri , S.D. Odintsov , S. Zerbini , Oneloop effective action for non-local modified Gauss-Bonnet 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 six-dimensional Einstein-Gauss-Bonnet gravity . Phys. Lett. B 644 , 1 - 6 ( 2007 ). arXiv:hep-th/0611213 10. K. Bamba , Z.-K. Guo , N. Ohta , Accelerating cosmologies in the Einstein-Gauss-Bonnet theory with dilaton . Progr. Theor. Phys . 118 , 879 - 892 ( 2007 ). arXiv:0707.4334 11. A. Toporensky , P. Tretyakov , Power-law anisotropic cosmological solution in 5+1 dimensional Gauss-Bonnet 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 Gauss-Bonnet 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 Bianchi-I 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 Gauss-Bonnet cosmologies in (n + 1) dimensions, governed by an n-dimensional Finslerian 4-metric . Gravit. Cosmol. 16 ( 2 ), 118 - 125 ( 2010 ). arXiv:0909.5462 17. V.D. Ivashchuk , On cosmological-type solutions in multidimensional model with Gauss-Bonnet 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 Einstein-Gauss-Bonnet flat anisotropic cosmology . Mod. Phys. Lett. A 29 , 1450093 ( 11 pages) ( 2014 ). arXiv:1401.2962 20. D. Chirkov , S.A. Pavluchenko , A. Toporensky , Non-constant volume exponential solutions in higher-dimensional 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 Gauss-Bonnet 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 Einstein-Gauss-Bonnet 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 Einstein-Gauss-Bonnet 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 Einstein-GaussBonnet cosmology . arXiv:1605.00041 26. 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 ). arXiv: 1607 .01244v2 27. V.D. Ivashchuk , On stable exponential solutions in EinsteinGauss-Bonnet 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 Einstein-GaussBonnet 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 high-redshift 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 Brans-Dicke frames in multidimensional cosmology . Gen. Relat. Gravit . 32 , 79 - 104 ( 2000 ). arXiv:gr-qc/9808073 34. V.D. Ivashchuk , V.N. Melnikov , Multidimensional gravity with Einstein internal spaces . Gravit. Cosmol . 2 ( 3 ), 211 - 220 ( 1996 ). arXiv:hep-th/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 Yang-Mills 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 ephemerides-EPM2011 and their use in scientific research . Astron. Vestnik 47 ( 5 ), 419 - 435 ( 2013 ). arXiv:1308.6416


This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1140%2Fepjc%2Fs10052-017-4974-7.pdf

K. K. Ernazarov, 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 \(\Lambda \) -term, The European Physical Journal C, 2017, DOI: 10.1140/epjc/s10052-017-4974-7