Stable exponential cosmological solutions with three different Hubble-like parameters in EGB model with a (cid:2) -term

We consider a D -dimensional Einstein-Gauss-Bonnet model with a cosmological term (cid:2) and two non-zero constants: α 1 and α 2 . We restrict the metrics to be diagonal ones and study a class of solutions with exponential time dependence of three scale factors, governed by three non-coinciding Hubble-like parameters: H (cid:2)= 0, h 1 and h 2 , obeying mH + k 1 h 1 + k 2 h 2 (cid:2)= 0 and corresponding to factor spaces of dimensions m > 1, k 1 > 1 and k 2 > 1, respectively ( D = 1 + m + k 1 + k 2 ). We analyse two cases: i) m < k 1 < k 2 and ii) 1 < k 1 = k 2 = k , k (cid:2)= m . We show that in both cases the solutions exist if α = α 2 /α 1 > 0 and α(cid:2) > 0 satisﬁes certain restrictions, e.g. upper and lower bounds. In case ii) explicit relations for exact solutions are found. In both cases the subclasses of stable and non-stable solutions are singled out. For m > 3 the case i) contains a subclass of solutions describing an exponential expansion of 3-dimensional subspace


Introduction
In this paper we consider D-dimensional Einstein-Gauss-Bonnet (EGB) model with a -term. To some extent this model is unique among the other higher-dimensional extensions of General Relativity (GR) with second order in curvature terms. The reason is the following one: the equations of motion for this model are of the second order (in derivatives) like it takes place in the Einstein gravity. It is well known that the so-called Gauss-Bonnet term appeared in (super)string theory as a first order correction (in α ) to the (super)string effective action (e.g. heterotic one) [1][2][3][4]. a e-mail: ivashchuk@mail.ru (corresponding author) Currently, EGB gravitational model in diverse dimensions and its modifications, see  and Refs. therein, are rather popular objects for studying in cosmology. They are used for possible explanation of accelerating expansion of the Universe (i.e. solving the dark energy problem), which follow from supernova (type Ia) observational data [31][32][33]. One may expect that the second order form of the equations of motion for these models will lead us to solutions which are in some sense close to those coming from GR and its higher dimensional extensions (e.g. avoiding the ghosts branches at least).
The D-dimensional EGB model is a particular case of the Lovelock model [34]. The equations of motion for the Lovelock model have also at most second order derivatives of the metric (as it takes place in GR). We note that at present there exist several modifications of Einstein and EGB actions which correspond to F(R), R + f (G), f (R, G) theories (e.g. for D = 4), where R is scalar curvature and G is Gauss-Bonnet term. These modifications are under intensive studying devoted to cosmological, astrophysical and other applications, see [28][29][30] and references therein.
In this paper we restrict ourselves to diagonal metrics and study (mainly) a class of cosmological 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 > 1, k 1 > 1 and k 2 > 1, respectively, with a restriction imposed: S 1 = m H + k 1 h 1 + k 2 h 2 = 0, and D = 1 + m + k 1 + k 2 . This restriction forbids the solutions with constant volume factor. We note that in generic anisotropic case with Hubblelike parameters h 1 , . . . , h n obeying S 1 = n i=1 h i = 0 (n = D − 1) the number of different real numbers among h 1 , . . . , h n should not exceed 3 [25].
Here we study two cases: i) m < k 1 < k 2 and ii) 1 < k 1 = k 2 = k, k = m. We show that in both cases the solutions exist only if α = α 2 /α 1 > 0, > 0 and B i e 2v i t dy i ⊗ dy i , (2.3) where B i > 0 are arbitrary constants, i = 1, . . . , n, and M 1 , . . . , M n are one-dimensional manifolds (either R or S 1 ) and n > 3. The equations of motion for the action (2.1) give us the set of polynomial equations [23] Here are, respectively, the components of two metrics on R n [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 forth-order polynomial equations. We note that for = 0 and n > 3 the set of equations (2.4) and (2.5) has an isotropic solution v 1 = · · · = v n = 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 v 1 , . . . , v n when = 0. This is valid also for = 0 if n i=1 v i = 0 [25]. Here we consider a class of solutions to the set of equations (2.4), (2.5) of the following form: where H is the Hubble-like parameter corresponding to an mdimensional factor space with m > 1, h 1 is the Hubble-like parameter corresponding to an k 1 -dimensional factor space with k 1 > 1 and h 2 is the Hubble-like parameter corresponding to an k 2 -dimensional factor space with k 2 > 1. In Sect. 6 we split the m-dimensional factor space for m > 3 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 + k 1 + k 2 )dimensional internal space.  .2) M 4 = · · · = M n = S 1 and we set the internal scale factors corresponding to the present time t 0 : a j (t 0 ) = (B k ) 1/2 ex p(v j t 0 ), k = 4, . . . , n, (see (2.3)) to be small enough in comparison with the scale factor of "our" space for t = t 0 : We consider the ansatz (2.7) with three Hubble-like parameters H , h 1 and h 2 which obey the following restrictions: In Ref. [26] the set of (n + 1) polynomial equations (2.4), (2.5) under ansatz (2.7) and restrictions (2.8) imposed was reduced to a set of three polynomial equations (of fourth, second and first orders, respectively) where E is defined in (2.4) and where here and in what follows This reduction is a special case of a more general prescription (Chirkov-Pavluchenko-Toporensky trick) from Ref. [20]. Moreover, it was shown in Ref. [26] that the following relations take place Let us denote Then restrictions (2.8) read (2.17) Here we should exclude from our consideration the case Indeed, for m = k 1 = k 2 > 1 we get from restriction (2.17): 1+x 1 +x 2 = 0, while (2.18) gives us the relation 1+x 1 +x 2 = 0, which is incompatible with the previous one. We get from (2.10) and (2.12) that where We note that relation (2.20) is obeyed for αP < 0. Let us prove that Indeed, using relation (2.18), or m+k 1 x 1 +k 2 x 2 = 1+x 1 +x 2 , we get Hence, the solutions under consideration take place only if α > 0. (2.24) The calculations gives us the following relation for the vector v from (2.7) (2.26) Eur. Phys. J. C (2020) 80:543 This may be obtained by using the relation from Ref. [17] where and Here we use the notation 31) or, equivalently, (2.32) Thus, we are led to polynomial equation in variables x 1 , x 2 of fourth order or less (depending upon λ). We call relations (2.18), (2.32), as a master equations. The set of these equations may solved in radicals. Indeed, solving eq. (2.18) and substituting into eq. (2.32) we obtain another (master) which is of fourth order or less depending upon the value of λ. It may solved in radicals for all m > 1, k 1 > 1 and k 2 > 1. Here we do not try to write the explicit solution for general setup. It seems more effective for any given dimensions m, k 1 and k 2 to find the solutions just by using Maple or Mathematica. An example of solution with k 1 = k 2 will be considered below.
In what follows we use the identity following from (2.23) and (2.33).

The case k 1 = k 2
Here we put the following restriction k 1 = k 2 . We write relation (2.31) as Using relation (2.33) we rewrite the restrictions (2.17) (respectively) as follows

Extremum points
The calculations give us and . Thus, the points of extremum of the function f (x 1 ) are excluded from our consideration due to restrictions (2.8).
For the values

11)
We note that (3.19) which are valid for natural numbers m, l, k obeying: m > 1, l > 1, k > 1 and either m = l, or m = k, or l = k. This is proved in "Appendix". We also note that the following symmetry identities take place for the functions λ i (m, k 1 , k 2 ), i = 1, 2, 3, The function λ 4 (m, k 1 , k 2 ) is symmetric with respect to variables since the functions v(m, k 1 , k 2 ) and w(m, k 1 , k 2 ) are symmetric. For .
We find that (in all cases) the function λ = f (x 1 ) is monotonically increasing in the interval (X 1 = 1, +∞) from λ 1 to λ ∞ and it is monotonically decreasing in the interval (X 3 , X 1 ) from λ 3 to λ 1 .
In the case (A + ) the function λ = f (x 1 ) is monotonically increasing in the intervals (−∞, X 4 ) and (X 2 , X 3 ) from λ ∞ to λ 4 and from λ 2 to λ 3 , respectively, while it is monotonically decreasing in the interval (X 4 , X 2 ) from λ 4 to λ 2 (see Fig. 1). In this case the points X 1 and X 2 are points of local minimum and points X 3 and X 4 are points of local maximum.
For the case (A − ) the function λ = f (x 1 ) is monotonically increasing in the intervals (−∞, X 2 ) and (X 4 , X 3 ) from λ ∞ to λ 2 and from λ 4 to λ 3 , respectively, while it is monotonically decreasing in the interval (X 2 , X 4 ) from λ 2 to λ 4 (see Fig. 2). The points X 1 and X 4 are points of local minimum and points X 2 and X 3 are points of local maximum. In this case λ 2 > λ ∞ .
In the case (A 0 ) the function λ = f (x 1 ) is monotonically increasing in the intervals (−∞, X 3 ) from λ ∞ to λ 3 , respectively (see Fig. 3). For this case the point X 1 is the point of local minimum, the point X 3 is a point of local maximum and the point X 2 = X 4 is a point of inflection.
Using the inequalities (3.38), (3.39) and (3.51) we get from the behaviour of the function f (x 1 ) mentioned above that X 3 is the point of absolute maximum and X 1 is the point of absolute minimum, i.e.
for all x 1 ∈ R. Due to (3.2) the points X 1 , X 2 , X 3 , X 4 are forbidden for our consideration. We get for all x 1 = X 1 , X 2 , X 3 , X 4 . Let us denote the set of definition of the fuction f for our consideration (−∞, ∞) * ≡ {x|x ∈ R, x = X 1 , X 2 , X 3 , X 4 }. Since the function f (x 1 ) is continuous one the image of the function f (due to interme- Thus, we a led the following proposition. The case H = 0. It may verified that in the case H = 0 the solutions under consideration take place only if α > 0, and where . These relations imply α > 0 and (3.59) The substitution of these values of h 1 and h 2 , and H = 0 into equation (2.9) gives us (due to (2.25) and (2.26)) relation (3.57).

The case k 1 = k 2
Here we consider the case m > 1, k 1 = k 2 = k > 1 and H = 0. We get from (2.18) In this case relation (2.23) implies The solutions under consideration take place for and α > 0 (see Sect. 2). Let us denote α > 0. It follows from (2.20) Due to (4.4) we have The substitution of relations (4.1), (4.2) into formulae (2.29), (2.30) gives us Using (4.5) we rewrite relation (2.31) as This relation may be written as quadratic relation where where Proof For m > k we have a sum of two positive terms in (4.15) and hence F > 0 in this case. For k > m, we denote k = m + p, p > 0. We obtain Due to m > 1 and p > 0 we have a sum of three positive terms in (4.17) and hence F > 0 for k > m.
The solution to eq. (4.10) reads We are seeking real soutions which obey two restrictions Here the case D = 0 is excluded from the consideration since as it will be shown later it implies either x 1 = 1 or x 2 = 1, which contradict restrictions (2.17). The inequality (4.19) may be rewritten as λ < λ 1 for m > k, (4.21) λ > λ 1 for m < k, (4.22) where For definition of λ 1 (m, k, l) see (3.9). The set of two equations (4.1) and (4.2) have the following solutions where ε 2 = ±1 and . (4.28) Now we explain why the case D = 0 was excluded from our consideration. Let us put D = 0. Then we get from (4.18) and hence which implies either x 2 = 1 for ε 2 = 1 or x 1 = 1 for ε 2 = −1. But this is forbiden by first two inequalities in (2.17). Moreover, it is not difficult to verify that relations (4.24), (4.25) and (4.28) imply all four inequalities in (2.17). Indeed, the violation of first two inequalities in (2.17) lead us either to x 1 = 1 or x 2 = 1 which may be valid only for E from (4.30) and ε 2 = −1 or ε 2 = 1, respectively. But due to definition (4.26), relation (4.30) implies (4.29) and hence D = 0, which contradict to relations (4.24), (4.25). The violation of the third inequality gives us x 1 = x 2 which imply E = 0, but this is forbidden by (4.28). Now, let us verify the last inequality in (2.17). In our case it reads From (4.24), (4.25) we obtain The relation is (4.31) is satisfied due to (4.32) and m = k. Now we analyse the inequalities in (4.28). We introduce new parameter ε 1 =ε 1 sign(m − k). (4.33) Then relation (4.18) reads as follows , (4.34) Let us consider the case ε 1 = −1. The second inequality in (4.28) X < Using the definition of D in (4.14) we obtain Relations (4.36) read as follows where It may be verified that where λ ∞ (k, l) is defined in (3.22). Using (4.23) and (4.40) we rewrite relations (4.37), (4.38) as follows (4.42) Now, we put ε 1 = 1. The inequality X > 0 is satisfied in this case. We should treat the inequality X < (4.44) Relations (4.44) read as follows It may be verified that where λ 3 (m, k, l) is defined in (3.11). Using (4.23) and (4.48) we rewrite relations (4.45), (4.46) as follows (4.50) We note that that for m < k (it proved in the previous section), while for m < k and

54)
where λ 1 = λ 1 (k, k), λ 3 = λ 3 (k, k) are defined in (3.9) and (3.11 The restrictions on λ for our solution may be explained just graphically as it was done in the previous section for k 1 Here x 2 (x 1 ) = − m−1 k−1 − x 1 and restrictions (2.17) reads as follows The fourth inequality in (2.17) is obeyed identically (it was checked above).
The points X 1 , X 2 , X 3 are points of extremum of the function f (x 1 ). They are excluded from our consideration due to restrictions (4.57). The function f (x 1 ) tends to λ ∞ as x 1 tends to ±∞.
For 1 < k < m the function has two points of maximum at X 1 and X 2 with The graphical representation of f (x 1 ) for m = 5 and k = 4 is depicted at Fig. 5.

The analysis of stability
Here we study the stability of the solutions under consideration by using the results of refs. [23,25,26]. We put the restriction det(L i j (v)) = 0 (5.1) on the matrix We remind that for general cosmological setup with the metric we have the set of equations [23] where h i =β i , and it is unstable if (and only if) In order to study the stability of solutions we should verify the relation (5.1) for the solutions under consideration. This verification was done (in fact) in Ref. [26]. The proof of Ref. [26] is based on first three relations in (2.8) and inequalities k 1 > 1, k 2 > 1 and m > 1. We note the relation (2.14) was also used in this proof.
Thus, the any solution under consideration is stable when relation (5.8) is obeyed while it is unstable when relation (5.9) is satified.

and it is unstable if and only if H
The exact solutions obtained in this section obey first three relations in (2.8) (since x 1 = 1, x 2 = 1 and x 1 = x 2 ) and hence the key restriction (5.1) is satisfied.
The stability condition (5.8) in this case reads as, For H > 0 (or ε 0 = 1, see (4.6)) our special solutions are stable for k > m and they are unstable for k < m. For H < 0 (or ε 0 = −1) the solutions are stable for k < m and they are unstable for k > m.
The case H = 0. Let us consider the solutions with H = 0 and h 1 , h 2 from (3.58), (3.59), which are valid for k 1 = k 2 , α > 0 and from (3.57). Here k 1 > 1 and k 2 > 1. We obtain where ± is sign parameter in (3.58), (3.59). It follows from our analysis above that the solution with ±(k 2 − k 1 ) > 0 is stable. This takes place when either k 2 > k 1 and the sign "+ is chosen in (3.58) and (3.59), or if k 2 < k 1 and the sign "− is selected. For ±(k 2 − k 1 ) < 0 the solution is unstable.
Here the restriction m > 1 (which is used for the proof of (5.1)) is also assumed.

Solutions corresponding to zero variation of G
Here we consider the special solutions to equations (2.9), (2.10), (2.11) with H > 0, 3 < m < k 1 < k 2 [26] (for m = 3 see [36]) Here 6 and = (m, k 1 , k 2 ), (6.4) where (6.5) These solutions describe accelerated exponential expansion of "our" 3d subspace and constant internal space volume factor, or zero variation of the effective gravitational constant (in Jordan frame) obeying the most stringent limitation on G-dot obtained by the set of ephemerides [37], when the following splitting of the Hubble-like parameters is keeping in ). (6.6) It follows from Proposition 1 that (m, k 1 , k 2 ) > 0. Moreover, in this case we have Due to graphical analysis from Sect. 3 we get from (6.7) the following bounds Remark It may be also shown that the effective gravitational constant G (in Jordan frame), calculated for our solutions, obeys the limitation on G-dot from Ref. [37], when belongs to some vicinity of (m, k 1 , k 2 ), i.e. | − (m, k 1 , k 2 )| < δ for some (small enough) δ > 0.

Hubble-like parameters vs. constants of the model
The initial contants of the model are α 1 = 0, α 2 = 0 and . The solutions for Hubble-like parameters H = 0, h 1 and h 2 , which were analyzed above, depend upon α = α 2 /α 1 > 0 and λ = α. In this section we consider for simplicity the generic case H = 0. The parameter α has the dimension of L 2 (L is a length), while λ is dimensionless one.
Here we discuss the existence of certain combinations of Hubble-like parameters, which either do not depend upon the parameters (or constants) of the model, i.e. α and λ, or depend only upon one of these constants. Such combinations (or functions) of H = 0, h 1 and h 2 do exist.
Indeed, it follows from (2.11) that the Hubble-like parameters for the solutions under consideration obey the following identity The third basic relation is just (3.1) which we rewrite here as where f (x 1 ) is the rational function defined in (3.1).

Fig. 6
The graphical representation (in Hubble-like variables H, h 1 , h 2 ) of intersection of plane (see (7.1)) and ellipsoid (see (7.2)) for m = 3, k 1 = 4, k 2 = 5 and α = 1 In the 3d space of Hubble-like parameters H, h 1 , h 2 , relation (7.1) describes a plane while (7.2) corresponds to an ellipsoid. The intersection of this plane and ellipsoid gives us an ellipse E. For m = 3, k 1 = 4, k 2 = 5 and α = 1 this intersection is depicted at Fig. 6. For H = 0 and m < k 1 < k 2 the solutions for (H, h 1 , h 2 ) are described by 1-dimensional mani- correspond to H > 0 and relations h 1 /H = X 1 , h 2 /H = X 2 , h 3 /H 3 = X 3 , h 4 /H 4 = X 4 , respectively (see (3.3), (3.4), (3.5), (3.6)). Thus, the manifold E sol is an 1dimensional manifold, which is obtained from the ellipse E by deleting 10 points. It is a disjoint union of ten arcs. Any of these arcs is parametrized by the pair (λ, s), where s is the number of the arc and λ is local coordinate given by (7.3).
Analogous consideration may be done for the case m = k 1 = k 2 : in this case one should delete 8 points from E to obtain E sol .

Conclusions
We have considered the D-dimensional Einstein-Gauss-Bonnet (EGB) model with a -term (or EGB model) and two (non-zero) constants α 1 and α 2 . The metric was chosen to be diagonal "cosmological" one. Here we were dealing (mainly) with a class of solutions with exponential time dependence of three scale factors, governed by three noncoinciding Hubble-like parameters H = 0, h 1 and h 2 , corresponding to factor spaces of dimensions m > 1, k 1 > 1 and k 2 > 1, respectively, with the restriction imposed: We have studied the solutions in two cases: i) m < k 1 < k 2 and ii) 1 < k 1 = k 2 = k = m. (The solutions under consideration with k 1 = k 2 = m are absent.) We have shown that in both cases the solutions exist only if: α = α 2 /α 1 > 0, λ = α > 0 and the dimensionless parameter of the model λ obeys certain restrictions, e.g. upper and lower bounds depending upon m, k 1 and k 2 (see Proposition 1). In the case ii) we have found explicit exact solutions (see Proposition 2).
Our consideration used the so-called Chirkov-Pavluchenko-Toporensky splitting trick from Ref. [20] (see also [26]) which allowed us to reduce the problem under consideration to master equation λ = f (x 1 ) (2.31), where x 1 = h 1 /H . This master equation is equivalent to polynomial equation (2.34) for x 1 which is of fourth order (in generic case) or less depending upon λ. Thus, the master equation may be solved in radicals for all m > 1, k 1 > 1 and k 2 > 1. Our restrictions on λ were obtained by analysing the equation λ = f (x 1 ) with the use of the formulas for the derivative d f/dx 1 , i.e. (3.7) and (4.55) in cases i) and ii), respectively. In the case i) m < k 1 < k 2 the extremum points of the function f (x 1 ) are just four non-coinciding points: X 1 , X 2 , X 3 , X 4 (see (3.3), (3.4), (3.5), (3.6)) which are exactly four values of x 1 forbidden by restrictions H = h 1 , H = h 2 , h 1 = h 2 , S 1 = m H + k 1 h 1 + k 2 h 2 = 0, respectively. In the case ii) 1 < k 1 = k 2 = m we have three forbidden points: The stability of the solutions (as t → +∞) in a class of cosmological solutions with diagonal metrics was analyzed for both cases ((i) and (ii)) and subclasses of stable and non-stable solutions were singled out. We have proved that in the case i) the solutions with H > 0 are stable for x 1 = h 1 /H > X 4 = m−k 2 k 2 −k 1 and unstable for x 1 < X 4 (see Proposition 3). It was proved that in the case ii) the solutions with H > 0 are stable for k > m and unstable for k < m (see Proposition 4). The stability conditions for H < 0 are equivalent to instability conditions for H > 0 and vice versa. The solutions of first class i) contains a subclass of stable solutions describing an exponential expansion of 3-dimensional subspace with Hubble-like parameter H > 0 and zero vari-ation of the effective gravitational constant G (in the Jordan frame) [26] (see Sect. 6).
Some of the results obtained in this paper may be considered as non-trivial and unexpected ones. Indeed, let us compare the solutions governed by three different Hubblelike parameters H > 0, h 1 , h 2 with the solutions from Ref. [27] obtained for two non-coinciding Hubble-like parameters H > 0 and h corresponding to factor spaces of dimensions m > 2 and l > 2 with m H + lh = 0. Here we have found that our solutions take place only for α > 0 and > 0, while in the case of Ref. [27] we have two branches with (a) α > 0, −∞ < α < λ + (m, l) and (b) α < 0, |α| > λ − (m, l), where λ ± (m, l) > 0. The solutions from Ref. [27] with α > 0 exist for any ∈ (−∞, 0], while in our case such solutions are absent. We note that the absence of solutions for = 0 may be considered as a special non-trivial result. For two different Hubble parameters such solutions (with = 0 and α > 0) were described in Ref. [38]. As it is proved here, in the case of three Hubble-like parameters (with the restrictions imposed above) the allowed gap for is bounded (at the top and the bottom).
Here we have also considered (for a completeness) the case H = 0 and have found that the solutions exist only for k 1 = k 2 , α > 0 and fixed value of > 0 from (3.57). In this case we have two opposite in sign solutions for (h 1 , h 2 ) with one solution being stable and the second one -unstable.
For possible physical (e.g. cosmological) applications one may keep in mind a dimensional reduction of the model under consideration to d = 4 which lead us to 4d Horndeski type model with a set of scalar fields. In this case one will obtain (1 + 3)-dimensional inflationary (cosmological) solution with Hubble parameter H > 0 and several scalar fields (coming from scale factors) with linear dependence upon the time variable (governed by h 1 and h 2 ). The effective cosmological term 0 = 3H 2 will have a nontrivial dependence upon the "bare" multidimensional cosmological constant , the dimensions of factor spaces m, k 1 and k 2 and the parameter α (for any root of polynomial equation for x 1 ).