Exponential Cosmological Solutions with Three Different Hubble-Like Parameters in (1 + 3 + k₁ + k₂)-Dimensional EGB Model with a Λ-Term

Abstract

A D-dimensional Einstein–Gauss–Bonnet model with a cosmological term $\Lambda$, governed by two non-zero constants: ${\alpha }_1$ and ${\alpha }_2$, is considered. By restricting the metrics to diagonal ones, we study a class of solutions with the exponential time dependence of three scale factors, governed by three non-coinciding Hubble-like parameters: $H>0$, $h_1$, and $h_2$, obeying $3H+k_1{h}_1+k_2{h}_2\ne 0$ and corresponding to factor spaces of dimensions: 3, $k_1>1$, and $k_2>1$, respectively, with $D=4+k_1+k_2$. The internal flat factor spaces of dimensions $k_1$ and $k_2$ have non-trivial symmetry groups, which depend on the number of compactified dimensions. Two cases: (i) $3<k_1<k_2$ and (ii) $1<k_1=k_2=k$, $k\ne 3$, are analyzed. It is shown that in both cases, the solutions exist if $\alpha ={\alpha }_2/{\alpha }_1>0$ and $\alpha \Lambda >0$ obey 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. Case (i) contains a subclass of solutions describing an exponential expansion of $3d$ subspace with Hubble parameter $H>0$ and zero variation of the effective gravitational constant G.

1. Introduction

Here, we deal with a D-dimensional Einstein–Gauss–Bonnet (EGB) model with a $\Lambda$ term. This model is unique among the other higher dimensional extensions of General Relativity (GR) with the second order in the curvature terms since the equations of motion for this model are of the second order (in derivatives) as takes place in Einstein gravity. It is well known that the so-called Gauss–Bonnet term appeared in (super)string theory as a first order correction (in ${\alpha }^{\prime }$) to the (super)string effective action (e.g., the heterotic one) [1,2,3,4].

At present, the EGB gravitational model in diverse dimensions and its modifications (see [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33] and the references therein) are rather popular objects for study in cosmology, e.g., for a possible explanation of the accelerating expansion of the Universe (i.e., solving the dark energy problem), which follow from supernova (type Ia) observational data [34,35,36]. It may be expected that the second order form of the equations of motion for these models will lead us to solutions that are in some sense close to those coming from GR and its higher dimensional extensions (e.g., avoiding the ghost branches at least).

For given D-dimensional manifold M with the metric g, the EGB model is a particular case of the Lovelock model [37] with the action:

$$S_{Lovelock}={\int }_M{d}^{D}z\sqrt{\left|g\right|}\{\sum _{k=0}^N{\alpha }_k{\mathcal{L}}_k[g]\},$$

(1)

where $N=[D/2]$, ${\alpha }_k$ are constants (${\alpha }_0=-2\Lambda$) and ${\mathcal{L}}_k[g]$ is the $k^{\mathrm{th}}$ term, for which a certain polynomial of Riemann tensor of $k^{\mathrm{th}}$ power: ${\mathcal{L}}_0[g]=1$, ${\mathcal{L}}_1[g]=R[g]$ (scalar curvature), ${\mathcal{L}}_2[g]=\mathcal{G}=R_{mnpq}{R}^{mnpq}-4R_{mn}{R}^{mn}+R^2$ (the Gauss–Bonnet term), etc. For even dimensions $D=2m$, the integral ${\int }_M{d}^{D}z\sqrt{\left|g\right|}{\mathcal{L}}_m[g]$ is topologically invariant, i.e., it has a zero variation with respect to the metric and contributes to the equations of motion. For the Riemannian metric g defined on a closed (oriented) manifold M, this invariant is proportional to the Euler characteristic $\chi (M)$ of this manifold; this was proven by S.S.Chern in [38] (the so-called Chern or Chern–Gauss–Bonnet theorem). The equations of motion for the Lovelock model have at most second order derivatives of the metric as takes place for the EGB model. It should be noted that at present, there exist several modifications of Einstein and EGB actions, which correspond to the $F(R)$, $R+f(\mathcal{G})$, and $f(R,\mathcal{G})$ theories (e.g., for $D=4$). These modifications are intensively studied in the literature devoted to cosmological, astrophysics and other applications; see [31,32,33] and the references therein.

In this paper, we restrict ourselves to diagonal metrics and study a class of cosmological solutions with the 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 3, $k_1>1$, and $k_2>1$, respectively, with a restriction imposed: $S_1=3H+k_1{h}_1+k_2{h}_2\ne 0$ and $D=4+k_1+k_2$. This restriction forbids the solutions with a constant volume factor. We note that in the generic anisotropic case with Hubble-like parameters $h_1,\cdots ,h_n$ obeying $S_1={\sum }_{i=1}^n{h}_i\ne 0$ ($n=D-1$), the number of different (real) numbers among $h_1,\cdots ,h_n$ does not exceed three [26].

Here, we study two cases: (i) $3<k_1<k_2$ and (ii) $1<k_1=k_2=k$, $k\ne 3$. We show that in both cases, the solutions exist only if $\alpha ={\alpha }_2/{\alpha }_1>0$, $\Lambda >0$, and $\Lambda$ obey certain restrictions, e.g., inequalities of the form: $0<{\lambda }_{*}(k_1,k_2)<\Lambda \alpha <{\lambda }_{**}(k_1,k_2)$ (in (super)string inspired models, $\alpha$ is positive and corresponds to Regge slope parameter ${\alpha }^{\prime }$, which is inverse proportional to the tension of the (super)string; non-zero $\Lambda$ terms appear for non-critical (super)strings). The solutions under consideration are reduced to solutions of the polynomial master equation of fourth order or less, which may be solved in radicals for all $k_1>1$ and $k_2>1$. In Case (ii) $1<k_1=k_2\ne 3$, we present explicit exact solutions for Hubble-like parameters.

Here, we use our previous results from [25,26] in studying the stability of the solutions under consideration. We single out (for both Cases (i) and (ii)) the subclasses of stable and non-stable solutions (see Section 5). In the first case (i), we consider as an example a subclass of solutions describing the exponential expansion of the $3d$ subspace with Hubble parameter $H>0$ and zero variation of the effective gravitational constant G, which may be extracted from [29] for $m=3$, $3<k_1<k_2$, $\alpha >0$, and a fixed value of $\Lambda$ (depending on $k_1,k_2$, and $\alpha$) (see Section 6).

We note that earlier, the work in [30] dealt with exponential cosmological solutions with two non-coinciding Hubble parameters $H>0$ and h obeying $S_1=3H+l{h}_1\ne 0$ and corresponding to three- and l-dimensional factor spaces ($l>2$) in the EGB model with a $\Lambda$ term. In this case, there were two sets of solutions obeying: (a) $\alpha >0$, $\Lambda <{\alpha }^{-1}{\lambda }_{+}(l)$ and (b) $\alpha <0$, $\Lambda >\left|\alpha \right|{\lambda }_{-}(l)$, with ${\lambda }_{\pm }(l)>0$. Thus, the case of two (non-coinciding) Hubble-like parameters from [30] is drastically different from the case of three (non-coinciding) Hubble-like parameters, which is studied in this paper.

Remark 1.

We note that the solutions under consideration have non-trivial groups of symmetries (isometry groups) depending on the number of compactified dimensions of internal manifolds $M_1$ and $M_2$ of dimensions $k_1$ and $k_2$, respectively. Here, we assume (from the physical set up) that the three-dimensional factor space $M_0$ is ${\mathbb{R}}^3$. This gives us the six-dimensional isometry subgroup $G_0=\mathrm{Isom}({\mathbb{R}}^3)$, which is a semidirect product of $O(3)$ (subgroup of rotations and reflections) and ${\mathbb{R}}^3$ (subgroup of translations). For the choice $M_1={\mathbb{R}}^{l_1}\times {(S_1)}^{k_1-l_1}$ and $M_2={\mathbb{R}}^{l_2}\times {(S_1)}^{k_2-l_2}$, we get two extra isometry subgroups $G_1=\mathrm{Isom}({\mathbb{R}}^{l_1})\times {(U(1))}^{k_1-l_1}\times S_{k_1-l_1}$ and $G_2=\mathrm{Isom}({\mathbb{R}}^{l_2})\times {(U(1))}^{k_2-l_2}\times S_{k_2-l_2}$, where $S_r$ is the group of permutations of r elements, which correspond here to permutations of tori coordinates. Since we consider the case of non-coinciding Hubble-like parameters, the isometry group of the metric for any solution under consideration seems to be isomorphic to the direct product of subgroups $G_0$, $G_1$, and $G_2$.

2. The Cosmological Model

We start with the action of the model of the following form:

$$S={\int }_M{d}^{D}z\sqrt{\left|g\right|}\{{\alpha }_1(R[g]-2\Lambda )+{\alpha }_2{\mathcal{L}}_2[g]\},$$

(2)

where $g=g_{MN}d{z}^M\otimes d{z}^N$ is the metric defined on a manifold M, $dimM=D$, $\left|g\right|=|det(g_{MN})|$, $\Lambda$ is the cosmological term, $R[g]$ is scalar curvature,

$${\mathcal{L}}_2[g]=R_{MNPQ}{R}^{MNPQ}-4R_{MN}{R}^{MN}+R^2$$

is the standard Gauss–Bonnet term, and ${\alpha }_1$, ${\alpha }_2$ are nonzero constants.

We deal with the product manifold:

$$M=\mathbb{R}\times M_1\times \dots \times M_n$$

(3)

equipped with the metric:

$$g=-dt\otimes dt+\sum _{i=1}^n{B}_i{e}^{2v^{i}t}d{y}^i\otimes d{y}^i,$$

(4)

where $B_i>0$ are arbitrary constants, $i=1,\cdots ,n$, $M_1,\cdots ,M_n$ are one-dimensional manifolds (compact or noncompact ones), and $n>3$.

The action (2) leads us to the set of equations of motion, which are polynomial equations [25]:

$$\begin{array}{c}\hfill E=G_{ij}{v}^i{v}^j+2\Lambda -\alpha G_{ijkl}{v}^i{v}^j{v}^k{v}^l=0,\end{array}$$

(5)

$$\begin{array}{c}\hfill Y_i=\left[2G_{ij}{v}^j-\frac{4}{3}\alpha G_{ijkl}{v}^j{v}^k{v}^l\right]\sum _{i=1}^n{v}^i-\frac{2}{3}{G}_{ij}{v}^i{v}^j+\frac{8}{3}\Lambda =0,\end{array}$$

(6)

$i=1,\dots ,n$, where $\alpha ={\alpha }_2/{\alpha }_1$. Here:

$$G_{ij}={\delta }_{ij}-1,G_{ijkl}=G_{ij}{G}_{ik}{G}_{il}{G}_{jk}{G}_{jl}{G}_{kl}$$

(7)

are, respectively, the components of two metrics on ${\mathbb{R}}^n$ [16,17]. The first one is a usual two-metric (so-called “minisupermetric”), and the second one is a Finslerian four-metric. For $n>3$, we get a set of forth order polynomial equations.

For $\Lambda =0$ and $n>3$, the set of Equations (5) and (6) has an isotropic solution $v^1=\cdots =v^n=H$ only if $\alpha <0$ [16,17]. This result was generalized in [19] to the case $\Lambda \ne 0$ (see also [26]).

It was shown earlier that there are no more than three different numbers among $v^1,\cdots ,v^n$ when either $\Lambda =0$ [16,17] or $\Lambda \ne 0$ and ${\sum }_{i=1}^n{v}^i\ne 0$ [26].

Here, we consider a class of solutions to the set of Equations (5) and (6) of the following form:

$$v=(H,H,H,\stackrel{k_1}{\overbrace{h_1,\dots ,h_1}},\stackrel{k_2}{\overbrace{h_2,\dots ,h_2}}),$$

(8)

where H is the Hubble-like parameter corresponding to a three-dimensional factor space, $h_1$ is the Hubble-like parameter corresponding to a $k_1$-dimensional factor space with $k_1>1$, and $h_2$ is the Hubble-like parameter corresponding to a $k_2$-dimensional factor space with $k_2>1$.

Here, we exclude from our consideration the case:

$$k_1=k_2=3.$$

(9)

The reason will be explained below.

We put:

$$H>0,$$

(10)

since we are interested in accelerated expansion of the $3d$ subspace.

We consider the ansatz (8) with three Hubble parameters H, $h_1$, and $h_2$, which obey the following restrictions:

$$H\ne h_1,H\ne h_2,h_1\ne h_2,S_1=3H+k_1{h}_1+k_2{h}_2\ne 0.$$

(11)

The last restriction means that the total volume factor should not be constant.

By using the approach of [20], the set of $(n+1)$ polynomial Equations (5) and (6), under the ansatz (8) and restrictions (11) imposed, was reduced in [29] to a set of three polynomial equations of the fourth, second, and first orders, respectively:

$$\begin{array}{c}\hfill E=0,\end{array}$$

(12)

$$\begin{array}{c}\hfill Q=-\frac{1}{2\alpha },\end{array}$$

(13)

$$\begin{array}{c}\hfill L=H+h_1+h_2-S_1=0,\end{array}$$

(14)

where E is defined in (5) and:

$$Q=Q_{h_1{h}_2}=S_1^2-S_2-2S_1(h_1+h_2)+2(h_1^2+h_1{h}_2+h_2^2).$$

(15)

Here and in what follows:

$$S_k\equiv \sum _{i=1}^n{(v^i)}^k.$$

(16)

Let us denote:

$$x_1=h_1/H,x_2=h_2/H.$$

(17)

Then, the restrictions (11) read:

$$x_1\ne 1,x_2\ne 1,x_1\ne x_2,3+k_1{x}_1+k_2{x}_2\ne 0.$$

(18)

Equation (14) in x variables reads as follows:

$$2+(k_1-1)x_1+(k_2-1)x_2=0.$$

(19)

We see that for $k_1=k_2=3$, we get from the restriction (18): $1+x_1+x_2\ne 0$, while (19) gives us the relation $1+x_1+x_2=0$, which is incompatible with the previous one. This is the reason for the exclusion of the case of equal dimensions in (9).

We get from (13) and (15) that:

$$H^2=-\frac{1}{2\alpha \mathcal{P}},$$

(20)

where:

$$\begin{array}{c}\hfill \mathcal{P}=\mathcal{P}(x_1,x_2)=\mathcal{P}(x_1,x_2,k_1,k_2)=\\ \hfill {(3+k_1{x}_1+k_2{x}_2)}^2-(3+k_1{x}_1^2+k_2{x}_2^2)\\ \hfill -2(3+k_1{x}_1+k_2{x}_2)(x_1+x_2)+2(x_1^2+x_1{x}_2+x_2^2).\end{array}$$

(21)

We note that the relation (20) is obeyed for $\alpha \mathcal{P}<0$. Let us prove that:

$$\mathcal{P}<0.$$

(22)

Indeed, using the relation (19), or $3+k_1{x}_1+k_2{x}_2=1+x_1+x_2$, we get:

$$\begin{array}{c}\hfill \mathcal{P}={(1+x_1+x_2)}^2-(3+k_1{x}_1^2+k_2{x}_2^2)\\ \hfill -2(1+x_1+x_2)(x_1+x_2)+2(x_1^2+x_1{x}_2+x_2^2)\\ \hfill =-2+(1-k_1)x_1^2+(1-k_2)x_2^2<0,\end{array}$$

(23)

for $k_1>1$, $k_2>1$.

Hence, the solutions under consideration take place only if:

$$\alpha >0.$$

(24)

The calculations gives us the following relation for the vector: v from (8)

$$G_{ij}{v}^i{v}^j=3H^2+k_1{h}_1^2+k_2{h}_2^2-{(3H+k_1{h}_1+k_2{h}_2)}^2,$$

(25)

and:

$$\begin{array}{c}\hfill G_{ijkl}{v}^i{v}^j{v}^k{v}^l=24H^3(k_1{h}_1+k_2{h}_2)\\ \hfill +36H^2[k_1(k_1-1)h_1^2+2k_1{k}_2{h}_1{h}_2+k_2(k_2-1)h_2^2]\\ \hfill +12H[k_1(k_1-1)(k_1-2)h_1^3+3k_1(k_1-1)k_2{h}_1^2{h}_2\\ \hfill +3k_1{k}_2(k_2-1)h_1{h}_2^2+k_2(k_2-1)(k_2-2)h_2^3]\\ \hfill +k_1(k_1-1)(k_1-2)(k_1-3)h_1^4+4k_1(k_1-1)(k_1-2)k_2{h}_1^3{h}_2\\ \hfill +6k_1(k_1-1)k_2(k_2-1)h_1^2{h}_2^2+4k_1{k}_2(k_2-1)(k_2-2)h_1{h}_2^3\\ \hfill +k_2(k_2-1)(k_2-2)(k_2-3)h_2^4.\end{array}$$

(26)

This may be obtained by using the relation from [17]:

$$G_{ijkl}{v}^i{v}^j{v}^k{v}^l=S_1^4-6S_1^2{S}_2+3S_2^2+8S_1{S}_3-6S_4.$$

(27)

Due to (5), (26), and (26), Equation (12) reads:

$$\begin{array}{c}\hfill 2\Lambda =-G_{ij}{v}^i{v}^j+\alpha G_{ijkl}{v}^i{v}^j{v}^k{v}^l\\ \hfill =H^2{V}_1+\alpha H^4{V}_2,\end{array}$$

(28)

where:

$$\begin{array}{c}\hfill V_1=V_1(x_1,x_2)=V_1(x_1,x_2,k_1,k_2)\\ \hfill =-3-k_1{x}_1^2-k_2{x}_2^2+{(3+k_1{x}_1+k_2{x}_2)}^2\end{array}$$

(29)

and:

$$\begin{array}{c}\hfill V_2=V_2(x_1,x_2)=V_2(x_1,x_2,k_1,k_2)\\ \hfill =24(k_1{x}_1+k_2{x}_2)+36\left({[k_1]}_2{x}_1^2+2k_1{k}_2{x}_1{x}_2+{[k_2]}_2{x}_2^2\right)\\ \hfill +12\left({[k_1]}_3{x}_1^3+3{[k_1]}_2{k}_2{x}_1^2{x}_2+3k_1{[k_2]}_2{x}_1{x}_2^2+{[k_2]}_3{x}_2^3\right)\\ \hfill +{[k_1]}_4{x}_1^4+4{[k_1]}_3{k}_2{x}_1^3{x}_2+6{[k_1]}_2{[k_2]}_2{x}_1^2{x}_2^2+4k_1{[k_2]}_3{x}_1{x}_2^3+{[k_2]}_4{x}_2^4.\end{array}$$

(30)

Here, we use the notation ${[N]}_k=N(N-1)\dots (N-k+1)$.

Using (20), we get:

$$\lambda =\alpha \Lambda =-\frac{V_1}{4\mathcal{P}}+\frac{V_2}{8{\mathcal{P}}^2},$$

(31)

or equivalently,

$$V_2(x_1,x_2)-2\mathcal{P}(x_1,x_2)V_1(x_1,x_2)-8{(\mathcal{P}(x_1,x_2))}^2\lambda =0.$$

(32)

Thus, we are led to the polynomial equation in variables $x_1,x_2$ of the fourth order or less (depending on $\lambda$).

We call the relations (32) and (19) master equations. The set of these equations may be solved in radicals. Indeed, solving Equation (19):

$$x_2=x_2(x_1)=-\frac{2}{k_2-1}-\frac{k_1-1}{k_2-1}{x}_1$$

(33)

and substituting into Equation (32), we obtain another (master) equation in $x_1$:

$$V_2(x_1,x_2(x_1))-2\mathcal{P}(x_1,x_2(x_1))V_1(x_1,x_2(x_1))-8{(\mathcal{P}(x_1,x_2(x_1)))}^2\lambda =0,$$

(34)

which is of fourth order or less depending on the value of $\lambda$ and may be solved in radicals for all $k_1>1$ and $k_2>1$. Here, we do not try to write the explicit solutions $k_1$ and $k_2$ for the general setup. It seems more effective for any given dimensions $k_1>1$ and $k_2>1$ to find the solutions just by using Maple or Mathematica. An example of a solution with $k_1=k_2$ will be considered below.

In what follows, we use the identity:

$$\begin{array}{c}\hfill -(k_2-1)\mathcal{P}(x_1,x_2(x_1))=(k_1-1)(k_1+k_2-2)x_1^2\\ \hfill +4(k_1-1)x_1+2(1+k_2),\end{array}$$

(35)

following from (23) and (33).

3. The Case ${\mathbf{k}}_{\mathbf{1}}\ne {\mathbf{k}}_{\mathbf{2}}$

Here, we put the following restriction $k_1\ne k_2$. We write the relation (31) as:

$$\lambda =f(x_1)\equiv -\frac{V_1(x_1,x_2(x_1))}{4\mathcal{P}(x_1,x_2(x_1))}+\frac{V_2(x_1,x_2(x_1))}{8{(\mathcal{P}(x_1,x_2(x_1)))}^2}.$$

(36)

Using the relation (33), we rewrite the restrictions (18) (respectively) as:

$$x_1\ne X_1,x_1\ne X_2,x_1\ne X_3,x_1\ne X_4,$$

(37)

where:

$$\begin{array}{c}\hfill X_1=1,\end{array}$$

(38)

$$\begin{array}{c}\hfill X_2=-\frac{k_2+1}{k_1-1},\end{array}$$

(39)

$$\begin{array}{c}\hfill X_3=-\frac{2}{k_1+k_2-2},\end{array}$$

(40)

$$\begin{array}{c}\hfill X_4=\frac{3-k_2}{k_2-k_1}.\end{array}$$

(41)

Extremum Points

The calculations give us:

$$\frac{df}{d{x}_1}=\frac{C(k_1,k_2)(x_1-X_1)(x_1-X_2)(x_1-X_3)(x_1-X_4)}{{\left(-(k_2-1)\mathcal{P}(x_1,x_2(x_1))\right)}^3},$$

(42)

where:

$$C(k_1,k_2)=2{(k_1-1)}^2(k_2-k_1)(k_1+k_2-2)$$

(43)

and $X_1,X_2,X_3,X_4$ are defined in (38)–(41). Thus, the points of the extremum of the function $f(x_1)$ are excluded from our consideration due to the restrictions (11).

For the values ${\lambda }_i=f(X_i)$, $i=1,2,3,4$, we get:

$$\begin{array}{c}\hfill {\lambda }_1={\lambda }_1(k_1,k_2)=\frac{u(k_2,3+k_1)}{8(k_1+k_2)(k_1+1)(k_2-1)},\end{array}$$

(44)

$$\begin{array}{c}\hfill {\lambda }_2={\lambda }_2(k_1,k_2)=\frac{u(k_1,3+k_2)}{8(k_1+k_2)(k_2+1)(k_1-1)},\end{array}$$

(45)

$$\begin{array}{c}\hfill {\lambda }_3={\lambda }_3(k_1,k_2)=\frac{u(3,k_1+k_2)}{16(k_1+k_2-2)(k_1+k_2)},\end{array}$$

(46)

$$\begin{array}{c}\hfill {\lambda }_4={\lambda }_4(k_1,k_2)=\frac{v(3,k_1,k_2)}{8w(3,k_1,k_2)},\end{array}$$

(47)

where:

$$\begin{array}{c}\hfill u(m,l)=l{m}^2+(l^2-8l+8)m+l(l-1),\end{array}$$

(48)

$$\begin{array}{c}\hfill v(m,l,k)=(k+l)m^2+(m+l)k^2+(m+k)l^2-6mlk,\\ \hfill w(m,l,k)=(k+l-2)m^2+(m+l-2)k^2+(m+k-2)l^2\end{array}$$

(49)

$$\begin{array}{c}\hfill +2ml+2mk+2lk-6mlk.\end{array}$$

(50)

We note that:

$${\lambda }_i={\lambda }_i(k_1,k_2)>0,$$

(51)

for all $k_1>1$, $k_2>1$, $i=1,2,3,4$.

For $i=1,2,3$, this relation follows from:

$$u(m,l)>0,$$

(52)

for $m>1$, $l>1$. Indeed, for $m\ge 4$, $l\ge 4$, we get $u(m,l)=ml(m+l-8)+8m+l^2-l>0$ and $u(4,3)=26$, $u(3,4)=24$, $u(3,3)=12$, $u(3,2)=8$, $u(2,3)=4$, $u(2,2)=2$. For $i=4$, the relation (51) follows from the inequalities:

$$\begin{array}{c}\hfill v(m,l,k)>0,\end{array}$$

(53)

$$\begin{array}{c}\hfill w(m,l,k)>0,\end{array}$$

(54)

which are valid for natural numbers $m,l,k$ obeying: $m>1$, $l>1$, $k>1$, and either $m\ne l$, or $m\ne k$, or $l\ne k$. This is proven in Appendix A.

We also note that the following symmetry identities take place for the functions ${\lambda }_i(k_1,k_2)$, $i=1,2,3$,

$$\begin{array}{c}\hfill {\lambda }_1(k_1,k_2)={\lambda }_2(k_2,k_1),\end{array}$$

(55)

$$\begin{array}{c}\hfill {\lambda }_3(k_1,k_2)={\lambda }_3(k_2,k_1).\end{array}$$

(56)

The function ${\lambda }_4(k_1,k_2)$ is symmetric with respect to variables since the functions $v(k_1,k_2)$ and $w(k_1,k_2)$ are symmetric.

For $x_1\to \pm \infty$, we get:

$${\lambda }_{\infty }=\underset{x_1\to \infty }{lim}f(x_1)=\frac{(k_1+k_2-6)k_1{k}_2+k_1^2+k_2^2+k_1+k_2}{8(k_1-1)(k_2-1)(k_1+k_2-2)}.$$

(57)

It may be readily verified that:

$${\lambda }_{\infty }={\lambda }_{\infty }(k_1,k_2)={\lambda }_{\infty }(k_2,k_1)>0$$

(58)

for all $k_1>1$ and $k_2>1$. Indeed, $(k_1+k_2-6)k_1{k}_2+k_1^2+k_2^2+k_1+k_2=(k_1+k_2-4)k_1{k}_2+{(k_1-k_2)}^2+k_1+k_2>0$ for $k_1\ge 2$ and $k_2\ge 2$.

The points of the extremum obey the following relations:

$$\begin{array}{c}\hfill X_2-X_1=-\frac{k_1+k_2}{k_1-1},\end{array}$$

(59)

$$\begin{array}{c}\hfill X_3-X_1=-\frac{k_1+k_2}{k_1+k_2-2},\end{array}$$

(60)

$$\begin{array}{c}\hfill X_3-X_2=\frac{(k_1+k_2)(k_2-1)}{(k_1+k_2-2)(k_1-1)},\end{array}$$

(61)

$$\begin{array}{c}\hfill X_4-X_1=-\frac{3+k_1-2k_2}{k_1-k_2},\end{array}$$

(62)

$$\begin{array}{c}\hfill X_4-X_2=\frac{(3-2k_1+k_2)(k_2-1)}{(k_1-1)(k_2-k_1)},\end{array}$$

(63)

$$\begin{array}{c}\hfill X_4-X_3=\frac{(6-k_1-k_2)(k_2-1)}{(k_1+k_2-2)(k_2-k_1)}.\end{array}$$

(64)

It follows from the definitions of $X_i$ and (59)–(61) that:

$$X_2<X_3<0<X_1=1$$

(65)

for all $k_1>1$ and $k_2>1$.

The corresponding relations for ${\lambda }_i-{\lambda }_j$ have the following form:

$$\begin{array}{c}\hfill {\lambda }_2-{\lambda }_1=\frac{(k_2-k_1)(k_1+k_2)}{2(k_1-1)(k_2-1)(k_1+1)(k_2+1)},\end{array}$$

(66)

$$\begin{array}{c}\hfill {\lambda }_3-{\lambda }_1=\frac{(k_1-1)(k_2-3)(k_1+k_2)}{8(k_2-1)(1+k_1)(k_1+k_2-2)},\end{array}$$

(67)

$$\begin{array}{c}\hfill {\lambda }_3-{\lambda }_2=\frac{(k_2-1)(k_1-3)(k_1+k_2)}{8(k_1-1)(1+k_2)(k_1+k_2-2)},\end{array}$$

(68)

$$\begin{array}{c}\hfill {\lambda }_4-{\lambda }_1=\frac{(k_1-1){(2k_2-k_1-3)}^3}{2(k_1+1)(k_2-1)(k_1+k_2)\omega },\end{array}$$

(69)

$$\begin{array}{c}\hfill {\lambda }_4-{\lambda }_2=\frac{(k_2-1){(2k_1-3-k_2)}^3}{2(k_2+1)(k_1-1)(k_1+k_2)\omega },\end{array}$$

(70)

$$\begin{array}{c}\hfill {\lambda }_4-{\lambda }_3=\frac{(k_1-1)(k_2-1){(6-k_1-k_2)}^3}{8(k_1+k_2-2)(k_1+k_2)\omega },\end{array}$$

(71)

where $\omega =\omega (3,k_1,k_2)$; see (50).

Here and in what follows, we put:

$$3<k_1<k_2.$$

(72)

Using (66), (68), and (72), we get:

$$0<{\lambda }_1<{\lambda }_2<{\lambda }_3.$$

(73)

It follows from (63), (70), and (72) that:

$$(A_{+})X_4<X_2,{\lambda }_4>{\lambda }_2,\mathrm{for}2k_1-3-k_2>0,$$

(74)

$$(A_{-})X_4>X_2,{\lambda }_4<{\lambda }_2,\mathrm{for}2k_1-3-k_2<0,$$

(75)

and:

$$(A_0)X_4=X_2,{\lambda }_4={\lambda }_2,\mathrm{for}2k_1-3-k_2=0.$$

(76)

The graphical representations of the function $\lambda =f(x_1)$ for $(k_1,k_2)=(5,6),(4,6),(4,5)$ are given in Figure 1, Figure 2 and Figure 3, respectively. These three sets obey the inequalities (74)–(76), respectively.

For ${\lambda }_i-{\lambda }_{\infty }$, we obtain:

$$\begin{array}{c}\hfill {\lambda }_1-{\lambda }_{\infty }=\frac{z_1}{2(k_1-1)(k_1+k_2-2)(1+k_1)(k_1+k_2)},\end{array}$$

(77)

$$\begin{array}{c}\hfill {\lambda }_2-{\lambda }_{\infty }=\frac{z_2}{2(k_2-1)(k_1+k_2-2)(1+k_2)(k_1+k_2)},\end{array}$$

(78)

$$\begin{array}{c}\hfill {\lambda }_3-{\lambda }_{\infty }=\frac{z_3}{8(k_1-1)(k_2-1)(k_1+k_2-2)(k_1+k_2)},\end{array}$$

(79)

$$\begin{array}{c}\hfill {\lambda }_4-{\lambda }_{\infty }=\frac{z_4}{2(k_1-1)(k_2-1)(k_1+k_2-2)\omega },\end{array}$$

(80)

with $\omega =\omega (3,k_1,k_2)$ (see (50)), where:

$$\begin{array}{c}\hfill z_1=2k_1+2k_2-3-2k_1{k}_2-k_2^2+2k_1^2,\end{array}$$

(81)

$$\begin{array}{c}\hfill z_2=2k_1+2k_2-3-2k_1{k}_2-k_1^2+2k_2^2,\\ \hfill z_3=-9{(k_1-k_2)}^2-((k_1^2-6k_1+6)k_1+(k_2^2-6k_2+6)k_2\\ \hfill +3(k_1+k_2)k_1{k}_2-4)-2(k_1+k_2)\end{array}$$

(82)

$$\begin{array}{c}\hfill +(12-6k_1-6k_2+{(k_1+k_2)}^2)k_1{k}_2,\end{array}$$

(83)

$$\begin{array}{c}\hfill z_4=-{(k_2-k_1)}^2(4(k_1+k_2)+{(k_1-k_2)}^2-2k_1{k}_2-6),\end{array}$$

(84)

It follows from (77) and (79) and inequalities $z_1<0$, $z_3>0$, proven in Appendix A, that:

$${\lambda }_1<{\lambda }_{\infty }<{\lambda }_3.$$

(85)

For our new restriction (72), we obtain from (43):

$$C(k_1,k_2)>0.$$

(86)

Using this inequality and the relation (42), we find that the function $\lambda =f(x_1)$ is monotonically increasing in the interval $(X_1=1,+\infty )$ from ${\lambda }_1$ to ${\lambda }_{\infty }$ and that it is monotonically decreasing in the interval $(X_3,X_1)$ from ${\lambda }_3$ to ${\lambda }_1$.

In the case $(A_{+})$, the function $\lambda =f(x_1)$ is monotonically increasing in the intervals $(-\infty ,X_4)$ and $(X_2,X_3)$ from ${\lambda }_{\infty }$ to ${\lambda }_4$ and from ${\lambda }_2$ to ${\lambda }_3$, respectively, while it is monotonically decreasing in the interval $(X_4,X_2)$ from ${\lambda }_4$ to ${\lambda }_2$ (see Figure 1). In this case, the points $X_1$ and $X_2$ are the points of the local minimum, and the points $X_3$ and $X_4$ are the points of the local maximum.

For the case $(A_{-})$, the function $\lambda =f(x_1)$ is monotonically increasing in the intervals $(-\infty ,X_2)$ and $(X_4,X_3)$ from ${\lambda }_{\infty }$ to ${\lambda }_2$ and from ${\lambda }_4$ to ${\lambda }_3$, respectively, while it is monotonically decreasing in the interval $(X_2,X_4)$ from ${\lambda }_2$ to ${\lambda }_4$ (see Figure 2). The points $X_1$ and $X_4$ are the points of the local minimum, and the points $X_2$ and $X_3$ are the points of the local maximum.

In the case $(A_0)$, the function $\lambda =f(x_1)$ is monotonically increasing in the intervals $(-\infty ,X_3)$ and from ${\lambda }_{\infty }$ to ${\lambda }_3$, respectively (see Figure 3). For this case, the point $X_1$ is the point of the local minimum, the point $X_3$ is the point of the local maximum, and the point $X_2=X_4$ is the point of inflection.

Due to the inequalities (72) and (85), we get that $X_3$ is the point of the absolute maximum and $X_1$ is the point of the absolute minimum, i.e.,

$${\lambda }_1\le \lambda =f(x_1)\le {\lambda }_3$$

(87)

for all $x_1\in \mathbb{R}$. Due to (37), the points $X_1,X_2,X_3,X_4$ are forbidden for our consideration, and hence:

$${\lambda }_1<\lambda =f(x_1)<{\lambda }_3,\lambda =f(x_1)\ne {\lambda }_2,{\lambda }_4,{\lambda }_{\infty }$$

(88)

for all $x_1\ne X_1,X_2,X_3,X_4$. Let us denote the set of definitions of the function f for our consideration ${(-\infty ,\infty )}_{*}\equiv \{x|x\in \mathbb{R},x\ne X_1,X_2,X_3,X_4\}$. Since the function $f(x_1)$ is a continuous one, the image of the function f (due to the intermediate value theorem) is:

$$f({(-\infty ,\infty )}_{*})={({\lambda }_1,{\lambda }_3)}_{*}\equiv \{\lambda \in ({\lambda }_1,{\lambda }_3),\lambda \ne {\lambda }_2,{\lambda }_4,{\lambda }_{\infty }\}.$$

(89)

Thus, we are led to the following proposition.

Proposition 1.

The solutions to Equations (5) and (6) for ansatz (8) with $3<k_1<k_2$ obeying the inequalities $H>0$, $H\ne h_1$, $H\ne h_2$, $h_1\ne h_2$, $S_1=3H+k_1{h}_1+k_2{h}_2\ne 0$ do exist if and only if $\alpha >0$,

$${\lambda }_1<\lambda =\alpha \Lambda <{\lambda }_3$$

(90)

and:

$$\alpha \Lambda \ne {\lambda }_2,{\lambda }_4,{\lambda }_{\infty },$$

(91)

where ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$, and ${\lambda }_4$ are defined in (44)–(47), respectively. In this case, $x_1=h_1/H\ne X_1,X_2,X_3,X_4$ (see (38)–(41)), $x_2=h_2/H=x_2(x_1)$ is given by (33), $x_1$ obeys the polynomial master equation (34) (of fourth order or less), and $H^2$ is given by (20) and (21).

4. The Case ${\mathbf{k}}_{\mathbf{1}}={\mathbf{k}}_{\mathbf{2}}$

Here, we consider the case $k_1=k_2=k>1$. We get from (19):

$$2+(k-1)(x_1+x_2)=0.$$

(92)

In this case, the relation (23) implies:

$$\mathcal{P}=-2+(1-k)(x_1^2+x_2^2).$$

(93)

We note that the solutions under consideration take place for:

$$k\ne 3.$$

(94)

Indeed, for $k=3$, the relation (92) reads: $1+x_1+x_2=0$, which contradicts the restriction $1+x_1+x_2\ne 0$ from (18).

Let us denote:

$$X\equiv \alpha H^2,$$

(95)

$\alpha >0$. It follows from (20):

$$X\mathcal{P}=-\frac{1}{2}.$$

(96)

Due to (95) and $H>0$, we have:

$$H=\sqrt{X/\alpha }.$$

(97)

The substitution of the relations (92) and (93) into Formulae (29) and (30) gives us:

$$\begin{array}{c}\hfill V_1=[2(3-k)+\mathcal{P}k(k-1)]/{(k-1)}^2,\end{array}$$

(98)

$$\begin{array}{c}\hfill V_2=[-4k(3-k)(k+1)+3{\mathcal{P}}^2{(k-1)}^{2}k]/{(k-1)}^3.\end{array}$$

(99)

Using (96), we rewrite the relation (31) as:

$$2\lambda =2\alpha \Lambda =X{V}_1+X^2{V}_2.$$

(100)

This relation may be written as a quadratic relation:

$$A{X}^2+BX+C=0,$$

(101)

where:

$$\begin{array}{c}\hfill A=4k(3-k)(k+1),\end{array}$$

(102)

$$\begin{array}{c}\hfill B=-2(3-k)(k-1),\end{array}$$

(103)

$$\begin{array}{c}\hfill C=-\frac{1}{4}k{(k-1)}^2+2\lambda {(k-1)}^3.\end{array}$$

(104)

Due to (94), we get $A\ne 0$. The discriminant $D=B^2-4AC$ has the following form:

$$D=2(3-k){(k-1)}^2(F-8\lambda f)$$

(105)

where:

$$\begin{array}{c}\hfill F=F(k)=2(3-k)+(m+k-2)(m+2k-3)k,\end{array}$$

(106)

$$\begin{array}{c}\hfill f=f(k)=2(k+1)k(k-1)>0.\end{array}$$

(107)

It is proven in Appendix A that $F=F(k)>0$ for all $k>1$, $k\ne 3$.

The solution to Equation (101) reads:

$$X=(-B+{\overline{\epsilon }}_1\sqrt{D})/(2A),{\overline{\epsilon }}_1=\pm 1.$$

(108)

We are seeking real solutions, which obey two restrictions:

$$\begin{array}{c}\hfill D>0,\end{array}$$

(109)

$$\begin{array}{c}\hfill X>0.\end{array}$$

(110)

Here, the case $D=0$ is excluded from the consideration since, as we will show below, it implies either $x_1=1$ or $x_2=1$, which contradicts the restrictions (18).

The inequality (109) may be rewritten as:

$$\begin{array}{c}\hfill \lambda <{\lambda }_1\mathrm{for}k<3,\end{array}$$

(111)

$$\begin{array}{c}\hfill \lambda >{\lambda }_1\mathrm{for}k>3,\end{array}$$

(112)

where:

$${\lambda }_1={\lambda }_1(k,k)=F(k)/(8f(k)).$$

(113)

(For a definition of ${\lambda }_1(k,l)$ see (44).)

The set of the two equations (92) and (93) has the following solutions:

$$\begin{array}{c}\hfill x_1=-({\epsilon }_2\sqrt{E}+2)/(2k-2),\end{array}$$

(114)

$$\begin{array}{c}\hfill x_2=-(-{\epsilon }_2\sqrt{E}+2)/(2k-2),\end{array}$$

(115)

where ${\epsilon }_2=\pm 1$ and:

$$\begin{array}{c}\hfill E=-4k-2\mathcal{P}(k-1)=(k-1)X^{-1}-4k.\end{array}$$

(116)

Here, we put:

$$E>0,$$

(117)

since $E=0$ implies the identity $x_1=x_2$, which is excluded by the restrictions (18). The relations (110) and (117) may be written as:

$$0<X<\frac{k-1}{4k}.$$

(118)

Now, we explain why the case $D=0$ was excluded from our consideration. Let us put $D=0$. Then, we get from (108):

$$X=(-B)/(2A)=(k-1)/(4k(1+k))$$

(119)

and hence:

$$E=4k^2,$$

(120)

which implies either $x_2=1$ for ${\epsilon }_2=1$ or $x_1=1$ for ${\epsilon }_2=-1$. However, this is forbidden by the first two inequalities in (18).

Moreover, it is not difficult to verify that the relations (114), (115), and (118) imply all four inequalities in (18). Indeed, the violation of the first two inequalities in (18) leads us either to $x_1=1$ or $x_2=1$, which may be valid only for E from (120) and ${\epsilon }_2=-1$ or ${\epsilon }_2=1$, respectively. However, due to the definition (116), the relation (120) implies (119), and hence, $D=0$, which contradict the relations (114) and (115). The violation of the third inequality in (18) gives us $x_1=x_2$, which implies $E=0$, but this is forbidden by (118). Now, let us verify the last inequality in (18). In our case, it reads:

$$x_1+x_2\ne -\frac{3}{k}.$$

(121)

From (114) and (115), we obtain:

$$x_1+x_2=-\frac{2}{k-1}.$$

(122)

The relation (121) is satisfied due to (122) and $k\ne 3$.

Now, we analyze the inequalities in (118). We introduce a new parameter:

$${\epsilon }_1={\overline{\epsilon }}_1\mathrm{sign}(3-k).$$

(123)

Then, the relation (108) reads as follows:

$$X=\frac{k-1}{4k(1+k)}+{\epsilon }_1\frac{\sqrt{D}}{2\left|A\right|},$$

(124)

${\epsilon }_1=\pm 1$.

Let us consider the case ${\epsilon }_1=-1$. The second inequality in (118) $X<\frac{k-1}{4k}$ is obeyed since $2(1+k)>2$. Now, we consider the first inequality $X>0$. We get:

$$0<\sqrt{D}<2|3-k|(k-1).$$

(125)

Using the definition of D in (105), we obtain:

$$0<2(3-k){(k-1)}^2(F-8\lambda f)<4{|3-k|}^2{(k-1)}^2.$$

(126)

The relations (126) read as follows:

$$\begin{array}{c}\hfill F_{-}<8\lambda f<F\mathrm{for}k<3,\end{array}$$

(127)

$$\begin{array}{c}\hfill F<8\lambda f<F_{-}\mathrm{for}k>3,\end{array}$$

(128)

where:

$$F_{-}\equiv F-2(3-k).$$

(129)

It may be verified that:

$$\frac{F_{-}}{8f}=\frac{k}{8(k-1)}={\lambda }_{\infty }={\lambda }_{\infty }(k,k),$$

(130)

where ${\lambda }_{\infty }(k,l)$ is defined in (57). Using (113) and (130), we rewrite the relations (127) and (128) as follows:

$$\begin{array}{c}\hfill {\lambda }_{\infty }<\lambda <{\lambda }_1\mathrm{for}3>k,\end{array}$$

(131)

$$\begin{array}{c}\hfill {\lambda }_1<\lambda <{\lambda }_{\infty }\mathrm{for}3<k.\end{array}$$

(132)

Now, we put ${\epsilon }_1=1$. The inequality $X>0$ is satisfied in this case. We should treat the inequality $X<\frac{k-1}{4k}$. We obtain:

$$0<\sqrt{D}<2|3-k|k(k-1),$$

(133)

or:

$${0<2(3-k)(F-8\lambda f)<|3-k|}^2{(2k)}^2.$$

(134)

The relations (134) read as follows:

$$\begin{array}{c}\hfill F_{+}<8\lambda f<F\mathrm{for}k<3,\end{array}$$

(135)

$$\begin{array}{c}\hfill F<8\lambda f<F_{+}\mathrm{for}k>3,\end{array}$$

(136)

where:

$$F_{+}\equiv F-2(3-k)k^2.$$

(137)

It may be verified that:

$$\frac{F_{+}}{8f}={\lambda }_3={\lambda }_3(k,k),$$

(138)

where ${\lambda }_3(k,l)$ is defined in (46). Using (113) and (138), we rewrite the relations (135) and (136) as follows:

$$\begin{array}{c}\hfill {\lambda }_3<\lambda <{\lambda }_1\mathrm{for}k<3,\end{array}$$

(139)

$$\begin{array}{c}\hfill {\lambda }_1<\lambda <{\lambda }_3\mathrm{for}k>3.\end{array}$$

(140)

We note that:

$${\lambda }_1<{\lambda }_{\infty }<{\lambda }_3$$

(141)

for $k>3$, while:

$${\lambda }_3<{\lambda }_{\infty }<{\lambda }_1$$

(142)

for $k<3$ (or $k=2$). The inequalities in (142) follow from $F_{+}<F_{-}<F$ for $k<3$.

Proposition 2.

The solutions to Equations (5) and (6) for ansatz (8) with $1<k_1=k_2=k$, $k\ne 3$, obeying the inequalities $H>0$, $H\ne h_1$, $H\ne h_2$, $h_1\ne h_2$, $S_1=3H+k{h}_1+k{h}_2\ne 0$, do exist if and only if $\alpha >0$,

$${\lambda }_1<\lambda =\alpha \Lambda <{\lambda }_3$$

(143)

for $k>3$ and:

$${\lambda }_3<\lambda =\alpha \Lambda <{\lambda }_1$$

(144)

for $k=2$, where ${\lambda }_1={\lambda }_1(k,k)$, ${\lambda }_3={\lambda }_3(k,k)$ are defined in (44) and (46). In this case, H obeys the relation (97) with X from (124), $x_1=h_1/H$ and $x_2=h_2/H$ are given by the relations (114) and (115) and λ obeys (131) and (132) for ${\epsilon }_1=-1$ and (139) and (140) for ${\epsilon }_1=1$ with ${\lambda }_{\infty }=\frac{k}{8(k-1)}$.

The restrictions on $\lambda$ for our solution may be explained just graphically as was done in the previous section for $k_1\ne k_2$. Indeed, for $k_1=k_2=k\ne 3$, $H\ne 0$, we have the same relation (36) $\lambda =f(x_1)$, where now:

$$\frac{df}{d{x}_1}=\frac{\overline{C}(k)(x_1-X_1)(x_1-X_2)(x_1-X_3)}{{\left(-(k-1)\mathcal{P}(x_1,x_2(x_1))\right)}^3}$$

(145)

with:

$$\overline{C}(k)=2{(k_1-1)}^2(k_1+k_2-2)(k-3).$$

(146)

Here, $x_2(x_1)=-\frac{2}{k-1}-x_1$, and the restrictions (18) read as follows:

$$x_1\ne X_1=1,x_1\ne X_2=-\frac{1+k}{k-1},x_1\ne X_3=-\frac{1}{k-1},$$

(147)

see (38)–(40). The fourth inequality in (18) is obeyed identically (it was checked above).

The points $X_1,X_2,X_3$ are the points of the extremum of the function $f(x_1)$. They are excluded from our consideration due to the restrictions (147). The function $f(x_1)$ tends to ${\lambda }_{\infty }$ as $x_1$ tends to $\pm \infty$.

Using the relations (145) and (146) and $\mathcal{P}(x_1,x_2(x_1))<0$, we get two cases.

For $k>3$, the function has two points of the minimum at $X_1$ and $X_2$ with ${\lambda }_1=f(X_1)=f(X_2)={\lambda }_2<{\lambda }_{\infty }$ and the point of the maximum at $X_3$ with ${\lambda }_3=f(X_3)>{\lambda }_{\infty }$. See the graphical representation of $f(x_1)$ for $k=4$ in Figure 4. We note that the solution with $k=4$ was presented recently in [39].

For $k=2$, the function has two points of the maximum at $X_1$ and $X_2$ with ${\lambda }_1=f(X_1)=f(X_2)={\lambda }_2>{\lambda }_{\infty }$ and one point of the minimum at $X_3$ with ${\lambda }_3=f(X_3)<{\lambda }_{\infty }$. The graphical representation of $f(x_1)$ for $k=2$ is given in Figure 5.

5. The Analysis of Stability

Here, we study the stability of the solutions under consideration by using the results of [25,26,29] (for another approach, see also [22]).

Let us put the restriction:

$$det(L_{ij}(v))\ne 0$$

(148)

on the matrix:

$$L=(L_{ij}(v))=(2G_{ij}-4\alpha G_{ijks}{v}^k{v}^s),$$

(149)

where $v=(v^i)$ is given in (8).

We remind that for the general cosmological setup with the metric:

$$g=-dt\otimes dt+\sum _{i=1}^n{e}^{2{\beta }^i(t)}d{y}^i\otimes d{y}^i,$$

(150)

we have the set of equations [25]:

$$\begin{array}{c}\hfill E=G_{ij}{h}^i{h}^j+2\Lambda -\alpha G_{ijkl}{h}^i{h}^j{h}^k{h}^l=0,\end{array}$$

(151)

$$\begin{array}{c}\hfill Y_i=\frac{d{L}_i}{dt}+(\sum _{j=1}^n{h}^j)L_i-\frac{2}{3}(G_{sj}{h}^s{h}^j-4\Lambda )=0,\end{array}$$

(152)

where $h^i={\dot{\beta }}^i$,

$$L_i=L_i(h)=2G_{ij}{h}^j-\frac{4}{3}\alpha G_{ijkl}{h}^j{h}^k{h}^l,$$

(153)

$i=1,\dots ,n$.

Due to results of [26], a fixed point solution $(h^i(t))=(v^i)$ ($i=1,\cdots ,n$; $n>3$) to Equations (151) and (152) obeying the restrictions (148) is stable under perturbations:

$$h^i(t)=v^i+\delta h^i(t),$$

(154)

$i=1,\dots ,n$, as $t\to +\infty$, if (and only if):

$$S_1(v)=\sum _{i=1}^n{v}^i>0.$$

(155)

and it is unstable if (and only if):

$$S_1(v)=\sum _{i=1}^n{v}^i<0.$$

(156)

In order to study the stability of the solutions, we should verify the relation (148) for the solutions under consideration. This verification was done (in fact) for the more general case (with dimension $m>2$ instead of 3) in [29]. Adapting to our case ($m=3$), the proof of [29] is based on the first three relations in (11) and inequalities $k_1>1$, $k_2>1$, and $3>1$.

Thus, any solution under consideration is stable when the relation (155) is obeyed, while it is unstable when the relation (156) is satisfied.

Let us consider the case $3<k_1<k_2$. The relation (155) reads as:

$$3+k_1{x}_1+k_2{x}_2=1+x_1+x_2>0,$$

(157)

or equivalently,

$$x_1>X_4=\frac{3-k_2}{k_2-k_1}.$$

(158)

Here, Equation (19) was used. The non-stability condition (156) reads as:

$$3+k_1{x}_1+k_2{x}_2=1+x_1+x_2<0,$$

(159)

or, equivalently, as

$$x_1<X_4.$$

(160)

Proposition 3.

The solutions to Equations (5) and (6) for ansatz (8) with $3<k_1<k_2$, obeying the inequalities $H>0$, $H\ne h_1$, $H\ne h_2$, $h_1\ne h_2$, $S_1=3H+k_1{h}_1+k_2{h}_2\ne 0$ are stable if and only if $x_1>X_4=\frac{3-k_2}{k_2-k_1}$ and unstable if and only if $x_1<X_4$.

Now, we consider the case $1<k_1=k_2=k$, $k\ne 3$. The exact solutions obtained in this section obey (11) (since $x_1\ne 1$, $x_2\ne 1$ and $x_2\ne x_3$), and hence, the key restriction (148) is satisfied.

The stability condition (155) in this case reads as follows:

$$H(3+k_1{x}_1+k_2{x}_2)=H(1+x_1+x_2)=H\left(1-\frac{2}{k-1}\right)>0,$$

(161)

(see (122)) or equivalently,

$$k>3.$$

(162)

The non-stability condition (156) reads as:

$$k<3,$$

(163)

or $k=2$ (since $k>1$).

Thus, we are led to the proposition.

Proposition 4.

The solutions to Equations (5) and (6) for ansatz (8) with $1<k_1=k_2=k$, $k\ne 3$, obeying the inequalities $H>0$, $H\ne h_1$, $H\ne h_2$, $h_1\ne h_2$, and $S_1=3H+k{h}_1+k{h}_2\ne 0$ are stable for $k>3$ and unstable for $k=2$.

The example of the stable solution with $k=4$ was presented in [39].

6. Solutions Corresponding to Zero Variation of $\mathit{G}$

Here, we consider the special solutions to equations: (12)–(14) with $H>0$, $3<k_1<k_2$ [29]

$$h_1=\frac{2k_2}{k_2-k_1}H,h_2=\frac{2k_1}{k_1-k_2}H.$$

(164)

Here:

$$H=|k_1-k_2{|(-2\alpha P)}^{-1/2},$$

(165)

$\alpha >0$,

$$\begin{array}{c}\hfill P=P(k_1,k_2)=-(k_1+k_2)(3(k_1+k_2-2)+\\ \hfill k_1(2k_2-5)+k_2(2k_1-5)+6)<0,\end{array}$$

(166)

and:

$$\begin{array}{c}\hfill \Lambda =\Lambda (k_1,k_2)=\Lambda (k_2,k_1)=\frac{1}{4\alpha {(k_1-2k_1{k}_2+k_2)}^2(k_1+k_1)}\times \\ \hfill \times [k_1{k}_2(2(k_1^2+k_2^2)+(k_1+k_2)(3+2k_1{k}_2)-8k_1{k}_2)-3(k_1^3+k_2^3)].\end{array}$$

(167)

These solutions describe the accelerated exponential expansion of “our” $3d$ subspace and constant internal space volume factor or zero variation of the effective gravitational constant, which has the following form for the ansatz (8) [29]:

$$G=\mathrm{const}\times exp(-k_1{h}_{1}t-k_2{h}_{2}t).$$

(168)

It follows from Proposition 1 that $\Lambda (k_1,k_2)>0$ for $3<k_1<k_2$. Moreover, in this case, we have:

$$x_1=\frac{2k_2}{k_2-k_1}>1.$$

(169)

Due to the graphical analysis from Section 3, we get from (169) the following bounds:

$$0<{\lambda }_1(k_1,k_2)<\Lambda (k_1,k_2)\alpha <{\lambda }_{\infty }(k_1,k_2),$$

(170)

for all $3<k_1<k_2$. Due to Proposition 3 the corresponding cosmological solution is stable since $x_1>X_4$ ($X_4=\frac{3-k_2}{k_2-k_1}<0$). The stability of this solution was proven earlier in [29].

7. Conclusions and Discussion

We considered the D-dimensional Einstein–Gauss–Bonnet (EGB) model with a $\Lambda$ term and two non-zero constants ${\alpha }_1$ and ${\alpha }_2$. The metric was chosen to be a diagonal “cosmological” one. We dealt with a class of solutions with the 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 3, $k_1>1$, and $k_2>1$, respectively, with $3H+k_1{h}_1+k_2{h}_2\ne 0$ and $D=4+k_1+k_2$.

We studied the solutions in two cases: (i) $3<k_1<k_2$ and (ii) $1<k_1=k_2=k\ne 3$. (the solutions under consideration with $k_1=k_2=3$ were absent). We showed that in both cases, the solutions exist only if: $\alpha ={\alpha }_2/{\alpha }_1>0$, $\lambda =\alpha \Lambda >0$, and the dimensionless parameter of the model $\lambda$ obeys certain restrictions, e.g., upper and lower bounds depending on $k_1$ and $k_2$ (see Proposition 1). In Case (ii), we found explicit relations for exact solutions (see Proposition 2).

It should be stressed here that our consideration was based on the Chirkov–Pavluchenko–Toporensky splitting trick from [20] (adapted to our case in [29]) and reducing the problem to master equation $\lambda =f(x_1)$ (31) on the ratio $x_1=h_1/H$. This master equation is equivalent to the polynomial Equation (34) for $x_1$ which is of fourth order (in the generic case) or less depending on the value of $\lambda$, and hence, it may be solved in radicals for all $k_1>1$ and $k_2>1$. The key relation in our analysis for (i) $3<k_1<k_2$ is the formula (42) for the derivative $df/d{x}_1$. In Case (ii) $1<k_1=k_2\ne 3$, an analogous relation is given by Formula (145). Our restrictions on $\Lambda$ for $3<k_1<k_2$ were obtained by using the methods of mathematical analysis without any use of an algebraic approach to the equations of fourth (or third) order. It seemed that this (analytical) way was more economic, straightforward, and transparent for this task of cosmological origin. The latter may be illustrated by the following remarkable coincidence: in Case (i) $3<k_1<k_2$, the points of the extremum for our (smooth and bounded) function $f(x_1)$ were just four non-coinciding points: $X_1,X_2,X_3,X_4$ from (38)–(41). These points were exactly four values of $x_1$ forbidden by restrictions $H\ne h_1$, $H\ne h_2$, $h_1\ne h_2$, $S_1=3H+k_1{h}_1+k_2{h}_2\ne 0$. In Case (ii) $1<k_1=k_2\ne 3$, we had three forbidden points: $X_1,X_2,X_3$. It should be mentioned that an analogous approach was used earlier for the case of two Hubble-like parameters H, h from [30], but in that case, the master equation $\lambda =f(x)$, with $x=h/H$, had a simpler form, with the unbounded smooth function $f(x)$ defined for $x\ne x_{-},x_{+}$.

In both cases, the stability of the solutions (as $t\to +\infty$) in a class of cosmological solutions with diagonal metrics was analyzed, and subclasses of stable and non-stable solutions were singled out. In Case (i), we proved that the solutions were stable for $x_1=h_1/H>X_4=\frac{3-k_2}{k_2-k_1}$ and unstable for $x_1<X_4$ (see Proposition 3). In Case (ii), the solutions were stable for $k>3$ and unstable for $k=2$ (this was proven in Proposition 4).

The first class of solutions (i) contained a subclass of stable solutions describing an exponential expansion of $3d$ subspace with Hubble-like parameter $H>0$ and zero variation of the effective gravitational constant G, which was a special case of the solutions studied in [29].

To a certain extent, the results obtained in this paper may be considered as unexpected ones. Indeed, if we compared these solutions (for three different Hubble-like parameters) with the analogous solutions from [30] obtained for two (non-coinciding) Hubble-like parameters H and h corresponding to factor spaces of dimensions three and $l>2$ with $3H+lh\ne 0$, we found that our solutions took place only for $\alpha >0$ and $\Lambda >0$, while in the case of [30], we had two branches with (a) $\alpha >0$, $-\infty <\Lambda <{\alpha }^{-1}{\lambda }_{+}(l)$ and (b) $\alpha <0$, $\Lambda >\left|\alpha \right|{\lambda }_{-}(l)$, where ${\lambda }_{\pm }(l)>0$. The solutions from [30] do exist for any $\Lambda \in (-\infty ,0]$ (for $\alpha >0$), while in our case, such solutions were absent (The absence of solutions for $\Lambda =0$ may be considered as a special “by product” result of this paper. In the case of two different Hubble parameters, such solutions do exist for $\alpha >0$; see Ref. [40].). Moreover, in our case, the allowed gap for $\Lambda$ was bounded (at the top and the bottom), while in the case of [30], the gap for $\Lambda$ was unbounded (either at the top or at the bottom).

For possible physical (e.g., cosmological) applications, one may keep in mind a dimensional reduction of the model under consideration to $d=4$, which leads us to the $4d$ Horndeski-type model with a set of scalar fields. In this case, one will obtain a $(1+3)$-dimensional inflationary (cosmological) solution with Hubble parameter H and several scalar fields (coming from scale factors) with linear dependence on the time variable (governed by $h_1$ and $h_2$). The effective cosmological term ${\Lambda }_0=3H^2$ will have a nontrivial dependence on the “bare” multi-dimensional cosmological constant $\Lambda$, the dimensions of factor spaces $k_1$ and $k_2$, and the parameter $\alpha$ (for any root of polynomial equation for $x_1$).