# Stable exponential cosmological solutions with 3- and *l*-dimensional factor spaces in the Einstein–Gauss–Bonnet model with a \(\Lambda \)-term

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## Abstract

A *D*-dimensional gravitational model with a Gauss–Bonnet term and the cosmological term \(\Lambda \) is studied. We assume the metrics to be diagonal cosmological ones. For certain fine-tuned \(\Lambda \), we find a class of solutions with exponential time dependence of two scale factors, governed by two Hubble-like parameters \(H >0\) and *h*, corresponding to factor spaces of dimensions 3 and \(l > 2\), respectively and \(D = 1 + 3 + l\). The fine-tuned \(\Lambda = \Lambda (x, l, \alpha )\) depends upon the ratio \(h/H = x\), *l* and the ratio \(\alpha = \alpha _2/\alpha _1\) of two constants (\(\alpha _2\) and \(\alpha _1\)) of the model. For fixed \(\Lambda , \alpha \) and \(l > 2\) the equation \(\Lambda (x,l,\alpha ) = \Lambda \) is equivalent to a polynomial equation of either fourth or third order and may be solved in radicals (the example \(l =3\) is presented). For certain restrictions on *x* we prove the stability of the solutions in a class of cosmological solutions with diagonal metrics. A subclass of solutions with small enough variation of the effective gravitational constant *G* is considered. It is shown that all solutions from this subclass are stable.

## 1 Introduction

In this paper we study a *D*-dimensional gravitational model with Gauss–Bonnet term and cosmological term \(\Lambda \), i.e. we deal with the so-called Einstein–Gauss–Bonnet model (in short, EGB-, or more precisely EGB\(\Lambda \)-model). The so-called Gauss–Bonnet term appeared in string theory as a correction to the string effective action [1, 2, 3, 4, 5].

At the moment there is a certain interest to Einstein–Gauss–Bonnet (EGB) gravitational model and its modifications, see [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] and Refs. therein. They are intensively studied in cosmology, e.g. for possible explanation of accelerating expansion of the Universe which follow from supernovae (type Ia) observational data [31, 32, 33].

Here we consider the cosmological solutions with diagonal metrics. They are governed by \(n = 3 + l> 5\) scale factors which depend upon the synchronous time variable. We deal with solutions which have exponential dependence of scale factors. We present a class of such solutions with two scale factors, which correspond to factor spaces of dimensions 3 and \(l > 2\), and are described by two Hubble-like parameters \(H >0\) and *h*, respectively. Here the total dimension is \(D = 1 + 3 + l\) . Any of these solutions is presented in parametrized form: the cosmological constant \(\Lambda \) is fine-tuned, it depends upon the ratio \(h/H = x\), *l* and a ratio two coupling constants. Any solution describes an exponential expansion of 3*d* factor space with Hubble parameter \(H > 0\) [34].

Here we study the stability of the solutions in a class of cosmological solutions with diagonal metrics and single out a subclass of stable solutions. Our analysis is based on earlier results of Refs. [25, 26] (see also the approach of Ref. [23]).

We also consider a subclass of solutions which correspond to a small enough variation of the effective gravitational constant *G* in the Jordan frame [35, 36] (see also [37, 38, 39] and Refs. therein). We show that all these solutions are stable.

## 2 The setup

*M*, \({\dim M} = D\), \(|g| = |\det (g_{MN})|\), \(\Lambda \) is the cosmological term,

*R*[

*g*] is scalar curvature and

In what follows we deal with anisotropic solutions. The isotropic solutions with \(v^1 = \cdots = v^n = H\), \(\alpha < 0\) (and \(n > 3\)) were considered in Refs. [17, 18] and [20] for \(\Lambda =0\) and \(\Lambda \ne 0\), respectively. As it was shown in Refs. [17, 18] there are no more than three different numbers among \(v^1,\ldots ,v^n\) when \(\Lambda =0\). This is valid also in the case \(\Lambda \ne 0\), when the additional restriction \(\sum _{i = 1}^{n} v^i \ne 0\) is imposed [26].

## 3 Solutions with two Hubble-like parameters

*H*is the Hubble-like parameter corresponding to the 3-dimensional factor space and

*h*is the Hubble-like parameter corresponding to the

*l*-dimensional factor space, \(l > 2\).

*l*-dimensional internal factor space is described by the Hubble-like parameter

*h*.

It is widely known that the 4*d* effective gravitational constant \(G = G_{eff}\) in the Brans–Dicke–Jordan (or simply Jordan) frame [35] (see also [36]) is proportional to the inverse volume scale factor of the internal space, see [37, 39] and references therein.

*H*and

*h*obeying two restrictions imposed

*x*from (3.8)

*l*. In this case the Hubble-like parameters read

*l*-dimensional Euclidean space.

Let us consider the behaviour of the function \(\lambda (x,l)\) in the vicinity of the points \(x_{-}(l)\) and \(x_{+}(l)\). Here the following proposition is valid.

### Proposition 1

In the proof of the Proposition 1 the following lemma is used.

### Lemma

The Lemma is proved in the Appendix A.

### The proof of Proposition 1

*l*and \(x \ne x_{\pm }(l)\). First, we find the extremum points which obey \(\frac{\partial }{\partial x} \lambda (x,l) = 0\). The calculations give us

*w*(

*l*) is defined in (3.43)].

*x*for fixed \(l>2\). We calculate \(n(\Lambda , \alpha )\), which is the number of solutions (in variable

*x*) of the relation \(\Lambda \alpha = \lambda (x,l)\). In what follows we use relations (3.33), (3.34), (3.35), (3.36), (3.37), (3.39), (3.46), (3.47), (3.48), (3.49), (3.50), (3.51), (3.52), (3.53), (3.54), (3.55), (3.56), (3.57), corresponding to points of extremum \(x_i\) and \(\lambda _i\) (\(i = a,b,c,d\)) and relations (3.28), (3.32).

First, we consider the case \(\alpha > 0\) and \(x_{-}< x < x_{+}\). We keep in mind that the solution \(x = x_d\) is excluded.

**(A)** \(2< l < 6\). We get \(x_b< x_d < x_c\) and \(\lambda _d < \lambda _c\), \(\lambda _d < \lambda _b\). Points \(x_b, x_c\) are points of local maximum and \(x_d\) is a point of local minimum.

We split this case on two subcases: \((A_{0})\) \(l=3\) and \((A_{-})\) \(3< l < 6\).

*i*the point \((x_i, \lambda _i)\), where \(i = a,b,c,d\).

**(B)**\(l > 6\). We have \(x_d< x_b < x_c\) and \(\lambda _b< \lambda _d < \lambda _c\). Points \(x_c\) and \(x_d\) are points of local maximum (\(x_c\) is a point of maximum on interval \((x_{-}, x_{+})\)) and \(x_b\) is a point of local minimum. We find

The function \(\lambda (x)\) for \(\alpha < 0\) and \(l =3\) is presented at Fig. 6.

*Master equation*The Eq. (3.12) may be written in the following form

*x*) for \(\lambda \ne \lambda _{\infty }(l)\) and of third order for \(\lambda = \lambda _{\infty }(l)\). For any \(l > 2\) the master equation can be solved in radicals.

*Example for*\(l = 3\) As an example we consider the solution for \(l = 3\). In this case \(x_{\pm } = - 2 \pm \sqrt{3}\), \(x_b = - 2\), \(x_c = - 1/2\), \(x_d = - 1\) and \(\lambda _a = - 5/16\), \(\lambda _{\infty } = - 3/4\), \(\lambda _b = \lambda _c = 1/4\), \(\lambda _d = 3/16\). The solutions obey \(x \ne x_{\pm }\). The master Eq. (3.64) for \(m = l = 3\) reads

## 4 Stability analysis

*v*from (3.1), obeying relations (3.3), the matrix

*L*has a block-diagonal form

Thus, we proved that relations (4.18) and (4.19) are valid. Hence the restriction (4.6) is satisfied for our solutions.

Thus we are led to the following proposition.

### Proposition 2

The cosmological solutions under consideration obeying \(x = h/H \ne x_i\), \(i = a,b,c,d\), where \(x_a =1\), \(x_b = - \frac{2}{l - 2}\), \(x_c = - \frac{1}{l -1}\), \(x_d = - \frac{3}{l}\), are stable if (i) \(x > x_d\) and unstable if (ii) \(x < x_d\).

We note that for the anisotropic solutions under consideration the points \(x= x_a\) and \(x= x_d\) are excluded. Nevertheless, the solutions are defined for special extremal cases, when \(x = x_b\) or \(x = x_c\), if \(x \ne x_d\). The stability analysis of these special solutions can not be deduced just from the equations for perturbations (see relations (B.3), (B.4) from the Appendix) in the linear approximation. They need a special consideration [30].

We note that for the example \(l =3\) from Sect. 3 we get stable cosmological solutions for \(x > -1\) and \(x \ne -1/2\), \(x \ne 1\).

Let us denote by \(n_{+}(\Lambda , \alpha )\) the number of non-special stable solutions which are given by Proposition 2 [see item (i)]. By using the results (e.g. figures) from the previous section we find for \(\alpha > 0\):

**(A)**\(2< l < 6\)

**(B)**\(l > 6\)

*x*obeying \(x > x_{+}\). The solution with \(x < x_{-}\) is unstable.

## 5 Solutions with small enough variation of *G*

*G*, which is proportional (in the Jordan frame) to the inverse volume scale factor of the anisotropic internal space, i.e.

*G*:

*G*-dot obtained in Ref. [40] (by the set of ephemerides)

*G*. The main condition for the stability \(x_0(\delta ,l) > x_d\) is satisfied since

Thus, we have shown that all (well-defined) solutions under consideration obeying the bounds (5.3) (coming from the physical bounds on variation of *G*) are stable. We note that the proof of this fact is also valid for less restrictive bounds for \(\delta \) than (5.3).

## 6 Conclusions

Here we have considered the Einstein–Gauss–Bonnet (EGB) model in dimension \(D = 1 + 3 + l\), \(l > 2\), with the \(\Lambda \)-term and two non-zero constants \(\alpha _1\) and \(\alpha _2\). By using the ansatz with diagonal cosmological metrics, we have found, for certain fine-tuned \(\Lambda = \Lambda (x,l,\alpha )\), where \(\alpha = \alpha _2 / \alpha _1 \), a class of solutions with exponential time dependence of two scale factors, governed by two Hubble-like parameters \(H >0\) and \(h = x H\), corresponding to submanifolds of dimensions 3 and \(l > 2\), respectively. The parameter \(x = h/H\) obey the restrictions \(x \ne x_a =1\), \(x \ne x_d = - 3/l\) and \(\mathcal {P}(x,l) = 2 + 4(l - 1) x + (l - 1)(l - 2)x^2 \ne 0\). Moreover, it should be imposed: \(\mathcal {P}(x,l) < 0\) for \(\alpha > 0\) and \(\mathcal {P}(x,l) > 0\) for \(\alpha < 0\). For fixed \(\Lambda , \alpha \) and \(l > 2\) the equation \(\Lambda (x,l,\alpha ) = \Lambda \) is equivalent to a polynomial equation of either fourth or third order and hence may be solved in radicals.

Any of solutions describes an exponential expansion of 3*d* subspace (our space) with the Hubble parameter \(H > 0\) and either contraction or expansion (with Hubble-like parameter *h*), or stabilization (\(h = 0\)) of *l*-dimensional internal subspace.

By using results of Ref. [26] we have proved that the cosmological solution (under consideration) is stable as \(t \rightarrow + \infty \), if it obey the following restrictions: \(x > x_d = -\, 3/l\), \(x \ne x_b = - \frac{2}{l - 2}\) and \( x \ne x_c = - \frac{1}{l - 1}\). Here the points \(x_a, x_b, x_c, x_d\) are points of extremum of the function \(\lambda (x,l) = \alpha \Lambda (x,l,\alpha )\) for any \(l > 2\).

We have also shown that all (well-defined) solutions with small enough variation of the effective gravitational constant *G* (in the Jordan frame), obeying physical bounds, are stable.

## Notes

### Acknowledgements

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

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