# Exponential cosmological solutions with two factor spaces in EGB model with \(\Lambda = 0\) revisited

## Abstract

We study exact cosmological solutions in *D*-dimensional Einstein–Gauss–Bonnet model (with zero cosmological term) governed by two non-zero constants: \(\alpha _1\) and \(\alpha _2\) . We deal with exponential dependence (in time) of two scale factors governed by Hubble-like parameters \(H >0\) and *h*, which correspond to factor spaces of dimensions \(m >2\) and \(l > 2\), respectively, and \(D = 1 + m + l\). We put \(h \ne H\) and \(mH + l h \ne 0\). We show that for \(\alpha = \alpha _2/\alpha _1 > 0\) there are two (real) solutions with two sets of Hubble-like parameters: \((H_1, h_1)\) and \((H_2, h_2)\), which obey: \( h_1/ H_1< - m/l< h_2/ H_2 < 0\), while for \(\alpha < 0\) the (real) solutions are absent. We prove that the cosmological solution corresponding to \((H_2, h_2)\) is stable in a class of cosmological solutions with diagonal metrics, while the solution corresponding to \((H_1, h_1)\) is unstable. We present several examples of analytical solutions, e.g. stable ones with small enough variation of the effective gravitational constant *G*, for \((m, l) = (9, l >2), (12, 11), (11,16), (15, 6)\).

## 1 Introduction

Currently, the Einstein–Gauss–Bonnet (EGB) model and related theories, see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] and Refs. therein, are under intensive studies in cosmology, aimed at explanation of accelerating expansion of the Universe [13, 14]. Here we study the EGB model with zero cosmological term in *D* dimensions (\(D = n+1\)). This model contains Gauss–Bonnet term, which arises in (super)string theory as a correction to the (super)string effective action (e.g. heterotic one) [15, 16, 17]. The model is governed by two nonzero constants \(\alpha _1\) and \(\alpha _2\) which correspond to Einstein and Gauss–Bonnet terms in the action, respectively. In this paper we continue our studies of the EGB cosmological model from Ref. [8]. We deal with diagonal metrics governed by \(n >3\) scale factors and consider the following ansatz for scale factors \(a_i(t)\) (*t* is synchronous time variable): \( a_1(t) = \dots = a_m(t) = \exp (Ht)\) and \(a_{m+1}(t) = \dots = a_{m+l}(t) = \exp (ht)\), where \(n =m + l\), \(m > 2\), \(l > 2\). We put here \(H >0\) in order to describe exponential accelerated expansion of 3*d* subspace with Hubble parameter *H* [18].

In contrary to our earlier publication [8], where a lot of numerical solutions with small enough value of variation of the effective gravitational constant *G* were found, here we put our attention mainly to the search of analytical exponential solutions with two factor spaces of dimensions *m* and *l*. Here we show that the anisotropic cosmological solutions under consideration with two Hubble-like parameters \(H>0\) and *h* obeying restrictions \(h \ne H\), \(mH + l h \ne 0\) do exist only if \(\alpha = \alpha _2/ \alpha _1> 0\). In this case we have two solutions with Hubble-like parameters: \((H_1 > 0, h_1<0)\) and \((H_2 > 0, h_2<0)\), respectively, such that \(x_1 = h_1/H_1< - m/l < x_2 = h_2/H_2\). By using results of Refs. [10, 11] (see also approach of Ref. [9]) we show that the solutions with Hubble-like parameters \((H_2, h_2)\) are stable (in a class of cosmological solutions with diagonal metrics), while those corresponding to \((H_1, h_1)\) are unstable.

Here we also present examples of analytical solutions for: (i) \(m =l\); (ii) \(m=3\), \(l =4\); (iii) \(m = 9\), \(l >2\); (iv) \(m = 12\), \(l = 11\); (v) \(m=11\), \(l=16\) and (vi) \(m =15\), \(l =6\). It should be noted that analytical solutions in cases (iii) and (iv) were considered numerically in Ref. [8] in a context of solutions with a small (enough) variation of *G* (in Jordan frame, see Ref. [20]), e.g. obeying the most severe restrictions on variation of *G* from Ref. [19]. The stable solutions with zero variation of *G* in cases (v) and (vi) were found earlier in [8], while the stability of these solutions was proved in Ref. [10].

## 2 The set up

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

*R*[

*g*] is scalar curvature,

## 3 Solutions governed by two Hubble-like parameters

*H*is the Hubble-like parameter corresponding to an

*m*-dimensional factor space with \(m > 2\), while

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

*l*-dimensional factor space, \(l > 2\). The splitting in (3.1) was done just for cosmological applications. Here we split the

*m*-dimensional factor space into the product of 3

*d*subspace (“our” space) and \((m-3)\)-dimensional subspace, which is a part of \((m-3 +l)\)-dimensional “internal” space.

*d*subspace, we impose the following restriction

*m*-dimensional subspace is expanding with the Hubble parameter \(H >0\). The behaviour of scale factor corresponding to

*l*-dimensional subspace is governed by Hubble-like parameter

*h*.

*H*and

*h*

*x*for any \(l > 2\). One can solve the Eq. (3.23) in radicals for any \(m > 2\) and \(l > 2\). The general solution is presented in Appendix.

Here we use the following proposition from Ref. [21].

### Proposition 1

*m*and

*l*[21]:

Now, let us consider the case \(\alpha < 0\). We have: \(x < x_{-}\) or \(x > x_{+}\). Due to to the relations (3.21), (3.30) and Proposition 1, the function \(\lambda (x)\) is monotonically decreasing in two intervals: (i) in the interval \((-\infty ,x_{-})\) from \( \lambda _{\infty } \) to \(- \infty \) and (ii) in the interval \((x_{a} = 1, +\infty )\) from \( \lambda _{a} \) to \( \lambda _{\infty } \). The function \( \lambda (x)\) is monotonically increasing in the interval \((x_{+}, x_{a})\) from \(- \infty \) to \(\lambda _{a} \). Here \(x_{a}= 1\) is a point of local maximum of the function \(\lambda (x) \), which is excluded from the solution and \(0> \lambda _{a} > \lambda _{\infty }\).

By using the behaviour of the function \(\lambda (x,m,l)\), which was considered above, one can readily prove the following proposition.

### Proposition 2

For any \(m > 2\), \(l > 2\) there are only two real solutions \(x_1, x_2 \) to the master equation \(\lambda (x) = \lambda (x,m,l) = 0\) (see (3.12)) for \(\alpha > 0\). These solutions obey \(x_{-}< x_1< - \frac{m}{l}< x_2< x_{+} < 0\) (see (3.16)). For \(\alpha < 0\) the solutions to master equation are absent.

### Proof

First, let us consider the case \(\alpha < 0\). In this case it follows from our analysis above that \(\lambda (x) < \lambda _{\infty }\) for \(x < x_{-}\) and \(\lambda (x) < \lambda _{a}\). Since \(\lambda _{\infty }< \lambda _a < 0\), we get in the case \(\alpha < 0\): \(\lambda (x)< \lambda _{a} < 0\). Hence the equation \(\lambda (x) = 0\) does not have solutions.

Now we consider the case \(\alpha > 0\). We are seeking the solutions to equation \(\lambda (x) = 0\) in the interval \((x_{-}, x_{+})\), where our function is smooth (and continuous). Let us denote: \(x_{*} = \mathrm{min}(x_b, x_c, x_d)\) and \(x_{**} =\mathrm{max}(x_b, x_c, x_d)\). The interval \([x_{*},x_{**}]\) should be excluded from our consideration since \(\lambda (x) \ge \mathrm{min}(\lambda _a,\lambda _b,\lambda _c) > 0\) for \(x \in [x_{*},x_{**}]\). (Here we use the fact that the smooth (e.g. continuous) function on the closed interval \([x_{*},x_{**}]\) has a minimum which should be equal to \(\lambda (x_{*})\) or \(\lambda (x_{**})\) or a value of the function in a point of local minimum (e.g. point of extremum) of the form \(\lambda (x_{i})\), \(i = b,c, d\). In any case this minimum coincides with \(\lambda (x_{i})\) for some \(i = b,c, d\).) Now we consider the interval \((x_{-},x_{*})\). The function \(\lambda (x)\) is monotonically increasing in the interval \((x_{-},x_{*})\). Due to relation (3.21) there exists a point \(x_{*,-} \in (x_{-},x_{*})\) such that \(\lambda (x_{*,-}) < -1\) and hence any point *x* in the interval \((x_{-}, x_{*,-}]\) obey \(\lambda (x) < -1\). Thus, we exclude the interval \((x_{-}, x_{*,-}]\) from our consideration. Now we consider the interval \([x_{*,-}, x_{*}]\), where \(\lambda (x_{*,-}) < -1\) and \(\lambda (x_{*}) > 0\). Due to intermediate value theorem there exists a point \(x_1 \in (x_{*,-}, x_{*}) \subset (x_{-}, x_{*})\) such that \(\lambda (x_{1}) = 0\). This point is unique since the function is monotonically increasing in this interval. By analogous arguments one can readily prove the existence of unique point \(x_2 \in (x_{*}, x_{+})\) such that \(\lambda (x_{2}) = 0\). By our definitions above we obtain \(x_{-}<x_1< x_{*} \le x_d = - \frac{m}{l} \le x_{**}< x_2< x_{+} < 0\). This completes the proof of the proposition.

Thus, we are led to the following (physical) result: the anisotropic cosmological solutions under consideration with two Hubble-like parameters \(H>0\) and *h* obeying restrictions (3.3) do exist only if \(\alpha > 0\). In this case we have two solutions with Hubble-like parameters: \((H_1 > 0, h_1 <0)\) and \((H_2 > 0, h_2 <0)\) such that \(h_1/H_1< - m/l< h_2/H_2< 0\). \(\square \)

## 4 Stability analysis and variation of *G*

### Proposition 3

[21] The cosmological solutions from [21], which obey \(x = h/H \ne x_i\), \(i = a,b,c,d\), where \(x_a =1\), \(x_b = - \frac{m -1}{l - 2}\), \(x_c = - \frac{m-2}{l -1}\), \(x_d = - \frac{m}{l}\), are stable, if (i) \(x > x_d\) and unstable, if (ii) \(x < x_d\).

Here it should be noted that our anisotropic solutions with non-static volume factor are not defined for \(x= x_a\) and \(x= x_d\). Meanwhile, they are defined when \(x = x_b\) or \(x = x_c\), if \(x \ne x_d\).

### Proposition 4

The cosmological solution under consideration for \(\alpha >0 \) corresponding to the big root of master equation \(x_2 \) is stable, while the solution corresponding to the small root \(x_1\) is unstable.

*G*(in the Jordan frame), which is inversely proportional to the volume scale factor of the (anisotropic) internal space (see [8] and references therein), i.e.

*G*-dot (by the set of ephemerides) [19] \({\dot{G}}/G = (0.16 \pm 0.6) \cdot 10^{-13} \ year^{-1}\), which are allowed at 95% confidence (2-\(\sigma \)) level, and the value of the Hubble parameter (at present) [18] \(H_0 = (67,80 \pm 1,54) \ km/s \ Mpc^{-1} = (6.929 \pm 0,157) \cdot 10^{-11} \ year^{-1}\), with 95% confidence level.

*x*-parameter corresponding to dimensionless parameter of variation of

*G*from (4.3). Then, we have

*G*, which were obtained in Ref. [8], are stable. The stability two of them was proved in Ref. [10].

### Remark

It follows from our consideration that a more wide class of solutions with \(\delta < 3\) consists of stable solutions.

## 5 Examples of solutions

Here we present certain examples of analytical solutions in the model under consideration. These solutions may be readily verified by using Maple or Mathematica. They are given by \(x = x_1, x_2\) and relations (3.6), (3.7), (3.8).

### 5.1 The solutions for \(m=l\)

### 5.2 The solution for \(m=3\) and \(l =4\)

### 5.3 The series of solutions for \(m=9\) and \(l > 2\)

*l*. By using (1 /

*l*)-decomposition we get

### 5.4 The solutions for \(m=12\) and \(l = 11\)

### 5.5 The solutions for \(m=11\) and \(l = 16\)

*G*in Jordan frame, i.e. \(\delta = 0\). The stability of the corresponding cosmological solution was proved earlier in [10].

### 5.6 The solutions for \(m=15\) and \(l = 6\)

*G*(\(\delta = 0\)). The stability of the corresponding cosmological solution was proved in [10].

## 6 Conclusions

*D*-dimensional Einstein–Gauss–Bonnet (EGB) model with two non-zero constants \(\alpha _1\) and \(\alpha _2\). By using the ansatz with diagonal cosmological metrics, we have studied a class of solutions with exponential time dependence of two scale factors, governed by two Hubble-like parameters \(H >0\) and

*h*, corresponding to submanifolds of dimensions \(m > 2\) and \(l > 2\), respectively, with \(D = 1 + m + l\). The equations of motion were reduced to the master equation \(\lambda (x,m,l) = 0\) (see (3.12) or (3.23)), where the parameter \(x = h/H\) obeys the restrictions: \(x \ne 1\), \(x \ne - m/l\) and \(x \ne x_{\pm }\) (\(x_{-}< x_{+} < 0\)) are defined in (3.16). By using our earlier analysis from Ref. [21] we have proved that the master equation has real solutions only for \(\alpha > 0\). In this case there are two solutions: \(x_1\), \(x_2\), which satisfy

Any cosmological solution corresponding to \(x_1\) or \(x_2\) (for \(\alpha > 0\)) describes an exponential expansion of 3-dimensional subspace (“our” space) with the Hubble parameter \(H > 0\) and anisotropic behaviour of \((m-3+ l)\)-dimensional internal space: expanding in \((m-3)\) dimensions (with Hubble parameter *H*) and contracting in *l* dimensions (with Hubble-like parameter *h*).

By using our earlier results from Ref. [21] we have proved that the solution corresponding to \(x_2\) is stable in a class of cosmological solutions with diagonal metrics, while the solution corresponding to \(x_1\) is unstable.

We have presented several examples of exact solutions (in terms of \(x = h/H\)) in the following cases: (i) \(m =l\); (ii) \(m=3\), \(l =4\); (iii) \(m = 9\), \(l >2\); (iv) \(m = 12\), \(l = 11\); (v) \(m=11\), \(l=16\) and (vi) \(m =15\), \(l =6\). In case (iii) we have also proved the asymptotical relation for variation of *G*: \({\dot{G}}/(GH) = 3/l + o(1/l)\), as \(l \rightarrow \infty \), which is valid for stable solutions.

## Notes

### Acknowledgements

The publication has been prepared with the support of the “RUDN University Program 5-100” (recipient V.D.I., mathematical model development). The reported study was funded by RFBR, project number 19-02-00346 (recipient A.A.K., simulation model development).

## References

- 1.H. Ishihara, Cosmological solutions of the extended Einstein gravity with the Gauss–Bonnet term. Phys. Lett. B
**179**, 217 (1986)ADSMathSciNetCrossRefGoogle Scholar - 2.N. Deruelle, On the approach to the cosmological singularity in quadratic theories of gravity: the Kasner regimes. Nucl. Phys. B
**327**, 253–266 (1989)ADSMathSciNetCrossRefGoogle Scholar - 3.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 MathSciNetCrossRefGoogle Scholar - 4.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 ADSMathSciNetCrossRefGoogle Scholar - 5.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 MathSciNetCrossRefGoogle Scholar - 6.D. Chirkov, S. Pavluchenko, A. Toporensky, Exact exponential solutions in Einstein–Gauss–Bonnet flat anisotropic cosmology. Mod. Phys. Lett. A
**29**, 1450093 (2014). arXiv:1401.2962 ADSMathSciNetCrossRefGoogle Scholar - 7.D. Chirkov, S.A. Pavluchenko, A. Toporensky, Non-constant volume exponential solutions in higher-dimensional Lovelock cosmologies. Gen. Relativ. Gravit.
**47**, 137 (2015). arXiv:1501.04360 - 8.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 (2015). Erratum: Eur. Phys. J. C 76, 584 (2016). arXiv:1503.00860 - 9.S.A. Pavluchenko, Stability analysis of exponential solutions in Lovelock cosmologies. Phys. Rev. D
**92**, 104017 (2015). arXiv:1507.01871 ADSMathSciNetCrossRefGoogle Scholar - 10.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)ADSMathSciNetCrossRefGoogle Scholar - 11.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 - 12.V.D. Ivashchuk, A.A. Kobtsev, Stable exponential cosmological solutions with \(3\)- and \(l\)-dimensional factor spaces in the Einstein–Gauss–Bonnet model with a Lambda-term. Eur. Phys. J. C
**78**, 100 (2018)ADSCrossRefGoogle Scholar - 13.A.G. Riess et al., Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astron. J.
**116**, 1009–1038 (1998)ADSCrossRefGoogle Scholar - 14.S. Perlmutter et al., Measurements of omega and lambda from 42 high-redshift supernovae. Astrophys. J.
**517**, 565–586 (1999)ADSCrossRefGoogle Scholar - 15.B. Zwiebach, Curvature squared terms and string theories. Phys. Lett. B
**156**, 315 (1985)ADSCrossRefGoogle Scholar - 16.E.S. Fradkin, A.A. Tseytlin, Effective action approach to superstring theory. Phys. Lett. B
**160**, 69–76 (1985)ADSCrossRefGoogle Scholar - 17.D. Gross, E. Witten, Superstrings modifications of Einstein’s equations. Nucl. Phys. B
**277**, 1 (1986)ADSMathSciNetCrossRefGoogle Scholar - 18.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 - 19.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 Google Scholar - 20.M. Rainer, A. Zhuk, Einstein and Brans–Dicke frames in multidimensional cosmology. Gen. Relativ. Gravit.
**32**, 79–104 (2000). arXiv:gr-qc/9808073 ADSMathSciNetCrossRefGoogle Scholar - 21.V.D. Ivashchuk, A.A. Kobtsev, Stable exponential cosmological solutions with \(m\)- and \(l\)-dimensional factor spaces in the Einstein–Gauss–Bonnet model with a \(\Lambda \)-term. Gen. Relativ. Gravit.
**50**, 119 (2018). arXiv:1712.09703v4 - 22.V.D. Ivashchuk, A.A. Kobtsev, Exact exponential cosmological solutions with two factor spaces of dimension \(m\) in EGB model with a \(\Lambda \)-term. Int. J. Geom. Methods Mod. Phys.
**16**(2), 1950025 (2019)ADSMathSciNetCrossRefGoogle Scholar

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