Stable exponential cosmological solutions with two factor spaces in the Einstein–Gauss–Bonnet model with a \(\Lambda \)-term

Abstract

We study D-dimensional Einstein–Gauss–Bonnet gravitational model including the Gauss–Bonnet term and the cosmological term \(\Lambda \). 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 \(m >2\) and \(l > 2\), respectively. These solutions contain a fine-tuned \(\Lambda = \Lambda (x, m, l, \alpha )\), which depends upon the ratio \(h/H = x\), dimensions of factor spaces m and l, and the ratio \(\alpha = \alpha _2/\alpha _1\) of two constants (\(\alpha _2\) and \(\alpha _1\)) of the model. The master equation \(\Lambda (x, m, l,\alpha ) = \Lambda \) is equivalent to a polynomial equation of either fourth or third order and may be solved in radicals. The explicit solution for \(m = l\) is presented in “Appendix”. Imposing certain restrictions on x, we prove the stability of the solutions in a class of cosmological solutions with diagonal metrics. We also consider a subclass of solutions with small enough variation of the effective gravitational constant G and show the stability of all solutions from this subclass.

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Notes

  1. 1.

    The second relation (1.2) was extended in ref. [28] to \(\Lambda |\alpha | \ge |\lambda _a|\) by adding into consideration the case \(H=h\) [16, 22]. In ref. [28] the cosmological constant \(\Lambda _P\) is related to our one as \(\Lambda _P = 2 \Lambda \) and the internal space dimension l is denoted as D.

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Acknowledgements

The publication has been 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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Correspondence to V. D. Ivashchuk.

Appendices

Appendix

The analytical solution for \(m=l\)

For any \(m = l > 2\) the master Eq. (3.33) reads

$$\begin{aligned} A x^4+B x^3 + C x^2 + B x + A = 0, \end{aligned}$$
(A.1)

where

$$\begin{aligned} A= & {} 8 \lambda (m-2)^2(m-1) + m(m+1)(m-2), \end{aligned}$$
(A.2)
$$\begin{aligned} B= & {} 32 \lambda (m-2)(m-1)^2 + 4 m (m-1)^2, \end{aligned}$$
(A.3)
$$\begin{aligned} C= & {} 16 \lambda (m-1)(3m^2-8m+6) + 2m(m-1)(3m -4). \end{aligned}$$
(A.4)

It may be readily solved in radicals, by using the substitution \(y = x + \frac{1}{x}\) [33]. For \(A \ne 0\) we obtain

$$\begin{aligned} \begin{aligned} x= \frac{1}{4A} \left( - B + \nu _1 \sqrt{E - 2 B \nu _2 \sqrt{d}} + \nu _2 \sqrt{d} \right) , \end{aligned} \end{aligned}$$
(A.5)

where \(\nu _1 = \pm \,1\), \(\nu _2 = \pm \,1\) and

$$\begin{aligned} d = 8A^2 - 4 C A + B^2, \qquad E = - 8A^2 - 4CA + 2 B^2. \end{aligned}$$
(A.6)

We get

$$\begin{aligned} d= & {} 16 m^2(2m^2-7m+7) - 128 m(m-1)(m-2) (2m-3) \lambda , \end{aligned}$$
(A.7)
$$\begin{aligned} E= & {} 1024 \lambda ^2(m-2)^2(m-1)^2(2m-3) \nonumber \\&+\,128 \lambda (m-2)(m-1)m(4m-7) \nonumber \\&-\,16 m^2 (2 m^3- 11 m^2+15 m-4). \end{aligned}$$
(A.8)

For \(A = 0\), the solution reads

$$\begin{aligned} \begin{aligned} x= \frac{1}{2B} \left[ - C \pm \sqrt{C^2 -4 B^2 } \right] , \quad \mathrm{or} \ x=0, \end{aligned} \end{aligned}$$
(A.9)

where

$$\begin{aligned} \begin{aligned} B = - 8 m (m-1), \qquad C = - \frac{4m}{m-2} (4 m^2-10m+7). \end{aligned} \end{aligned}$$
(A.10)

The special solution for \(m=3\) was considered recently in ref. [27].

The proof of the Lemma

Here we give the proof of the Lemma from Sect. 2. The calculations (by using Mathematica) lead us to following relations

$$\begin{aligned} {\mathcal {R}}_{\pm }(m,l)={\mathcal {R}}(x_{\pm }(m,l),m,l)=\frac{A(m,l) \pm B(m,l)\sqrt{\Delta (m,l)}}{C(l)} \end{aligned}$$
(B.11)

where

$$\begin{aligned} A(m,l)&=-2(m-1)(l+m-3)A_{*}(m,l),\\ A_{*}(m,l)&=l^2m^2+4lm^2-4m^2+l^3m-4l^2m-8lm+8m-2l^3+8l^2-4l,\\ B(m,l)&=8l(m-1)^2(l+m-3)>0,\\ \Delta (m,l)&=(m-1)(l-1)(l+m-3)>0,\\ C(l)&= (l - 2)^3 (l -1) > 0. \end{aligned}$$

In order to prove \({\mathcal {R}}_{-}(m,l) <0\) it is sufficient to prove that \(A_{*}(m,l) > 0\) for \(m>2\) and \(l>2\).

Let \(m \ge 4\). Then we group \(A_*(m,l)\) as the sum of the non-negative terms:

$$\begin{aligned} A_*(m,l)= & {} (l^2m^2-4l^2m)_1+(4lm^2-4m^2-8lm)_2 \\&+(l^3m-2l^3)_3+(8m)_4+(8l^2-4l)_5; \end{aligned}$$

where

$$\begin{aligned} (.)_1&=l^2m^2-4l^2m=l^2m(m-4)\ge 0, \\ (.)_2&=4lm^2-4m^2-8lm=2(l-2)m^2+2lm(m-4)>0, \\ (.)_3&=l^3(m-2)>0, \\ (.)_4&>0, \\ (.)_5&=4l(2l-1)>0. \end{aligned}$$

Thus, we get \(A_{*}(m,l) > 0\) for \(m \ge 4\) and \(l > 2\). For \(m=3\) we have \(A_*(3,l)=l^3+5l^2+8l-12 \ge 84 \) (as \(l\ge 3\)). Thus, \({\mathcal {R}}_{-}(m,l) < 0\) (\(m>2\), \(l>2\)) is proved.

Now we prove \({\mathcal {R}}_{+}(m,l) < 0\) (\(m>2\), \(l>2\))). By using the identities (3.18), (3.25) and definitions of \({\mathcal {R}}_{\pm }(m,l)\) we obtain

$$\begin{aligned} {\mathcal {R}}_{+}(m,l)= & {} {\mathcal {R}}(x_{+}(m,l),m,l) = (x_{+}(m,l))^4 {\mathcal {R}}\left( \frac{1}{x_{+}(m,l)},l,m\right) \nonumber \\= & {} (x_{+}(m,l))^4 {\mathcal {R}}(x_{-}(l,m),l,m) = (x_{+}(m,l))^4 {\mathcal {R}}_{-}(l,m) < 0.\nonumber \\ \end{aligned}$$
(B.12)

By this we complete the proof of the Lemma.

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Ivashchuk, V.D., Kobtsev, A.A. Stable exponential cosmological solutions with two factor spaces in the Einstein–Gauss–Bonnet model with a \(\Lambda \)-term. Gen Relativ Gravit 50, 119 (2018). https://doi.org/10.1007/s10714-018-2447-4

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Keywords

  • Gauss-Bonnet
  • Cosmology
  • Variation of G