Eur. Phys. J. C (2019) 79:824
https://doi.org/10.1140/epjc/s10052-019-7329-8
Regular Article - Theoretical Physics
Exponential cosmological solutions with two factor spaces in EGB
model with = 0 revisited
V. D. Ivashchuk
1,2,a
,A.A.Kobtsev
3
1
Institute of Gravitation and Cosmology, Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya Street,
Moscow 117198, Russian Federation
2
Center for Gravitation and Fundamental Metrology, VNIIMS, 46 Ozyornaya Street, Moscow 119361, Russian Federation
3
Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Troitsk 142190, Russian Federation
Received: 12 April 2019 / Accepted: 20 September 2019 / Published online: 8 October 2019
© The Author(s) 2019
Abstract We study exact cosmological solutions in D-
dimensional Einstein–Gauss–Bonnet model (with zero cos-
mological term) governed by two non-zero constants: α
1
and
α
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 = H and mH + lh = 0. We show that for
α = α
2
/α
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 α<0 the (real)
solutions are absent. We prove that the cosmological solu-
tion corresponding to (H
2
, h
2
) is stable in a class of cosmo-
logical 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 [1–12] and Refs. therein, are under inten-
sive studies in cosmology, aimed at explanation of accel-
erating expansion of the Universe [13,14]. Here we study
the EGB model with zero cosmological term in D dimen-
sions (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) [15–17].
The model is governed by two nonzero constants α
1
and α
2
which correspond to Einstein and Gauss–Bonnet terms in
the action, respectively. In this paper we continue our stud-
a
e-mail: ivashchuk@mail.ru
ies 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 syn-
chronous time variable): a
1
(t) = ··· = a
m
(t) = exp(Ht)
and a
m+1
(t) =···=a
m+l
(t) = exp(ht ), where n = m +l,
m > 2, l > 2. We put here H > 0 in order to describe expo-
nential accelerated expansion of 3d subspace with Hubble
parameter H [18].
In contrary to our earlier publication [8], where a lot
of numerical solutions with small enough value of varia-
tion of the effective gravitational constant G were found,
here we put our attention mainly to the search of analyti-
cal exponential solutions with two factor spaces of dimen-
sions m and l. Here we show that the anisotropic cosmo-
logical solutions under consideration with two Hubble-like
parameters H > 0 and h obeying restrictions h = H ,
mH + lh = 0 do exist only if α = α
2
/α
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].
123
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