We study exact cosmological solutions in -dimensional Einstein-Gauss-Bonnet model (with zero cosmological term) governed by two non-zero constants: and . We deal with exponential dependence (in time) of two scale factors governed by Hubble-like parameters and , which correspond to factor spaces of dimensions and , respectively, and . We put and . We show that for there are two (real) solutions with two sets of Hubble-like parameters: and , which obey: , while for the (real) solutions are absent. We prove that the cosmological solution corresponding to is stable in a class of cosmological solutions with diagonal metrics, while the solution corresponding to is unstable. We present several examples of analytical solutions, e.g. stable ones with small enough variation of the effective gravitational constant , for .
Exponential cosmological solutions with two factor spaces in EGB model with revisited
V. D. Ivashchuk and A. A. Kobtsev
Institute of Gravitation and Cosmology, Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St., Moscow 117198, Russian Federation,
Center for Gravitation and Fundamental Metrology, VNIIMS, 46 Ozyornaya St., Moscow 119361, Russian Federation,
Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Troitsk, 142190, Russian Federation
Currently, the Einstein-Gauss-Bonnet (EGB) model and related theories, see - 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 dimensions (). 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) -. The model is governed by two nonzero constants and 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. . We deal with diagonal metrics governed by scale factors and consider the following ansatz for scale factors ( is synchronous time variable): and , where , , . We put here in order to describe exponential accelerated expansion of subspace with Hubble parameter .
In contrary to our earlier publication , where a lot of numerical solutions with small enough value of variation of the effective gravitational constant were found, here we put our attention mainly to the search of analytical exponential solutions with two factor spaces of dimensions and . Here we show that the anisotropic cosmological solutions under consideration with two Hubble-like parameters and obeying restrictions , do exist only if . In this case we have two solutions with Hubble-like parameters: and , respectively, such that . By using results of refs. [10, 11] (see also approach of ref. ) we show that the solutions with Hubble-like parameters are stable (in a class of cosmological solutions with diagonal metrics), while those corresponding to are unstable.
Here we also present examples of analytical solutions for: i) ; ii) , ; iii) , ; iv) , ; v) , and vi) , . It should be noted that analytical solutions in cases iii) and iv) were considered numerically in ref.  in a context of solutions with a small (enough) variation of (in Jordan frame, see ref. ), e.g. obeying the most severe restrictions on variation of from ref. . The stable solutions with zero variation of in cases v) and vi) were found earlier in , while the stability of these solutions was proved in ref. .
2 The set up
We start with the following action of the model
Here is the metric defined on the manifold , , , is the cosmological term, is scalar curvature,
is the Gauss-Bonnet term and , are nonzero constants.
We deal with warped product manifold
with the (cosmological) metric
where are one-dimensional manifolds (either or ) and .
Here we put
, where and are constants.
3 Solutions governed by two Hubble-like parameters
where is the Hubble-like parameter corresponding to an -dimensional factor space with , while is the Hubble-like parameter corresponding to an -dimensional factor space, . The splitting in (3.1) was done just for cosmological applications. Here we split the -dimensional factor space into the product of subspace (“our” space) and -dimensional subspace, which is a part of -dimensional “internal” space.
Keeping in mind a possible description of an accelerated expansion of a subspace, we impose the following restriction
Due to ansatz (3.1), the -dimensional subspace is expanding with the Hubble parameter . The behaviour of scale factor corresponding to -dimensional subspace is governed by Hubble-like parameter .
Relation (3.5) implies for and :
We rewrite (3.3) as follows
The relation (3.9) lead us to inequality
Here the following identity is valid
where are roots of the quadratic equation , obeying
Using (3.9) we get
For the following relation is valid
Equation (3.12) may be rewritten in the following form
This equation is of fourth order in for any . One can solve the equation (3.23) in radicals for any and . The general solution is not presented here (it has a rather cumbersome form).
Here we use the following proposition from ref. .
Proposition 1 . For ,
as , where and hence
In what follows we use the relations for the extremum points of the function () from :
which follow from the identity 
Here and the points belong to the interval for all and . The location of the point depends upon and :
The values , , were calculated in . They obey
First, we consider the case and .
For in cases and we have two points of local maximum and one point of local minimum among and , see Figure 1, while in cases and we have one point of local maximum and one point of inflection, see Figure 2. Due to relations (3.30), (3.31) the function is monotonically increasing in the interval , and it is monotonically decreasing in the interval .
Now, let us consider the case . We have: or . Due to to the relations (3.21), (3.30) and Proposition 1, the function is monotonically decreasing in two intervals: i) in the interval from to and ii) in the interval from to . The function is monotonically increasing in the interval from to . Here is a point of local maximum of the function , which is excluded from the solution and .
The functions for , respectively, and are presented at Figure 3.
By using the behaviour of the function , which was considered above, one can readily prove the following proposition.
Proof. First, let us consider the case . In this case it follows from our analysis above that for and . Since , we get in the case : . Hence the equation does not have solutions.
Now we consider the case . We are seeking the solutions to equation in the interval , where our function is smooth (and continuous). Let us denote: and . The interval should be excluded from our consideration since for . (Here we use the fact that the smooth (e.g. continuous) function on the closed interval has a minimum which should be equal to or or a value of the function in a point of local minimum (e.g. point of extremum) of the form , . In any case this minimum coincides with for some .) Now we consider the interval . The function is monotonically increasing in the interval . Due to relation (3.21) there exists a point such that and hence any point in the interval obey . Thus, we exclude the interval from our consideration. Now we consider the interval , where and . Due to intermediate value theorem there exists a point such that . 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 such that . By our definitions above we obtain . 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 and obeying restrictions (3.3) do exist only if . In this case we have two solutions with Hubble-like parameters: and such that .
4 Stability analysis and variation of
Now, we consider the stability of cosmological solutions in a class of solutions with the metric (2.3)
In ref.  we have proved the following proposition, which is valid for exponential solutions with two factor spaces and Hubble-like parameters obeying (3.2) and (3.3) in the EGB model with a -term:
Here it should be noted that our anisotropic solutions with non-static volume factor are not defined for and . Meanwhile, they are defined when or , if .
Proposition 4. The cosmological solution under consideration for corresponding to the big root of master equation is stable, while the solution corresponding to the small root is unstable.
Here we analyze the solutions by using the restriction on variation of the effective gravitational constant (in the Jordan frame), which is inversely proportional to the volume scale factor of the (anisotropic) internal space (see  and references therein), i.e.
By using (4.2) we get
Here we use, as in ref. , the following bounds on the value of the dimensionless variation of the effective gravitational constant:
They come from the most stringent bounds on -dot (by the set of ephemerides)  , which are allowed at 95% confidence (2-) level, and the value of the Hubble parameter (at present)  , with 95% confidence level.
Let us consider the solution with -parameter corresponding to dimensionless parameter of variation of from (4.3). Then, we have
Let us consider a solution with a small enough parameter , which satisfies restrictions (4.4). It obeys (4.7) and hence we obtain from (4.6) since , while . Thus, this solution is stable due to Proposition 4. Hence, all solutions with small enough variation of , which were obtained in ref. , are stable. The stability two of them was proved in ref. .
Remark. It follows from our consideration that a more wide class of solutions with 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 and relations (3.6), (3.7), (3.8).
5.1 The solutions for
, . In our notations and .
For we get:
see , and
5.2 The solution for and
For the case , the master equation (3.22) has two real solutions
. In our notations and . (Approximate values are following ones: and .)
5.3 The series of solutions for and
Now we consider the case , . The master equation (3.22) in this case reads
It has two real solutions for any
Now, we study the behaviour of solutions and for big values of . By using -decomposition we get
for . These relations just follow from the formulae
as , where
The solutions give us unstable cosmological soutions (as ), while lead us to stable ones.
Let us consider the second series of solutions. Here, one can obtain more subtle relation instead of (5.25)
as . This relation implies the following asymptotic formula for the parameter of dimensionless variation of the effective gravitational constant in Jordan frame (see (4.2))
as . Thus, we get
5.4 The solutions for and
Let us consider the case and . We get
where . Approximate numerical values for and read
The cosmological solution corresponding to is stable and gives the -parameter (from (4.2))
5.5 The solutions for and
For and we get two solutions. The first solution to the master equation, corresponging to unstable cosmological solution, reads
or numerically, . The second one was obtained in ref. :
It gives a zero variation of the effective gravitational constant in Jordan frame, i.e. . The stability of the corresponding cosmological solution was proved earlier in .
5.6 The solutions for and
We have considered the -dimensional Einstein-Gauss-Bonnet (EGB) model with two non-zero constants and . 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 and , corresponding to submanifolds of dimensions and , respectively, with . The equations of motion were reduced to the master equation (see (3.14) or (3.23)), where the parameter obeys the restrictions: , and () are defined in (3.17). By using our earlier analysis from ref.  we have proved that the master equation has real solutions only for . In this case there are two solutions: , , which satisfy
The master equation may be solved in radicals, since it is equivalent to a polynomial equation of fourth order (for ).
Any cosmological solution corresponding to or (for ) describes an exponential expansion of 3-dimensional subspace (“our” space) with the Hubble parameter and anisotropic behaviour of -dimensional internal space: expanding in dimensions (with Hubble parameter ) and contracting in dimensions (with Hubble-like parameter ).
By using our earlier results from ref.  we have proved that the solution corresponding to is stable in a class of cosmological solutions with diagonal metrics, while the solution corresponding to is unstable.
We have presented several examples of exact solutions (in terms of ) in the following cases: i) ; ii) , ; iii) , ; iv) , ; v) , and vi) , . In case iii) we have also proved the asymptotical relation for variation of : , as , which is valid for stable solutions.
The publication was financially supported by the Ministry of Education and Science of the Russian Federation (the Agreement number 02.a03.0008). It was also partially supported by the Russian Foundation for Basic Research, grant Nr. 19-02-00346.
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