Exact $(1 + 3 + 6)$-dimensional cosmological-type solutions in gravitational model with Yang-Mills field, Gauss-Bonnet term and $\Lambda$-term

TL;DR

This paper derives exact 10-dimensional cosmological solutions in a gravitational model with Yang-Mills, Gauss-Bonnet, and cosmological constant terms, analyzing their stability and static counterparts.

Contribution

It presents new exact exponential solutions in a 10D gravitational-Yang-Mills-Gauss-Bonnet model with specific product manifolds, including stability analysis.

Findings

Exact exponential cosmological solutions with two Hubble-like parameters.

Static solutions with similar properties.

Identification of stability regions for these solutions.

Abstract

We consider $10$-dimensional gravitational model with $SO(6)$ Yang-Mills field, Gauss-Bonnet term and $\Lambda$-term. We study so-called cosmological type solutions defined on product manifold $M = R \times R^3 \times K$, where $K$ is $6d$ Calabi-Yau manifold. By putting the gauge field 1-form to be coinciding with 1-form spin connection on $K$, we obtain exact cosmological solutions with exponential dependence of scale factors (upon $t$-variable), governed by two non-coinciding Hubble-like parameters: $H >0$, $h$, obeying $ H + 2 h \neq 0$. We also present static analogs of these cosmological solutions (for $H \neq 0$, $h \neq H$ and $ H + 2 h \neq 0$). The islands of stability for both classes of solutions are outlined.

Equations

$$g^{(10)}=-wd\chi \otimes d\chi +a_3^2(\chi )g^{(3)}+a_6^2(\chi )g^{(6)}.$$

$$S=\frac{1}{2{\kappa }^2}\int d^{10}x\sqrt{-g}\{R+\alpha [\mathrm{tr}{F}_{MN}{F}^{MN}$$

$$+R_{MNPQ}{R}^{MNPQ}-4R_{MN}{R}^{MN}+R^2]\},$$

$$\displaystyle R_{MN}-\frac{1}{2}g_{MN}R=-2\alpha\biggl{(}{\rm tr}F_{MP}F_{N}^{\ \ P}-\frac{1}{4}g_{MN}\ {\rm tr}F_{PQ}F^{PQ}\biggr{)}\qquad\qquad$$

$$\displaystyle+\alpha\biggl{[}\frac{1}{2}g_{MN}(R_{MNPQ}R^{MNPQ}-4R_{MN}R^{MN}+R^{2})-2RR_{MN}$$

$$\displaystyle+4R_{MP}R_{N}^{\ \ P}+4R_{MPNQ}R^{PQ}-2R_{M}^{\ \ PQS}R_{NPQS}\biggr{]},$$

$$D_M{F}^{MP}=0.$$

$$M=\mathbb{R}\times {\mathbb{R}}^3\times K,$$

$$g^{(10)}=-dt\otimes dt+a_3^2(t)g^{(3)}+a_6^2(t)g^{(6)},$$

$$A={\omega }^{(6)}.$$

$$g_{mn}^{(6)}=e_m^a{e}_n^b{\delta }_{ab},$$

$${\omega }_m^{(6)}=\mid \mid {\omega }_{bm}^{(6)a}\mid \mid \equiv \mid \mid e_n^a{\nabla }_m^{(6)}{e}_b^n\mid \mid \subset so(6),$$

$$F={\Omega }^{(6)},$$

$$D_m({\omega }^{(6)}){\Omega }^{(6)mn}=0,$$

$${\nabla }_m^{(6)}{R}^{(6)mnpq}=0,$$

$$H(t)\equiv {\dot{a}}_3/a_3,h(t)\equiv {\dot{a}}_6/a_6,$$

$$B_0+2\Lambda -\alpha B_1=0,$$

$$\frac{d{L}_H}{dt}+(3H+6h)L_H-L_0=0,$$

$$\frac{d{L}_h}{dt}+(3H+6h)L_h-L_0=0.$$

$$L_0=B_0-2\Lambda -\frac{1}{3}\alpha B_1,$$

$$L_H=2B_H-\frac{4}{3}\alpha A_H,$$

$$L_h=2B_h-\frac{4}{3}\alpha A_h,$$

$$B_0=3H^2+6h^2-(3H+6h{)}^2,$$

$$B_H=H-(3H+6h),$$

$$B_h=h-(3H+6h),$$

$$B_1=144H^{3}h+1080H^2{h}^2+1440H{h}^3+360h^4$$

$$A_H=36H^{2}h+180H{h}^2+120h^3,$$

$$A_h=6H^3+90H^{2}h+180H{h}^2+60h^3.$$

$$R_{pqmn}^{(6)}{R}^{(6)q{p}^{\prime }mn}=g^{(6)m{p}^{\prime }}{g}^{(6)nq}\mathrm{tr}{F}_{mn}{F}_{pq},$$

$$R_{qmn}^{(6)p}=e_a^p{e}_q^b{\Omega }_{bmn}^{(6)a}.$$

$$H(t)=H=\mathrm{const},h(t)=h=\mathrm{const}.$$

$$a_3(t)=\exp (Ht),a_6(t)=\exp (ht).$$

$$B_0+2\Lambda -\alpha B_1=0,$$

$$(3H+6h)L_H-L_0=0,$$

$$(3H+6h)L_h-L_0=0,$$

$$H>0.$$

$$3H+6h\ne 0,H\ne h,$$

Full text

Key words: cosmology; Gauss-Bonnet; Calabi-Yau; Yang-Mills; stability; exact solutions

1 Introduction

Here we deal with a so-called Einstein-Gauss-Bonnet-Yang-Mills-$\Lambda$ gravitational model in dimension $D=10$. The action of the model contains scalar curvature, Gauss-Bonnet, cosmological term ($\Lambda$-term) and Yang-Mills term with a value in $so(6)$ Lie algebra. The model includes non-zero constant $\alpha$, coupled to the sum of Yang-Mills and Gauss-Bonnet terms. The equations of motion for this model are of second order (like it takes place in General Relativity). The so-called Gauss-Bonnet term has appeared in (super)string theory as a second order correction in curvature to the effective (super)string effective action [1, 2, 3] for a heterotic string [4].

At present, Einstein-Gauss-Bonnet (EGB) gravitational models, e.g. with cosmological term and extra matter fields and its modifications [5]-[26], are under intensive studyies in astrophysics and cosmology. The main goal in these studies is a solution of dark energy problem. One can study such models for possible explanation of accelerating expansion of the Universe, which was supported by supernovae (type Ia) observational data [27, 28].

We note that at present there exist several modifications of Einstein and EGB actions which correspond to $F(R)$, $R+f({\cal G})$, $f({R,\cal G})$, $f({R,\cal G})$ , $f({R,\cal G},T_{..}T^{..})$ theories (e.g. for $D=4$), where $R$ is scalar curvature and ${\cal G}$ is Gauss-Bonnet term. These modifications are under intensive studying devoted to cosmological, astrophysical and other applications, see [29]-[35] and references therein.

Another point of interest is a search of possible local manifestation of dark energy related to wormholes, black holes, etc. The most important results for black holes in models with Gauss-Bonnet term are related with the Boulware-Deser-Wheeler solution [36, 37] and its generalizations [38, 39, 40, 41], see also Refs. [42, 43, 44] and references therein. For certain applications of brane-world models with Gauss-Bonnet term, see Refs. [45, 46] and related bibliography. For wormholes solutions in Einstein-Gauss-Bonnet models with certain fields see Refs. [47, 48] and references threin.

In this article we deal with the so-called cosmological type solutions with the $10d$ metric

[TABLE]

defined on product manifold ${\mathbb{R}}\times{\mathbb{R}}^{3}\times K,$ where $w=\pm 1$, ${\mathbb{R}}^{3}$ is flat $3d$ manifold (“our” space) with the metric $g^{(3)}$ and $K$ is $6d$ Ricci-flat Calabi-Yau manifold (internal space) of $SU(3)$ holonomy group with the metric $g^{(6)}$. The warped product model is governed by two scale factors, depending upon one variable $\chi$. It is the synchronous time variable for cosmological case $w=1$: $\chi=t$, while it coincides with space-like variable for $w=-1$: $\chi=u$. The presence of Yang-Mills field makes this ansatz consistent if we choose the Lie algebra for Yang-Mills field to be equal (at least) to $so(6)$, which contains $su(3)$ subalgebra, corresponding to $SU(3)$ group of golonomy of $6d$ Calabi-Yau manifold. For Yang-Mills field we consider the following ansatz: we put here the gauge field 1-form to be equal to spin connection 1-form on $K$ (see Section 2): $A=\omega^{(6)}$. In such ansatz the gauge field plays a role of compensator which “waves out” the terms with non-zero Riemann tensor of Calabi-Yau metric $g^{(6)}$.

Originally such idea of compensation was used by Wu and Wang [49] (see also [50]) in $(1+3+6)$-dimensional cosmological model based on $10d$ Yang-Mills ($SO(32)$- and/or $E_{8}\times E_{8}$-) supergravity theory “upgrated” by additions of Chern-Simons and Gauss-Bonnet terms (of superstring origin). The work of Wu and Wang was influenced greatly by the well-known paper of Candelas et al. [51] devoted to vacuum configurations in ten-dimensional $O(32)$ and $E_{8}\times E_{8}$ supergravity and superstring theory that have unbroken $N=1$ supersymmetry in four dimensions.

It should be noted that compactifications of $11$-dimensional supergravity on ($6d$) Calabi-Yau manifold were considered in Refs. [52, 53]. Moreover, $4d$ and $6d$ Calabi-Yau manifolds also appeared in partially sypersymmetric solutions of $D=11$ supergravity with $M$-branes, see Refs. [54, 55, 56] and and references threin.

In Section 3 we obtain exact cosmological solutions with exponential dependence of scale factors (upon $t$-variable), governed by two non-coinciding Hubble-like parameters: $H>0$ and $h$, corresponding to factor spaces of dimensions $3$ and $6$, respectively, when the following restriction $3H+6h\neq 0$ is used (excluding the solutions with constant volume factor).

In Section 4 we obtain static solutions ($w=-1$) for non-coinciding Hubble parameters $H\neq 0$, $h$, which obey $3H+6h\neq 0$. We also study stability (in certain restricted sence) of the obtained solutions in cosmological case for $t\to+\infty$ (see Section 3) and in static case for $u\to\pm\infty$ (see Section 4) by using results of Ref. [22] (see also approach of Ref. [19]) and single out the subclasses of stable/non-stable solutions.

2 The $10$-dimensional model

2.1 The action and equations of motion

We take the action of the model as

[TABLE]

where $\kappa^{2}$ is $10$-dimensional gravitational constant, $\alpha\neq 0$ is constant, $g_{MN}$ are components of the metric, $F_{MN}$ are components of the Yang-Mills field strengths corresponding to $2$-form with the value in the Lie algebrs $so(6)$: $F=\frac{1}{2}F_{MN}dx^{M}\wedge dx^{N}=dA+A\wedge A,$ where $A=A_{M}dx^{M}$ is the $1$-form with the value in $so(6)$ ($F_{MN}=\partial_{M}A_{N}-\partial_{N}A_{M}+[A_{M},A_{N}]$).

The action (2.1) leads us to the following equations of motions:

[TABLE]

[TABLE]

Here we use notation for covariant gauge derivative: $D_{M}=D_{M}(A)=\nabla_{M}+[A_{M},.]$.

2.2 Cosmological ansatz.

Let us consider ten-dimensional manifold

[TABLE]

where $K$ is a $6d$ Calabi-Yau manifold, i.e., a compact 6-dimensional Kähler Ricci-flat manifold with the metric, which has $SU(3)$ holonomy group. For corresponding Lie algebra we have $su(3)\subset so(6)$.

We start with the cosmological case, i.e. we consider the set of equations (2.2), (2.3) on the manifold (2.4) with the following ansatz for fields

[TABLE]

Here $g^{(3)}=dx^{1}\otimes dx^{1}+dx^{2}\otimes dx^{2}+dx^{3}\otimes dx^{3}$, i.e. we deal with a flat Euclidean metric on ${\mathbb{R}}^{3}$ and $g^{(6)}=g^{(6)}_{mn}(y)dy^{m}\otimes dy^{n}$ is Calabi-Yau metric on $K$.

By $\omega^{(6)}=\omega^{(6)}_{m}(y)dy^{m}$ we denote the spin connection 1-form on $K$ with the value in the Lie algebra $so(6)$ 111In fact it belongs to subalgebra $su(3)\subset so(6)$. corresponding to the (local) basis of co-vectors $e^{a}_{\ m}$ on $K$ , which diagonalizes the metric $g^{(6)}$:

[TABLE]

[TABLE]

where covariant derivative $\nabla^{(6)}_{m}$ corresponds to the metric $g^{(6)}$ and $e^{n}_{\ b}$ is “inverse” (dual) basis of vector fields obeying $e^{a}_{\ n}e^{n}_{\ b}=\delta^{a}_{b}$.

It follows from (2.6) that

[TABLE]

where $\Omega^{(6)}=d\omega^{(6)}+\omega^{(6)}\wedge\omega^{(6)}$ is the curvature two-form on $K$ with the value in $so(6)$.

The spin connection $\omega^{(6)}$ on $K$ obeys the identity

[TABLE]

where $D_{m}(\omega^{(6)})=\nabla^{(6)}_{m}+[\omega^{(6)}_{m},.\ ]$. The identity (2.10) is equivalent to the following identity for the Riemann tensor on $K$

[TABLE]

which is valid for any Kähler Ricci-flat manifold [57].

The Yang-Mills equations (2.3) are satisfied identically due to eqs. (2.5), (2.6) and (2.10).

Let us denote

[TABLE]

where in this section we denote $\dot{a}\equiv\frac{da}{dt}$.

Then, equations (2.2) for the ansatz (2.5)-(2.6) may be written as follows:

[TABLE]

Here

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Equations (2.13)-(2.15) are obtained from (2.2) using the Ricci flatness of $K$ and the equality for Riemann tensor of internal space $K$ with the metric $g^{(6)}$

[TABLE]

which follows from (2.9) and well-known identity

[TABLE]

3 Cosmological solutions

Here we consider the case when Hubble-like parameters are constant, i.e.

[TABLE]

For scale factors we get the exponential dependence on $t$

[TABLE]

We get a set of polynomial equations

[TABLE]

where polynomials $B_{0},B_{1},L_{0},L_{H},L_{h}$ are defined above.

We set

[TABLE]

This relation is used for a description of an accelerated expansion of the $3$-dimensional subspace (which may describe our Universe). The evolution of the $6$-dimensional internal factor space is described by the Hubble-like parameter $h$.

It follows from Ref. [22, 24] (for a more general splitting scheme see paper by Chirkov, Pavluchenko and Toporensky [18]) that if we consider Hubble-like parameters $H$ and $h$ obeying two restrictions imposed

[TABLE]

we reduce relations (3.3), (3.4), (3.5) to the following set of equations

[TABLE]

[TABLE]

Using equation (3.9), we get

[TABLE]

where

[TABLE]

and

[TABLE]

Due to restrictions (3.7) we have for $x$ from (3.12)

[TABLE]

The relation (3.9) is valid only if

[TABLE]

Substituting relation (3.10) into (3.8) we obtain

[TABLE]

From (3.15) we get

[TABLE]

where $x_{\pm}$ are roots of the quadratic equation ${\cal P}(x)=0$. They obey

[TABLE]

According to eq. (3.13) we get (in this cosmological case)

[TABLE]

and

[TABLE]

The graphical representation of the function $\lambda=f(x)$ is given at Figure 1.

The function $f(x)$ obeys

[TABLE]

and

[TABLE]

It has a maximum at $x_{c}=-\frac{1}{5}$ with the value $f(x_{c})=\lambda_{c}\equiv 3/10$ and an inflection point at $x_{d}=-\frac{1}{2}$ with the value $f(x_{d})=\lambda_{d}\equiv 3/16$. It has also a point of local maximum at $x_{a}\equiv 1$ with the value $f(x_{a})=\lambda_{a}\equiv-3/14$.

Equation (3.16) is equivalent to the following master equation

[TABLE]

For

[TABLE]

the solution to this (fourth order) master equation reads

[TABLE]

where $\varepsilon_{1}=\pm 1$, $\varepsilon_{2}=\pm 1$,

[TABLE]

and

[TABLE]

It follows from Figure 1 that for given parameters $\Lambda$ and $\alpha$, obeying restrictions $\alpha\neq 0$ and (3.24), real solutions in formula (3.25) appear for suitably chosen $\varepsilon_{1}=\pm 1$ and $\varepsilon_{2}=\pm 1$ if

[TABLE]

for $\alpha>0$ and

[TABLE]

for $\alpha<0$.

For example, for $\lambda=\lambda_{c}=3/10$ relation (3.25) gives us $x=x_{c}=-1/5$ if we put $\varepsilon_{1}=\pm 1$ and $\varepsilon_{2}=-1$.

In exceptional case

[TABLE]

we have a cubic master equation (3.23) which has three real roots

[TABLE]

$k=1,2,3$, or, numerically

[TABLE]

Graphical analysis. The graphical representation $\Lambda|\alpha|$ upon $x=h/H$ (in this cosmological case) is presented at Figure 2. In drawing this figure we use the relation

[TABLE]

in agreement with (3.11) and (3.13). It follows from Figure 2 that real solutions take place if

[TABLE]

for $\alpha>0$ and

[TABLE]

for $\alpha<0$. (The point $x_{a}=1$ is excluded from our consideration.)

Stability.

Using results of Ref. [22, 24] we obtain that cosmological solutions under consideration obeying $x=h/H\neq x_{i}$, $i=a,c,d$, where $x_{a}=1$, $x_{c}=-\frac{1}{5}$, $x_{d}=-\frac{1}{2}$, are stable if i) $x>x_{d}=-1/2$ and unstable if ii) $x<x_{d}=-1/2$. 222For isotropic cosmological solutions with $H=h$, see Refs. [17, 22] for generic $\Lambda$ and [14, 15] for $\Lambda=0$.

We note that the the points $x_{a}=1$ and $x_{d}=-1/2$ are excluded from our consideration due to restrictions (3.7) while the point of maximum $x_{c}=-\frac{1}{5}$ is excluded since the analysis of Ref. [22] was based on the equations for perturbations for $\delta H(t)$, $\delta h(t)$ in the linear approximation which can be resolved when $x_{c}\neq-\frac{1}{5}$. In special case $x_{c}=-\frac{1}{5}$ higher order terms in pertubations should be considered.

Let us denote by $n_{s}$ the number of non-special stable solutions. By using Figure 2 we find just graphically for $\alpha>0$

[TABLE]

(here $\lambda_{i}=f(x_{i})$) while for $\alpha<0$ we obtain

[TABLE]

Thus, for $\alpha>0$ and small enough value of $\Lambda$ there exists at least one stable solution with $x\in(x_{-},x_{+})$, while for $\alpha<0$ and big enough value of $\Lambda$ there exists at least one stable solution with $x$ obeying $x>x_{+}$. The solutions with $x<x_{-}$ are unstable.

In cosmological case real solutions corresponding to $\Lambda=0$ exists only if $\alpha>0$. We obtain from (3.25) for $\varepsilon_{1}=\pm 1$, $\varepsilon_{2}=1$, two solutions

[TABLE]

The first cosmological solution (for $x_{-}$) is unstable, while the second one (for $x_{+}$) is stable in agreement with Ref. [25].

Remark. It should be noted that here as in the Ref. [22] we deal with restricted stability problem. We do not consider the general setup for perturbations $\delta g_{MN}(t,x)$ and $\delta A_{M}(t,x)$ but only consider the cosmological perturbations of scale factors $\delta a_{3}(t)$, $\delta a_{6}(t)$ in the framework of our ansatz (2.5), (2.6) with fixed $g^{(10)}$ and $\omega^{(6)}$. Analogous remark should be addressed to our analysis of static solutions in the next section.

Zero variation of $G$. The cosmological solution with $x=0$, or $h=0$, takes place if $\alpha<0$ and

[TABLE]

We get $\Lambda>0$. The scale factor $a_{6}$ is constant in this case and we are led to zero variation of the effective $4d$ gravitational constant (in Jordan frame). This solution is stable. Moreover, we get $H^{2}=-1/(4\alpha)$, which implies for the effective $4$-dimensional cosmological constant $\Lambda_{eff}=3H^{2}=\Lambda$. In general case $\Lambda_{eff}$ is a nontrivial function of $\Lambda$ and $\alpha$ given by (3.10) and generic solution for $x$ from (3.25) (or from (3.35) in special case).

We note that for $\alpha<0$ and for $\Lambda$ from (3.43) there exists another real solution corresponding to certain $x_{*}<x_{-}$, which is unstable.

4 Static analogs of cosmological solutions

Now with deal with the static case by considering the set of equations (2.2), (2.3) on the manifold (2.4) with the following ansatz

[TABLE]

where $u$ is a spatial coordinate and $g^{(3)}=-dt\otimes dt+dx^{1}\otimes dx^{1}+dx^{2}\otimes dx^{2},$ is flat pseudo-Eucleadean metric on ${\mathbb{R}}^{3}$, $g^{(6)}$ is the Calabi-Yau metric on $K$ and $\omega^{(6)}$ is spin connection 1-form on $K$ defined in a previous section.

The Yang-Mills equations are satisfied identically as in the previous case.

Now, we denote

[TABLE]

where in this section we denote $\dot{a}\equiv\frac{da}{du}$.

Then, equations (2.2) in the ansatz (4.1)-(4.2) may be written as follows:

[TABLE]

where

[TABLE]

$B_{0},B_{H},B_{h},B_{1}$ are defined in (2.19), (2.20), (2.21), (2.22), and $A_{H},A_{h}$ are defined in (2.23), (2.24), respectively.

As we see, the equations of motion for “Hubble-like” parameters (4.3) in static case may be obtained from cosmological ones (of Section 3) just by replacement

[TABLE]

The dimensionless parameter $\lambda=\Lambda\alpha$ is invariant under this replacement.

Here we consider the case when “Hubble-like” parameters are constant, i.e.

[TABLE]

or, equivalently,

[TABLE]

We get a set of polynomial equations

[TABLE]

where polynomials $\bar{L}_{0},\bar{L}_{H},\bar{L}_{h}$ are defined by relations (4.7), (4.8), (4.9), respectively.

Here we consider slightly more general case

[TABLE]

As in previous section we impose the conditions (3.7) and reduce relations (4.13), (4.14), (4.15) to the set of two equations [26]

[TABLE]

[TABLE]

Using equation (4.18) and restriction (4.16) we get

[TABLE]

where $\varepsilon_{0}=\pm 1$, and quadratic polynomial ${\cal P}={\cal P}(x)$ ($x=h/H$) is defined in (3.11).

Here we get

[TABLE]

instead of (3.13).

According to eq. (4.20) we get in the static case

[TABLE]

and

[TABLE]

(The real numbers $x_{\pm}$ are defined in (3.17).)

We obtain that the main equations (3.16) and (3.23) for the ratio $x=h/H$ are unchanged in static case.

Thus, for $\lambda\neq-21/80$ and restrictions (3.7), (4.16) imposed, we obtain exact solutions for $H$ and $h$, which are given by formulae (3.25), (4.19) and (4.20). For $\lambda=-21/80$ we should use (3.35) instead of (3.25).

Graphical analysis. The graphical representation of $\Lambda|\alpha|$ upon $x=h/H$ in the static case is presented at Figure 3. Here we use the relation

[TABLE]

in agreement with (4.20). It follows from Figure 3 that real solutions take place if

[TABLE]

for $\alpha<0$ and

[TABLE]

for $\alpha>0$.

In static case real solutions for $\Lambda=0$ exists only if $\alpha<0$.

Stability.

Using results of Ref. [26] we can analyse the stability of static solutions under consideration obeying $x=h/H\neq x_{i}$, $i=a,c,d$, where $x_{a}=1$, $x_{c}=-\frac{1}{5}$, $x_{d}=-\frac{1}{2}$.

The solutions are stable for $H>0$ and $u\to+\infty$ if i) $x>x_{d}=-1/2$ and unstable if ii) $x<x_{d}=-1/2$. For $H>0$ and $u\to-\infty$ they stable for i) $x<x_{d}=-1/2$ and unstable if ii) $x>x_{d}=-1/2$.

The solutions are stable for $H<0$ and $u\to+\infty$ if i) $x<x_{d}=-1/2$ and unstable if ii) $x>x_{d}=-1/2$. For $H<0$ and $u\to-\infty$ they stable for i) $x>x_{d}=-1/2$ and unstable if ii) $x<x_{d}=-1/2$.

5 Conclusions

Here we have considered Einstein-Gauss-Bonnet-Yang-Mills-$\Lambda$ gravitational model in dimension $D=10$ with non-zero constant $\alpha$ coupled to a sum of Yang-Mills and Gauss-Bonnet terms.

We have studied so-called cosmological type solutions with the metrics (1.1) defined on product manifolds $M={\mathbb{R}}\times{\mathbb{R}}^{3}\times K$, where $w=\pm 1$, ${\mathbb{R}}^{3}$ is flat $3d$ subspace with the metric $g^{(3)}$, and $K$ is $6d$ Ricci-flat Calabi-Yau manifold with the metric $g^{(6)}$. The gauge field 1-form was considered to be coinciding with spin connection 1-form on $K$: $A=\omega^{(6)}$.

For $w=+1$, $\chi=t$, we have obtained exact cosmological solutions with exponential dependence of scale factors (upon $t$-variable), governed by two non-coinciding Hubble-like parameters: $H>0$, $h$, corresponding to factor spaces of dimensions $3$ and $6$, respectively, when the following restriction: $3H+6h\neq 0$ is used (excluding the solutions with constant volume factor).

Static analogs of cosmological solutions ($w=-1$, $\chi=u$ ) with exponential dependence of scale factors and non-coinciding “Hubble-like” parameters $H\neq 0$ and $h$, obeying $3H+6h\neq 0$, are also presented here.

We have also outlined the stability of the solutions in cosmological case (for $t\to+\infty$, Section 3) and in static case for ($u\to\pm\infty$, Section 4) and have singled out “islands” of stable/non-stable solutions.

Some cosmological applications of the model ($w=1$) may be of interest in context of dark energy problem and problems of stability/variation of gravitational constant. For static case ($w=-1$) possible applications of the obtained solutions may be a subject of a further research, aimed at a search of topological black hole solutions (with flat horizon) or wormhole solutions which are coinciding asymptotically (for $u\to\pm\infty$) with our solutions.

Acknowledgments

This research was funded by RUDN University, scientific project number FSSF-2023-0003.