1. Introduction
gr-qc/yymmnnnGrav. & Cosmol.13, 137 (2007)

AN -BRANE SOLUTION WITH ACCELERATION
AND SMALL ENOUGH VARIATION OF

J.-M. Alimi, V.D. Ivashchuk and V.N. Melnikov

Laboratoire de l’Univers et de ses Theories CNRS UMR8102, Observatoire de Paris 92195, Meudon Cedex, France

Centre for Gravitation and Fundamental Metrology, VNIIMS, 46 Ozyornaya St., Moscow 119361, Russia

Institute of Gravitation and Cosmology, Peoples’ Friendship University of Russia, 6 Miklukho-Maklaya St., Moscow 117198, Russia

An -brane solution with two non-composite electric branes and a set of scalar fields is considered. The intersection rule for branes corresponds to the Lie algebra . The solution contains five factor spaces with the fifth one interpreted as “our” 3-dimensional space. It is shown that there exists a time interval where accelerating expansion of “our” 3-dimensional space is compatible with small enough value of effective gravitational “constant” variation.

11footnotetext: e-mail: jean-michel.alimi@obspm.fr22footnotetext: e-mail: ivas@vniims.ru33footnotetext: e-mail: melnikov@phys.msu.ru

1. Introduction

As is well known [1, 2, 3, 4, 5, 6, 7], cosmological models in scalar-tensor and multidimensional theories are a framework for describing possible time variations of fundamental physical constants due to scalar fields which are present explicitly in STT or are generated by extra dimensions in multidimensional theories. In [8], we have obtained solutions for a system of conformal scalar and gravitational fields in 4D and calculated the presently possible relative variation of at the level of less than .

Later, in the framework of a multidimensional model with a perfect fluid and two factor spaces (our 3D space of Friedmann open, closed and flat models and an internal 6D Ricci-flat space) we have obtained the same limit for such variation of [9].

We have also estimated the possible variations of the gravitational constant in the framework of a generalized (Bergmann-Wagoner-Nordtvedt) scalar-tensor theory of gravity on the basis of field equations, without using their special solutions. Specific estimates were essentially related to values of other cosmological parameters (the Hubble and acceleration parameters, the dark matter density etc.), but the values of compatible with modern observations did not exceeded per year [10].

In [11], we continued the studies of models in arbitrary dimensions and obtained relations for in a multidimensional model with a Ricci-flat internal space and a multicomponent perfect fluid. A two-component example, dust + 5-brane, was considered explicitly. It was shown that is smaller than . Expressions for were also considered in a multidimensional model with an Einstein internal space and a multicomponent perfect fluid [12]. In the case of two factor spaces with non-zero curvatures without matter, a mechanism for prediction of small was suggested. The result was compared with our exact (1+3+6)-dimensional solutions obtained earlier.

A multidimensional cosmological model describing the dynamics of Ricci-flat factor spaces in the presence of a one-component anisotropic fluid was considered in [13]. The pressures in all spaces were supposed to be proportional to the density: , . Solutions with accelerated power-law expansion of our 3-space and a small enough variation of were found. These solutions exist for two branches of the parameter . The first branch describes superstiff matter with , the second one may contain phantom matter with , e.g., when grows with time.

Similar exact solutions, but nonsingular ones and with an exponential behaviour of the scale factors, were considered in [14] for the same multidimensional cosmological model describing the dynamics of Ricci-flat factor spaces in the presence of a one-component perfect fluid. Solutions with accelerated exponential expansion of our 3-space and small variation of were also found.

Here we continue our investigations of in multidimensional cosmological models. The main problem is to find an interval of the synchronous time where the scale factor of our 3D space exhibits an accelerated expansion according to the observational data [15, 16] while the relative variation of the effective 4-dimensional gravitational constant is small enough as compared with the Hubble parameter, see [17, 18, 12, 19, 20] and references therein.

As we have already mentioned, in the model with two non-zero curvatures [12] there exists an interval of where accelerated expansion of “our” 3-dimensional space co-exists with a small enough value of . In this paper we suggest an analogous mechanism for a model with two form fields and several scalar fields (e.g., phantom ones).

2. The model

We here deal with -brane solutions describing two electric branes and a set of scalar fields.

The model is governed by the action

(2.1)

Here is the -dimensional metric of pseudo-Euclidean signature , is a form of rank , is a non-degenerate symmetric matrix, is a vector of scalar fields, is a linear function, with and , and .

We consider the manifold

(2.2)

where are oriented Riemannian Ricci-flat spaces of dimensions , , and .

Let two electric branes be defined by the sets and . They intersect on . The first brane also covers while the second one covers . The first brane corresponds to the form and the second one to the form .

For the world-volume dimensions of branes we get

(2.3)

, and

(2.4)

is the brane intersection dimension.

We consider an -brane solution governed by the function

(2.5)

where is a time variable,

(2.6)

and

(2.7)

, is supposed to be non zero. Here . Thus .

The intersection rule is as follows:

(2.8)

This relation corresponds to the Lie algebra [21, 22]. Recall that , , where the “electric” vectors and the scalar products were defined in [23, 24] (see also [25, 21]). The relations and (2.8) follow just from the formula , where is the Cartan matrix for (with ).

We consider the following exact solution:

(2.9)
(2.10)
(2.11)
(2.12)

where

(2.13)
(2.14)

. Here denotes a volume form on (, ). We remind the reader that all Ricci-flat metrics have Euclidean signatures.

This solution is a special case of a more general solution from [26] corresponding to the Lie algebra . It may also be obtained as a special 1-block case of -brane solutions from [28].

3. Solutions with acceleration

Let us introduce the synchronous time variable by the following relation:

(3.1)

We put , and hence due to (2.6) which implies . Let us consider two intervals of the parameter :

(3.2)
(3.3)

In case (i), the function is monotonically increasing from to , for , where , while in case it is monotonically increasing from 0 to a finite value .

Let the space be “our” 3-dimensional space with the scale factor

(3.4)

For the first branch (i), we get the asymptotic relation

(3.5)

for , where

(3.6)

and, due to (3.2), . For the second branch (ii) we obtain

(3.7)

for , where due to (3.3), see (3.6).

Thus we get an asymptotic accelerated expansion of the 3D factor space in both cases (i) and (ii), and .

Moreover, it may be readily verified that the accelerated expansion takes place for all , i.e.,

(3.8)

Here and in what follows we denote .

Indeed, using the relation (see (3.1)), we get

(3.9)

and

(3.10)

that certainly implies the inequalities in (3.8).

Now we consider the variation of the effective . For our model, the 4-dimensional gravitational “constant” (in Jordan’s frame) is

(3.11)

where

(3.12)

are the scale factors of the “internal” spaces , respectively.

The function has a minimum at the point corresponding to

(3.13)

At this point, is zero. This follows from an explicit relation for the dimensionless variation of ,

(3.14)

where

(3.15)

is the Hubble parameter of our space.

The function monotonically decreases from to for and monotonically increases from to for . Here for the case (i) and for the case (ii).

The scale factors monotonically decrease from 1 to 0 for since the powers and are positive and . The scale factor monotonically increases from zero to for and monotonically decreases from to zero for , where is a point of maximum.

We should consider only solutions with accelerated expansion of our space and small enough variations of the gravitational constant obeying the present experimental constraint

(3.16)

Here, as in the model with two curvatures [12], is restricted to a certain range containing . It follows from (3.14) that in the asymptotical regions (3.5) and (3.7) , which is unacceptable due to the experimental bounds (3.16). This restriction is satisfied for a range containing the point where .

A calculation of in the linear approximation near gives the following approximate relation for the dimensionless parameter of relative variation of :

(3.17)

where (compare with an analogous relation in [12]). This relation gives approximate bounds for values of the time variable allowed by the restriction on . (For another mechanism of obtaining an acceleration in a multidimensional model with a “perfect fluid” see [13, 14]).

It should be stressed that the solution under consideration with , and takes place when the configuration of branes, the matrix and the dilatonic coupling vectors , obey the relations (2.7) and (2.8) with . This is not possible when is positive-definite, since in this case . In the next section we will give an example of a setup obeying (2.7) and (2.8) by introducing “phantom” fields.

4. Example

Let us consider the following example: , i.e. the ranks of forms are equal, and , , i.e. there are two ”phantom” scalar fields. We also put . Due to (2.3), .

Then the relations (2.7) and (2.8) read

(4.1)

and

(4.2)

where . Here we have used that and .

The relations (4.1) and (4.2) are compatible since it may be verified that they imply

(4.3)

i.e., the vectors and , belonging to the Euclidean space , and obeying the relations (4.1) and (4.2), do exist. The left-hand side of Eq. (4.3) gives , where is the angle between these two vectors.

The solution from Sec. 2 for the metric and two phantom fields in this special case reads

(4.4)
(4.5)

where

(4.6)

The relations for the form fields (2.11) and (2.12) remain the same. Recall that , . The relations (4.1) and (4.2) for the scalar products of dilatonic coupling vectors are assumed. Recall that all factor spaces are Ricci-flat and with and . Hence the spaces and are flat.

Here, as in the previous section, the metric (4.) describes an accelerated expansion of the fifth factor space for . The 4D effective gravitational “constant” , as a function of the synchronous time variable, monotonically decreases from to its minimum value for and monotonically increases from to for . Recall that is finite for and for . There exists a time interval containing where the variation of obeys the experimental bound .

5. Conclusions

We have considered an -brane solution with two non-composite electric branes and a set of scalar fields. The solution contains five factor spaces, and the fifth one, , is interpreted as “our” 3D space.

As in the of the model with two non-zero curvatures [12], we have found that there exists a time interval where accelerated expansion of “our” 3-dimensional space co-exists with a small enough value of obeying the experimental bounds.

Other results on , -dot and may be found in [29, 30, 31, 32, 33, 34].

Acknowledgement

This work was supported in part by the Russian Foundation for Basic Research grant Nr. 05-02-17478. VNM is grateful to colleagues from Observatoire de Paris — Meudon for hospitality during his visit in March, 2007.

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