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 MiklukhoMaklaya St., Moscow 117198, Russia
An brane solution with two noncomposite 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” 3dimensional space. It is shown that there exists a time interval where accelerating expansion of “our” 3dimensional space is compatible with small enough value of effective gravitational “constant” variation.
1. Introduction
As is well known [1, 2, 3, 4, 5, 6, 7], cosmological models in scalartensor 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 Ricciflat 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 (BergmannWagonerNordtvedt) scalartensor 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 Ricciflat internal space and a multicomponent perfect fluid. A twocomponent example, dust + 5brane, 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 nonzero 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 Ricciflat factor spaces in the presence of a onecomponent anisotropic fluid was considered in [13]. The pressures in all spaces were supposed to be proportional to the density: , . Solutions with accelerated powerlaw expansion of our 3space 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 Ricciflat factor spaces in the presence of a onecomponent perfect fluid. Solutions with accelerated exponential expansion of our 3space 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 4dimensional 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 nonzero curvatures [12] there exists an interval of where accelerated expansion of “our” 3dimensional space coexists 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 pseudoEuclidean signature , is a form of rank , is a nondegenerate 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 Ricciflat 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 worldvolume 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 Ricciflat metrics have Euclidean signatures.
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” 3dimensional 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) 
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 4dimensional 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 positivedefinite, 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), .
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 lefthand 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 Ricciflat 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 noncomposite 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 nonzero curvatures [12], we have found that there exists a time interval where accelerated expansion of “our” 3dimensional space coexists with a small enough value of obeying the experimental bounds.
Acknowledgement
This work was supported in part by the Russian Foundation for Basic Research grant Nr. 050217478. VNM is grateful to colleagues from Observatoire de Paris — Meudon for hospitality during his visit in March, 2007.
References

[1]
V.N. Melnikov, “Multidimensional Classical and Quantum Cosmology
and Gravitation. Exact Solutions and Variations of Constants.”
CBPFNF051/93, Rio de Janeiro, 1993;
V.N. Melnikov, in: “Cosmology and Gravitation”, ed. M. Novello, Editions Frontieres, Singapore, 1994, p. 147. 
[2]
V.N. Melnikov, “Multidimensional Cosmology and Gravitation”,
CBPFMO002/95, Rio de Janeiro, 1995, 210 p.;
V.N. Melnikov. In: “Cosmology and Gravitation. II”, ed. M. Novello, Editions Frontieres, Singapore, 1996, p. 465.  [3] V.N. Melnikov, “Exact Solutions in Multidimensional Gravity and Cosmology III.” CBPFMO03/02, Rio de Janeiro, 2002, 297 pp.

[4]
K.P. Staniukovich and V.N. Melnikov, “Hydrodynamics, Fields and
Constants in the Theory of Gravitation”, Energoatomizdat, Moscow,
1983, 256 pp. (in Russian). See English translation of first 5
sections in:
V.N. Melnikov, “Fields and Constraints in the Theory of Gravitation”, CBPF MO02/02, Rio de Janeiro, 2002, 145 pp.  [5] V.N. Melnikov, Int. J. Theor. Phys. 33, 1569 (1994).
 [6] V.N. Melnikov, “Gravity as a Key Problem of the Millennium”. Proc. 2000 NASA/JPL Conference on Fundamental Physics in Microgravity, NASA Document D21522, 2001, p. 4.14.17, Solvang, CA, USA.
 [7] V.D. Ivashchuk and V.N. Melnikov, Nuovo Cim. B 102, 131 (1988).
 [8] N.A. Zaitsev and V.N. Melnikov. In: Problems of Gravitation and Elementary Particle Theory (PGEPT), Energoatomizdat, Moscow, 1979, v. 10, p. 131 (in Russian). See also the English version in [3].
 [9] K.A. Bronnikov, V.D. Ivashchuk and V.N. Melnikov, Nuovo Cim. B 102, 209 (1988).
 [10] K.A. Bronnikov, V.N. Melnikov and M. Novello, “Possible time variations of in scalartensor theories of gravity”, Grav. & Cosmol. 8, Suppl. II, 1821 (2002).
 [11] V.D. Ivashchuk and V.N. Melnikov. “Problems of and multidimensional models”. In: Proc. JGRG11, Eds. J. Koga et al., Waseda Univ., Tokyo, 2002, pp. 405409.
 [12] H. Dehnen, V.D. Ivashchuk, S.A. Kononogov and V.N. Melnikov, “On time variation of in multidimensional models with two curvatures”, Grav. & Cosmol. 11, 340 (2005).
 [13] J.M. Alimi, V.D. Ivashchuk, S.A. Kononogov and V.N. Melnikov, “Multidimensional cosmology with anisotropic fluid: acceleration and variation of , Grav. & Cosmol. 12, 173 (2006); grqc/0611015.
 [14] V.D. Ivashchuk, S.A. Kononogov, V.N. Melnikov and M. Novello, “Nonsingular solutions in multidimensional cosmology with perfect fluid: acceleration and variation of , Grav. & Cosmol. 12, 273 (2006); hepth/0610167.
 [15] A.G. Riess et al, AJ 116, 1009 (1998).
 [16] S. Perlmutter et al, ApJ 517, 565 (1999).
 [17] R. Hellings, Phys. Rev. Lett. 51, 1609 (1983).
 [18] J.O. Dickey et al, Science 265, 482 (1994).
 [19] V. Baukh and A. Zhuk, “brane accelerating cosmologies”, Phys. Rev. D 73, 104016 (2006).
 [20] A.I. Zhuk, “Conventional cosmology from multidimensional models”, hepth/0609126.
 [21] V.D. Ivashchuk and V.N. Melnikov, “Multidimensional classical and quantum cosmology with intersecting branes”, J. Math. Phys. 39, 28662889 (1998); hepth/9708157.
 [22] V.D. Ivashchuk and V.N. Melnikov, “Exact solutions in multidimensional gravity with antisymmetric forms”, topical review, Class. Quantum Grav. 18, R82R157 (2001); hepth/0110274.
 [23] V.D. Ivashchuk and V.N. Melnikov, “Intersecting pbrane solutions in multidimensional gravity and Mtheory”, Grav. & Cosmol. 2, 297 (1996); hepth/9612089.
 [24] V.D. Ivashchuk and V.N. Melnikov. “Generalized intersecting pbrane solutions from sigmamodel”, Phys. Lett. B 403, 23–30 (1997).
 [25] V.D. Ivashchuk and V.N. Melnikov, “Sigmamodel for the generalized composite pbranes”, Class. Quantum Grav. 14, 30013029 (1997); Erratumibid., 15, 39414942 (1998); hepth/9705036.
 [26] I.S. Goncharenko, V.D. Ivashchuk and V.N. Melnikov, mathph/0612079.
 [27] V.D. Ivashchuk, “Composite fluxbranes with general intersections”, Class. Quantum Grav. 19, 30333048 (2002); hepth/0202022.
 [28] V.D. Ivashchuk, “Composite Sbrane solutions related to Todatype systems”, Class. Quantum Grav. 20, 261276 (2003); hepth/0208101.
 [29] V.N. Melnikov. In: “Gravitational Measurements, Fundamental Metrology and Constants”, eds. V. de Sabbata and V.N. Melnikov, Kluwer Academic Publ., Dordtrecht, 1988, p. 283.
 [30] V.N. Melnikov, “Variations of and SEE Project. Proc. Rencontre de Moriond99: “Gravitational Waves and Experimental Gravity”. Editions Frontieres, 1999.
 [31] V.N. Melnikov. “Time variations of in different models”. Proc. 5th Int. Friedmann Seminar, Joao Pessoa, Brazil, Int. J. Mod. Phys., A 17, 43254334 (2002).
 [32] V.N. Melnikov. , SEE and multidimensional cosmology”. In: “Gravitation, Cosmology and Relativistic Astrophysics”, Kharkov Univ., Ukraine, 2004, pp. 2119.
 [33] V.N. Melnikov. “Gravity and cosmology as key problems of the millennium”. In: “Albert Einstein Century Int. Conf.”, eds. J.M. Alimi and A. Fuzfa, AIP Conf. Proc. MelvilleNY, 2006, v. 861, p. 109126.
 [34] V.N. Melnikov. “Integrable cosmological models in DD and variations of fundamental constants”. In: Proc. VII AsiaPacific International Conference on Gravitation and Astrophysics, ICGA7, eds. J.M. Nester, C.M. Chen and J.P. Hsu, World Scientific, 2006, p. 53.