On multidimensional cosmological solutions with scalar fields and 2-forms corresponding to rank-3 Lie algebras: acceleration and small variation of G

Abstract

By means of a simple model we investigate the possibility of an accelerated expansion of a 3-dimensional subspace in the presence of the variation of the effective 4-dimensional constant obeying the experimental constraint. Multidimensional cosmological solutions with m 2-form fields and l scalar fields are presented. Solutions corresponding to rank-3 Lie algebras are singled out and discussed. Each of solutions contain two factor spaces: one-dimensional space M_1 and Ricci-flat space M_2. A 3-dimensional subspace of M_2 is interpreted as "our" space. We show that there exists a time interval where accelerated expansion of our 3D space is compatible with a small enough variation of the effective gravitational constant G(\tau). This interval contains \tau_0 which is the point of minimum of G(\tau) (here \tau is the synchronous time variable). Special solutions with three phantom scalar fields are analyzed. It is shown that in the vicinity of the point \tau_0 the time variation of G(\tau) decreases in the sequence of Lie algebras A_3, C_3 and B_3 when the solutions with asymptotically power-law behavior of scale-factors for \tau \to \infty are considered. Exact solutions with asymptotically exponential accelerated expansion of 3D space are also considered.