We perform a systematic analysis of nonminimally coupled cosmologies in (n+1)-dimensional homogeneous and isotropic spacetimes, searching for Nöther’s symmetries and generalizing the results of our previous works. We obtain (i) the absence of symmetries when the spatial curvature constant k is nonzero and n=2, 3, but their existence for all the other n; (ii) the existence of such symmetries for every number of spatial dimensions (except n=1) when k=0. In this latter case, we are able to find a general transformation through which we recover the string-dilaton effective action in (n+1) dimensions and the major peculiarity of string cosmology: the scale factor duality. Furthermore, the symmetry fixes a relation among the coupling F(ϕ), the potential V(ϕ) of the scalar field ϕ, the number of spatial dimensions and the spatial curvature constant. When this is the case, it is possible to find a constant of motion and then get the general solution of the dynamics. Finally, in the framework of the so-called Induced Gravity Theories, we are able to obtain the Newton constant at the present time (t→∞) depending on the number of spatial dimensions and directly related to the constant of motion existing in such a model.
Effective action and phase transitions of scalar field on the fuzzy sphere
