Noether symmetries in cosmology

We study Einstein equations for a homogeneous and isotropic metric coupled with a scalar field phi, firstly introduced for allowing G variability and now widely used in connection with the inflationary paradigm. The still unsolved problem of choosing the coupling function F(phi) between gravity and field and the potential V(phi) for the scalar field is considered. Our approach looks for such functions which allow exact integration of the equations or, at least, more treatable ones, giving us the possibility of a more precise analysis of all the physical situations in which the knowledge of the background plays an important role. We look for Noether symmetries of the Lagrangian, the cosmological Einstein equations being second-order differential equations derivable from it. The existence of infinitesimal transformations leaving the Lagrangian invariant imply very strong conditions for V(phi) and F(phi), and it is possible in most cases to derive exactly their forms, which are usually physically interesting. Many are the topics investigated by means of such a method. After dealing with quadratic and quartic potentials, attention is focused to nonflat and or multidimensional cosmology. The invariance properties of string-dilaton-like Lagrangian are examined, and some considerations are made on quantum cosmology. Finally, applications to the large scale structures of the universe are considered.