14 M ay 1 99 5 . G R - Q C - 9 50 50 16 Semiclassical Gravity and Large { Scale Structure

The possible cosmological e ects of primordial non{adiabatic uctuations in the matter{gravity quantum system are explored. In particular, both the metric and a scalar matter eld are expanded around their homogeneous values and the corrections induced on the scalar eld uctuation spectrum by the non{adiabatic terms are perturbatively estimated. Finally, results of a preliminary numerical simulation to investigate the e ects on large{scale structure formation are presented. The invariance of the Einstein action under arbitrary space{time transformations has as a consequence that time does not appear in the corresponding Hamiltonian formulation and such a feature is maintained in the canonical quantization of gravity within the superspace approach [1] [2]. It has been observed, however, that the introduction of matter allows one to also introduce the concept of time: time parametrizes how matter follows gravity. In particular, it has been noted that the semiclassical wave function for gravity provides a parametrization for the evolution of matter in which the latter follows the former adiabatically [3] [4]. In order to illustrate the above the matter{gravity wave function is factorized into two parts: one involving only gravitational degrees of freedom and the other involving both the gravitational and the matter degrees of freedom. Correspondingly, the Wheeler{De Witt (WD) equation in which time is absent is split into two pieces: one describing gravitation in an e ective potential given by the mean energy{momentum tensor of matter and the other describing the matter whose evolution is parametrized by time which is derived from the semiclassical approximation to the gravitational wave function [4]. The above is contingent on the hypothesis that the Planck mass is much larger than the mass of any matter eld or any inverse length scale used to describe matter.