Classical solutions from quantum regime for barotropic FRW model

It is a general belief that the existence of an initial singularity can be remove through the employment of a quantum theory of gravity. However, there is no consistent theory of gravity until now, and in this sense the problem of the initial singularity remains of actuality. It is well known that is possible to construct a quantum model for the universe as a whole, through the Wheeler-DeWitt (WDW) equation, based in the ADM decomposition of the gravity sector, which leads to a hamiltonian approach of general relativity, from which a canonical quantization procedure can be applied (Halliwel, 1991). Moreover, in the hamiltonian formalism, the notion of time is lost (Isham, 1992), but there are some proposals by which this notion of time can be recovered (Schutz, 1970, 1971) coupling of the gravity sector to a perfect fluid. In this scheme, called Schutz’s formalism, a quantization procedure is possible, and the canonical momentum associated with the perfect fluid appear linearly in the WDW equation, permiting to rewrite this equation in the form a Schrödinger equation with a time coordinate associated with the matter field. Recently, a convincing quantum gravity origin for inflation should however be proved. According to Bojowald (Bojowald, 2002) a non-perturbative approach would produce the most reliable answer as to whether or not inflation can be derived from quantum gravity. At the less ambitious level of minisuperspace quantum cosmology the generality of inflation as well as the quantum creation of the universe should be treated by analyzing the WheelerDeWitt equation and its physical solutions, the so-called wave functions of the universe (Gibbons and Grishchuk, 1989). In this way, the behaviour of the scale factor may be determined in two different forms from the quantum regime: when the wave function is time dependent, we can calculate the expectation value of the scale factor, in the spirit of the many worlds interpretation of quantum mechanics ( Tipler, 1986)