Model universe with variable space dimension: Its dynamics and wave function - art. no. 123512

Assuming the space dimension is not constant, but varies with the expansion of the universe, a Lagrangian formulation of a toy universe model is given . After a critical review of previous works, the field equations are derive d and discussed. It is shown that this generalization of the FRW cosmology is not unique. There is a free parameter in the theory, C, with which we ca n fix the dimension of space, say, at the Planck time. Different possibilit ies for this dimension are discussed. The standard FRW model corresponds to the limiting case C --> + infinity+. Depending on the free parameter of th e theory, C, the expansion of the model can behave differently from the sta ndard cosmological models with constant dimension. This is explicitly studi ed in the framework of quantum cosmology. The Wheeler-DeWitt equation is wr itten down. It turns out that in our model universe, the potential of the W heeler-DeWitt equation has different characteristics relative to the potent ial of the de Sitter minisuperspace. Using the appropriate boundary conditi ons and the semiclassical approximation, we calculate the wave function of our model universe. In the limit of C --> + infinity, corresponding to the case of constant space dimension, our wave function does not have a unique behavior. It can either lead to the Hartle-Hawking wave function or to a mo dified Linde wave function, or to a more general one, but not to that of Vi lenkin. We also calculate the probability density in our model universe. It is always more than the probability density of the de Sitter minisuperspac e in three-space as suggested by Vilenkin, Linde, and others. In the limit of constant space dimension, the probability density of our model universe approaches that of the Vilenkin and Linde probability density, being exp(-2 \S-E\), where SE is the Euclidean action. Our model universe indicates ther efore that the Vilenkin wave function is not stable with respect to the var iation of space dimension. [S0556-2821(99)03322-6].