Model universe with variable space dimension: Its dynamics and wave function

Abstract

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 derived 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 can fix the dimension of space say at the Planck time. Different possibilities for this dimension are discussed. The standard FRW model corresponds to the limiting case C → +∞. Depending on the free parameter of the theory, C, the expansion of the model can behave differently to the standard cosmological models with constant dimension. This is explicitly studied in the framework of quantum cosmology. The Wheeler-De Witt equation is written down. It turns out that in our model universe, the potential of the Wheeler-DeWitt equation has different characteristics relative to the potential of the de Sitter minisuperspace. Using the appropriate boundary conditions and the semiclassical approximation, we calculate the wave function of our model universe. In the limit of C → +∞, corresponding to the case of constant space dimension, our wave function has not a unique behavior. It can either leads to the Hartle-Hawking wave function or to a modified Linde wave function, or to a more general one, but not to that of Vilenkin. We also calculate the probability density in our model universe. It is always more than the probability density of the de Sitter minisuperspace in 3-space as suggested by Vilenkin, Linde, and others. In the limit of constant space dimension, the probability density of our model universe approaches to Vilenkin and Linde probability density being exp(−2|S E |), where S E is the Euclidean action. Our model universe indicates therefore that the Vilenkin wave function is not stable with respect to the variation of space dimension.

Dynamics of chaotic inflation with variable space dimension

“…In [2,3], there are some clarifications about the idea of variable space dimension, and some motivations for choosing a model Universe with variable space dimension. There are also some reasonably raised questions: what causes the number of spatial dimension to decrease as the Universe expands?…”

“…The way of generalizing the standard cosmological model to the variable space dimension used in [6] is questioned, and another way of writing the field equations is proposed [3]. It has been pointed out that the model discussed in [6] has no upper bound for the space dimension.…”

“…[3] studied critically previous works in [6], and derived new Lagrangians and field equations. They also discussed the model Universe with variable space dimension from the view point of quantum cosmology, and obtained a general wave function.…”

“…They also discussed the model Universe with variable space dimension from the view point of quantum cosmology, and obtained a general wave function. In the limit of constant space dimension, the wave function of the model approaches the tunneling Vilenkin wave function or the modified Linde wave function [3].…”

“…Interest in a speculative model in which the number of space dimension decreases continuously as the Universe expands has increased during the past few years [1][2][3][4][5].…”