Cosmology with minimal length uncertainty relations

TL;DR

This paper investigates how a minimal length scale, suggested by quantum gravity theories, affects classical and quantum cosmological models of de Sitter and Anti de Sitter universes, leading to modified inflationary and oscillatory behaviors.

Contribution

It extends the generalized uncertainty principle to cosmological models, analyzing its impact on classical inflation and quantum wave functions in de Sitter and Anti de Sitter universes.

Findings

GUP induces an inflationary expansion in classical de Sitter cosmology.

GUP alters the oscillatory nature of Anti de Sitter cosmology.

Quantum solutions show deviations from standard cosmology with GUP effects.

Abstract

We study the effects of the existence of a minimal observable length in the phase space of classical and quantum de Sitter (dS) and Anti de Sitter (AdS) cosmology. Since this length has been suggested in quantum gravity and string theory, its effects in the early universe might be expected. Adopting the existence of such a minimum length results in the Generalized Uncertainty Principle (GUP), which is a deformed Heisenberg algebra between minisuperspace variables and their momenta operators. We extend these deformed commutating relations to the corresponding deformed Poisson algebra in the classical limit. Using the resulting Poisson and Heisenberg relations, we then construct the classical and quantum cosmology of dS and Ads models in a canonical framework. We show that in classical dS cosmology this effect yields an inflationary universe in which the rate of expansion is larger than the usual dS universe. Also, for the AdS model it is shown that GUP might change the oscillatory nature of the corresponding cosmology. We also study the effects of GUP in quantized models through approximate analytical solutions of the Wheeler-DeWitt (WD) equation, in the limit of small scale factor for the universe, and compare the results with the ordinary quantum cosmology in each case.