Einstein static universe, GUP, and natural IR and UV cut-offs

Abstract

We study the Einstein static universe in the framework of Generalized Uncertainty Principle constructed by the Snyder non-commutative space. It is shown that the deformation parameter can induce an effective energy density subject to GUP which obeys the holographic principle (HP) and plays the role of a cosmological constant. Using the holographic feature of this effective energy density, we introduce natural IR and UV cut-offs which depend on the GUP based effective equation of state. Moreover, we propose a solution to the cosmological constant problem. This solution is based on the result that the Einstein equations just couple to the tiny holographic based surface energy density (cosmological constant) induced by the deformation parameter, rather than the large quantum gravitational based volume energy density (vacuum energy) having contributions of order $M_P^4$.