Abstract
The non-commutativity of the space-time had important implications for the very early Universe, when its size was of the order of the Planck length. An important implication of this effect is the deformation of the standard dispersion relation of special relativity. Moreover, in the Planck regime gravity itself must be described by a quantum theory. We consider the implications of the modified dispersion relations for a photon gas, filling the early Universe, in the framework of loop quantum cosmology, a theoretical approach to quantum gravity. We consider three types of deformations of the dispersion relations of the photon gas, from which we obtain the Planck scale corrections to the energy density and pressure. The cosmological implications of the modified equations of state are explored in detail for all radiation models in the framework of the modified Friedmann equation of loop quantum cosmology. By numerically integrating the evolution equations we investigate the evolution of the basic cosmological parameters (scale factor, Hubble function, radiation temperature, and deceleration parameter) for a deformed photon gas filled Universe. In all models the evolution of the Universe shows the presence of a (nonsingular) bounce, corresponding to the transition from a contracting to an expanding phase.
Highlights
Cosmology is undergoing presently a very rapid development
In the present paper we investigate the LQC-modified Friedmann equations, representing effective cosmological evolution equation derived from Loop Quantum Gravity (LQG), together with the assumption that the Universe was initially filled with a photon gas described the deformed statistical mechanics induced by the non-commutativity of the space-time
In the present section we will consider the cosmological evolution of some classes of loop quantum cosmological models, whose dynamics is described by the modified Friedmann equations, under the assumption that the matter content of the early Universe consisted of radiation only
Summary
Cosmology is undergoing presently a very rapid development. Very precise observational data, obtained from the study of both cosmic microwave background (CMB) [1–3], and the matter distribution in the Universe [4,5], did offer the possibility of testing sophisticated cosmological models. Space-time non-commutativity leads to modified energy-momentum dispersion relations, with important implications in physics, cosmology, and astrophysics [100– 104]. Evolution equations that very closely approximate the full quantum dynamics of sharply peaked states at all times were derived, and the analytical and numerical methods used to study other forms of matter fields were improved It was suggested in [107] that in the limit of high energies loop quantum gravity may lead to the deformation of the local Poincaré algebra. In the present paper we investigate the LQC-modified Friedmann equations, representing effective cosmological evolution equation derived from LQG, together with the assumption that the Universe was initially filled with a photon gas described the deformed statistical mechanics induced by the non-commutativity of the space-time. Another approach to investigating the relation between non-commutative space-time and modified dispersion relation is using non-commutative geometry [114,115]
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