Effects of Quantum Gravity on a Vector Field Cosmological Model

The modification of laws of physics at short intervals is an important result of the theory of quantum gravity. For instance, commutative relations of standard quantum mechanics change on scales of length- called Planck length. It should be noted that these changes can be neglected at low energy levels but they are considerable only at high energy levels such as the initial universe. In this regard, the principle of uncertainty of standard quantum mechanics is changed with modified relations of uncertainty including a visible minimum of Planck order. Early moments of the universe, which included the inflation period, was a period with noticeable effects of quantum gravity due to the high energy level, and as such, the effects can be studied during this period. To do this, characteristics of the inflation period can be examined according to initial parameters of the universe such as the initial fluctuations in the formation of the universe structure and the spectral index. On the other hand, vector cosmology models have been taken into consideration by researchers. These models include an action in which a vector field (in addition to the scalar field) is included to investigate effects of violation of the Lorentz invariance in observations.
The present paper investigated effects of quantum gravity (with effects on non-commutative geometry and generalization of the uncertainty principle) on parameters of a vector cosmological model. The vector model was used as this scenario had acceptable adaptation to parameters of cosmology after inflation (e.g. the transition from the Phantom boundary, etc.) (Nozari and Sadatian, 2009). Furthermore, the present study could test this vector model for determining parameters of the inflation period based on effects of quantum gravity. According to calculations in the present paper, we concluded that, first: the density of scalar perturbations decreased in the vector model based on effects of quantum gravity (the reduction of standard model was more considerable), and second: due to the ignorance of effects quantum gravity, the scalar spectral index parameter remained invariant as observations indicate, but due to large enough gravitational effects (depending on amount of  β), the spectral index parameter is not maintained its invariance scale. According to obtained modification in the present study, the quantum gravity can be tested for the density of scalar perturbation (which can be measured by observing the spectrum of cosmic microwave background radiation).
In order to compare our results with other studies, we can refer to (Zhu et al, 2014) where they examined the spectral index in accordance with high-order correction mechanism. It also indicated that a single asymmetric approximation does not lead to a considerable error value for the spectral index, and the invariance scale is maintained. Furthermore, the paper (Hamber and Sunny Yu, 2019) found the same results for invariance scale of the spectral index according to the Wilson normalization analysis method. Therefore there was no need to have common assumptions in the inflation period.
Finally, it should be noted that despite a great number of studies on effects of quantum gravity, the reviewed model of this paper considers a state in which the effects can be investigated at all stages of the universe evolution from inflation till now.