n-Dimensional FLRW Quantum Cosmology

TL;DR

This paper extends quantum cosmology formalism to n-dimensional FLRW universes, demonstrating singularity resolution and analyzing the behavior of physical quantities across different dimensions.

Contribution

It generalizes the Schutz formalism and quantum cosmology techniques to arbitrary dimensions, revealing unique features of four-dimensional spacetime and the behavior as dimensions increase.

Findings

Quantum singularities are resolved in n-dimensional FLRW models.

The ratio of pressure to density approaches classical values only in four dimensions.

As n approaches infinity, the pressure-to-density ratio tends to 1.

Abstract

We introduce the formalism of quantum cosmology in a Friedmann-Lema\^itre-Robertson-Walker (FLRW) universe of arbitrary dimension filled with a perfect fluid with $p=\alpha\rho$ equation of state. First we show that the Schutz formalism, developed in four dimensions, can be extended to a n-dimensional universe. We compute the quantum representant of the scale factor $a(t)$, in the Many-Worlds, as well as, in the de Broglie-Bohm interpretation of quantum mechanics. We show that the singularities, which are still present in the n-dimensional generalization of FLRW universe, are excluded with the introduction of quantum theory. We quantize, via the de Broglie-Bohm interpretation of quantum mechanics, the components of the Riemann curvature tensor in a tetrad basis in a n-dimensional FLRW universe filled with radiation ($p=\frac{1}{n-1}\rho$). We show that the quantized version of the Ricci scalar are perfectly regular for all time $t$. We also study the behavior of the energy density and pressure and show that the ratio $<p>_L/<\rho>_L$ tends to the classical value $1/(n-1)$ only for $n=4$, showing that $n=4$ is somewhat privileged among the other dimensions. Besides that, as $n\to\infty$, $<p>_L/<\rho>_L\to 1$.