$n$-dimensional FLRW quantum cosmology

We introduce the formalism of quantum cosmology in a Friedmann-Lemaître-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 $n\to \infty$, $⟨p{⟩}_L/⟨\rho {⟩}_L\to 1$.

Figure 1

The scale factor for $n=4$, 10, 20, and 40 (for $\alpha =1/(n-1)$). We note that the big-bang is still present.

Figure 2

The Bohmian trajectory of the scale factor for $n=4$, 10, 20, 40, and 80, for $a_0=\gamma =1$. The expansion of the universe slower as $n$ grows up and as $n\to \infty$, we approach a static universe.

Figure 3

The local expectation value of the Ricci scalar $R(t)$ for $n=4$, 10, 20, 40 ($a_0=\gamma =1$). They are perfectly regular for all values of $t$ indicating that the singularity is excluded by the introduction of quantum cosmology.

Figure 4

The ratio $⟨p{⟩}_L/⟨\rho {⟩}_L$ for $n=4$, 10, 20, 40, and $a_0=\gamma =1$. In four dimensions the classical behavior of the fluid is recovered while $n\to \infty$ $⟨p{⟩}_L/⟨\rho {⟩}_L\to 1$.

Figure 5

The graphic of ${\kappa }_n[⟨\rho {⟩}_L+(n-1)⟨p{⟩}_L]$ for $n=4$, 10, 20, and 40. We see that the strong energy condition is violated when $t$ is small and returns to which is classically expected as $t$ gets bigger.