Quantum singularities in the FRW universe revisited

The components of the Riemann tensor in the tetrad basis are quantized and, through the Einstein equation, we find the local expectation value in the ontological interpretation of quantum mechanics of the energy density and pressure of a perfect fluid with equation of state $p=\frac{1}{3}\rho$ in the flat Friedmann-Robertson-Walker quantum cosmological model. The quantum behavior of the equation of state and energy conditions are then studied, and it is shown that the energy conditions are violated since the singularity is removed with the introduction of quantum cosmology, but in the classical limit both the equation of state and the energy conditions behave as in the classical model. We also calculate the expectation value of the scale factor for several wave packets in the many-worlds interpretation in order to show the independence of the nonsingular character of the quantum cosmological model with respect to the wave packet representing the wave function of the Universe. It is also shown that, with the introduction of nonnormalizable wave packets, solutions of the Wheeler-DeWitt equation, the singular character of the scale factor, can be recovered in the ontological interpretation.

Figure 1

The local expectation value of the non-null curvature components in the tetrad basis and of the Ricci scalar. They are perfect for all values of $t$. Here, we have chosen $\sigma =1$ and $a_0=0.1$.

Figure 2

The local expectation value of the energy density and pressure for $\sigma =1$ and $a_0=0.1$.

Figure 3

The local expectation value of $p/\rho$ for $\sigma =1$ and $a_0=0.1$. The function explodes in $t$ between 20 and 25 because $\rho$ becomes zero at this point.

Figure 4

The energy conditions for $\sigma =1$ and $a_0=0.1$. We see that every one of them are violated at the quantum level.

Figure 5

The local expectation value of the non-null curvature components in the tetrad basis and of the Ricci scalar for a wave packet satisfying the Dirichlet boundary condition for $\sigma =1$ and $a_0=0.1$.

Figure 6

The energy conditions for a wave packet satisfying $\Psi (a,0)=0$ for $\sigma =1$ and $a_0=0.1$.

Figure 7

The expectation value of the scale factor for (1) $\Psi (a,0)=e^{-a^2}$, (2) $\Psi (a,0)=a^2{e}^{-a^2}$, (3) $\Psi (a,0)=a^4{e}^{-a^2}$, (4) $A(E)=e^{-E}$, and (5) $A(r)=e^{-r^2}{I}_{1/2}(r)$. We see that they have qualitatively the same behavior, i.e., asymptotically they behave as straight lines which pass trough the origin.