1.3 Structure of the thesis
2.2.5 Singularity avoidance
If we had a Hilbert space at hand we would also have a clear probability interpretation and hence a clear notion of singularity avoidance. This is, however, so far not the case and we have to rely on criteria that we can make us of without these notions.
The most prominent criterion for singularity avoidance is the DeWitt criterion also called the DeWitt boundary condition: A singularity is said to be avoided if Ψ→ 0 in the vicinity
2.2. QUANTUM COSMOLOGY 63
of the classical singularity. However, our discussion above shows up problems with this criterion. Since different representations of the wave function are related by eΨ = Ω2−d2 Ψ this
criterion is not conformally invariant. When d > 2 it is in general not true that Ψ → 0 is equivalent to Ψe → 0. Moreover, there does not seem to be a privileged representative of the wave function for the imposition of this criterion.
If we decided stick to the Hawking-Page probability interpretation we would use the following definition: A singularity is said to be avoided if ?|Ψ|2 → 0 in the vicinity of the
classical singularity. But since ?|Ψ|2 is not conformally invariant this criterion suffers from
the same problem as the DeWitt criterion.
The following two criteria satisfy the demand to be invariant under both coordinate transformations and conformal transformations:
Criterion 1. A singularity is said to be avoided if J [Ψ, Ψ] → 0 in the vicinity of the classical singularity.
The problem with this criterion is that J [Ψ, Ψ]≡ 0 if Ψ is real. We remark that a similar criterion based, however, on the Schr¨odinger current was used in [85].
Criterion 2. A singularity is said to be avoided if?|Ψ|d−22d → 0 in the vicinity of the classical
singularity.
Note that|Ψ|d−22d → |Ψ|2 asd→ ∞. Unlike criterion 1 the second criterion does not seem
to suffer from any problems. The only issue that appears is that there is no clear physical interpretation of the quantity ?|Ψ|d−22d .
Is it a problem, that the criteria 1 and 2 are formulated by using densities instead of usual scalars/tensors? We can for example compare the situation with the case of the non- relativistic wave function of a particle in a Coulomb potential in 3–dimensional Euclidean space. Here the wave function Ψ(r, ϑ, ϕ) and the square of its absolute value do not vanish value at r = 0. The probability density ?|Ψ(r, ϑ, ϕ)|2 = |Ψ(r, ϑ, ϕ)|2r2sin(ϑ)dr∧ dϑ ∧ dϕ,
however, vanishes and implies a zero probability of the particle being at the singular point r = 0. This comparison suggests that the criteria should indeed be formulated in terms of densities.
Another criterion first proposed by D¸abrowski and Kiefer [24] states:
Criterion 3. A wave packet is said to avoid the singularity if the wave packet spreads in the vicinity of the classical singularity.
A spreading of the wave packets indicates the breakdown of the eikonal approximation and therefore the classical limit. The singularity theorems by Hawking and Penrose do then
no longer apply. In the examples we consider in this work we shall see that wave packets spread close to the initial singularity if the dimension d of minisuperspace is larger than 2. The spreading can be linked to a decrease of the amplitude of Ψ which might lead to an avoidance of the singularity by both criteria 1 and 2. In particular the examples we consider later on give the impression that there is a correlation between the criteria 1 and 3.
Singularity avoidance in other approaches to Quantum Cosmology
Let us give a short overview over the status of singularity avoidance in other approaches to Quantum Cosmology.
Loop Quantum Cosmology (LQC) is the symmetry reduced minisuperspace version of Loop Quantum Gravity (LQG) [86]. It is often found in LQC that singularities are avoided by replacing them with a bounce. These results are mostly based on the effective quantum corrected equations of motion arising from LQC (see e.g. [87]). More rigorous results at the level of the full LQC equations are only known in the context of the isotropic models [88, 89]. The situation in LQC, however, seems to be anything but settled [13]. Other results [90] that take perturbations into account, for example, indicate that instead of a bounce a transition into an Euclidean regime takes place (similar to Hartle’s and Hawking’s no-boundary proposal).
In the Bohmian approach to Quantum Cosmology [91] the wave function of the Universe Ψ is interpreted as a pilot wave. This interpretation is universal in the sense that it can be applied to any approach to Quantum Cosmology. In the case of the Wheeler-DeWitt equation it can be applied as follows: The wave function is written as Ψ = |Ψ|eiS, with S being a real valued phase (in general this is not the Hamilton-Jacobi function). In addition to the Wheeler-DeWitt equation a guidance equation is postulated:
˙ qA=
GAB∂
BS . (2.178)
The momentum conjugate to qA is then defined via p
A = GABq˙B = ∂AS.13 It follows now
from the Wheeler-DeWitt equation that the Bohmian dynamics are described by the quantum corrected Hamiltonian constraint
HQ =
1 2G
ABp
ApB+V + Q = 0 , (2.179)
where Q = −12|Ψ|−1(− ξR)|Ψ| is called the quantum potential. The quantum potential is
13We remark that this momentum coincides with the smeared out version of the WKB time ∂
∂τ[Ψ∗] that
2.2. QUANTUM COSMOLOGY 65
thought of as a quantum correction to the classical potential. The presence of Q can lead to a bouncing scenario for the Bohmian trajectory qA(t). This feature is, however, not generic
and it depends strongly on the choice of the wave packet Ψ.
Singularity avoidance was also studied in the context of gravitational collapse models. Quantization was performed for a reduced system describing the spherically symmetric collapse of a thin null dust shell [92]. This approach lead to a Schr¨odinger equation. As a direct consequence of the demand for a unitary time evolution of the quantum state it was found that the singularity was replaced by a bounce. Similar results were obtained for the marginally–bound Lemaˆıtre-Tolman-Bondi model [93], where the decoupling of the single dust shells allows to treat the dynamics of each single dust shell like a minisuperspace model. These results are also directly transferable to the Oppenheimer-Snyder model [94]. Bounces which lead to a Black hole to white hole transitions and to an accompanying avoidance of the singularity are also believed to occur in LQG. [95] offers a review on quantum bounces in the context of gravitational collapse.
Another approach to Quantum Cosmology is provided by the so-called affine coherent state quantization. This approach was for example employed in [96] and [97] where it was shown that it indeed leads to singularity avoidance in the case of the Bianchi IX model.
The authors of [85] studied the resolution of Big Bang type singularities in FLRW models filled with dust. To this end they used a criterion which is similar to the criterion 1 (vanishing of KG current) in this thesis, based, however, on the Schr¨odinger probability current. It was found that the singularity can be avoided for certain classes of factor orderings. The resolution of the singularity can be understood to be caused by a repulsive potential that is generated by a particular class of factor orderings.
We remark that in some of the above mentioned approaches it is often made use of the fact that matter acts as a clock. After quantization one can then obtain a Schr¨odinger type equation. This has the advantage that a clear probability interpretation emerges. One has, however, to deal with a multiple choice problem, that is, the choice of a clock is not unique and different choices lead to in-equivalent quantum theories [98].
The paper [99] employs a relational approach to quantization. This approach allows for the construction of a Hilbert space as well. By applying such a quantization to the Bianchi I model the author of [99] showed that the probability to reach the singularity was zero for a specific wave packet. The singularity is here replaced by a bounce as well.
The framework used in [100] yields effective quantum corrected dynamics based on a certain moment decomposition of the quantum state of the universe. It has been applied to the Bianchi I model. The results obtained there indeed indicate that the singularity is
avoided.