9th International Workshop DICE2018 : Spacetime - Matter - Quantum Mechanics IOP Publishing

IOP Conf. Series: Journal of Physics: Conf. Series 1275 (2019) 012011 doi:10.1088/1742-6596/1275/1/012011

Singularity avoidance in anisotropic quantum cosmology

Claus Kiefer and Nick Kwidzinski
Institut für Theoretische Physik, Universität zu Köln, Zülpicher Straße 77, 50937 Köln, Germany

Abstract. We discuss the fate of the classical singularities in quantum cosmological models. We state our
criteria of singularity avoidance and apply them to Friedmann-Lemaı̂tre models models as well as, in more
detail, to the anisotropic case of a Bianchi I universe. We find that the classical singularities are generally
avoided in these cases.

1. Introduction
A major issue in any quantum theory is the fate of the classical singularities. Since so far no final theory
of quantum gravity exists, this problem can only be addressed within particular approaches [1]. In our
contribution, we shall address singularity avoidance in the framework of canonical (metrical) quantum
general relativity (quantum geometrodynamics).
In Section 2, we state and discuss our general criteria for singularity avoidance. We employ a
generalization of the DeWitt criterion that demands the quantum gravitational state to approach zero
in the region of the classical singularity. In Section 3, we briefly review the situation for Friedmann-
Lemaı̂tre models. In Section 4, we extend the disucssion to an anisotropic model – the Bianchi I universe.
Sections 2 and 4 are based on our recent paper [2]. Section 5 contains our conclusions and an outlook.

2. Criteria for singularity avoidance

It is well known that singularities are ubiquitous in general relativity. In their general form, the singularity
theorems do not display the physical nature of singularity. They usually state incompleteness for timelike
or null geodesics. Since cases of physical importance mostly deal with curvature singularities, it seems
appropriate to include them explicitly in the definition. In the words of Ellis et al. ([3], p. 145):

We shall define a singularity as a boundary of spacetime where either the curvatures diverges
. . . or geodesic incompleteness occurs.

We shall adopt this definition here. (Scalar) curvature singularities are either Ricci tensor divergences
or Weyl tensor divergences or a mixture of the two. The first are related to unbounded matter (or field)
densities, while the second are related to gravitational field divergences in the vacuum. Both cases will
be of relevance here.
In general relativity, singularity theorems can be formulated and proved in a mathematically rigorous
manner. This is possible because the theory itself has a sound mathematical foundation. This is not the
case in quantum gravity. First, there exist various approaches to such a theory, without any consensus so
far [1]. Second, even within particular approaches, the mathematical foundation is limited. Nevertheless,
it is of importance to study the fate of classical singularities in quantum gravity approaches, even if this
is only possible heuristically. The reason for this is to gain hints on the viability of approaches and to
get an idea about the conceptual nature of singularity avoidance.

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9th International Workshop DICE2018 : Spacetime - Matter - Quantum Mechanics IOP Publishing
IOP Conf. Series: Journal of Physics: Conf. Series 1275 (2019) 012011 doi:10.1088/1742-6596/1275/1/012011

Our approach here is quantum geometrodynamics. This is, in a sense, the most conservative approach,
because its central equations (Wheeler-DeWitt equation and quantum momentum constraints) are
obtained by formulating quantum wave equations that lead to the Einstein equations1 in the semiclassical
(WKB) limit. The kinematic quantity here is the wave functional of all three-geometries (and non-
gravitational fields). This configuration space is called superspace.
In his pioneering paper on canonical quantum gravity, DeWitt came up with a heuristic proposal for
singularity avoidance [4]. Assuming that the wave functional Ψ in quantum gravity is at least loosely
related to the probability interpretation in ordinary quantum theory, he suggested to demand that Ψ = 0
in regions where classical singularities are situated. He writes ([4], p. 1129):
Provided it does not turn out to be ultimately inconsistent, this condition,2 . . . yields two
important results. Firstly, it makes the probability amplitude for catastrophic 3-geometries
vanish, and hence gets the physicist out of his classical predicament. Secondly, it may permit
the Cauchy problem for the “wave equation” . . . to be handled in a manner very similar to that
of the ordinary Schrödinger equation.
In his second point, DeWitt expresses the hope that Ψ = 0 together with the specification of Ψ on
one three-geometry (instead of two, as is natural for a second-order equation) suffices to determine Ψ for
all three-geometries. This is, however, mathematically far from clear. It is even imaginable that only the
trivial solutions remains from these two boundary conditions. The specification of Ψ on one boundary
only is related, in spirit, to the no-boundary proposal by Hartle and Hawking (see e.g. [1], p. 279).
The DeWitt condition has frequently been used in the context of Friedmann-Lemaı̂tre cosmology,
where the configuration space is finite-dimensional and called minisuperspace [1, 6]. It also arises as a
consequence of unitary evolution for quantum collapsing dust shells; see, for example, [5] in which it is
shown how a wave packet representating a collapsing shell will develop into a superposition of collapsing
and expanding shell, thereby avoiding the black and while hole singularities.
In the rest of this section, we shall formulate the DeWitt criterion for a general anisotropic model
in quantum cosmology. In this, we make use of the fact that the quantum cosmological wave function
is, in fact, defined over a conformal manifold, which leads us to a generalization of the original DeWitt
criterion [2]. This generalization becomes relevant for models with dimension of configuration space
greater than two; see below the example for a Bianchi I universe.
To be specific, let us consider the action for general (diagonal) Bianchi class A models with
unspecified matter part (see for the following [2] and the references therein):
" #
Z
−α̇ 2 + β̇ 2 + β̇ 2 (3) R
+ −
SEH + Sm = dt N e3α + + Sm . (1)
2N 2 12

The minisuperspace, M, of these models is parametrized here by the coordinates q = {α, β+ , β− , φ},
where α ≡ ln a, β+ , and β− are the ‘Misner variables’, and φ denotes matter field degrees of freedom;
(3)R denotes the three-dimensional Ricci scalar. Units are chosen such that 3c6 V /4πG = 1, where V
0 0
is the volume of three-dimensional space (assumed to be compact here).
The central equation of the canonical theory is the Hamiltonian constraint, which follows from (1) by
variation with respect to the lapse function N ,
1
H = G IJ pI pJ + V = 0. (2)
2
Here, GIJ are the components of the DeWitt metric, G IJ the components of its inverse, and V is the
minisuperspace potential which contains contributions from the three-curvature and from the matter part.
The pI are the momenta canonically conjugate to the q above.
1
More precisely, the four Einsteins equations which are constraints.
2
He refers to Ψ = 0.

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9th International Workshop DICE2018 : Spacetime - Matter - Quantum Mechanics IOP Publishing
IOP Conf. Series: Journal of Physics: Conf. Series 1275 (2019) 012011 doi:10.1088/1742-6596/1275/1/012011

Under a re-scaling of the lapse, N → N e = Ω2 N , the transformation of the Hamiltonian constraint

follows from the invariance of the total Hamiltonian H,
   
1 −2 IJ −2 1 IJ
H = NH = N eH e=N e Ω G pI pJ + Ω V ≡ N e Ge pI pJ + V
e . (3)
2 2

For this reason, we can interpret the minisuperspace as a conformal manifold (M, [GIJ dq I ⊗ dq J ]). This
conformal structure motivates the choice of a factor ordering that renders the Wheeler–DeWitt equation
conformally covariant. Explicitly,

~2
 
− ( − ξR) + V Ψ = 0, (4)
2
d−2
where R denotes the Ricci scalar constructed from GIJ and ξ = with d = dim(M). For the
4(d−1) ,
equation to be conformally covariant the wave function has to transform as Ψ → Ψ e = Ω− d−2
2 Ψ. This is

why the original DeWitt criterion (Ψ → 0) is only a good criterion when dealing with two-dimensional
minisuperspaces.
We are now interested in the formulation of a conformally invariant DeWitt criterion. For this we note
that the following scalar density is invariant under a conformal transformation in minisuperspace:
2d 2d
?|Ψ| d−2 = |Ψ| d−2 dvol. (5)

Here, dvol contains the square root of the (absolute value of the) determinant of the DeWitt metric, and
? denotes the Hodge star. The generalized DeWitt criterion then reads:
2d
A singularity is said to be avoided if ?|Ψ| d−2 → 0 in the vicinity of the singularity.
We emphasize that this criterion, as the original one, is a sufficient criterion only, not a necessary one. As
one knows from quantum mechanics (e.g. from the solutions for the energy eigenstates of the hydrogen
atom from the Dirac equation), one can have singularity avoidance even for diverging wave functions.
One can think of other criteria. In the first paper along this direction, [7], singularity avoidance
was infered from the breakdown of the semiclassical approximation when approaching the region of the
singularity. Then, the notion of a classical trajectory loses its meaning and the classical theorems cannot
be applied. In a sense, the DeWitt criterion also obeys this criterion, because the vanishing of the wave
function signals the breakdown of the semiclassical approximation.
In [2], we have also applied the criterion that implies singularity avoidance if the Klein-Gordon flux
vanishes in the vicinity of the classical singularity3 . In the explicit examples discussed there, this leads
in general to results different from the DeWitt criterion. Because some aspects of the Klein-Gordon flux
are problematic in this context (e.g. it is not positive definite), we shall not consider this criterion here. It
must, however, be remarked that the validity of the DeWitt criterion usually demands the superposition
of different semiclassical components for the Klein-Gordon current (see Eq. (6.29) in [4]).

3. Singularity avoidance in Friedmann-Lemaı̂tre cosmology

Friedmann-Lemaı̂tre models are homogeneous and isotropic. From the gravitational side, the scale
factor a(t) is the only dynamical quantity. In order to represent matter and render the model non-trivial,
usually a scalar field φ(t) with an appropriate potential V (φ) is added. We thus have a two-dimensional
minisuperspace. The metric is given by

ds2 = −N 2 (t)dt2 + a2 (t)dΩ23 , (6)

3
This was motivated by the fact that the Klein-Gordon current is a conformally invariant (d − 1)-form.

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9th International Workshop DICE2018 : Spacetime - Matter - Quantum Mechanics IOP Publishing
IOP Conf. Series: Journal of Physics: Conf. Series 1275 (2019) 012011 doi:10.1088/1742-6596/1275/1/012011

where dΩ23 is the three-dimensional spatial line element (here chosen to be the three-sphere). After
quantization, the Wheeler-DeWitt equation reads

~2 ∂ ~2 ∂ 2 Λa3
   
1 ∂ 3
a − 3 2 −a+ + 2a V (φ) Ψ(a, φ) = 0. (7)
2 a2 ∂a ∂a a ∂φ 3

Here, the Laplace-Beltrami factor ordering is chosen, which for a two-dimensional configuration space
is equivalent to the above conformal factor ordering.
The standard singularities in general relativity are big bang and (for a recollapsing universe) big
crunch. Motivated by the observation of a currently accelerating universe, one can introduce a scalar
field with a suitable potential as the origin of such an acceleration. The presence of such a ‘dark energy’
can lead to further singularities, such as big rip, big brake, big freeze, and others; see, for example,
[8] and the references therein. In a series of papers (see e.g. [9] and the references therein) it was
possible to investigate the applicability of the DeWitt criterion in detail. The main result is that this is
in generally possible: the Wheeler-DeWitt equation possesses solutions with Ψ = 0 in the region of the
classical singularity. Let us quote one specific example. In [10], the avoidance of a classical big-brake
singularity4 was demonstrated. The result is illustrated in Figure 1.

Ψ(τ, φ)
15

0.8

10
0.7
0.6
0.5
5
0.4
0.3
φ

0
0.2
0.1
-5
0
2
-10 4
6 10
τ 8
0
5
-15 10 -5
0 2 4
a
6 8 10
12 -10
φ

(a) Classical trajectory. From [10]. (b) Wave packet; here, τ = a6 . From [10].

Figure 1: The avoidance of a classical singularity in quantum cosmology.

The left part (a) shows the classical trajectory φ(a) in configuration space. The big brake occurs at
φ = 0. The right part b) shows a plot of an exact solution of the Wheeler-DeWitt equation close to
φ = 0; this solution goes to zero there and thus satisfies the DeWitt criterion.

4. Singularity avoidance in Bianchi I quantum cosmology

The simplest anisotropic model is the vacuum Bianchi I model, also called Kasner universe. The metric
can be written as

ds2 = −dt2 + t2px dx2 + t2py dy 2 + t2pz dz 2 with (8)

p2x + p2y + p2z = 1 and px + py + pz = 1. (9)
4
In a big-brake singularity the universe comes to an abrupt halt in the future, with the density ρ finite, but the pressure p
diverging.

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9th International Workshop DICE2018 : Spacetime - Matter - Quantum Mechanics IOP Publishing
IOP Conf. Series: Journal of Physics: Conf. Series 1275 (2019) 012011 doi:10.1088/1742-6596/1275/1/012011

Details on this model (and other anisotropic models) can be found, for example, in [3]. In the Kasner
universe one has a singularity which can be described by the divergence of the Weyl squared scalar. After
quantization, one obtains the Wheeler-DeWitt equation, which in the above conformal factor ordering
reads (with ~ = 1),
∂2 ∂2 ∂2
 
− 2+ 2 + ∂β 2 Ψ(α, β+ , β− ) = 0. (10)
∂α ∂β+ −
This is identical, in its form, to the classical wave equation in d = 1 + 2 dimensions. Using decay
rate estimates for such an equation, one finds that the DeWitt criterion is fulfilled for a small universe
(α → −∞) as well as for a big universe (α → +∞):

?|Ψ|6 → 0 as α → ±∞. (11)

The initial singularity is thus avoided by this criterion. The decay of the amplitude |Ψ| is induced by a
spreading of the wave packet as shown in Figure 2. It is at first glance surprising that this criterion applies

Figure 2: Plot of equipotential surfaces of |Ψ| for a wave packet Ψ solving the Wheeler-DeWitt equation
(10). The thin black line is the classical trajectory.

to the future evolution, too: whereas classically the universe expands forever, quantum cosmologically
the wave packets cease to evolve, and the semiclassical approximation breaks down. This demonstrates
that quantum effects can, in principle, occur at any scale, as a mere consequence of the superposition
principle. Other examples are the above mentioned quantum effects near the big brake as well as quantum
effects near the turning point of a classically recollapsing universe [11].
Let us now add matter to this model. The simplest way is to treat matter by an effective potential
only, without considering a kinetic term for it [2]. Using the ansatz for a barotropic fluid, one finds that
the situation for a small universe is similar to the quantum Kasner model, and that the DeWitt criterion
holds as before: the initial singularity is avoided. For a big universe, one can have classically an ever
expanding universe or a big-rip singularity, depending on the exact form of the equation of state. In both
cases, the DeWitt criterion holds, so the wave packets do not reflect either of these two cases. A plot of
a corresponding wave packet is shown in Figure 3.
At a fundamental level, matter should be described dynamically, for example by using a scalar field.
Let us here only address the case of a phantom field, that is, a field with negative kinetic term that can
mimic dark energy [2]. Classically, this leads to a big-rip singularity in the future (universe expanding

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9th International Workshop DICE2018 : Spacetime - Matter - Quantum Mechanics IOP Publishing
IOP Conf. Series: Journal of Physics: Conf. Series 1275 (2019) 012011 doi:10.1088/1742-6596/1275/1/012011

Figure 3: Plot of equipotential surfaces of |D−1/2 Ψ| for a wave packet Ψ solving the Wheeler-DeWitt
equation with an effective dust potential. The factor D−1/2 provides a suitable rescaling for the purpose
of visualization. The thin black line is the classical trajectory, which describes an isotropizing Bianchi I
universe. The wave packet spreads in the region close to the big-bang singularity similarly to the Kasner
model.

to infinity in a finite time). Unlike the situation shown in Figure 1, the field φ diverges in this limit. The
conformally covariant Wheeler–DeWitt equation in the limit approaching the big rip is found to read
 2
∂2 ∂2 ∂2

∂ 6(α+|φ|)
− 2 − ∂β 2 + ∂φ2 + V0 e Ψ(α, β+ , β− , φ) = 0. (12)
∂α2 ∂β+ −

A detailed analysis of this equation and its solutions reveals that the DeWitt criterion is fulfilled and the
big rip is thus avoided [2].
Figure 4 shows a plot of a wave packet using numerical integration. It gives the results for α = 10 and
increasing values of |φ|. For increasing |φ|, the wave packet assumes an annular shape and propagates
outwards with decreasing amplitude; it is peaked in the direction of negative β± .

5. Conclusion and Outlook

In this contribution, we have discussed the fate of classical singularities in Friedmann-Lemaı̂tre as well
as Bianchi I models of quantum cosmology. Employing a generalized form of the DeWitt criterion, we
have seen that these singularities can generally be avoided.
Singularity avoidance is also discussed in other approaches such as loop quantum cosmology. There,
the resolution is usually treated as a non-singular bounce in quantum-corrected cosmological models;
see, for example, the recent discussion of Bianchi models in [12]. It would, of course, be of great interest
to perform a detailed comparison of this approach with the present one.
Future work should address the issue of singularity avoidance for models of increasing complexity.
The Bianchi I models discussed here belong to the class of asymptotically velocity dominated (AVTD)
models, for which the kinetic terms become dominant in the vicinity of the initial singularity. Our results
for these models should thus be representative for more general models which admit such an AVTD
behaviour.
The general approach to a classical spacelike singularity is been assumed to be of the Belinsky-
Khalatnikov-Lifshits (BKL) type; see, for example, [13] for a recent discussion. This corresponds to

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9th International Workshop DICE2018 : Spacetime - Matter - Quantum Mechanics IOP Publishing
IOP Conf. Series: Journal of Physics: Conf. Series 1275 (2019) 012011 doi:10.1088/1742-6596/1275/1/012011

1.25
2

1.00
ϕ=10.75
β+ 0
0.75

0.50
-2 ϕ=14.25, x5

0.25

-4
ϕ=18, x10

-4 -2 0 2 4
β-

3
Figure 4: Density plot for |Ψ|e 2 (α+|φ|) , showing the position of the wave packet for different values of
|φ|, each with a different scaling to visualize the decaying wave in a single graphic. From [2].

an evolution that behaves as if a separate mixmaster model were attached to each spatial point. The
discussion of the Wheeler-DeWitt equation for such a situation should give a quantum picture of the
BKL behaviour and should answer the question whether the singularities are avoided in the generic
case. If this could be achieved, one would have gained a complete picture of the situation in quantum
geometrodynamics, given our present level of understanding.

Acknowledgments
We are grateful to Dennis Piontek for collaboration on the quantum Bianchi I model.

References
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[gr-qc]
[3] Ellis G F R, Maartens R and MacCallum M A H 2012 Relativistic Cosmology (Cambridge: Cambridge University Press)
[4] DeWitt B S 1967 Quantum Theory of Gravity. 1. The Canonical Theory Phys. Rev. 160 1113
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