# Singularity avoidance in Bianchi I quantum cosmology

## Abstract

We extend recent discussions of singularity avoidance in quantum gravity from isotropic to anisotropic cosmological models. The investigation is done in the framework of quantum geometrodynamics (Wheeler–DeWitt equation). We formulate criteria of singularity avoidance for general Bianchi class A models and give explicit and detailed results for Bianchi I models with and without matter. The singularities in these cases are big bang and big rip. We find that the classical singularities can generally be avoided in these models.

## 1 Introduction

A major issue in any quantum theory of gravity is the fate of the classical singularities. So far, such a theory is not available in final form, although various approaches exist in which this question can be sensibly addressed [1, 2]. It is clear that such an investigation cannot yet be done at the level of mathematical rigor comparable to the singularity theorems in the classical theory (see e.g. [3]). Nevertheless, focusing on concrete approaches and concrete models, one can state criteria of singularity avoidance and discuss their implementation. This is what we shall do here.

We restrict our analysis of singularity avoidance to quantum geometrodynamics, with the Wheeler–DeWitt equation as its central equation [1]. Although this may not be the most fundamental level of quantum gravity, it is sufficient for addressing the issue of singularity avoidance. Quantum geometrodynamics follows directly from general relativity by rewriting the Einstein equations in Hamilton–Jacobi form and formulating quantum equations that yield the Hamilton–Jacobi equations in the semiclassical (WKB) limit. It thus makes as much sense to addressing singularity avoidance here than it does to addressing it in quantum mechanics at the level of the Schrödinger equation. Singularity avoidance has also been discussed in loop quantum gravity [2, 4, 5], with various results, but we will not consider this here.

Singularity avoidance was already addressed by DeWitt in his pioneering paper on canonical quantum gravity [6]. He suggested to impose the condition \(\Psi \rightarrow 0\) for the quantum-gravitational wave functional \(\Psi \) when approaching the region of a classical singularity. The wave functional is effectively defined on the configuration space of all *three*-dimensional geometries, also called superspace [1, 7]. The “DeWitt criterion” of vanishing wave function then means that \(\Psi \) must approach zero when approaching a singular three-geometry (which itself is not part of superspace, but can be envisaged as its boundary). It is important to emphasize that this criterion is a sufficient but not a necessary one: singularities can be avoided for non-vanishing or even diverging \(\Psi \) (recall the ground state solution of the Dirac equation for the hydrogen atom, which diverges).

DeWitt had in mind cosmological singularities such as big bang or big crunch. The DeWitt criterion applies, of course, also to the singularities that classically arise from gravitational collapse. In simple models of quantum geometrodynamics, their avoidance can be rigorously addressed. One example is the collapse of a null dust shell, which classically develops into a black hole, but quantum gravitationally evolves into a re-expanding shell, with \(\Psi =0\) in the region of the classical singularity [8, 9, 10]. In general, however, such cases are too difficult to allow for an exact mathematical treatment, so most investigations so far were restricted to Friedmann–Lemaître–Robertson–Walker (FLRW) cosmology. The first detailed discussion of singularity avoidance in Wheeler–DeWitt quantum cosmology was performed for the big-rip singularity that occurs in the presence of phantom matter [11].^{1} Other applications followed, which generally focused on singularities occurring in dark-energy models but also on the big bang; see, for example, [13, 14, 15, 16] and the references therein. The question was also investigated for *f*(*R*) quantum cosmology [17, 18, 19]. In most cases, the DeWitt criterion was applied. In our paper, too, the main focus will be on this criterion, although we shall employ a second criterion that makes use of a current density.

In the present paper, we make a step forward and discuss the issue of singularity avoidance for *anisotropic* cosmologies. The simplest case is the Bianchi I model (see e.g. [20]), to which we restrict our investigation here. The more interesting Bianchi IX model is reserved for a future investigation. Anisotropic models are characterized by the fact that the dimension of their configuration space (minisuperspace) is bigger than two already for the pure gravitational case. This will be important for the formulation of the DeWitt criterion. We address the anisotropic case here mainly for structural reasons, in order to see how criteria of singularity avoidance apply there. We do not expect anisotropies to play a crucial role in the late universe, although such anisotropies may be relevant in the early universe.

Our article is organized as follows. In Sect. 2, we formulate our criteria for singularity avoidance. In this, a generalization is made that takes into account the conformal structure of minisuperspace. Section 3 then addresses the vacuum Bianchi I (Kasner) model. There, we encounter only the big bang singularity. Sections 4 and 5 are devoted to Bianchi I models with matter: an effective matter potential is used in Sect. 4, and a dynamical (phantom) scalar field is used in Sect.5. While in Sect. 4 both the big bang and big rip singularities are addressed, Sect. 5 focuses on the big rip singularity. We shall find that singularities can be avoided in all relevant cases. Section 6 presents a short conclusion and an outlook.

## 2 Criteria for singularity avoidance

In this section, we formulate the criteria of singularity avoidance at the level of a general (diagonal) Bianchi class A model. These will be applied in detail to the Bianchi I model in the following sections.

^{2}and \(\phi \) denotes matter field degrees of freedom; \({}^{(3)}R\) is the three-dimensional Ricci scalar. Units are chosen such that \(3c^6V_0/4\pi G=1\), where \(V_0\) is the volume of three-dimensional space (assumed to be compact here).

*N*yields the Hamiltonian constraint,

*H*according to

*k*the conformal weight of \(\mathcal {T}\) and denote it by \(\hbox {w}(\mathcal {T})=k\). Because of the conformal nature of configuration space, a spacetime which satisfies Einstein’s equations can, in fact, be regarded as a sheaf of geodesics on this space [23].

^{3}this is achieved by

Let us now turn to the discussion of the criteria for singularity avoidance. As mentioned in the Introduction, the first one goes back to DeWitt [6], who suggested to take \(\Psi \rightarrow 0\) near the region of the classical singularity as a sufficient criterion for quantum avoidance. In a heuristic sense, this corresponds to “probability zero” for the singularity. Application of this criterion is based on the idea that the (square of) the wave function is related to probability, as is the case in quantum mechanics. In quantum gravity, this is far from clear [1]. The main reason is the absence of an external time parameter in the Wheeler–DeWitt equation. Only in the semiclassical (Born-Oppenheimer) approximation, where an approximate time parameter emerges, can one impose the usual probability interpretation in a straightforward manner. Nevertheless, we shall stick heuristically to this idea also in the full theory. Peaks of the wave function have often been interpreted as giving predictions in cosmology; see, for example, [27] and the references therein.

In the semiclassical limit with only one WKB component, an interpretation using probabilties in minisuperspace was suggested in [28]; see also [29], pp. 186–190. Because the DeWitt criterion rests on the heuristic notion of a probability, we find it appropriate to include in this section some remarks on the formulation of this proposal in the language of conformal minisuperspace.

*S*is a solution to the Hamilton–Jacobi equation

*D*is the van Vleck factor which satisfies the linear transport equation

*B*a thin ‘pencil’ drawn out by the classical solutions, that is, integral curves of the vector field \(\mathcal {G}^{IJ}(\partial _I S)\partial _J\). It was shown in [28] that

*B*.

^{4}The contribution of

*B*to \(\int _{A \cap B} \star |\Psi |^2 \) is therefore proportional to the coordinate-time that the classical solutions filling out the pencil

*B*spend in the region

*A*. Note that \(\text {w}(\star D)=\text {w}(\star |\Psi |^2)=2\). This reflects the fact that the integral \(\int N \hbox {d}t \) on the right-hand side of (10) depends on the representation of the lapse which we choose before the quantization of the Hamiltonian constraint in order to obtain the Wheeler–DeWitt equation (7). In this sense, a conformal rescaling \(\mathcal {G}_{IJ}\rightarrow \Omega ^2 \mathcal {G}_{IJ}\), \(\Psi \rightarrow \Omega ^{\hbox {w}(\Psi )}\Psi \) corresponds to a time reparametrization at the quantum level. Equation (10) can also help us to interpret the behavior of wave packets in regions of minisuperspace where the WKB approximation is valid.

The DeWitt criterion was successfully applied to cosmological models in a series of recent papers; see, for example, [16] and references therein. These examples deal mostly with two-dimensional minisuperspaces where \(\hbox {w}(\Psi )=0\) and the usual Laplace–Beltrami operator coincides with the conformal Laplacian. In dimensions \(d\ge 3\), however, the DeWitt criterion is not conformally invariant. Moreover, there does not seem to be a privileged representative of the wave function for the imposition of the criterion. For the reasons mentioned above, we seek here a generalization of the DeWitt criterion to guarantee its conformal invariance.

### Criterion 1

A singularity is said to be avoided if \(\star |\Psi |^\frac{2d}{d-2}\rightarrow 0\) in the vicinity of the singularity.

This is the conformally invariant version of the DeWitt criterion [6].

### Criterion 2

A singularity is said to be avoided if \(\mathbf {J}[\Psi ,\Psi ]\rightarrow 0\) in the vicinity of the singularity.

Another criterion, which was introduced in the discussion of the quantum fate of the big-rip singularity in [11], is the following:

### Criterion 3

A wave packet is said to avoid the singularity if it spreads in the vicinity of the singularity.

The spreading of wave packets indicates the breakdown of the semiclassical approximation. Classical cosmology and in particular the classical singularity theorems then cease to hold. The notion of a classical spacetime can no longer be applied, which leads to the end of classical predictability before reaching the singularity. This criterion is fulfilled, for example, in the big-rip case studied in [11].

We note that the second criterion suffers from the problem that it is not applicable in the case of real wave functions, which often arise as solutions to the (real) Wheeler–DeWitt equation. An example of a real wave function is the no-boundary (Hartle-Hawking) state. In contrast to this, the “tunneling wave function” is complex, and criterion 2 can be applied.^{5}

The Klein–Gordon flux is not positive definite and can thus in general not be interpreted as a probability flux. Exceptions are situations where only one WKB branch is present; this has led to the proposal that the Klein–Gordon current only be applied to such cases [26]. The case of one WKB wave function can also be interpreted as a decohered branch of a real wave function. In the following, we shall thus mainly concentrate on the first criterion, which is the natural generalization of the DeWitt criterion to higher-dimensional minisuperspaces.

## 3 Kasner solution

*x*,

*y*, and

*z*.

*f*and \(\xi \) parametrize a family of operator orderings. After the transformation \(\Psi \rightarrow \widetilde{\Psi }=\hbox {e}^{f\alpha }\Psi \) we obtain

Let us now turn to the formulation of the criteria for singularity avoidance. There, the minisuperspace dimension *d* will be crucial. Solutions to the free wave equation in \(1+1\) dimension can propagate only into two directions. Wave packets are not subject to spreading and their amplitudes do not decay. In higher dimensions, however, the wave can propagate into infinitely many directions. This leads to a spreading and a resulting decay of the amplitude of the wave. The above statement can be made more precise in the form of *decay rate estimates*.

*f*and

*g*are smooth functions \(\mathbb {R}^{d-1}\rightarrow \mathbb {R}\) with compact support. Then there exist \(C_{1/2}>0\) such that

*t*and the standard scalar product. In quantum cosmology, there is no consensus about the choice of inner product. If one used a norm motivated by the conformally invariant DeWitt criterion, that is, an integral over \(\beta _+\) and \(\beta _-\) of \(\star |\Psi |^{6}\), this is not conserved in \(\alpha \); one can even estimate that the integral goes to zero for \(\alpha \rightarrow +\infty \), so the situation is very different from the quantum mechanical free particle: there is

*no*unitarity with respect to the timelike variable \(\alpha \). In the spirit of the DeWitt criterion, one could call this a quantum avoidance of the late-time evolution. Figure 1 displays the behavior of the wave packet for \(\alpha \rightarrow -\infty \) and \(\alpha \rightarrow +\infty \).

This is an example where quantum effects are not restricted to small scales of Planck size. Because the superposition principle is universally valid in quantum cosmology, quantum effects can arise in principle at any scale. One example is the turning point of a classically recollapsing universe [31], where destructive interference has to occur in order to guarantee a recollapsing wave packet.

One could argue that a natural inner product for the Wheeler–DeWitt equation is the Klein–Gordon inner product, which provides unitarity with respect to \(\alpha \). The vanishing of the Klein–Gordon current corresponds to our criterion 2 of singularity avoidance and is fulfilled in the present case, both for \(\alpha \rightarrow -\infty \) and \(\alpha \rightarrow +\infty \), see (24).

A somewhat different approach to the quantization of Bianchi I vacuum models with and without cosmological constant was developed in [32]. There, the dynamics was reduced such that the wave function depends only on one degree of freedom, the determinant of the scale factor *a*. How a singularity avoidance in this approach relates to the singularity avoidance discussed here, is an interesting question that is beyond the scope of our investigation.

In the next section, we will investigate if and how the situation changes if matter is added to the model.

## 4 Bianchi I model with an effective matter potential

In this section, we treat matter in a phenomenological way. The representation of matter by a dynamical scalar field \(\phi \) is relegated to the next section. In anisotropic models, anisotropic pressures can be used, but we address for simplicity the case of a barotropic fluid.

*N*leads to

*a*(

*t*) is shown for different

*w*in Fig. 2a.

*a*, the hypergeometric function asymptotically equals 1, and we get for \(a\rightarrow 0\):

*w*. For large

*a*and \(w\ne -1\), the hypergeometric function can be simplified, too, and one gets from (26) in the limit \(a\rightarrow \infty \):

*and*big rip.

*a*, see also Fig. 2b. Thus in contrast to the vacuum solution, this universe isotropizes for late times. For small

*a*, the asymptotic behavior corresponds to (20), which is again independent of the matter content. This property is sometimes called “matter doesn’t matter”. Since the Kasner behavior is recovered in the limit \(a\rightarrow 0\), this approach to the singularity is referred to as asymptotically velocity term dominated (AVTD) [30].

*k*. We conclude that the quantum Kasner behavior is recovered in this limit (which follows as a solution of (23)).

The decay of the mode functions in this representation can be intuitively understood by inspecting the Hawking–Page formula (10): the representation of the wave function \(\Psi \) we are working with is related to the gauge \(N=\hbox {e}^{3\alpha }\) by the corresponding representation of the DeWitt metric. In this gauge, classical solutions reach \(\alpha =\infty \) in a finite time *t*. Hence they spend less and less time *t* in the region of minisuperspace where \(\alpha \) is large. In this sense the decay of the density \(\sqrt{-\mathcal {G}}D\) is implied by (10).

*w*, that is, there is no difference between the cases \(w \ge -1\) and \(w < -1\), although the latter case leads to a big rip. The big rip is thus only avoided by criterion 1.

## 5 Bianchi I model with a phantom field

*p*are as follows:

*a*(

*t*) and \(\beta _\pm (t)\) as in the previous section. We use here \(\kappa _\pm \) instead of \(p_\pm \), because these constants will be used in the construction of \(V(\phi )\). Combining (36), (25), \(p=w\rho \), and \(\rho (a)\) we get for the classical solution in configuration space,

*a*and diverges logarithmically for large

*a*.

*a*by \(\phi \) and using (37) we find

*a*and therefore large \(|\phi |\); that is, we investigate the limit when approaching the big rip. This gives

The linearization (46) breaks down when the absolute value of \(\alpha -|\phi |\) becomes large. We then perform a numerical integration of the full wave packet^{6} \(|\Psi |\hbox {e}^{\frac{3}{2}(\alpha +|\phi |)}\). Figure 5a shows the results for \(\alpha =10\) and different values of \(|\phi |\). For \(|\phi |\approx 10.75\), the wave packet has a global maximum which corresponds to the analytical result (47). For increasing \(|\phi |\), the wave packet assumes an annular shape and propagates outwards with decreasing amplitude. The wave is peaked in the direction of negative \(\beta _\pm \). For decreasing \(|\phi |\), one has a similar behavior with the maximum moving into the opposite direction. Note that this annular waves also appear for the corresponding wave packet of the Kasner solution. The dispersion takes place due to the additional degrees of freedom introduced by the anisotropy.

In Fig. 5b we display the maxima of the wave packet for different \(|\phi |\) together with a Gaussian fit to the peak region. One can see that close to the peak the wave packet decreases like a Gaussian, but decays much weaker (not even exponentially) further away. The amplitude of the full wave packet \(|\Psi |\) including the van Vleck determinant will increase for decreasing \(|\phi |\) such that the peak along the classical trajectory will be at best a local maximum. This might be interpretable as a transition from a semiclassical into a full quantum regime.

## 6 Conclusion

In this paper, we have extended previous investigations of singularity avoidance from isotropic to anisotropic models. We have, in particular, adapted the avoidance criterion to the covariant structure of minisuperspace, which becomes relevant for dimensions higher than two. We have found that the DeWitt criterion can, in general, be fulfilled, but not so the vanishing of the Klein–Gordon current. For the reasons mentioned, however, we attribute more relevance to the DeWitt criterion.

The flux criterion was recently applied in [38] to investigating the fate of (big bang and big crunch) singularities in FLRW models with Brown–Kuchař dust. Singularity avoidance was then found for a certain class of factor orderings.

In our paper, singularity avoidance was found by studying properties of the quantum cosmological Wheeler–DeWitt equation. The structure of this differential equation was used to disclose the fate of both the big bang and the big rip singularities. The tacit assumption in this is that information about the fates of different types of singularities can be obtained from one and the same differential equation (in the same way as information about different types of classical singularities can be obtained from the same Friedmann–Lemaître equations). A full mathematical treatment should address the properties of the Wheeler–DeWitt equation and its boundary conditions in much more detail, pointing out structural differences between the singularities, but this is beyond the scope of this paper.

While the general criteria were formulated for general Bianchi class A models, detailed investigations were made for the Bianchi I model with and without matter. Bianchi I models admit the prototype of an asymptotically velocity term dominated (AVTD) model. Our results of singularity avoidance should thus be representative for such a kind of singularity with a sufficiently large number of degrees of freedom. Other Bianchi models such as Bianchi VIII and Bianchi IX exhibit an oscillatory behavior when approaching the singularity. The singularity can, however, become AVTD if, for example, a scalar field is added [30].

Because Bianchi IX models are generally considered as reflecting the generic behavior towards a spacelike singularity, future investigations of singularity avoidance should attempt to address these models in as much detail as possible. For this, it would be desirable to have mathematical theorems available such as those discussed here for the Kasner solution. In [39] one finds an existence and uniqueness theorem (Theorem 8.6 there), but for the Bianchi IX potential no decay rate estimates seem to exist. The Wheeler–DeWitt equation for the vacuum Bianchi IX (mixmaster) model was solved numerically in [40] by using the ‘hard wall approximation’. The results found in this analysis strongly indicate the decay of wave packet amplitudes.

It is generally believed that the approach to a spacelike singularity at the level of the full Einstein equations can be described by the Belinsky–Khalatnikov–Lifshits (BKL) scenario; see, for example, [30, 41] and references therein. This corresponds to a decoupling of spatial points, in which every spatial point exhibits the dynamics of a separate Bianchi IX model. The eventual goal will be to present a quantum-gravitational analysis of this situation, from which one should be able to draw general conclusions about singularity avoidance. An investigation in the framework of affine quantization was made recently in [42]. Attempts in this direction using the Wheeler–DeWitt equation will be the subject of future investigations.

## Footnotes

## Notes

### Acknowledgements

We are grateful to Anne Franzen and Tim Schmitz for helpful discussions.

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