Superpositions of the cosmological constant allow for singularityresolution and unitary evolution in quantum cosmology

Sean Gryb1, 2, ∗ and Karim P. Y. Thébault1, †

1Department of Philosophy, University of Bristol2H. H. Wills Physics Laboratory, University of Bristol

(Dated: January 17, 2018)

A novel approach to quantization is shown to allow for superpositions of the cosmological constantin isotropic and homogeneous mini-superspace models. Generic solutions featuring such superpo-sitions display: i) a unitary evolution equation; ii) singularity resolution; iii) a cosmic bounce.Explicit cosmological solutions are constructed. These exhibit characteristic bounce features in-cluding a ‘super-inflation’ regime with universal phenomenology that can naturally be made to beinsensitive to Planck-scale physics.

PACS numbers: 04.20.CvKeywords: quantum cosmology, singularity resolution, problem of time

Introduction The ‘big bang’ singularity and the cos-mological constant are well established features of clas-sical cosmological models [1]. In the context of quan-tum cosmology, the singularity is typically understoodas a pathology that can be expected to be ‘resolved’ byPlanck-scale effects. Most contemporary approaches toresolving the singularity are based upon cosmic bouncescenarios [2]. In contrast, the cosmological constant re-ceives very much the same treatment in classical andquantum cosmological models: it is a constant of na-ture classically, and thus quantum solutions are supers-selected to eigenstates labelled by its classical value. Cos-mological time evolution is unlike either the singularityor cosmological constant. Whereas, its classical treat-ment is relatively unproblematic, quantum cosmologiesbased upon the standard Dirac quantization techniquesare described by a ‘frozen formalism’ that lacks a fun-damental evolution equation [3–5]. In this letter we usea simple model to demonstrate that by treating the cos-mological constant differently in quantum cosmologicalmodels, there is a prospect to produce a bounce scenariothat simultaneously resolves the classical singularity andrestores fundamental quantum time evolution.

In the following sections we apply a novel quantiza-tion scheme [6–8] to a class of isotropic and homogeneousmini-superspace models. For these models our scheme isdemonstrated to allow for a superpositions of the cosmo-logical constant in a manner connected to the unimodularapproach to gravity [9, 10]. Three particularly notewor-thy features result directly from including solutions withsuperposition of cosmological constant. First, our modelfeatures an evolution equation for the entire quantumstate that is guaranteed to be unitary. This is in con-trast to internal time approaches to representing evolu-tion in quantum cosmology [11–18]. Second, the mecha-nism for singularity avoidance obtained does not involvethe introduction of a Planck-scale cutoff [19]. Rather,observable operators evolve unitarily and remain finitebecause they are ‘protected’ by the uncertainty princi-

ple. Third, characteristic features of the cosmologicalbounce persist into a ‘super-inflation’ regime that con-tains universal phenomenology that can be rendered in-sensitive to the underlying Planck-scale physics in verynature way. In particular, the model displays a ‘cosmicbeat’ phenomenon and associated ‘bouncing envelope’.The cosmic beats can be identified with Planck-scale ef-fects and, under certain parameter constraints, are negli-gible compared with the effective envelope physics. Un-der these same constraints, the bouncing envelope per-sists into the super-inflation regime where it is insensitiveto the beat effects in a manner that is closely analogousto Rayleigh scattering. Significantly, this ‘Rayleigh’ limitis only available when superpositions of the cosmologi-cal constant are allowed. This behaviour constitutes aremarkable unique feature of the bouncing unitary cos-mologies identified. Two companion papers provide fur-ther, more detailed, interpretation and analysis of bothgeneral and particular cosmological solutions. [20, 21]

Model and Observables Consider an homogeneousand isotropic FLRW universe with zero spatial curvature(k = 0); scale factor, a; massless free scalar field, φ; andcosmological constant, Λ. The field redefinitions

v =

√2

3a3 ϕ =

√3κ

2φ , (1)

where κ = 8πG, give a convenient parameterization ofthe configuration space, C(v, ϕ), in terms of relative spa-tial volumes, v, and the dimensionless scalar field, ϕ. Thetime evolution of the system is given in terms of coordi-nate time, t, related to the proper time, τ , via the lapsefunction dτ = Ndt. The dimensionless lapse, Ñ , andcosmological constant, Λ̃, can be defined as

Ñ =

√3

2

κ~2vNV0

Λ̃ =V 20κ2~2

Λ , (2)

using the reference volume V0 of some fiducial cell andthe (at this point) arbitrary angular momentum scale ~.

2

In terms of these variables, the mini-superspace Hamil-tonian is

H = Ñ

[1

2~2

(−π2v +

1

v2π2ϕ

)+ Λ̃

], (3)

where πv and πϕ are the momenta conjugate to v andϕ respectively. The utility of these variables is re-vealed by their suggestion of a coordinate-independentformulation of H in terms of the Rindler metric ηAB =~2diag(−1, v2), where A,B = 1, 2. Using generalized co-ordinates, qA, and momenta, pA, the Hamiltonian can beexpressed as

H = Ñ

[1

2ηABpApB + Λ̃

]. (4)

The configuration space, C, is Rindler space defined asthe set of points contained in (and including) the forwardlight-cone of Minkowski space centred on the origin.

Rindler space is geodesically incomplete becausegeodesics cannot be extended past its boundary, ∂C, theRindler horizon at v = 0. This boundary leads to themost physically important properties, both classical andquantum, of this cosmological model.

It is important to distinguish between the geodesicincompleteness of the configuration space and that ofthe space-time metric, which are logically distinct. Inthis model, ∂C corresponds to the region in configura-tion where we it was shown in [20] that: i) the expan-sion parameter of some congruence of geodesics in space-time becomes negative and unbounded, implying that thespace-time geodesics terminate in finite proper time; andii) there is a curvature pathology in space-time signaledby a divergence in all Kretschmann invariants. This im-plies a classical singularity in both relevant senses of thePenrose–Hawking singularity theorems.

The importance of the boundary in the quantum the-ory relates to the existence of self-adjoint representationsof the operator algebra. Consider the Hilbert space,H = L2(C,dθ) of square integrable functions on C underthe Borel measure dθ = d2q

√−η, where η = det ηAB .

This space is spanned by all functions (Φ,Ψ) ∈ C satis-fying

〈Φ,Ψ〉 ≡∫C

d2q√−ηΦ†Ψ

3

states), we obtain the self-adjoint operators

µ̂Ψ = µΨ π̂µ = −i~e−µ∂

∂µ(eµΨ) (10)

ϕ̂Ψ = µΨ π̂ϕ = −i~∂Ψ

∂ϕ. (11)

whose eigenstates

ψπµ =1√2π~

e−i~µπµ−µ ψk =

1√2π~

e−i~ϕk (12)

are orthonormal under the measure dθ = dϕdµe2µ andspan H. It is straightforward to verify that the operatorsabove satisfy the self-adjointness condition (7) while themomentum operator π̂v in the vϕ-chart defined by (6)does not. This recovers the well-known result, studiedin detail by Isham [22], that the momentum operatoris not well-defined on R+. We believe these geometricmethods provide deeper insight into this problem andits conventional solution.

Unitary Quantum Cosmology Application of rela-tional quantization [6–8] leads to a Schrödinger-type evo-lution equation for the system of the form

ĤΨ = i~∂tΨ , (13)

where the eigenvalues of Ĥ are to be identified with the(dimensionless) cosmological constant Λ̃. The classicalHamiltonian (4) suggests the real and symmetric chart-independent Hamiltonian operator

Ĥ ≡ 12� , (14)

where � is the d’Alambertian operator on Rindler space.Unlike p̂A, Ĥ is a scalar on C. Diffeomorphisms onC therefore bijectively induce changes of basis of self-adjoint representations of Ĥ. Integration by parts canbe used to show that Ĥ is equal to its dual provided∮

∂Cdl(A√χηAB

(Φ†∂B)Ψ−Ψ∂B)Φ†

)= 0 (15)

for all states in (Φ,Ψ) ∈ H.A theorem by Von-Neumann (see [23] theorem X.3)

guarantees that self-adjoint extensions of the real, sym-metric operator Ĥ exist. Given an explicit self-adjointrepresentation of Ĥ, the time evolution is guaranteed tobe unitary by Stone’s theorem [24, p.264]. The deficiencysubspaces of Ĥ can be calculated from the square integralsolutions to (18) when Λ̃→ ±i. These are easily seen tobe one dimensional. We, therefore, expect a U(1) familyof self-adjoint extensions, which we parametrize by thelog-periodic, positive reference scale Λref. The coordinateinvariance of Ĥ implies that it is sufficient to constructrepresentations in a particular basis. To find these exten-sions explicitly and to construct the general solution to

(13), we compute the eigenstates of Ĥ (with eigenvaluesΛ̃) in the vϕ-chart. Using the separation Ansatz

Ψ±Λ (v, ϕ) = ψΛ,k(v)ν±k (k) , (16)

we find

ν±k (ϕ) =1√2π~

e±i~kϕ , (17)

and

vd

dv

(v

d

dvψΛ,k

)+

(2Λ̃v2 +

k2

~2

)ψΛ,k = 0 . (18)

The latter equation is Bessel’s differential equation forpurely imaginary orders, ik/~.

The solutions of Bessel’s equation are qualitatively dif-ferent depending on the sign of Λ. For Λ < 0, solu-tions are modified Bessel functions of the first (expon-tially growing) and second kind (exponentially decaying)kind. The self-adjointness condition, (15), leads us toreject the growing solutions, leaving only the decaying‘bound’ modes, Kik. The asymptotic expansion of theBessel functions about v = 0 further implies (see [20])that only discrete values of Λ̃ are allowed. These followthe geometric series

Λ̃n = Λ̃ref e2nπ~/k (∀n ∈ N) , (19)

which is seeded by the self-adjoint extension parame-ter Λ̃ref. The general normalized ‘bound’ eigenstates arethen

ψboundΛ,k =

√4~|Λ̃| sinh (πk/~)

πkKik/~(

√2Λ̃v) . (20)

For Λ > 0, solutions are the oscillating Bessel functionsof the first, Jik/~, and second kind, Yik/~. The condition(15) can be satisfied by analyzing the behaviour of theBessel functions near v = 0 from the perspective of theconformal completion, (C0, η0). There, the Bessel func-tions behave as ordinary sines and cosines whose phase

difference, θ = k2~ log(

Λ̃Λ̃ref(k)

), parametrizes the U(1)

space of self-adjoint extensions (see [20]).1 The generalnormalized solutions are continuous in Λ̃ and are explic-itly given by

ψundoundΛ,k =Re[e−iθJik/~(

√2Λ̃v)

]∣∣cosh (πk2~ + iθ)∣∣ . (21)

The 2π periodicity in θ implies a πk/~ log-periodicityin Λref that is consistent with the bound spectrum (19).

1 Note that, in general, different choices of θ can be made fordifferent values of k.

4

The general solution to (13) is then,

Ψ(v, ϕ, t) =1√2

∑±

∫ ∞−∞

dk

[ ∞∑n=−∞

eiΛ̃nt/~A±n (k)Ψ±,bound−Λn,k

+

∫ ∞0

dΛ e−iΛ̃t/~B±(Λ̃, k)Ψ±,unboundΛ,k

], (22)

for the suitably normalized coefficients A±n and B±(Λ̃).

Singularity Resolution In our view, the basic condi-tion for non-singular behaviour in a quantum theory isthat the expectation value of all observable operators, asevaluated on all possible states in Hilbert space, remainsfinite. Given a classical theory in which some classicalobservable is pathological, it is a necessary and sufficientcondition for singularity resolution, in our sense, that theexpectation value of all elements of the quantum observ-able algebra be always finite. This definition is equiva-lent to the requirement, in the sense of [25, 26], that theevolution on the quantum phase space be everywhere fi-nite. Thus, in requiring finiteness of expectation values,we are not implicitly relying upon extending the corre-spondence principle into the quantum bounce regime, butrather simply insisting that a physically reasonable quan-tum theory can be defined at all times.

It is straightforward to demonstrate that our modelsatisfies the finite-expectation-value condition for singu-larity avoidance. Given the self-adjointness of Ĥ, theunitary evolution equation (13) implies the generalisedEhrenfest theorem:

∂

∂t

〈Ô(t)

〉=

1

i~

〈[Ô(t), Ĥ]

〉+〈∂Ô(t)

∂t

〉. (23)

Provided that Ô is a bounded member of a well-definedquantum observable algebra, the commutator on theRHS is bounded and the evolution of the expectationvalue of all Ô will be well-behaved. Thus, for quantumcosmology with a unitary evolution equation, the condi-tion for singularity avoidance ultimately amounts to theusual requirement for a well-defined quantum theory.

Modeling Constraints The choice of physically rele-vant particular solutions is under-constrained by obser-vational data. Here we assume that constraints placedupon the model that are not based upon observationaldata should be minimally specific in the precise sense de-fined in [21]. Below, we will use this as a guiding principleto briefly justify the choices made for the free-parametersof the model. For much greater detail on the justificationfor these choices, see [21].

Observational data imply that the current universe iswell-approximated by a semi-classical state with a def-inite positive Λ. If the bound negative Λ states hadsignificant support at large v, then linearity would im-ply that these bound states would be currently observ-able. Since they are not, this restricts the bound part of

the wavefunction to be confined to a region of configura-tion space where v is much smaller than it is currently.Since we wish to be minimally specific with regard non-observational constrains, we set the bound part of thewavefunction to vanish by setting A±n (k) = 0.

We can characterise the semi-classical regime in a min-imally specific way by the vanishing of higher order gen-eralized moments of the wavefunction [26]. This is equiv-alent to requiring that the non-Gaussianties of the wave-function are very small in a particular basis. The min-imally specific choice of basis is that which is maxi-mally stable.2 This is provided by considering the large-vasymptotic Killing vectors of the classical configurationspace, which allow us to select a preferred basis givenin terms of the eigenstates of π̂ϕ and π̂v. Because, inthis asymptotic limit, H = 12~2π

2v , we take the semi-

classical state to be expressed in terms of Gaussians of k

(the eigenvalues of π̂ϕ) and ω =√

2Λ̃~ (the approximateeigenvalues of π̂v in the large-v limit).

Requiring Λ and πϕ to be well-resolved implies thatthe absolute value of the means of the scalar densi-ties B±(k, Λ̃) = ω~B

±(k, ω) must be much larger thanthe variances, otherwise the quantum mechanical uncer-tainty, given by σω and σk respectively, would make themindistinguishable from zero. This leads to:

ω

~B±(k, ω) =

(~2

2πσωσk

)1/2exp

{− (ω − ω0)

2

4σ2ω

− i~

(ω − ω0)v0 −(k − k±0 )2

4(σ±k )2− i~ (k − k

±0 )ϕ

±∞

}, (24)

where ω0 � σω > 0 and |k±0 | � σ±k > 0.

Two further minimally specific choices consistent withobservation are: i) to select t = 0 as the time of mini-mal dispersion by appeal to time-translational invariance;and ii) to assume a semi-classical regime for t → ±∞.Given current observational constraints, the quantumbounce wipes out the vast majority of the informationabout pre-bounce physics. The minimally specific as-sumption is, therefore, to impose the maximum amountof time-reflection symmetry around the bounce. This isachieved by: i) setting the phase shift between in- andout-going π̂ϕ-eigenstates to zero by setting B

+ = B− us-ing a single mean, k0, and variance; σk, and offset, ϕ∞;ii) requiring the bounce time to occur at t = 0 by set-ting v0 = 0; and iii) fixing the self-adjoint extensions tominimize the phase-shift between in- and out-going Ĥ-eigenstates (the specific choice that accomplishes this isspecified below).

2 In fact, what is ultimately needed is a super-selection principlefor such a basis, which would require a way to model an ‘environ-ment’ for this system. Lacking this, we note that our stabilitycriterion is at least consitent with definitions of environmentallyinduced super-selection arizing from decoherence.

5

We can use the global ‘boost’ isometry of C to restrictto ϕ∞ = 0 without loss of generality. The parameterpairs (k0, σk) and (ω0, σω) can only be independently de-fined via reference to an external scale. We can avoidhaving to specify such a scale by noticing that the Gaus-sians of (24) depend only on the ratios k0/σk and ω0/σω,which are well defined parameters of the model, whenv0 = ϕ∞ = 0.

Fixing the self-adjoint extensions by specifying θ doesrequire introduction of an external reference scale how-ever. Inspection of (21) reveals a dependence on k/~,which suggests k0/~ as the third parameter of the model.We will discuss the physical interpretation of this scalein relation to Planck-scale effects in the final section. Forour present purpose, it suffices to note that the choice:

Λref =V 20κ2~2

ω202~2

, (25)

is minimally specific since it does not involve introducingany new parameters. It is also the choice that allows θ tobe as close to zero as possible and, therefore, maximizestime-reflection symmetry.

de Sitter and Rayleigh limits Let us designate the

limit in which ω0/σω|k0|σk � 1 as the de Sitter limit and thelimit in which ω0/σω � 1 as the Rayleigh limit. In theRayleigh limit, Planck-scale effects will be found to benegligible in a manner analogous to the negligibility ofmolecular effects in Rayleigh scattering. In the de Sit-ter limit, the energy of the cosmological constant domi-nates over that of the scalar field when quantum effectsdue to Rayleigh scattering take over. The effective dy-namics is, therefore, dominated by the quantization ofa de Sitter geometry.3 This can be modelled by takingC(k) = δ(k/~). However, care must be taken becauseIm {J0} = 0, so it can no longer be taken as a linearlyindependent solution. Fortunately, Y0 is integrable andprovides an adequate second solutions. The self-adjointextensions are arbitrary phases, α, between these, andthe general wavefunction is

Ψ(v, t) =

∫ ∞0

dωω

~2eiω

2t/~3E(ω/~)(cos α2J0

(ωv~)

− sin α2Y0(ωv~)). (26)

In the combined Rayleigh and de Sitter limits, v and ωare approximately canonically conjugate. At the bouncewhen 〈v〉 is at a minimum, the wavefunction (26) willhave most of its support in the region v ∼ σv ∼ ~/σω.4

3 Because we have imposed spatial curvature equal to zero, therelevant geometry is the de Sitter half-plane, which has an initialsingularity.

4 The last approximation holds because, as we will see, the wave-function remains reasonably close to Gaussian during the bouncein this limit.

Thus,

vω ∼ ω0σω� 1 . (27)

In this limit, the Bessel functions can be expanded togive

cos α2J0 (ωv) + sinα2Y0 (ωv) ≈

√2

πωvcos (ωv −∆/2) ,

(28)where ∆ = π2 −α. Inserting a Gaussian function for E(ω)leads to

Ψ(v, t) = N∑±A±eiS

±, (29)

where N =(

2π

)1/4√ σω1+2iσ2ωt

.

A± = exp

{−σ

2ω (v ∓ ω0t)

2

1 + 4σ4ωt2

}

S± =±ω0v − ω

20t2 + 2σ

2ωv

2t

1 + 4σ4ωt2

∓∆/2 . (30)

The amplitudes A± and phases S± are those of free in-and out-going Gaussian wavepackets phase shifted by ∆.

The total Born amplitude in the vϕ-basis is the sum ofthe Born amplitudes of both in-, (A+)2, and out-, (A−)2,going envelopes plus an interference term of the form:

2A+A− cos

(2ω0v −∆1 + 4σ4ωt

2

). (31)

The interference indicates that the beat frequency is pro-portional to ω0 in v-space when A

+ and A− overlap. Thisbeat frequency implies that there are many beats in asingle envelope of size ∼ σv, and confirms that the beateffects should be attributed to the micro- (i.e., Planck-scale) physics of the system. It also follows that the in-terference term can be approximately ignored when com-puting expectation values, which are integrals over v. Wecan, therefore, use a variety of analytic techniques tocompute the mean

〈v̂〉 ≈√

2

πe−ω

20t

2/2σ2vσv + ω0t erf

(ω0t√2σv

), (32)

and variance

Var(v̂)2 = σ2v + ω20t

2 − 〈v̂〉2 , (33)

of v̂, where σv(t) ≡√

1+4σ4ωt2

2σω. This behaviour can be

checked against numerically computed expectation val-ues (see [20]), and shows excellent agreement. FIG. 1illustrates the general behaviour.

6

1 2 3 4t/tenv0.5

1.0

1.5

2.0

2.5

FIG. 1: 〈v̂〉 (t) for ω0/σω = 5 with confidence intervalcomputed from Var(v̂). The expectation value (solid),〈v̂〉, follows the classical curve (dashed) until v ∼ ~/σω,when quantum effects due to Rayleigh scattering takeover. The minimum, vmin =

1√2πσω

of 〈v̂〉 is reached atthe bounce time, t = 0. (Note: tenv ≡ 1/2σωω0.)

Bouncing Cosmology Given the log-periodicity ofΛref, the limit

e|k0|/~ � eω0/σω (34)

implies that for any choice of Λref, there is an equivalentone imperceptibly close to Λ0. The limit (34), therefore,implies that the self-adjoint extension behaviour becomesuniversal. Combined with the Rayleigh limit, (34) is suchthat the scalar field momentum is sufficiently large inunits of ~ and is reasonably dominant, at early times,over the effects of the cosmological constant.

In this limit, the normalization of the unboundeigenstates, (21), simplifies to sech

(πk2~), which is ω-

independent. If we regularize the Gaussian of E(ω) interms of the function

E(ω) ≈(

~2√2πσωω

)1/2(ω

ω0

)ω20/4σ2ω× exp

{−2ω

20

σ2ω

[(ω

ω0

)2− 1

]}, (35)

which is a good approximation to a Gaussian in theRayleigh limit, the ω0-space integrals can be evaluatedanalytically in terms of confluent hypergeometric func-tions F1 1 . The explicit form of the result of this integra-tion is unilluminating and can be found explicitly in [21].The remaining integral reduces to a Fourier transform ink. The Fourier transform can be evaluated using the FastFourier Transform (FFT) algorithm after cutting off thek-integral at ±6σk from k0 and sampling at the Nyquistfrequency, fs. Modest oversampling (i.e., 2fs) allows forstandard spline interpolations of the Fourier transformedfunction. Plotting and numerical integrations of variousfunctions of Ψ can be performed using these interpola-tions in reasonable computation times.

To analyze the resulting solutions, we consider the ef-fect of the three independent parameters k0/~, ω0/σk,and k0/σk separately. The choice of self-adjoint extension(25) minimizes the phase difference between in- and out-going modes due to non-zero k0/~. We, therefore, expectthis choice to lead to a negligible correction to the beatfrequency as predicted by the interference term, (31), ofthe de Sitter model. Explicit comparison of the Bornamplitudes of the wavefunction in the vϕ-basis for mod-est parameter values5 confirms this expectation. Thisprovides numerical evidence that the Rayleigh limit, un-derstood analytically in the de Sitter solution, persistswhen k0~ 6= 0.

The parameter ω0/σω should be expected to controlthe beat frequency according to (31), given that the over-lap between in- and out-going envelopes occurs in the re-gion v ∼ ~/σω. To verify this, we can plot (see FIG 2)the Born amplitude of the wavefunction in the vϕ-basisat t = 0, where the overlap is maximum. Comparisonof the beat frequency for different values of ω0/σω is inexcellent agreement with the de Sitter results.

The parameter k0/σk controls how tightly the individ-ual envelopes stay peaked on the classical solutions. Thiscan be studied by varying the parameter s = k0/ω0 forfixed ω0/σω and k0/~ and parametrically plotting 〈v̂〉 /sand 〈ϕ̂〉. The advantage of this choice of parameteriza-tion of the quantum solutions in terms of s is that theclassical equations of motion can be written parametri-cally as

v

s= |cosech (ϕ− ϕ∞)| . (36)

Thus, the quantum curve for different choices of s can becompared with the same universal classical curve. FIG 3illustrates the relevant features. The expectation val-ues begin to diverge from their classical values in theregion v ∼ 1/σω as expected. The expectation value of ϕ̂reaches a maximum value, which increases as s increases.The expectation value of v̂ reaches a minimum at t = 0as expected.

Prospectus Following [27, 28], we can connect thephysics of our model to inflationary cosmology by consid-ering an effective Hubble parameter, He, given by a func-tion of the expectation value of φ̂. Because the Hubbleparameter, as a phase space function, is proportional toπv, this translates into computing the expectation valueof π̂v as a function of the expectation value of ϕ̂. Effectiveslow-roll parameters, �He and ηHe

�He(φ) =m2Pl4π

(H ′e(φ)

He(φ)

)2ηHe(φ) =

m2Pl4π

H ′′e (φ)

He(φ), (37)

can then be conveniently expressed in terms of the ex-pectation values computed in our model.

5 E.g., ω0/σω = 10, k0/σk = 10, ~ = 1, 2

7

(a) v|Ψ|2 for ω0/σω = 10, s = 1

(b) v|Ψ|2 for ω0/σω = 15, s = 1

FIG. 2: Comparison of Born amplitude at bounce timefor different choices of ω0 and s (for σω = σk = ~ = 1).

The beat physics is affected in the same way by ω0 as itwas in the de Sitter limit.

The curve of FIG. 3 shows a reasonably flat regionnear the maximum of 〈ϕ̂〉 indicating a modestly stablede Sitter-like epoch of super-inflation. The existenceof the Rayleigh limit suggests that this super-inflationepoch could be found to take place far below the Planckenergy. This leaves open the possibility that the super-inflation of our model could be connected to power-spectrum data of the CMB. Because analytic methodsbreak down in precisely the super-inflation regime, nu-merical techniques are required for such an identifica-tion. It is hoped that the existence of the Rayleigh limit,which is an exclusive feature of our model, may avoid in-stabilities and other issues found in existing models withsuper-inflation. These computations will be the subjectof future investigations.

The general features of our quantization can be ap-plied to the unitary quantization of anisotropic Bianchimodels [29]. While the extension to Bianchi I is almosttrivial, Bianchi IX models will lead to modified Besselequations. However, the asymptotic behaviour of the

0.0 0.2 0.4 0.6 0.8 1.0 1.20.0

0.5

1.0

1.5

2.0

FIG. 3: Plot of 〈v̂〉 /s versus 〈ϕ̂〉 for different values ofs. The top blue line represents s = 1, the bottom blue

line s = 2, and the yellow line is the classical curve withω0 = 10. Increasing s can be seen to decrease vmin andincrease ϕmax. Changing ω0 has negligible effect. Thefigure is symmetric upon the reflection ϕ→ −ϕ, which

represents t→ −t.

wavefunction near the singularity and near the late-timeattractors (i.e., the large v limit) will be identical tothe model treated here. Since the construction of theself-adjoint extensions depends on the behaviour of thewavefunction near v = 0 and since the existence of thesemi-classical approximation depends on the Gaussian-ity of the wavefunction near the late-time attractors, onemay expect that many of the qualitative features of thepresent model will carry forward to unitary solutions ofthe Bianchi IX model that persist semi-classically to thelate-time attractors. The Bianchi IX model may be par-ticularly valuable for studying general singularity resolu-tion in quantized GR in light of the BKL conjecture [30].Such a framework may be useful for studying singular-ity resolution of time-like singularities via, for example,black-to-white hole transitions.

Inclusion of a non-trivial potential for φ will have asimilar effect on the Bessel equation as the Bianchi IXmodel, and can be handled similarly.

8

FUNDING

We are very grateful for the support from the Insti-tute for Advanced Studies and the School of Arts at theUniversity of Bristol and to the Arts and Humanities Re-search Council (Grant Ref. AH/P004415/1). S.G. wouldlike to acknowledge support from the Netherlands Or-ganisation for Scientific Research (NWO) (Project No.620.01.784) and Radboud University. K.T. would like tothank the Alexander von Humboldt Foundation and theMunich Center for Mathematical Philosophy (Ludwig-Maximilians-Universität München) for supporting theearly stages of work on this project.

We are appreciative to audiences in Bristol, Berlin,Geneva, Harvard, Hannover, Nottingham and thePerimeter Institute for comments. We also thank Hen-rique Gomes, David Sloan, and Martin Bojowald for help-ful comments.

∗ sean.gryb@gmail.com† karim.thebault@bristol.ac.uk

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Superpositions of the cosmological constant allow for singularity resolution and unitary evolution in quantum cosmologyAbstractFundingAcknowledgmentsReferences