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Bouncing Unitary Cosmology I. Mini-Superspace General Solution

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Bouncing Unitary Cosmology I. Mini-Superspace General Solution Bouncing Unitary Cosmology I. Mini-Superspace General Solution Sean Gryb1, 2, ∗ and Karim P. Y. Th´ebault1, y 1Department of Philosophy, University of Bristol 2H. H. Wills Physics Laboratory, University of Bristol (Dated: January 17, 2018) We offer a new proposal for cosmic singularity resolution based upon a quantum cosmology with a unitary bounce. This proposal is illustrated via a novel quantization of a mini-superspace model in which there can be superpositions of the cosmological constant. This possibility leads to a finite, bouncing unitary cosmology. Whereas the usual Wheeler{DeWitt cosmology generically displays pathological behaviour in terms of non-finite expectation values and non-unitary dynamics, the finiteness and unitarity of our model are formally guaranteed. For classically singular models with a massless scalar field and cosmological constant, we show that well-behaved quantum observables can be constructed and generic solutions to the universal Schr¨odingerequation are singularity-free. Key features of the solutions of our model include a cosmic bounce due to quantum gravitational effects, a well-defined FLRW limit far from the bounce, and a super-inflationary epoch in the intermediate region. Furthermore, our model displays novel features including: i) superpositions of values of the cosmological constant; ii) a non-zero scattering length around the big bounce; and iii) bound ‘Efimov universe' states for negative cosmological constant. The last feature provides a new platform for quantum simulation of the early universe. A companion paper provides detailed interpretation and analysis of particular cosmological solutions. PACS numbers: 04.20.Cv Keywords: quantum cosmology, singularity resolution, problem of time CONTENTS I. INTRODUCTION I. Introduction1 The `big-bang' cosmic singularity can be characterised classically in terms of both the incompleteness of causal II. Relational Quantization3 (i.e., non-spacelike) past-directed curves and the exis- tence of a curvature pathology [1{4]. That the pathology III. Classical Mini-Superspace Cosmology4 in question is physically problematic is unambiguously A. Configuration Space Geometry4 demonstrated by the existence of scalar curvature invari- B. Classical Singular Behaviour5 ants that grow without bound in finite proper time along C. General Explicit Solutions7 the incomplete curves in question [5,6]. Such patholog- IV. Quantum Mini-Superspace Cosmology9 ical behaviour in observable classical quantities can be A. Algebra of Observables9 understood to signal the breakdown of general relativ- B. Hamiltonian 11 ity and, thus, the requirement for new theoretical tools. 1. `Bound States' (Λ < 0) 12 A reasonable hope is that the cosmic singularity prob- 2. `Unbound States' (Λ > 0) 13 lem might be resolved via the introduction of inflationary 3. Critical Case (Λ = 0) 15 mechanisms [7,8]. In particular, one might expect that C. Bouncing Unitary Cosmology 15 eternal inflation models could resolve the initial singular- D. Efimov Analogue Cosmology 16 ity [9]. However, the Penrose-Hawking singularity theo- rems can, in fact, be extended to show that a broad range V. Singularity Resolution by Quantum Evolution 16 of `physically reasonable' eternal inflationary spacetimes are necessarily geodesically past incomplete and there- VI. Conclusions 18 fore singular in the relevant sense [10]. There are thus reasons to expect a `big-bang' singularity to exist prior Funding 18 to the inflationary phase of such models. A complete de- scription of the early universe requires us to introduce Acknowledgments 18 new physics that goes beyond inflation. References 18 Most contemporary approaches to resolving the cosmic singularity problem are based upon cosmic bounce sce- narios. In such models the classical singularity is replaced by a new pre-big bang t ! −∞ epoch. Big bounce cos- ∗ [email protected] mologies have hitherto been proposed based upon stringy y [email protected] effects [11{15], path integral techniques [16, 17], loop ap- 2 proaches [18{21], and group field theory [22{24]. Whilst, the universe [34]. An equation of the same form was in bouncing cosmologies can be combined with inflation { fact derived some time ago in the context of unimodu- giving mixed scenarios { a particular attraction is that lar gravity [35]. That earlier treatment did not include the big bounce can potentially replace inflation as the a detailed analysis of generic or specific cosmological so- mechanism for solving many of the problems of standard lutions, an explicit construction of the observable oper- big bang cosmology [25]. The horizon problem, in par- ators, or an investigation of the fate of the singularity. ticular, is automatically solved in bouncing cosmologies, More problematically, the quantization did not include since the observed isotropy of the cosmic microwave back- the self-adjoint representation of the Hamiltonian nec- ground can be explained simply by causal interactions in essary to guarantee unitarity via appeal to Stone's the- the pre-big bang epoch. Bouncing cosmologies are thus orem. In our approach, the Hamiltonian will be given attractive as either a supplement to or replacement of the an explicit self-adjoint representation. Furthermore, for inflationary cosmological paradigm. classically singular models with a massless scalar field In this paper we propose an alternative quantization and cosmological constant Λ, well-behaved quantum ob- procedure for cosmology in which superpositions of the servables will be constructed and generic solutions to cosmological constant are allowed. This possibility al- our `universal Schr¨odingerequation' will be shown to lows for a new model of bouncing cosmology: a bouncing be singularity-free. Key features of the solutions of our unitary cosmology. Our approach is based upon canon- model include a cosmic bounce due to quantum gravi- ical quantization of isotropic and homogeneous mini- tational effects, a well-defined FLRW limit far from the superspace models but differers significantly from the bounce, and a super-inflationary epoch in the intermedi- two standard canonical treatments of such models. The ate region. Our bounce scenario is phenomenologically first, older approach is based upon a Dirac quantization distinct form those studied in the literature { e.g. those of mini-superspace expressed in terms of ADM variables found in LQC [18]. In particular, our model displays [26, 27], and leads to a Wheeler{DeWitt mini-superspace novel features including: i) superpositions of values of quantum cosmology. Simple models, where one consid- the cosmological constant; ii) a non-zero scattering length ers a massless scalar field with zero spatial curvature and around the big bounce; and iii) bound ‘Efimov universe' non-negative cosmological constant, can be shown ex- states for negative cosmological constant. The last fea- plicitly to contain observable operators with divergent ture provides a new platform for a `few-body' quantum expectation values [28]. In this strong sense, the big simulation [36, 37] of the early universe. bang singularity is not resolved in even the most basic The remainder of the paper is organised as follows. 1 class of mini-superspace quantum cosmologies. The sec- Section xII provides a brief overview of the alternative ond, newer approach is inspired by the techniques used relational quantization procedure upon which our treat- in Loop Quantum Gravity and is known as Loop Quan- ment of quantum mini-superspace is based. Section xIII tum Cosmology [18{21]. Loop Quantum Cosmology re- presents the main formal details of the classical mini- lies upon a `polymer' representation of the observable superspace model. Our initial focus, in Section xIII A, is algebra of mini-superspace and involves the introduction upon developing a coordinate-free representation of the of a Planck-scale cutoff for the problematic operators of configuration space and its boundary. In Section xIII B, the Wheeler{DeWitt theory. The same models that ex- we then consider the nature of the classical singularity hibit pathological behaviour under the Wheeler{DeWitt and introduce the `tortoise' coordinate chart that pro- treatment can be shown to feature operators with finite vides a conformal completion of the configuration space. expectation values when treated in Loop Quantum Cos- Finally, in Section xIII C, we give an explicit analysis of mology [28]. In this precise sense, the big bang singular- the general solutions in order to both isolate their patho- ity is resolved in Loop Quantum Cosmologies for models logical features and offer suggestions as to why one might where the Wheeler{DeWitt treatment fails. expect such features to persist in the quantum treatment. In this paper, we will offer a new proposal for singu- The considerable care taken in setting up the classical larity resolution in mini-superspace quantum cosmology analysis will reap benefits in the quantum theory. In that is based upon evolution. Our method does not make particular, our classical analysis will be instrumental for use of polymer methods. Rather, the observable opera- isolating and resolving the various quantum pathologies tors in our theory evolve unitarily and remain finite be- that are encountered. cause they are `protected' by the uncertainty principle. The unitary evolution of operators in our model is based We start, in Section xIV A, by solving the problem upon a novel approach to the quantization of ADM mini- of defining an algebra of self-adjoint quantum observ- superspace that leads to a Schr¨odinger-type equation for ables. Then, as detailed in Section xIV B, we construct a self-adjoint `Wheeler{DeWitt' Hamiltonian operator and analyse three families of eigenfunctions that are distin- guished based upon the value of the cosmological con- 1 Although there are good general reasons to expect singularities to persist in Wheeler{DeWitt mini-superspace cosmologies, forms stant. Particularly noteworthy is the Λ < 0 family which of singularity resolution have been shown to hold for models with has a mathematical form that mirrors that of bound Efi- Brown-Kuchaˇrdust fields [29, 30]. See [31{33] and xV for further mov states [38] found in few-body quantum physics. Pro- discussion.
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