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Quantum Propagation Across Cosmological Singularities

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Quantum Propagation Across Cosmological Singularities Quantum propagation across cosmological singularities Steffen Gielen Theoretical Physics, Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada and Canadian Institute for Theoretical Astrophysics (CITA), 60 St George Street, Toronto, Ontario M5S 3H8, Canada Neil Turok Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada (Dated: May 14, 2017) The initial singularity is the most troubling feature of the standard cosmology, which quantum effects are hoped to resolve. In this paper, we study quantum cosmology with conformal (Weyl invariant) matter. We show that it is natural to extend the scale factor to negative values, allowing a large, collapsing universe to evolve across a quantum \bounce" into an expanding universe like ours. We compute the Feynman propagator for Friedmann-Robertson-Walker backgrounds exactly, identifying curious pathologies in the case of curved (open or closed) universes. We then include anisotropies, fixing the operator ordering of the quantum Hamiltonian by imposing covariance under field redefinitions and again finding exact solutions. We show how complex classical solutions allow one to circumvent the singularity while maintaining the validity of the semiclassical approximation. The simplest isotropic universes sit on a critical boundary, beyond which there is qualitatively different behavior, with potential for instability. Additional scalars improve the theory's stability. Finally, we study the semiclassical propagation of inhomogeneous perturbations about the flat, isotropic case, at linear and nonlinear order, showing that, at least at this level, there is no particle production across the bounce. These results form the basis for a promising new approach to quantum cosmology and the resolution of the big bang singularity. PACS numbers: 98.80.Qc, 04.60.Kz, 98.80.Cq, 04.20.Dw I. INTRODUCTION the running of couplings, although these are no doubt important. Our focus is on the technical calculation of the causal (Feynman) propagator in some specific (and On the largest scales we can observe, our Universe quasirealistic) cosmologies. We also postpone a discus- has a remarkably simple structure. It is homogeneous, sion of the important question of the interpretation of isotropic, and spatially flat to very high accuracy. Fur- the propagator and its use to compute probabilities to thermore, the primordial curvature fluctuations which future work. Nevertheless, we believe our findings are seeded the formation of structure apparently took an instructive and form a useful starting point for such in- extremely minimal form: a statistically homogeneous, vestigations. Gaussian-distributed pattern of very small-amplitude The simplest example of conformal matter is a perfect curvature perturbations, with an almost perfectly scale- fluid of radiation. In the context of cosmology, this is invariant power spectrum. While inflationary models are extremely well motivated since the early Universe was, we capable of fitting the data, it is nonetheless tempting to believe, radiation dominated. Furthermore, if we add a look for a simpler and more fundamental explanation. single scalar field then at least at a classical level, minimal The early Universe was dominated by radiation, a form coupling is equivalent to conformal coupling under field of matter without an intrinsic scale. In fact, it is believed redefinitions. So this case too may be considered as an that any well-defined quantum field theory must possess example of conformal matter. It is then instructive to a UV fixed point, signifying conformal invariance at high extend the discussion to include an arbitrary number of energies. These lines of argument encourage us to investi- conformally and minimally coupled free scalar fields. gate a minimal early universe cosmology, namely a quan- tum universe filled with conformally invariant matter [1]. Since the matter Lagrangian of interest is, by assump- We start by studying the quantum propagation of ho- tion, conformally (Weyl) invariant at a classical level, it mogeneous background universes, uncovering a number makes sense to \lift" general relativity (GR) to a larger of surprising features. Then, we include inhomogeneous theory possessing the same symmetry. This is done by perturbations, treated semiclassically and perturbatively introducing an extra scalar field which is locally a pure at both linear and nonlinear order. We do not, in this gauge degree of freedom. The full theory is now clas- paper, propose a realistic scenario. Nor do we proceed sically Weyl invariant and it may be viewed with ad- far enough to study the effects of renormalization and vantage in various Weyl gauges. Its solutions contain 2 all solutions of GR but the theory allows for extended The theory we consider consists of Einstein gravity and more general solutions that do not possess a global plus radiation and a number of free scalar fields. All gauge fixing to GR. In particular, it turns out that while of these forms of matter satisfy the strong energy con- classical cosmological (homogeneous and isotropic) back- dition, and any cosmological solution necessarily pos- ground solutions are typically singular and geodesically sesses a big bang singularity. However, there is no singu- incomplete in Einstein gauge (the Weyl gauge in which larity in the Feynman propagator for closed, open and the gravitational action takes Einstein-Hilbert form, with flat Friedmann-Robertson-Walker (FRW) and for flat, a fixed Newton's gravitational constant G), they are reg- anisotropic Bianchi I cosmologies; the propagator is well ular and geodesically complete in a more general class of behaved across a bounce representing a transition from Weyl gauges, where G is no longer constant and can even a large, collapsing classical universe to a large, expand- change sign. In these gauges, generic cosmological back- ing one. We also study inhomogeneous perturbations of ground solutions pass smoothly through the \big bang the isotropic, radiation-dominated universe, studying the singularity" and into new regions of field space, includ- propagation of scalar and tensor perturbations across the ing \antigravity" regions where G is negative [2]. Strictly bounce, at linear and nonlinear order. We find that, in speaking, the lifted classical theory is still incomplete the semiclassical approximation at least and with strictly because these highly symmetrical backgrounds are un- conformal matter, the incoming vacuum state evolves stable to perturbations in the collapsing phase, leading into the outgoing vacuum state, with no particle pro- to diverging anisotropies which cannot be removed via a duction. The bounce may be viewed as an example of Weyl gauge choice. It was argued nevertheless that the quantum-mechanical tunneling, and we use complex clas- classical theory possesses a natural continuation across sical solutions as saddle points to the path integral, in a singularities of this kind [3]. manner which generalizes the use of instanton solutions to describe tunneling in more familiar contexts. Some of In this paper, following [1], we take a different tack. these results were anticipated in Ref. [1]; here we present We ask whether quantum effects might rescue the theory more details on their derivation, extend them to further from its breakdown at big bang-type singularities. First, cases not discussed in Ref. [1], and provide more mathe- we show that for the simplest types of conformal matter, matical and conceptual background. An alternate inter- the Feynman propagator for cosmological backgrounds pretation of the antigravity regions, not employing com- may be computed exactly, allowing us to explore many plex solutions, has been presented in Ref. [4]. issues with precision in this symmetry-reduced (minisu- The simplest example of a \perfect bounce" is provided perspace) context. Second, we extend the discussion to by a spatially flat, homogeneous and isotropic FRW uni- include anisotropies, resolving the ordering problems in verse, filled with a perfect radiation fluid. Adopting con- the quantum (Wheeler-DeWitt) Hamiltonian by impos- formal time (denoted by η), i.e., choosing the line element ing covariance under field redefinitions. We go further to be than the treatment of [1] by defining the quantum the- ory along the real axis in superspace, and discovering ds2 = a2(η)(−dη2 + d~x2) ; (1) some remarkable features. For a flat, isotropic universe, quantum effects indeed become large near the singular- one finds the scale factor a(η) / η. One way to see this is ity. Therefore, strictly speaking, one should not attempt from the trace of the Einstein equations. For the line el- to employ the real classical theory there. Instead, one ement (1), R = 6(d2a/dη2)=a3 and for conformal matter, can solve the quantum Wheeler-DeWitt-type equation the stress tensor is traceless. So the Einstein equations for the propagator in complexified superspace, along a imply a(η) / η. If spatial curvature is included, it is contour which avoids the singularity in taking one from subdominant at small a so that a still vanishes linearly the incoming collapsing universe to the outgoing expand- with η. Hence a(η) / η is a direct consequence of con- ing one. Provided the contour stays sufficiently far from formal invariance and cosmological symmetry. While the the singularity, the semiclassical approximation remains line element (1) clearly contains a big bang/big crunch valid all along it, so one can employ complex solutions singularity at η = 0, it is regular everywhere else, not of the classical theory to follow the quantum evolution only along the real η-axis but also in the entire complex across (or more accurately, around) the singularity. In η plane. Any complex η trajectory that connects large doing so, we find that while quantum effects are large negative and large positive a while avoiding a = 0 gives a near the singularity, they take a very special form such regular, complexified metric that asymptotes to a large, that they are \invisible" in the evolution between incom- Lorentzian universe in the past and the future, while cir- ing and outgoing states.
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