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Article: Gielen, S. orcid.org/0000-0002-8653-5430 and Turok, N. (2017) Quantum propagation across cosmological singularities. Physical Review D, 95 (10). ISSN 2470-0010 https://doi.org/10.1103/physrevd.95.103510

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[email protected] https://eprints.whiterose.ac.uk/ PHYSICAL REVIEW D 95, 103510 (2017) 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 (Received 11 March 2017; published 18 May 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.

DOI: 10.1103/PhysRevD.95.103510

I. INTRODUCTION these are no doubt important. Our focus is on the technical calculation of the causal (Feynman) propagator in some On the largest scales we can observe, our Universe has a specific (and quasirealistic) cosmologies. We also postpone a remarkably simple structure. It is homogeneous, isotropic, interpretation and spatially flat to very high accuracy. Furthermore, the discussion of the important question of the of primordial curvature fluctuations which seeded the for- the propagator and its use to compute probabilities to future mation of structure apparently took an extremely minimal work. Nevertheless, we believe our findings are instructive form: a statistically homogeneous, Gaussian-distributed and form a useful starting point for such investigations. pattern of very small-amplitude curvature perturbations, The simplest example of conformal matter is a perfect with an almost perfectly scale-invariant power spectrum. fluid of radiation. In the context of cosmology, this is While inflationary models are capable of fitting the data, it extremely well motivated since the early Universe was, we is nonetheless tempting to look for a simpler and more believe, radiation dominated. Furthermore, if we add a fundamental explanation. The early Universe was domi- single scalar field then at least at a classical level, minimal nated by radiation, a form of matter without an intrinsic coupling is equivalent to conformal coupling under field scale. In fact, it is believed that any well-defined quantum redefinitions. So this case too may be considered as an field theory must possess a UV fixed point, signifying example of conformal matter. It is then instructive to conformal invariance at high energies. These lines of extend the discussion to include an arbitrary number of argument encourage us to investigate a minimal early conformally and minimally coupled free scalar fields. universe cosmology, namely a quantum universe filled Since the matter Lagrangian of interest is, by assumption, with conformally invariant matter [1]. We start by studying conformally (Weyl) invariant at a classical level, it makes the quantum propagation of homogeneous background sense to “lift” general relativity (GR) to a larger theory universes, uncovering a number of surprising features. possessing the same symmetry. This is done by introducing Then, we include inhomogeneous perturbations, treated an extra scalar field which is locally a pure gauge degree of semiclassically and perturbatively at both linear and non- freedom. The full theory is now classically Weyl invariant linear order. We do not, in this paper, propose a realistic and it may be viewed with advantage in various Weyl scenario. Nor do we proceed far enough to study the effects gauges. Its solutions contain all solutions of GR but the of renormalization and the running of couplings, although theory allows for extended and more general solutions that

2470-0010=2017=95(10)=103510(28) 103510-1 © 2017 American Physical Society STEFFEN GIELEN and NEIL TUROK PHYSICAL REVIEW D 95, 103510 (2017) do not possess a global gauge fixing to GR. In particular, it cosmological solution necessarily possesses a big bang turns out that while classical cosmological (homogeneous singularity. However, there is no singularity in the Feynman and isotropic) background solutions are typically singular propagator for closed, open and flat Friedmann-Robertson- and geodesically incomplete in Einstein gauge (the Weyl Walker (FRW) and for flat, anisotropic Bianchi I cosmol- gauge in which the gravitational action takes Einstein- ogies; the propagator is well behaved across a bounce Hilbert form, with a fixed Newton’s gravitational constant representing a transition from a large, collapsing classical G), they are regular and geodesically complete in a more universe to a large, expanding one. We also study inho- general class of Weyl gauges, where G is no longer constant mogeneous perturbations of the isotropic, radiation- and can even change sign. In these gauges, generic cosmo- dominated universe, studying the propagation of scalar logical background solutions pass smoothly through the and tensor perturbations across the bounce, at linear and “big bang singularity” and into new regions of field space, nonlinear order. We find that, in the semiclassical approxi- including “antigravity” regions where G is negative [2]. mation at least and with strictly conformal matter, the Strictly speaking, the lifted classical theory is still incomplete incoming vacuum state evolves into the outgoing vacuum because these highly symmetrical backgrounds are unstable state, with no particle production. The bounce may be to perturbations in the collapsing phase, leading to diverging viewed as an example of quantum-mechanical tunneling, anisotropies which cannot be removed via a Weyl gauge and we use complex classical solutions as saddle points to choice. It was argued nevertheless that the classical theory the path integral, in a manner which generalizes the use of possesses a natural continuation across singularities of this instanton solutions to describe tunneling in more familiar kind [3]. contexts. Some of these results were anticipated in Ref. [1]; In this paper, following [1], we take a different tack. here we present more details on their derivation, extend We ask whether quantum effects might rescue the theory them to further cases not discussed in Ref. [1], and provide from its breakdown at big bang-type singularities. First, we more mathematical and conceptual background. An alter- show that for the simplest types of conformal matter, the nate interpretation of the antigravity regions, not employing Feynman propagator for cosmological backgrounds may be complex solutions, has been presented in Ref. [4]. computed exactly, allowing us to explore many issues with The simplest example of a “perfect bounce” is provided precision in this symmetry-reduced (minisuperspace) con- by a spatially flat, homogeneous and isotropic FRW text. Second, we extend the discussion to include anisot- universe, filled with a perfect radiation fluid. Adopting ropies, resolving the ordering problems in the quantum conformal time (denoted by η), i.e., choosing the line (Wheeler-DeWitt) Hamiltonian by imposing covariance element to be under field redefinitions. We go further than the treatment of [1] by defining the quantum theory along the real axis in ds2 a2 η −dη2 dx⃗ 2 ; 1 superspace, and discovering some remarkable features. For ¼ ð Þð þ Þ ð Þ a flat, isotropic universe, quantum effects indeed become one finds the scale factor a η ∝ η. One way to see this is large near the singularity. Therefore, strictly speaking, one from the trace of the Einsteinð Þ equations. For the line should not attempt to employ the real classical theory there. element (1), R 6 d2a=dη2 =a3 and for conformal matter, Instead, one can solve the quantum Wheeler-DeWitt-type the stress tensor¼ isð traceless.Þ So the Einstein equations equation for the propagator in complexified superspace, imply a η ∝ η. If spatial curvature is included, it is along a contour which avoids the singularity in taking one subdominantð Þ at small a so that a still vanishes linearly from the incoming collapsing universe to the outgoing with η. Hence a η ∝ η is a direct consequence of con- expanding one. Provided the contour stays sufficiently far formal invarianceð andÞ cosmological symmetry. While the from the singularity, the semiclassical approximation line element (1) clearly contains a big bang/big crunch remains valid all along it, so one can employ complex singularity at η 0, it is regular everywhere else, not only solutions of the classical theory to follow the quantum along the real η-axis¼ but also in the entire complex η plane. evolution across (or more accurately, around) the singu- Any complex η trajectory that connects large negative and larity. In doing so, we find that while quantum effects are large positive a while avoiding a 0 gives a regular, large near the singularity, they take a very special form such complexified metric that asymptotes¼ to a large, Lorentzian that they are “invisible” in the evolution between incoming universe in the past and the future, while circumventing the and outgoing states. In the final section of this paper, we big bang singularity. treat inhomogeneous perturbations, at linear and nonlinear Such complex solutions have been discussed in quantum order, showing how they may be followed smoothly and cosmology for a long time as saddle points of the path unambiguously across (or, more accurately, around) the big integral, for instance in the no-boundary proposal of Hartle bang singularity, in a similar manner. and Hawking [5,6]. The crucial difference in our proposal is The theory we consider consists of Einstein gravity plus that the complex solutions connect two large, Lorentzian radiation and a number of free scalar fields. All of these regions, identified with a collapsing incoming universe and forms of matter satisfy the strong energy condition, and any an expanding outgoing one. The Weyl-invariant lift of GR

103510-2 QUANTUM PROPAGATION ACROSS COSMOLOGICAL … PHYSICAL REVIEW D 95, 103510 (2017) provides a convenient simplification of the geometry on for FRW universes is actually regular at the singularity superspace, cleanly exhibiting its Lorentzian nature and the a 0, its asymptotic behavior for large arguments displays role of the scale factor a as a single timelike coordinate for interesting¼ pathologies both for closed and, in particular, both “gravity” regions. This leads us to a novel formalism open universes, so that only flat FRW universes seem to for quantum cosmology in which, rather than restricting to consistently admit a quantum bounce. In Sec. V, we extend positive a and imposing boundary conditions at a 0, a is the treatment to anisotropic universes of Bianchi I type, extended to the entire real line. The Feynman propagator¼ including for generality a number of free minimally turns out to have simple behavior at large negative and coupled scalar fields. We resolve the ordering problem positive a, describing a contracting and reexpanding uni- in the quantum Hamiltonian, and we are again able to verse, respectively, and connecting them through a quan- explicitly derive the Feynman propagator. A very special, tum bounce. The purpose of this paper is to flesh out the singular potential arises centered on a 0 which, in the ¼ details of this formalism, and show how it leads to a novel minisuperspace context, is harmless and actually invisible form of singularity avoidance in the context of an extremely in the scattering amplitude between incoming and outgoing simple (but not altogether unrealistic) cosmology. states. The coefficient of this singular potential turns out to Some of the features of our discussion are not new. The take a special value for the isotropic universe with zero or possibility of solving minisuperspace quantum cosmology one conformally coupled scalars, placing it on the edge of a models exactly by recasting their dynamics as those of a potential quantum instability, as we discuss. The addition relativistic free particle or harmonic oscillator was pointed of further conformally coupled scalars moves the theory out before (see, e.g., Refs. [7]). However, the crucial new away from this edge, however. This is an intriguing result feature in our work is the existence of regular solutions that deserves further attention, as it could be used to select (especially complex ones) that connect two large between isotropic and strongly anisotropic universes. In Lorentzian universes through a quantum bounce. This Sec. VI, we add inhomogeneities, treated linearly and feature relies on having a positive energy density in nonlinearly in the semiclassical approximation where radiation, a possibility which, as far as we know, was one employs complex classical solutions to the classical overlooked in previous work. In fact, our results suggest Einstein equations. We show how this is sufficient to that the fact that the early Universe was dominated by determine mixing between positive- and negative- radiation may be sufficient in itself for a semiclassical frequency modes, and hence to compute the particle “ ” quantum resolution of the big bang/big crunch singularity, production across the quantum bounce. We find no without the need for less well-motivated ingredients such as particle production, but instead verify that the perturbation exotic forms of matter [8], modified theories of gravity [9], expansion breaks down at late times due to the formation of or a proposed theory of quantum gravity [10]. To avoid any shocks in the fluid, a phenomenon which is now physically potential confusion, let us reemphasize that the theories we well understood [13]. Section VII concludes. consider consist of general relativity with a radiation fluid and a number of free scalars, and nothing more: our use of II. WEYL-INVARIANT COSMOLOGY the Weyl lift of GR does not introduce any additional We start by studying the cosmology of a universe filled degrees of freedom. with perfectly conformal radiation and a number M of The plan of our paper is as follows. In Sec. II,we conformally coupled scalar fields, with gravitational introduce the Weyl lift of GR plus radiation and scalars, dynamics governed by a lift of GR to a classically and show how the degree of freedom corresponding to the Weyl-invariant theory that contains an additional dilaton metric determinant can be isolated straightforwardly, lead- “ ” field ϕ. We stress again that the field ϕ is locally pure ing us to a Weyl-invariant notion of a, the scale factor. We gauge, and possesses no nontrivial dynamics of its own. then study homogeneous, isotropic FRW universes in This formalism for GR was developed in Ref. [2] and Sec. III, showing that in these cases, the Einstein-matter works in any number of dimensions D>2 (we set D 4 action corresponds to that of a massive relativistic particle shortly). The total action we consider is ¼ moving in Minkowski spacetime, either freely or subject to a quadratic, Lorentz-invariant potential. We discuss the 1 J classical and quantum dynamics of FRW universes, using S dDx p−g ∂ϕ 2 − ∂χ⃗ 2 − ρ j j ¼ Z  2 ðð Þ ð Þ Þ p−g the classical Hamiltonian analysis to define the Wheeler- ffiffiffiffiffiffi DeWitt quantum Hamiltonian. We discuss the Klein- D − 2 ffiffiffiffiffiffi ð Þ ϕ2 − χ⃗ 2 R − Jμ ∂ φ~ β ∂ αA : 2 Gordon-type inner product proposed by DeWitt [11],but þ 8 D − 1 ð Þ  ð μ þ A μ Þ ð Þ take the point of view that the fundamental quantity of ð Þ interest is really the causal (Feynman) propagator, which is The independent dynamical variables are the spacetime naturally defined as a path integral over four-geometries metric gμν (assumed to be Lorentzian throughout), the [12]. Accordingly, in Sec. IV we calculate the propagator “dilaton” ϕ, M physical scalar fields χ⃗ χ1; …; χM , and for various cases of interest. While the Feynman propagator a densitized particle number flux Jμ¼ðcharacterizingÞ the

103510-3 STEFFEN GIELEN and NEIL TUROK PHYSICAL REVIEW D 95, 103510 (2017) μ radiation fluid. The latter can be identified by J D 1 i j J μ ¼ S d x p−g − Gij ν ∂ν · ∂ν − ρ j j p−gnU , where n is the particle number density and ¼ Z   2 ð Þ p−g μ 2 ffiffiffiffiffiffi Uffiffiffiffiffiffiis the four-velocity vector field satisfying U −1; the ¼ 1 μ A ffiffiffiffiffiffi energy density ρ is only a function of n, concretely ρ n ∝ R − J ∂μφ~ βA∂μα ; 6 D ð Þ þ 16πG  ð þ Þ ð Þ nD−1 for radiation which is the case we are interested ~ in. There is a Lagrange multiplier φ which enforces where Gij ν is a positive definite metric of constant ∂ μ ð Þ particle number conservation μJ 0, and D − 1 further negative curvature on the gauge-fixed field space para- ¼ … Lagrange multipliers βA, with A 1; ;D− 1, enforcing metrized by the νi.Again,Eq.(6) shows that there are no μ∂ A ¼ 2 2 constraints J μα 0 that restrict the fluid flow to be ϕ − χ⃗ > 0 ¼ physical ghosts in the theory, at least as long as . directed along flow lines labeled by the fields αA which There are two different sectors in the space of field play the role of Lagrangian coordinates for the fluid. configurations where Einstein gauge is available, corre- In general, the fluid energy density ρ would also depend sponding to “future-directed” and “past-directed” (in field on the entropy per particle. For simplicity, we henceforth space) configurations, i.e., to ϕ > 0 or ϕ < 0. There are assume an isentropic fluid for which this entropy per also regions where ϕ2 − χ⃗ 2 becomes negative, identified particle is a constant. The fluid part of our action is then with “antigravity” in Ref. [2] as they would appear to the one given for isentropic fluids in Eq. (6.10) of Ref. [14], correspond to a negative G. We identify such regions with where further details on the construction of actions of imaginary values of the scale factor and show how the relativistic fluids and their corresponding Hamiltonian passage of the Universe through antigravity regions is a dynamics can be found. semiclassical representation of what is really a quantum The action (2) is invariant under a Weyl transformation bounce, similar to how quantum tunneling can be described that takes by complex classical trajectories. The antigravity regions do contain a ghost, as now ϕ; χ⃗ would be constrained to 2 ⃗ 2−D =2 ⃗ M−1;1 ð Þ gμν → Ω gμν; ϕ; χ → Ωð Þ ϕ; χ ; 3 de Sitter space dS which has a timelike direction. ð Þ ð Þ ð Þ These regions and their ghost excitations do not appear where Ω x is an arbitrary function on spacetime. Such a in the physical “in” and “out” states of the theory, which transformationð Þ also takes ρ → Ω−Dρ. Because of this local are defined in asymptotic timelike regions where 2 2 conformal symmetry, the field ϕ does not correspond to a ϕ − χ⃗ → ∞; nevertheless, the existence of these regions physical degree of freedom; indeed, if ϕ2 − χ⃗ 2 > 0 every- can cause pathologies in the quantum theory if initial where, one can gauge fix the conformal symmetry to gravity states can propagate into the antigravity regions, as recover the usual Einstein-Hilbert formulation of GR. It we see in Sec. IV B. is then clear that there is no physical ghost in the theory Setting D 4, to make this more precise it is now ¼ 2 even though ϕ appears in Eq. (2) with the wrong-sign useful to define a scale factor, or rather its square a , with kinetic term. the following properties: it should be Weyl invariant, so Let us make this explicit. For ϕ2 − χ⃗ 2 > 0, one can go to that it takes the same value in any conformal gauge. It “ ” should respect the O M; 1 isometry of the metric on the Einstein gauge by performing a conformal transforma- ð Þ tion (3) that takes space of scalar fields (defined by the kinetic terms) and so depend only on the combination ϕ2 − χ⃗ 2, the radiation D − 1 density ρ and the metric determinant g. It should have ϕ2 − χ⃗ 2 → constant ≕ 4 physical dimensions of an area (in the usual conventions 2 D − 2 πG ð Þ ð Þ ℏ c 1), and scale like the square of the scale factor for ¼ ¼ ’ an FRW universe in Einstein gauge in conformal time. where G is Newton s constant. Note that Eq. (4) does not These properties fix the “squared scale factor,” up to an entirely fix the gauge freedom as one can still perform a overall constant, to be global rescaling that takes the constant to a different one; the exact value of Newton’s constant is arbitrary and 2 1 −1 2 2 corresponds to a choice of units. Einstein gauge corre- a ≡ −g 4 ϕ − χ⃗ : 7 2ρ ð Þ ð Þ ð Þ sponds to constraining the (M 1)-vector formed by ϕ; χ⃗ þ ð Þ to a hyperboloid HM in (M 1)-dimensional field space at We use Eq. (7) as a natural definition in general gauges (the þ each point in spacetime. One can introduce an explicit motivation for the factor 1 becomes clear shortly). Note that i 2 parametrization of this hyperboloid by M coordinates ν , a2 is in general not positive; if we assume positive ρ and a Lorentzian metric, as we always do in the following, then in 1 1 ϕ ϕ ν ; …; νM ; χi χi ν ; …; νM ; 5 the antigravity regions a2 < 0 and so a is imaginary. This ¼ ð Þ ¼ ð Þ ð Þ definition of a differs from the one in Ref. [2] as it depends so that in this gauge the action (2) reads on the energy density of the radiation. In Ref. [2], there was

103510-4 QUANTUM PROPAGATION ACROSS COSMOLOGICAL … PHYSICAL REVIEW D 95, 103510 (2017) no such dependence. Instead, factors of Newton’s constant This leads to a diverging Weyl curvature which cannot were used to ensure the correct physical dimensions. be removed because it is conformally invariant. For a2 > 0, we fix the sign of a by choosing a time Nevertheless, in the presence of scalar fields (such as orientation in field space: a is defined to have the same sign the electroweak Higgs boson) there is generically no as ϕ. The Minkowskian field space parametrized by ϕ; χ⃗ Mixmaster chaos and one expects the classical evolution is then partitioned into two regions with real a andð oneÞ to become ultralocal and Kasner-like. There are a number region with imaginary a; see Fig. 1. Note that the entire of asymptotically conserved classical quantities, including light cone corresponds to a 0. This picture, as we have the Kasner exponents, suggesting a natural matching rule anticipated, gives physical meaning¼ to positive, negative across the singularity [3] but the issue has not been and imaginary a, generalizing the case of pure radiation, conclusively settled [15]. M 0, where there are no spacelike directions and a takes In this paper, we take a different approach. We show that values¼ along the real axis, as in the example discussed in the by extending the classical discussion to a quantum picture introduction. one can avoid the critical surface a 0 where the theory ¼ Apart from Einstein gauge, another gauge that we often becomes problematic, independently of any Weyl gauge employ, again following the framework introduced in choice. We give a description of nonsingular quantum Ref. [2],is“Weyl gauge” where the metric determinant bounces in terms of analytic continuation in a, where the g is fixed to a constant (typically −1). This gauge is Universe evolves from large negative a to large positive a available whenever the metric is nonsingular; in particular, along a contour in the complex a-plane which avoids it covers the entire field space pictured in Fig. 1, encom- a 0. We argue that as long as the quantum mechanics of ¼ passing both gravity and antigravity regions. It is often a the a degree of freedom make sense, the classical singu- convenient gauge to work in. In Weyl gauge, the expression larity at a 0 can be avoided without obstruction. ¼ for the scale factor reduces to a2 1 ϕ2 − χ⃗ 2 , where ¼ 2ρ ð Þ for homogeneous models by energy-momentum conserva- III. FRW BOUNCES tion ρ is constant. a is then proportional to the (signed) As a first step, we perform the familiar symmetry timelike O M;1 -invariant distance from the origin in field ð Þ reduction of our theory to homogeneous and isotropic space, making it a natural choice of time coordinate on FRW universes, with the metric assumed to be of the form superspace. For highly symmetric solutions such as FRW universes, 2 2 2 2 i j ds A t −N t dt hijdx dx 8 conformal symmetry can be used to eliminate curvature ¼ ð Þð ð Þ þ ÞðÞ singularities in the metric by moving them into 0’s of the where hij is a fixed metric on hypersurfaces of constant t, 2 2 quantity ϕ − χ⃗ . Since this quantity has no geometric 3 ð Þ which has constant three-curvature Rð Þ 6κ. Note our use interpretation, it is a priori reasonable for it to vanish or of a conformal lapse function N; the usual¼ definition of the change sign. However, following the dynamical evolution lapse would be N0 t A t · N t . We can now set the through such points is in general problematic because the function A t to oneð byÞ¼ a conformalð Þ ð Þ transformation, so that effective Newton constant diverges so gravity becomes the metricð becomesÞ nondynamical and all dynamics are strongly coupled. This is reflected, for example, in the in the scalar fields ϕ and χ⃗ . Also, with FRW symmetry behavior of tensor (gravitational wave) perturbations, Jμ phnδμ, and the action (2) reduces to which diverge as the effective Newton constant does. ¼ 0 ffiffiffi ⃗_2 _ 2 χ − ϕ κ 2 2 S V0 dt N ϕ − χ⃗ − ρ n − φ~ n_ ; ¼ Z  2N þ 2 ð Þ ð Þ  9 ð Þ : where denotes derivative with respect to t and V0 ¼ d3xph is the comoving spatial volume (which, as Rusual forffiffiffi minisuperspace models, must be assumed to be finite). We have simplified the last term including the _ A Lagrange multipliers which would be −n φ~ βAα_ since ð þA Þ the equations of motion involving βA and α are clearly redundant in FRW symmetry. As before, ρ n ∝ n4=3, and we can replace n by ρ as the independentð variable.Þ FIG. 1. Associating positive, negative and imaginary scale It is evident that Eq. (9) is the action for a factor a to different regions in field space. The light cone relativistic massive free particle (for κ 0) or a relativistic corresponds to the singularity a 0. massive particle in a harmonic potential¼ or a harmonic ¼

103510-5 STEFFEN GIELEN and NEIL TUROK PHYSICAL REVIEW D 95, 103510 (2017) “ ” upside-down potential (for κ ≠ 0) moving in (M 1)- 1 α α þ x_ x_α κx xα −1; 14 dimensional Minkowski spacetime. To make this more N2 þ ¼ ð Þ explicit, we can introduce new variables 1 N m_ 0; φ_ − x_αx_ κxαx 1 : 15 1 ¼ ¼ 2N α þ 2 ð α þ Þ ð Þ xα ≔ ϕ; χ⃗ ; α 0; …;M; m≔ 2V ρ; 10 2 0 p ρ ð Þ ¼ ð Þ The general solution to these equations is m constant, ffiffiffiffiffi ¼ so that Eq. (9) now takes the form α t α x1 x t exp ipκ dt0N t0 ð Þ¼pκ  Z0 ð Þ m 1 α α ffiffiffi S dt x_ x_ − N κx x 1 − φm_ 11 ffiffiffi α t 2 α α x2 ¼ Z  N ð þ Þ  ð Þ exp −ipκ dt0N t0 16 þ pκ  Z0 ð Þ ð Þ ffiffiffi where we have redefined the Lagrange multiplier for 1 ffiffiffi with x1 · x2 − ; the lapse function N t is arbitrary and simplicity, φ ≔ φ~ V0 dn=dm , and the Minkowski metric ¼ 4 ð Þ on the space of scalarð fields ηÞ diag −1; 1; 1; … is used φ t is determined from integrating Eq. (15).Forκ 0, the αβ ¼ ð Þ ð Þ ¼ to raise and lower indices. A crucial role is played by the general solution is simply a general timelike straight line in mass m which corresponds to (twice) the total energy in the Minkowski spacetime, → 0 radiation; the limit m would correspond to a massless t α α α relativistic particle moving in a potential, which is the case x t x1 dt0N t0 x2 17 well known in minisuperspace quantum cosmology with ð Þ¼ Z0 ð Þþ ð Þ scalar fields [7]. Having a positive mass, and hence timelike with x2 −1.Forκ 0, all solutions describe a bounce, trajectories as classical solutions, is one of the essential 1 similar to¼ the example¼ in the introduction: the Universe features of our model that leads to a bounce. With the comes in from negative real infinite a, goes through a 0 definition (10), the Weyl-invariant scale factor is simply followed in general by an “excursion” into imaginary a,¼ and a2 −x2, which explains the factor 1 in Eq. (7). The ¼ 2 crosses a 0 again before going off to real positive variable a is simply a time coordinate on superspace. One infinity. When¼ we go quantum, since the action is quadratic, can introduce it explicitly by setting the saddle point approximation is exact and the quantum dynamics is given purely in terms of these classical α α 2 x av ;v−1 12 solutions. When viewed as saddle points, these trajectories ¼ ¼ ð Þ can be deformed in the complex a-plane so that the so that vα is restricted to a hyperboloid HM (see Fig. 2). singularity a 0 is avoided. The situation is more subtle This parametrization, which isolates the physical scalar for κ < 0, where¼ there are spacelike as well as timelike fields as the variables vα, is useful below. solutions, and for κ > 0 where there is a turnaround in the Starting from Eq. (11), the classical equations of classical solutions and the Universe must recollapse due to motion are the spatial curvature. In Sec. IV, we see how the more complicated structure of solutions for κ ≠ 0 is reflected in a 1 d mx_α pathological behavior of the Feynman propagator for large mκxα 0; 13 N dt  N  þ ¼ ð Þ arguments. A. Canonical formalism In order to pass to the Hamiltonian formalism, following Dirac’s algorithm [16], one computes the canoni- cal momenta for the action (11) and finds ∂L m p x_ ;p≈−φ;p≈0;p≈0: 18 α ¼ ∂x_α ¼ N α m N φ ð Þ While the first equation can be inverted to express the α velocities x_ in terms of the momenta pα, the last three equations are primary constraints—we use Dirac’s notion of “weak equality” ≈ for equations that hold on the constraint surface. The second and fourth constraint would be second class, meaning one has to introduce a Dirac FIG. 2. The points of constant (real) a form a hyperboloid bracket and “solve” them. However, in this case, one can parametrized by vα. use the shortcut of simply identifying −φ with the

103510-6 QUANTUM PROPAGATION ACROSS COSMOLOGICAL … PHYSICAL REVIEW D 95, 103510 (2017) momentum conjugate to m and removing the separate ∂2 M ∂ 1 Δ 2 2 Ψ i − HM m 1 − κa a; ν ;m 0 variable pφ. This is equivalent to saying that the term −φm_ ∂a2 þ a ∂a a2 þ ð Þ ð Þ¼ in Eq. (11) is part of the symplectic form p q_ i so that one i 25 can read off pm −φ. ð Þ ¼ The Hamiltonian is then Δ where HM is the Laplace-Beltrami operator on M- dimensional hyperbolic space, i.e., on the space para- p2 m H N κx2 1 ξp : 19 metrized by the coordinates νi. For example, using 2 2 N ¼  m þ ð þ Þ þ ð Þ Beltrami coordinates νi xi=x0 one would have ¼ Preservation of pN ≈ 0 under time evolution gives the 2 ij i j i Δ M 1 − ν⃗ δ − ν ν ∂ ∂ − 2ν ∂ : 26 secondary, Hamiltonian constraint, H ¼ð Þ½ð Þ i j iŠ ð Þ This coordinate choice on superspace makes the role of p2 m C ≔ κx2 1 ≈ 0: 20 the timelike coordinate a explicit. The Wheeler-DeWitt 2m þ 2 ð þ Þ ð Þ equation can then be simplified by Fourier transforming from the νi coordinates to their conserved momenta ζ, N can then be treated as a Lagrange multiplier; it only enters linearly in the Hamiltonian, and its time evolution ∂2 M ∂ c under H is N_ N;H ξ where ξ is undetermined. − m2 1−κa2 Ψ a;ζi;m 0 27 ¼f g¼ ∂a2 þ a ∂a a2 þ ð Þ ð Þ¼ ð Þ Removing N;pN from the phase space (and setting ξ 0 in theð Hamiltonian),Þ we are left with the canonical ¼ α with pairs x ;pα and m; pm , subject to the constraint C, whichð triviallyÞ satisfiesð ÞC; H 0. C generates time 1 2 ⃗ 2 reparametrizations, f g¼ c ≡ − M − 1 − ζ 28 4 ð Þ ð Þ Npα α α corresponding to the eigenvalues of the Laplacian on HM δNx x ;NC ; 21 ¼f g¼ m ð Þ (for M ≥ 1). As we see in Sec. V below, the same form of the Wheeler-DeWitt equation applies when including δ p −Nmκx ; 22 N α ¼ α ð Þ anisotropies in a Bianchi I model or additional minimally coupled scalar fields, with the only change that the p2 1 δ p −N − κx2 1 ; 23 constant c receives additional contributions from con- N m ¼  2m2 þ 2 ð þ Þ ð Þ served anisotropy and scalar field momenta as well as from fixing ordering ambiguities. which correspond to the Lagrangian notion of time repar- A natural inner product on solutions of a second-order ametrization, by the equations of motion (13)–(15). equation like Eq. (25) is the Klein-Gordon-like norm

∂ M M i i B. Quantization Ψ Φ ≡ ia d νdmpg M ΨÃ a; ν ;m Φ a; ν ;m h j i Z H  ð Þ ∂a ð Þ Having set up the canonical formalism, we can ffiffiffiffiffiffiffiffi ∂ proceed with quantization in the standard way. The i i − ΨÃ a; ν ;m Φ a; ν ;m ; 29 Hamiltonian constraint is imposed as an operator equa- ∂a ð Þ ð Þ ð Þ tion restricting the set of physical states. In the x; m ð Þ M representation for the wave function, this is the Wheeler- with gHM being a constant negative curvature metric on H , DeWitt equation which is conserved under time evolution, i.e., independent of a, for solutions of Eq. (25). This inner product was 1 introduced by DeWitt [11] and has the well-known problem −□ m2 κx2 1 Ψ x; m 0: 24 2m ð x þ ð þ ÞÞ ð Þ¼ ð Þ (if it is used to define probabilities) that it is only positive on positive-frequency solutions to Eq. (25), when they Different m sectors simply decouple, as a consequence of exist. For some simple cosmological models, this subspace conservation of the total energy in the radiation, with no is well defined, and may be interpreted as the space of transitions between different m values allowed. An expanding quantum universes: if a is taken to be positive, alternative representation of wave functions is obtained a wave function describing an expanding universe must by separating the scale factor from the physical scalar be an eigenstate of p −i∂ with negative eigenvalue a ¼ a field degrees of freedom, as in Eq. (12), and introducing (note that pa −ma=N_ and so a>_ 0 means pa < 0), i.e., a set of coordinates νi, i 1; …;M, on the hyperboloid a positive-frequency¼ solution. This notion of positive HM. The Wheeler-DeWitt¼ equation then becomes frequency breaks down for cosmological models with

103510-7 STEFFEN GIELEN and NEIL TUROK PHYSICAL REVIEW D 95, 103510 (2017) recollapsing solutions, such as the FRW universe with describe the “emergence” of spacetime from quantum- κ > 0, where it is only well defined until one reaches the mechanical first principles [20]. turning point, and it is known that a decomposition into In this section, we show how to calculate the Feynman positive and negative frequencies of the type we are using propagator for FRW universes directly from the path here is not available in general [17]. The question of how integral; in particular, the path integral may be used to to define meaningful probabilities in quantum cosmology define the propagator without the need for an additional has, of course, been a matter of long debate (see, e.g., “iϵ” prescription and, furthermore, the propagator directly Refs. [5,18,19]). defines the positive- and negative-frequency vacuum We do not aim to resolve this debate here. The only use modes. As we have stressed, with FRW symmetry the we make of the DeWitt norm (29) is to help us construct action is quadratic and the saddle-point approximation is the Feynman propagator from mode function solutions of therefore exact for the path integration over the phase-space the Wheeler-DeWitt equation. The expansion rate −pa variables. However, we also have a constraint (the does play the role of an energy, which leads us to adopt Friedmann equation) which must be imposed via an addi- Feynman’s picture for quantum field theory in which tional integration over the lapse function or Lagrange positive energy (i.e., expanding) states are propagated multiplier N [21]. This integral is no longer Gaussian forward in proper time. The natural two-point function and has to be considered with more care. Things are we consider below in Sec. IV is hence the Feynman considerably simpler for the flat FRW case κ 0, where propagator. In what follows we alternate between the path we have seen that the dynamics are just those of¼ a massive integral and the Feynman propagator as basic formulations relativistic free particle in Minkowski spacetime. We of quantum cosmology, explicitly showing their equiva- therefore begin by reviewing how the Feynman propagator lence in simple cases. for a relativistic particle is calculated both from a path So far, this is a completely standard definition of a integral and as a Green’s function for the differential minisuperspace model in Wheeler-DeWitt quantum cos- equation satisfied by physical wave functions. We then mology. However, there is one crucial difference between extend these methods to treat general FRW universes for the our approach and previous treatments, in that we do not types of matter we consider. restrict the wave function to positive a, nor do we impose any boundary condition at a 0 [such as the popular choice Ψ a 0 0]. At fixed¼m, the domain of the wave A. Relativistic particle functionð is simply¼ Þ¼RM;1. Any boundary condition at a 0 Consider the action for a relativistic massive particle in would seem artificial from the viewpoint of classical¼ (M 1)-dimensional Minkowski spacetime, solutions such as classical FRW “bounces” which connect þ negative and positive a, as we have described, and is also m x_αx_βη S dt αβ − N ; 30 generally inconsistent with the wave function describing an ¼ 2 Z  N  ð Þ expanding Universe, i.e., a positive-frequency solution. In our proposal, the natural choice of wave functions corre- where m>0, xα t is the parametrized particle world line sponds to positive-frequency solutions that asymptote to and N t is the “einbein.ð Þ ” Classically, N may be eliminated plane waves at large a , where the Universe becomes usingð itsÞ equation of motion, obtaining the manifestly j j semiclassical, while we allow for irregular behavior in the reparametrization-invariant action S −m dtp−x_2. wave function at a 0. The examples we consider all ¼ ¼ Quantum mechanically, it is more convenientR to fixffiffiffiffiffiffiffiffi the admit a semiclassical Wentzel-Kramers-Brillouin (WKB) reparametrization invariance [see also the discussion of expansion in which one can deform the contour from the gauge fixing below Eq. (40)]: one can work in a gauge real a-axis into complex a, avoiding a 0 entirely. ¼ in which t varies over a fixed range, conveniently taken to be − 1

103510-8 QUANTUM PROPAGATION ACROSS COSMOLOGICAL … PHYSICAL REVIEW D 95, 103510 (2017) straightforward, with the only unusual factor being the prefactor of i in the second line, which arises from the τ functional integral over x0, which has the “wrong sign” kinetic term so that the overall phase factor contributed is iπ=4 −iπ=4 T eþ rather than the usual e . The second line of Eq. (31) is, of course, just the familiar Schwinger representation of the Feynman propagator, in which the exponent is the classical action evaluated on a solution of the equations of motion ẍ 0, satisfying the ¼ 1 1 -iR correct boundary conditions, i.e., x t x t 2 x0 2 −t , and the prefactor is given by theð Þ¼ usualð þ (regularized)Þþ ð Þ functional determinant. Note that this solution is only the saddle point for the functional integral over paths 2 2 x t , at fixed τ. The constraint x_ −τ arises sub- FIG. 3. Integration contours in the complex τ plane, for ð Þ ¼ sequently, as the condition for a saddle point in the the relativistic massive propagator, defined in Eq. (31).Asσ ≡ 2 2 exponent of the τ integral. In fact, once the saddle point − x − x0 is varied, from timelike values σ T with T real, to is chosen, the integration contour is then fixed (up to an spacelikeð Þ values σ −R2, with R real, by¼ passing beneath the equivalence class of contours yielding the same result) as origin in the complex¼ σ plane, the saddle point in τ, shown by the the complete extension of the corresponding steepest black point, moves correspondingly. In each case, the integral is descent contour. Integrating over negative proper time in defined by the associated steepest descent contour running from 0 Eq. (31) would reverse the notion of time ordering, whereas to ∞, also shown. integrating over both positive and negative τ would lead to a symmetrized two-point function in which one sums over saddle point for the τ integral moves as shown in both time orderings, i.e., the Hadamard propagator. Fig. 3, passing below the origin in the complex τ plane For M>0, the τ integral in Eq. (31) has a potential and down the imaginary τ-axis. At spacelike separations, divergence at τ 0. In fact, the integral converges at all σ −R2 < 0, the saddle point is at τ −iR, and the real values of σ ¼except σ 0. That divergence is real: the ¼ ¼ ¼ propagator falls exponentially with spacelike separation. Feynman propagator is singular for null-separated points. Notice that although the classical constraint x_2 −τ2 For other real values of σ, given that the integral converges remains satisfied at the saddle point, the saddle-point¼ for all σ in the lower-half σ plane, one may define the value for τ is imaginary, and hence classically disallowed. Feynman propagator as the boundary value of the function Hence, the propagation of a massive relativistic particle in defined by the integral, which is analytic in the lower-half σ spacelike directions may be viewed as a semiclassical plane. Traditionally, the mass m is also taken to have a quantum tunneling process, mediated by a complex small negative imaginary part, in order to make the τ classical solution. integral absolutely convergent at large positive values In order to match the path-integral definition of the ’ “ ” (Feynman s iϵ prescription). Equivalently, one may Feynman propagator to its definition as a Green’s function, distort the τ-contour to run to infinite values below the it is convenient to use a time slicing with maximal spatial real axis. The integral (31) may be directly expressed in symmetry. Here, the trivial time slicing defined by x0 is terms of a Hankel function, whose properties confirm these spatially homogeneous, so one can Fourier transform in the general arguments (see the appendix). spatial coordinates and reduce the problem to a single It is instructive, however, to evaluate the τ integral in timelike dimension. Defining Eq. (31) in the saddle-point approximation. First, consider 2 timelike separations, σ T > 0. The exponent in the τ M⃗ d k ⃗ ¼ ik· x⃗ −x⃗ 0 0 0 integral (31) is stationary at τ T and τ −T, but only G x x0 e ð ÞGk x ;x0 ; 32 ¼þ ¼ ð j Þ¼Z 2π M ð Þ ð Þ the former saddle point is relevant to the contour we want, ð Þ which is deformable into the positive τ-axis. The saddle one finds that G x0;x0 is given by point at τ T gives rise to a positive-frequency result, kð 0 Þ G ∼ e−imT ¼þat large T. The integration contour for τ may 0 0 2 ω2 m x −x k ∞ m −i ð 0 Þ τ then be taken to be the corresponding steepest descent 0 0 2 τ þm2 Gk x ;x0 i dτ e   33 contour, shown as the solid line in Fig. 3. Now consider ð Þ¼ Z0 rffiffiffiffiffiffiffiffiffi2πiτ ð Þ analytically continuing T through the lower right quadrant → 2 2 to the negative imaginary axis, T −iR. It follows that G where ωk ≡ pk m . −mR þ converges as G ∼ e at large R. Correspondingly, this By consideringffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the saddle-point approximation to continuation implies that σ T2 runs below the origin in Eq. (33), we see that the Fourier-transformed Feynman the complex σ plane to negative¼ values. The corresponding propagator asymptotically satisfies

103510-9 STEFFEN GIELEN and NEIL TUROK PHYSICAL REVIEW D 95, 103510 (2017)

0 0 0 −iωkx 0 0 k k Gk x ;x0 ∼ e ;x→ ∞;x0 fixed; 34 where ψ 1 and ψ 2 are two independent solutions to the ð Þ þ ð Þ ∂2 2 homogeneous equation 0 ωk ψ 0, and W ψ 1; ψ 2 0 ð þ Þ ¼ ð Þ¼0 0 0 iωkx0 0 0 ψ ψ − ψ ψ is the natural conserved (i.e., x - Gk x ;x0 ∼ eþ ;x0 → −∞;xfixed: 35 1 20 2 10 ð Þ ð Þ independent) inner product, or Wronskian. The dependence Such asymptotic expressions can be used to fix boundary of the Feynman propagator at large positive and negative k 0 conditions for the corresponding wave (Wheeler-DeWitt) times now determines the appropriate choices for ψ 1 x k 0 ð Þ equation, as we do shortly. and ψ 2 x : comparing Eqs. (34) and (35) with Eq. (39) we ð Þ 0 k iωkx Formally, G x x0 is the matrix element infer that, up to irrelevant constants, ψ 1 eþ and ∞ −iHτ ð j Þ −1 k −iω x0 ¼ x dτe x0 −i x H x0 , where we again ψ e k . Inserted into Eqs. (32) and (39), these give h j 0 j i¼ h j j i 2 assumeR the integral converges at infinite τ. Hence, the¼ usual expression for the Feynman propagator in “time- suitably defined, G x x0 should obey ordered” form. This shows how, in the example of the ð j Þ relativistic particle, the correct boundary conditions that M 1 HxG x x0 −iδ þ x − x0 ; 36 define the Feynman propagator as one particular solution of ð j Þ¼ ð Þ ð Þ Eq. (36) can be obtained from the asymptotic behavior of where Hx is the Hamiltonian in the x-representation, its path-integral definition. We now proceed similarly to H 1 −□ m2 . We can check Eq. (36) is indeed define the Feynman propagator for general FRW universes. x ¼ 2m ð x þ Þ satisfied by applying Hx to the last line of Eq. (31) and using the fact that the integrand is a product of free- B. FRW universes particle quantum -mechanical propagators, For our cosmological model, the Feynman propagator can be defined through a phase-space path integral, taking 1 −□ m2 G x x into account the integration over the lapse N [21], 2m ð x þ Þ ð j 0Þ M 1 α ∞ 2 G x; m x ;m Dx DP DmDp DN 2þ im x−x 0 0 α m d m ð 0Þ −τ i dτi e 2 ð τ Þ ð j Þ¼Z ¼ Z0 dτ 2πiτ  1=2 α M 1 2 × exp i dt x_ Pα mp_ m m 2þ im x−x ð 0Þ  Z−1=2  þ lim e 2 τ ; 37 → ð Þ ¼ τ 02πiτ ð Þ P Pα m − N α κxαx 1 : 40 2m 2 α where the limit should be taken along the appropriate  þ ð þ Þ ð Þ contour in the complex τ plane. The last line of Eq. (37) As in the previous example, due to the reparametrization is a representation of the (M 1)-dimensional delta invariance of the theory the parameter time (specified by t) function: separating it into a productþ of similar terms between the initial and final configurations is arbitrary, and α 1 1 for each coordinate x , we determine the coefficient of we choose it to run from − 2 to 2. In order for the path integral the corresponding delta function by Fourier transforming to be well defined, the gauge invariance under time repar- with respect to xα, and then taking the limit τ → 0.For ametrizations generated by the Hamiltonian constraint must 0 0 the timelike coordinate we obtain −iδ x − x0 , whereas be broken by fixing a specific gauge. One simple gauge ð ÞM _ for the spacelike coordinates we obtain δ x⃗ − x⃗ 0 . fixing, N 0, can be obtained by introducing a new field Π ð Þ ¼ Together, these results verify Eq. (36). and adding the term ΠN_ to the action; we refer to Ref. [22] The point is now that the Feynman propagator can also for details. Integrating over the field Π then reduces the be computed by directly solving Eq. (36) in terms of mode integration over N to an ordinary integral over the total functions, again because the Fourier transform allows conformal time between the initial and final configurations; reduction of the problem to one dimension. Writing both we make this explicit by again writing N as τ. the delta function and the propagator in Eq. (36) as Fourier The remaining path integrals may be computed exactly. transforms, one sees that G x x0 clearly satisfies Eq. (36) Path integration over m and p simply gives a delta ð j Þ m as long as function in m, as expected since m has trivial dynamics constraining it to be constant. One can then integrate over ∂2 ω2 G x0;x0 −2imδ x0 − x 0 : 38 P which yields ð 0 þ kÞ kð 0 Þ¼ ð 0 Þ ð Þ α G x; m x ;m This equation is solved by ð j 0 0Þ m x_2 δ m − m dτDx exp i dt − τ 1 κx2 2im ¼ ð 0Þ  2  τ ð þ Þ G x0;x0 − ψ k x 0 ψ k x0 θ x0 − x 0 Z Z kð 0 Þ¼ W ψ k; ψ k ð 1ð 0 Þ 2ð Þ ð 0 Þ ð 1 2Þ 41 k 0 k 0 0 0 ð Þ ψ 1 x ψ 2 x0 θ x0 − x ; 39 corresponding to M 1 decoupled harmonic oscillators. þ ð Þ ð Þ ð ÞÞ ð Þ þ

103510-10 QUANTUM PROPAGATION ACROSS COSMOLOGICAL … PHYSICAL REVIEW D 95, 103510 (2017) For κ 0, apart from the overall delta function, this is is easily calculated: the classical path which fixes the exactly the¼ expression Eq. (31). Accordingly, the path exponent generalizes to integral over x is just that of a free relativistic particle and the τ integral can be evaluated exactly; the result is x sin pκτ t 1 x sin pκτ 1 − t x t ð ð þ 2ÞÞ þ 0 ð ð2 ÞÞ ; 46 ð Þ¼ ffiffiffi sin pκτ ffiffiffi ð Þ 1 1−M 2 ð Þ 0 M 2 G x; m x0;m0 δ m − m0 −im 2πs HðM−Þ1 s ; ffiffiffi ð j Þ¼2 ð Þð Þ ð Þ 2 ð Þ which is unique for all x, x0 and τ [where we exclude special 42 cases for which sin pκτ 0], and the prefactor is given ð Þ¼ ð Þ by the usual regularizedffiffiffi functional determinant for the 2 2 harmonic oscillator, so that with s ≡ m − x − x0 − iϵ, where Hðα Þ x is a Hankel ð Þ ð Þ − function ofp theffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi second kind (see the appendix). The iϵ G x; m x ;m in its argument indicates that the expression is the boundary ð j 0 0Þ M 1 value of a function which is analytic in the lower half − ∞ mpκ 2þ 2 iδ m − m0 dτ x − x0 plane. As we have emphasized, we have derived ¼ ð Þ Z0 2iπ sin pffiffiffi κτ  ð Þ ð Þ this definition from the path integral, and the requirement m x2 x 2 cosffiffiffipκτ − 2x · x that the integral over proper time τ converges. × exp i pκ ð þ 0 Þ ð Þ 0 − τ :  2  sin pκffiffiffiτ  In order to understand the more involved case of spatial ffiffiffi ð Þ curvature κ ≠ 0, it is again helpful to recall how Eq. (42) ffiffiffi 47 can be obtained from the Wronksian method; for simplicity, ð Þ let us set M 0 and label x0 ≡ a which is our scale factor. The overall factor of i arises just as it did for the free Then the Feynman¼ propagator is a solution to relativistic particle, discussed in the previous subsection. As there, we are left with an ordinary integral over τ, and 2 need to establish the appropriate integration contour. 1 d m 0 G a; m a0;m0 −iδ m − m0 δ a − a0 The resulting integral in Eq. (47) is difficult to do directly 2m da2 þ 2  ð j Þ¼ ð Þ ð Þ but we can use its behavior at large positive a and large 43 ð Þ negative a0 to fix the mode functions ψ 1 and ψ 2 appearing in the Wronskian representation. As a consistency check, and can be written in the form we compare the resulting Green’s function to a numerical evaluation of the τ integral in Eq. (47) along a suitable 0 ψ 1 a0 ψ 2 a contour, finding perfect agreement in all cases. In this G a; m a0;m0 −2imδ m − m0 ð Þ ð Þ θ a − a0 ⃗ ð j Þ¼ ð Þ W ψ 1; ψ 2 ð Þ numerical evaluation, we choose x T;0 and x0 ð Þ ⃗ ¼ð Þ ¼ ψ 1 a ψ 2 a0 −T;0 , so that any classical real solution has to pass ð Þ ð Þ θ a0 − a 44 ð Þ a 0 þ W ψ 1; ψ 2 ð Þ ð Þ through the singularity at least once, which is the ð Þ situation of main relevance¼ for our study. For ease of comparison, we plot the flat case κ 0, with M 0, i.e., where W ψ 1; ψ 2 ψ 1 a ψ 20 a − ψ 10 a ψ 2 a is again ð Þ¼ ð Þ ð Þ ð Þ ð Þ G e−2imT, with m 7, in Fig.¼4 [here and¼ in the the (a-independent) Wronskian and ψ 1 a and ψ 2 a are ¼ ¼ two appropriate independent solutions toð theÞ homogeneousð Þ following we are of course plotting the function multiply- 1 d2 m ing the singular part δ m − m0 ]. equation 2 ψ a 0, found by matching Eq. (44) ð2m da þ 2 Þ ð Þ¼ Consider the saddleð pointsÞ in the τ integral for the with the asymptotic behavior of Eq. (31) at infinity, with curved-space propagator, and the associated steepest σ a − a 2 (and there are no spacelike directions to be ¼ð 0Þ descent contour. The condition for the exponent to be considered). As explained below Eq. (33), (31) asymptotes stationary with respect to τ is precisely the Hamiltonian −ima ima0 to e for large positive a at fixed a0, and to e for large constraint (Friedmann equation) τ−2x_2 1 κx2 0, negative a0 at fixed a; this fixes the modes in (44) as with x t given by Eq. (46). Real saddleþð pointsþ of theÞ¼ full ima −ima ψ 1 a e and ψ 2 a e (up to a normalization functionalð Þ integral, when they exist, are real solutions of the ð Þ¼ ð0 Þ¼ that is irrelevant for G ). Thus, one finds classical equations of motion, including the constraints. Given such saddle points, one defines the associated τ 0 −im a−a G a; m a0;m0 δ m − m0 e j 0j; 45 integration cycle as the complete extension of the steepest ð j Þ¼ ð Þ ð Þ descent contour. If this cycle can be deformed to the real τ- in agreement with Eq. (42) for M 0. axis while maintaining the convergence of the integral, then With this in mind, we can now go¼ beyond the simplest the saddle point contribution is relevant to the final result. flat case, and consider κ ≠ 0, where the dynamics of the We start with the case of negative κ, where, just as in the Universe corresponds to those of a relativistic oscillator or flat case, there is always a unique classical solution: for upside-down oscillator. The path integral over x in Eq. (41) timelike separated x and x0, the saddle point in τ is located

103510-11 STEFFEN GIELEN and NEIL TUROK PHYSICAL REVIEW D 95, 103510 (2017)

1.0 −1 −imp−κa2 ψ 3 a ∼ a 2e 2 ;a→ −∞; ð Þ j j −1 −imp−ffiffiffiffiκa2 ψ 4 a ∼ a 2e 2 ;a→ ∞; 50 0.5 ð Þ j j þ ð Þ ffiffiffiffi for the mode functions ψ 3 and ψ 4 appearing in the analog 0.2 0.4 0.6 0.8 1.0 1.2 of Eq. (44). This asymptotic behavior is also consistent with the requirement that, as Eq. (47) is invariant under 0.5 x → −x and x0 → −x0, ψ 3 and ψ 4 must satisfy

ψ 3 −a ψ 4 a : 51 1.0 ð Þ¼ ð Þ ð Þ Two independent solutions to Eq. (49) are given by FIG. 4. Feynman propagator for flat universes as a function of T, for m 7, showing real part (blue), imaginary part (dashed) 1=4 ¼ ψ a Di m −1 1 − i pm −κ a 52 and absolute value (black) which is constant here. ð Þ¼ 2p−κ 2ðð Þ ð Þ ÞðÞ ffiffiffiffi and its complex conjugate,ffiffiffi which asymptotically become on the real axis, and the steepest descent contour is the solid pure negative and positive frequency modes as a → ∞ but curve in Fig. 5. The singular behavior of the integrand at are a mixture as a → −∞. We therefore set τ 0 [cf. Eq. (37)] then ensures, just as in the argument ¼ 1=4 leading to Eq. (37), that Eq. (47) is a Green’s function for ψ 4 a D−i m −1 1 i pm −κ a 53 ð Þ¼ 2p−κ 2ðð þ Þ ð Þ ÞðÞ the Wheeler-DeWitt equation, ffiffiffiffi and ψ 3 a ψ 4 −a . Byffiffiffi computing their Wronskian we □ m obtain theð Þ¼ Greenð’s functionÞ − κx2 1 G −iδ m − m δM 1 x − x :  2m þ 2 ð þ Þ ¼ ð 0Þ þ ð 0Þ pim 1 im 48 G a; m a0;m0 Γ δ m − m0 ð Þ ð j Þ¼pπ −ffiffiffiffiffiffiκ 1=4 2 þ 2p−κ ð Þ ð Þ Once more we set M 0 for simplicity, and obtain the × ffiffiffiψ 3 a0 ψ 4 a θ a − affiffiffiffiffiffi0 a ↔ a0 : ’ ¼ ð ð Þ ð Þ ð Þþð ÞÞ Green s function from the Wronskian method, with the 54 modes determined from their behavior at large argument. ð Þ The Wheeler-DeWitt equation, at fixed m,is As we have said, this result can also be obtained from

2 numerical evaluation of Eq. (47). Again we choose x d ¼ m2 1 − κa2 ψ a 0; 49 T;0⃗ and x −T;0⃗ and also fix κ −1, M 0 and da2 þ ð Þ ð Þ¼ ð Þ 0 mð Þ7. The resulting¼ð functionÞ of T is¼ plotted in¼ Fig. 6. ¼ and is solved by parabolic cylinder functions [23], denoted Notice that the resulting propagator resembles the flat- space (κ 0) expression for small T, and the effects of by Dν z . In order to find the modes that generalize the ¼ 1 ð Þ ima spatial curvature become relevant only at scales x ∼ . plane waves eÆ used in the κ 0 case, we can again j j p κ ¼ j j study the asymptotic limits of Eq. (47) using the saddle- For positive κ, the behavior is rather different from κ ≤ffiffiffiffi0 point approximation, finding that we must have (with in that for positive κ the real, classical solutions are periodic κ < 0) in τ; for given x and x0, when one classical solution exists there will be an infinite number, and in general they should all contribute to the propagator. Again for consistency with the κ → 0 limit, we can choose the τ integration contour such that it only picks out the simplest saddle point, where the classical solution interpolating between x and x0 has no turning points. The corresponding saddle point and steepest descent contour, indicated by the dashed curve in Fig. 5, goes over to the unique κ 0 saddle point and steepest descent contour in Fig. 3 in¼ the flat limit κ → 0. Another issue is that for large timelike x and x0 there is no real classical solution at all; for large timelike argu- FIG. 5. Integration contours in the complex τ plane for κ ≤ 0 ments, the saddle point for τ becomes imaginary and, as (solid line) and for κ > 0 (dashed line), for the case where explained above, we choose the one on the negative x T;0⃗ and x −T;0⃗ . imaginary axis. By the same saddle-point approximation ¼ð Þ 0 ¼ð Þ

103510-12 QUANTUM PROPAGATION ACROSS COSMOLOGICAL … PHYSICAL REVIEW D 95, 103510 (2017) 1.0 4

0.5 3

2

0.2 0.4 0.6 0.8 1.0 1.2 1

0.5 0 0.2 0.4 0.6 0.8 1.0 1.2

1.0 1

FIG. 6. Feynman propagator for open universes as a function of FIG. 7. Feynman propagator for closed universes as a function T,form 7 and κ −1, showing real part (blue), imaginary of T, for m 7 and κ 1, showing real part (blue), imaginary ¼ ¼ part (dashed) and absolute value (black). part (dashed)¼ and absolute¼ value (black). As discussed in the text, for T ≥ 1 the semiclassical interpretation fails. as above, one then determines the asymptotic behavior of the chosen contour. With x T;0⃗ , x −T;0⃗ , as well the relevant mode functions ψ 5 and ψ 6, 0 as κ 1, M 0 and m 7¼ð, the resultingÞ ¼ð functionÞ of T is m 1 m 2 ¼ ¼ ¼ − −2 pκa plotted in Fig. 7. Again, it reduces to the κ 0 expression ψ 5 a ∼ a 2pκ e 2 ;a→ −∞; 55 ð Þ j j þ ð Þ e−2imT for small T. ¼ ffiffi ffiffi m 1 m 2 The integration contour in Fig. 5 is chosen in such a way − −2 pκa ψ 6 a ∼ a 2pκ e 2 ;a→ ∞; 56 ð Þ j j þ þ ð Þ that its main contribution, for small enough T, comes from ffiffi ffiffi the lowest positive real saddle point in τ, corresponding to again consistent with ψ 5 a ψ 6 −a . These results, as ð Þ¼ ð Þ the real classical solution that takes the smallest amount of well as the form of the propagator, can in fact be obtained proper time. As the arguments of the Feynman propagator by replacing p−κ → iκ in the expressions for the open approach x 1=pκ; 0⃗ and x −1=pκ; 0⃗ , this saddle case. [The factor a −m= 2pκ was dropped in the expres- 0 ffiffiffiffiffiffi ð Þ point moves¼ð towards τÞ π where¼ð it eventuallyÞ merges sions above sincej itj was a subleading oscillatory factor ffiffiffi pκ ffiffiffi ffiffi ¼ π −im= 2p−κ with another saddle point approaching τ from above, a ð Þ.] ffiffi ¼ pκ j j corresponding to two classical solutions that become It is straightforwardffiffiffiffi to obtain a complete analytic ffiffi expression for the κ > 0 propagator from parabolic cylinder indistinguishable in this limit. Our choice of integration functions by solving the homogeneous equation (49).Two contour then becomes ambiguous and is no longer defined independent solutions with κ > 0 are by consistency with the κ → 0 limit. As we extend the arguments to T > 1, where there is no longer a real 1=4 solution, thesej twoj saddle points separate again and start ψ 5 a D− m −1 −ip2mκ a ; ð Þ¼ 2pκ 2ð Þ ffiffiffiffiffiffiffi moving up and down in the imaginary direction. This is 1=4 ψ 6 a D− mffiffi −1 ip2mκ a : 57 akin to the situation for spacelike separations for the ð Þ¼ 2pκ 2ð Þ ð Þ ffiffiffiffiffiffiffi relativistic particle, and means that our saddle point needs π − R These are again complex conjugatesffiffi of each other, and here to be replaced by a saddle point on the line pκ i parallel also already satisfy ψ a ψ −a , unlike for the open to the negative imaginary τ-axis. For the purposes of this 5 6 ffiffi case κ < 0. On the otherð Þ¼ hand,ð theyÞ are not asymptotic paper, we are mainly interested in studying the propagator positive or negative frequency modes, but blow up expo- with arguments for which there is a classical solution, so nentially both at positive and at negative infinity. At small that a semiclassical picture of the propagator as given by a, up to corrections that vanish as κ → 0 they reduce to these solutions is meaningful. ima plane waves eÆ . From Eq. (57), the Wronskian method The exponential blowup of the Feynman propagator for gives the Green’s function large T follows from the asymptotic behavior of the integral (47) for large timelike x and x0. The corresponding mode pm 1 m functions increase exponentially for T > 1 when there are G a; m a0;m0 Γ δ m − m0 j j ð j Þ¼pπκffiffiffiffi1=4 2 þ 2pκ ð Þ no classical solutions, as can be verified explicitly from the asymptotics of the parabolic cylinder functions (57). The × ffiffiffiψ 5 a ψ 6 a θ affiffiffi− a a ↔ a : ð ð 0Þ ð Þ ð 0Þþð 0ÞÞ Feynman propagator is hence pathological for large time- 58 like separations, and does not define a suitable two-point ð Þ function on the entire superspace, because positive curva- As before, we have verified that this expression agrees with ture forces classical solutions to recollapse. An asymptotic the result of a numerical integration of the τ integral along description in terms of well-defined states that can be used

103510-13 STEFFEN GIELEN and NEIL TUROK PHYSICAL REVIEW D 95, 103510 (2017) to formulate a quantum theory of expanding universes does transition between incoming and outgoing gravity states not exist for positive spatial curvature, and this case does with a → ∞, while the κ 0 case leads directly to a not consistently describe the type of quantum bounce we perfect bounce.Æ We conclude¼ that, at least for the theories are interested in. Of course, this situation could be altered considered here, only flat FRW universes lead to a con- by the inclusion of a positive cosmological constant, as we sistent quantum theory, able to account for an expanding mention later. universe. The inclusion of a positive cosmological constant For κ < 0, classical solutions with pure radiation are well could rescue positively curved universes from this con- behaved, expanding to infinite volume in the future and clusion, provided the curvature is too small to cause a past and leading to a well-behaved Feynman propagator, recollapse. Nevertheless, the quantum pathology we have Eq. (50). However, as we mentioned in Sec. III, when identified for negatively curved FRW universes is in- conformally coupled scalars are introduced (M ≥ 1), the triguing, because it raises the possibility that the observed general solution (nearly flat) Universe lives on the corresponding critical boundary. This would be the case, for example, if we could α t α x1 identify the correct quantum measure on the space of closed x t exp p−κ dt0N t0 ð Þ¼p−κ  Z0 ð Þ universes, with sufficiently large cosmological constant to ffiffiffiffiffiffi α t prevent recollapse, and if this measure favored the flat case. ffiffiffiffiffiffix2 exp −p−κ dt0N t0 ; 59 We explore this possibility in future work. For the remain- þ p−κ  Z0 ð Þ ð Þ ffiffiffiffiffiffi der of this paper, however, we ignore spatial curvature. ffiffiffiffiffiffi 1 with x1 · x2 , has both spacelike antigravity and timelike ¼ 4 gravity solutions: choosing spacelike x1 and x2 that satisfy V. ADDING ANISOTROPIES AND the constraint, one finds a universe that comes in from one FREE SCALAR FIELDS antigravity direction, turns around before entering gravity We now extend the treatment to anisotropic cosmologies, and then disappears into a different (or the same) anti- choosing the simplest form of anisotropies, the Bianchi I gravity direction. Such spacelike solutions are far enough model: we still require the metric to be spatially homo- in the antigravity region that the curvature term dominates geneous, with an Abelian group of isometries acting on over the positive mass in the potential, m2 1 κx2 < 0, ð þ Þ constant time hypersurfaces, but no longer impose isotropy. leading to their acceleration towards spacelike infinity. The most convenient parametrization of such a metric Even though there are no real classical solutions that employs Misner variables [24], connect incoming gravity solutions to these far antigravity regions, and starting in a gravity region one is guaranteed to ds2 A2 t −N2 t dt2 e2λ1 t 2p3λ2 t dx2 asymptote into the other gravity region, quantum mechan- ð Þþ ð Þ 1 ¼ ð Þð ð Þ þ ffiffi 2λ1 t −2p3λ2 t 2 −4λ1 t 2 ically one expects the spacelike solutions to determine the e ð Þ ð Þdx2 e ð Þdx3 : 62 behavior of the Feynman propagator for spacelike separa- þ ffiffi þ Þ ð Þ tions. This is indeed confirmed by finding the saddle-point The Ricci tensor, and hence the Einstein tensor, for this approximation to Eq. (47) for large spacelike separations, metric are diagonal, which by the Einstein equations 2 e.g., for x → ∞ at fixed x0, forbids any anisotropy in the fluid, manifest in a velocity ui. We hence continue to assume that the fluid moves with 2 −1=2 imp−κx μ μ G x; m x0;m0 ∼ x e 2 : 60 the cosmological flow, J ∝ δ . ð j Þ j j ð Þ 0 ffiffiffiffi We can then again exploit the conformal freedom to set The propagator becomes oscillatory at spacelike separa- A t to 1. The Ricci scalar of (62) is then tions, so that a given initial state, e.g., a wave packet ð Þ centered around an initial state in the gravity region a<0, _2 _2 λ1 λ2 is propagated to large spacelike distances into the anti- R 6 þ ; 63 N2 gravity region. In this sense, the quantum theory is even ¼ ð Þ more pathological for open than for closed universes, where 2 giving the correct canonical normalization of the anisotropy one finds, again for x → ∞ at fixed x0, variables λi in the symmetry-reduced action,

− m −1 m 2 2pκ 2 − 2 pκx 2 G x; m x0;m0 ∼ x e ; 61 ⃗_ − _ 2 _2 _2 ð j Þ j j ð Þ χ ϕ λ1 λ2 2 ⃗ 2 ffiffi S V0 dt þ ϕ − χ − Nρ − φ~ n_ : ffiffi ¼  2N þ 2N ð Þ  i.e., exponential falloff for large spacelike separations. This Z is because for κ > 0, both timelike and spacelike classical 64 ð Þ solutions are bounded due to the potential, and never reach (timelike or spacelike) infinity. Neither κ < 0 nor κ > 0 can In terms of the variables xα and m defined in Eq. (10), this lead to a viable perfect bounce picture in terms of a action reads

103510-14 QUANTUM PROPAGATION ACROSS COSMOLOGICAL … PHYSICAL REVIEW D 95, 103510 (2017) 1 m 2 2 _2 _2 (the anisotropy variables are dimensionless while a scalar S dt x_ − x λ1 λ2 − N − φm_ : 65 ¼ Z  2 N ð ð þ ÞÞ   ð Þ field has dimensions of mass), showing the equivalence. Hence, we obtain a simple generalization of the theory As for the flat FRW universe which the Bianchi I universe we have discussed by also adding an arbitrary number of generalizes, this is the action of a free massive relativistic minimally coupled free scalar fields, which can represent particle. However, here the particle is not moving through a either physical scalar fields or anisotropy degrees of free- flat Minkowski spacetime but through a curved superspace, dom of the Bianchi I model. One has to lift the free scalar with metric fields to a Weyl-invariant theory by

2 α β 2 2 2 1 1 ds ηαβdx dx − x dλ1 dλ2 66 − d4xp−g ∂τ 2 → − d4xp−g ϕ2 − χ⃗ 2 ∂λ 2 ¼ ð þ ÞðÞ 2 Z ð Þ 2 Z ð Þð Þ ffiffiffiffiffiffi ffiffiffiffiffiffi or, if we again use the parametrization xα avα νi with 72 v2 −1 to separate the scale factor a, introducing¼ ð Þ coor- ð Þ dinates¼ νi on the hyperboloid HM, where λ is again dimensionless and conformally invariant. In Einstein gauge, the right-hand side of Eq. (72) clearly 2 2 2 2 2 2 ds −da a gHM a dλ1 dλ2 67 reduces to the right-hand side of Eq. (68). Going back to ¼ þ þ ð þ ÞðÞ Weyl gauge, the total action is then M where gHM is a constant negative curvature metric on H , as in Sec. III. The geometry of superspace at fixed a 1 K m 2 2 _2 M R2 S dt x_ − x λi − N − φm_ ; 73 corresponds to the maximally symmetric space H × ; ¼  2 N     ð Þ Z Xi 1 in the absence of anisotropies, the last term would vanish ¼ and one would simply use Milne coordinates for flat which is a simple generalization of Eq. (65). The K Minkowski spacetime. variables λi, i 1; …;K, can now correspond to anisotropy It is well known that the dynamics of anisotropies in the variables or¼ minimally coupled scalar fields with the Bianchi I model are equivalent to those of minimally unusual normalization (71). Equation (73) is now the coupled free scalar fields in this background. To see this, action of a particle moving in an M K 1 -dimensional we momentarily switch to Einstein gauge, in which the superspace with curved metric ð þ þ Þ metric is given by Eq. (62) with a general A t . The action ð Þ of a free scalar field in this background is K 2 α β 2 2 ds ηαβdx dx − x dλi 2 ¼ 1 4 2 2 τ_ Xi 1 − d xp−g ∂τ V0 dtA 68 ¼ 2 Z ð Þ ¼ Z 2N ð Þ K ffiffiffiffiffiffi 2 2 2 2 −da a gHM a dλi : 74 ¼ þ þ i 1 ð Þ (identical to the action of a free scalar in a flat FRW X¼ universe; a homogeneous scalar field does not feel the anisotropies). To see that this reduces to the kinetic terms By an elementary generalization of the procedure described in Sec. III A, Eq. (73) gives a Hamiltonian constraint for the anisotropies λi in Eq. (64), we note that since the scale factor (7) is Weyl invariant, one can express it both in 1 m Weyl and Einstein gauge, C ≔ gμν x; λ p p ≈ 0 75 2m ð Þ μ ν þ 2 ð Þ 1 3 2 2 2 ⃗ 2 A where gμν is the inverse metric on superspace, and p a ϕ − χ Weyl 69 μ ¼ 2ρ ð Þj ¼ 8ρ πG ð Þ α 0 Einstein includes conjugate momenta for both the variables x and the new degrees of freedom λ ; more concretely, −4 i with ρ ρ0A in Einstein gauge (in Weyl gauge, ρ ρ0 is ¼ ¼ constant). To change gauges, one hence has to replace 1 1 1 m C −p2 gij ν ζ ζ δijk k 76 2m a a2 HM i j a2 i j 2 4πG ¼  þ ð Þ þ  þ ð Þ A2 → ϕ2 − χ⃗ 2 ; 70 jEinstein 3 ð ÞjWeyl ð Þ in terms of the momentum pa conjugate to a, momenta ζi 4πG conjugate to the conformally coupled scalar field variables the factor 3 can be absorbed in the normalization of the i M ν living on H , and momenta ki conjugate to the free scalar fields, scalar fields and anisotropy variables. When quantizing this Hamiltonian constraint in order to 4πG λ ≔ τ 71 obtain the Wheeler-DeWitt equation, there is now an rffiffiffiffiffiffiffiffiffi3 ð Þ ambiguity, the well-known quantization ambiguity for a

103510-15 STEFFEN GIELEN and NEIL TUROK PHYSICAL REVIEW D 95, 103510 (2017) particle moving on a curved manifold [25]: if the Ricci K 2M K − 1 R ð þ Þ : 82 scalar for the superspace metric (74) is R, the general ¼ a2 ð Þ expression for the quantum Hamiltonian is As in Sec. III, one can simplify the Wheeler-DeWitt 1 m equation by Fourier transforming on HM × RK from ν H −□ ξR 77 ¼ 2m ð þ Þþ 2 ð Þ and λ to the momenta ζ and k. One then has where □ is the Laplace-Beltrami operator for the curved ∂2 M K ∂ c þ − m2 Ψ a;ζ;k;m 0 83 metric (corresponding to the operator ordering that ensures ∂a2 þ a ∂a a2 þ  ð Þ¼ ð Þ that the Hamiltonian is independent of the choice of coordinates on superspace) and ξ is, in general, a free with parameter. Halliwell [22] has given the following strong ξ argument for fixing : the (classical) Hamiltonian of 1 2 1 ⃗ 2 ⃗ 2 H c − M − 1 δM;0 − ζ − k − ξK 2M K − 1 ; minisuperspace models is really NC (see Sec. III) ¼ 4 ð Þ þ 4 ð þ Þ where the lapse function N is arbitrary,¼ and in particular can 84 be redefined arbitrarily, N → Ω−2N~ , where Ω x; λ is any ð Þ ð Þ function on minisuperspace. Under such a redefinition, the where we have explicitly included the case M 0 through constraint (75) becomes the Kronecker delta. This is precisely the same¼ functional form as the Wheeler-DeWitt equation for FRW universes, 1 m~ x; λ ~ μν Eq. (27), and so the extension of our formalism from FRW C ≔ g~ x; λ pμpν ð Þ ≈ 0 78 2m ð Þ þ 2 ð Þ symmetry to the Bianchi I model and the inclusion of minimally coupled scalars are completely straightforward. with g~μν Ω−2gμν and m~ x; λ Ω−2m, leading by the The constant c now gets contributions from the eigenvalues same argument¼ as above toð a quantumÞ¼ Hamiltonian of the Laplacian on HM × RK, as well as from the curvature 1 m~ x; λ on superspace. H~ −□~ ξR~ ð Þ ; 79 We can now obtain the general solution to Eq. (83) in the 2 2 ¼ m ð þ Þþ ð Þ usual way, by setting where now □~ and R~ are the Laplace-Beltrami operator and Ψ − M K =2 ~ a; ζ;k;m a ð þ Þ χ a; ζ;k;m 85 Ricci scalar for the conformally rescaled metric g on ð Þ¼ ð ÞðÞ superspace. Halliwell now asks that, since redefining the lapse is always possible classically, the solutions Ψ and Ψ~ to eliminate the first derivative. χ then satisfies the differ- ential equation to HΨ 0 and H~ Ψ~ 0 be related by a conformal trans- ¼ ¼ formation, Ψ~ ΩγΨ for some γ, and finds that this is only 2 ∂ c0 possible if one¼ fixes ξ to be the conformal coupling, − m2 χ a; ζ;k;m 0 86 ∂a2 a2 þ  ð Þ¼ ð Þ M K − 1 ξ þ 80 where c ≡ c 1 M K M K − 2 , which has two ¼ 4 M K ð Þ 0 þ 4 ð þ Þð þ Þ ð þ Þ independent solutions in terms of Bessel functions of the (recall that the dimension of our superspace manifold is first and second kind, M K 1; Ref. [22] gives an overall minus sign for ξ, þ þ presumably due to a different sign convention for the Ricci χ1 paJ1p1 4c ma ; χ2 paY1p1 4c ma : 87 ¼ 2 þ 0 ð Þ ¼ 2 þ 0 ð Þ ð Þ curvature). Demanding covariance under field redefinitions ffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffi ffiffiffiffiffiffiffiffiffi of the lapse function hence fixes ξ uniquely. Special cases A more convenient basis is given by the Hankel functions are M K 0 where there is no conformal coupling 1;2 þ ¼ Hð Þ of the first and second kind (which are just linear that can restore covariance under lapse redefinitions, and combinations of the Bessel functions), so that two linearly M K 1 where the Laplace-Beltrami operator is con- þ ¼ independent solutions of Eq. (83), for fixed ζ, k and m, are formally covariant and ξ 0. finally given by The Wheeler-DeWitt¼ equation for Ψ Ψ a; ν; λ;m , ¼ ð Þ corresponding to the classical constraint (76), then becomes − −1 2 2;1 ψ a a M K = Hð Þ ma : ;− ð þ Þ 1p1 4 88 þ ð Þ¼ 2 c0 ð Þ ð Þ ∂2 ∂ þ M K 1 2 þ − Δ M RK ξR m Ψ 0 81 ffiffiffiffiffiffiffiffiffi ∂a2 þ a ∂a a2 H × þ þ  ¼ ð Þ As indicated by the subscript ; −, these functions re- present positive- and negative-frequencyþ modes for the with Wheeler-DeWitt equation. Indeed, when extended to

103510-16 QUANTUM PROPAGATION ACROSS COSMOLOGICAL … PHYSICAL REVIEW D 95, 103510 (2017) negative a through the analytic continuation (see, e.g., 1 2 1 ⃗ 2 ⃗ 2 c0 − M − 1 δ 0 − ζ − k 2 iπν 1 1 M; Ref. [26]) Hðν Þ −z −e Hðν Þ z and Hðν Þ −z ¼ 4 ð Þ þ 4 2 2 −iπν 2 ð Þ¼ ð Þ ð Þ¼ M M − 2 K M − M − 1 −e Hðν Þ z , ψ ;− have the interesting property of cor- ð Þþ ð Þ 92 respondingð toÞ pureþ positive and pure negative frequency, þ 4 M K ð Þ ð þ Þ respectively, both at positive and negative infinite a, if we use the value (80) for ξ, that is, we fix the ordering − M K =2 −ima ψ a ∼ a ð þ Þ e ;a→ ∞; ambiguities by demanding coordinate covariance on super- þð Þ Æ − M K =2 ima space and covariance under redefinitions of the lapse ψ − a ∼ a e ;a→ ∞: 89 ð Þ ð þ Þ þ Æ ð Þ function, giving a purely quantum contribution in the second line of Eq. (92). If we ignore the trivial case M K 0,we That is, for the Wheeler-DeWitt equation (83) one finds that can rewrite Eq. (92) as ¼ ¼ an incoming positive-frequency mode simply continues to an outgoing positive-frequency mode, with the potential at ⃗ 2 1 a 0 not even leading to a phase shift. This complete −ζ − 4 M ≥ 1;K 0; ¼ 2 ¼ invisibility of the 1=a potential is a direct consequence of 2 8 −k⃗ − 1 K ≥ 1;M 0; the symmetry of Eq. (83) under a → λa and m → λ−1m, > 4 c0 > ¼ 93 ¼ > 2 ⃗ 2 1 ð Þ which forbids any phase shift. These special properties of a < −ζ − k − 4 K ≥ 1;M 1; 1=x2 potential, and its invisibility in a scattering process, ¼ −⃗ 2 − ⃗ 2 − 1 M−1 K ≥ 1 1 > ζ k 4 4ð M ÞK K ;M > : are well known in quantum mechanics. In the context of our > þ ð þ Þ perfect bounce scenario, they imply that the Universe can :> go through the singularity a 0 without any noticeable This is an intriguing result. The contributions coming ¼ impact on its evolution, when viewed asymptotically. This from anisotropy or scalar field momenta are both negative. is already true classically, where the classical solutions The numerical term is fixed by covariance. The first line bounce without any net time delay or advance: the classical corresponds to the situation of Sec. III, where no anisotropies Hamiltonian is equal to the constraint (76) times a lapse or minimally coupled scalars are present [from Eq. (27), function, removing the first derivative term changes c in Eq. (28) to c0 given here]. The similarity of the first three lines is not a 1 1 1 m coincidence; for M ≤ 1, the superspace metric (74) is H N −p2 gij ν ζ ζ k⃗ 2 : 90 ¼ 2m a þa2 HM ð Þ i j þa2 þ 2  ð Þ conformally flat. As we have imposed conformal coupling to the Ricci scalar on superspace in Eqs. (79) and (80),the The terms multiplying 1=a2 are again conserved and can dynamics must be equivalent to the flat superspace case of Sec. III. be replaced by a constant, −c0 with c0 < 0; classically the The value c − 1 is well known as a critical value in effect of anisotropies and momenta in the scalar fields 0 ¼ 4 always leads to an attractive potential for a, centered on the quantum mechanics of an inverse square potential. If 1 the singularity at a 0. The classical solutions to the c0 ≥−4, the negative classical potential is outweighed by equations of motion including¼ the constraint are then the kinetic energy due to the Heisenberg uncertainty principle, rendering the energy spectrum strictly positive.

2 c0 2 2 c0 2 There are various infrared regularized versions of the theory a N t − t0 τ − τ0 91 ¼ m2 þ ð Þ ¼ m2 þð Þ ð Þ in which the spectrum is made discrete by including a positive harmonic potential [27], with a taken either on the in terms of proper time τ Nt. These solutions are singular infinite line, the half-line a>0, or by imposing periodicity at a 0 and perform an¼ excursion into the antigravity in a, in which case the model becomes the Calogero- region¼ of imaginary a, just as the generic flat FRW Sutherland model (see, e.g., Ref. [28]). These are well- solutions described in Sec. III which would be of the exact defined, exactly solvable models which exhibit, among same form. The attractive potential at a 0 speeds up the other interesting phenomena, anomalous dimensions in the trajectory as it heads toward the singularity,¼ but this time operator product expansion [27]. 1 advance is canceled by the additional time it takes to cross If, however, c0 < − 4, any finite energy wave function antigravity. Indeed, both at large positive and negative a we has an infinite number of oscillations on the way to a 0. ¼ have simply a τ ≈ τ − τ0 . In quantum mechanics, standard arguments then imply an In the quantumð Þ ð theory,Þ ordering ambiguities in the infinite number of lower energy states, and hence a Hamiltonian constraint can alter the coefficient of the spectrum which is unbounded below. It has been claimed 1=a2 potential, making it repulsive in some cases. Indeed, that the theory is nevertheless renormalizable, although the the relevant coefficient of the potential is the one appearing renormalization group displays a limit cycle [29]. (There is in Eq. (86), a large literature on inverse square potentials in quantum

103510-17 STEFFEN GIELEN and NEIL TUROK PHYSICAL REVIEW D 95, 103510 (2017) mechanics, and even some experimental tests. See, e.g., For the remainder of the paper, we assume that the Ref. [30] for a recent discussion and further references.) quantum mechanics for a makes sense. As explained in At the minisuperspace level discussed here, negative Ref. [1], this allows us to calculate the propagation of the energy states are irrelevant because we are only interested Universe, and all inhomogeneous modes in it, by solving in solutions of the Wheeler-DeWitt equation with positive the theory on complex trajectories which bypass a 0 in energy, defined by m2. However, when we include inter- the complex a plane. Remarkably, as was also explained¼ in actions with other modes, such as the inhomogeneous Ref. [1], due to its scale-invariant property, the inverse 1 square potential, if present, is actually invisible in our final modes of gravitons or scalars, then for c0 < − 4 it is possible that the negative energy states for the scale factor a would results for “in-out” amplitudes. become excited, potentially signifying strong backreaction as the Universe passes through the quantum bounce. The A. Feynman propagator problem may be avoided in two ways. For M 0 or M 1 ¼ ¼ Having defined positive- and negative-frequency modes one can restrict consideration to background cosmologies by their asymptotics, given in Eq. (89) (and without using for which the zero-mode momenta of the anisotropy and any boundary condition at a 0), it is easy to obtain the scalar fields are strictly 0, in which case the quantum Feynman propagator for the anisotropic¼ case as a Green’s mechanics for a lies on the critical boundary where it (just) function for the Wheeler-DeWitt equation, by using the makes sense. Or, one can include additional conformally Wronskian method as before. coupled scalars, taking M>1 so that, from the last line With the quantum Hamiltonian given by Eq. (77), the of Eq. (93), the quantum mechanics of a is well defined for Feynman propagator satisfies a range of classical anisotropy and scalar field momenta. For K 2 (i.e., only anisotropies but no minimally coupled 2 −□ ξR m G x; λ;mx0; λ0;m0 scalars)¼ and M>4, the numerical contribution can be large ð þ þ Þ ð j Þ −1 M 1 K enough to make the potential repulsive at small momenta. −2im −g 2δ þ x − x0 δ λ − λ0 δ m − m0 94 ¼ ð Þ ð Þ ð Þ ð ÞðÞ If we consider classical solutions with this (order ℏ −1 squared) potential, an isotropic universe with no scalar where we must introduce a factor −g 2 for the nontrivial momenta would bounce off the repulsive potential and metric determinant on superspace.ð AgainÞ switching to the avoid the singularity altogether. Quantum mechanically, scale factor coordinate a, Eq. (94) is equivalent to however, if we extend the range of a to negative values, ∂2 ∂ then a tunnels through the barrier in a process which may M K 1 2 þ − Δ M RK ξR m G be described with complex classical solutions, as we ∂a2 þ a ∂a a2 H × þ þ  explained in Ref. [1]. δ a − a δM K ν − ν ; λ − λ 2 0 þ 0 0 The conclusion is that when anisotropy and scalar field − im ð M K Þ ð Þ δ m − m0 degrees of freedom are included, then for small numbers ¼ a þ pgHM ð Þ of conformal scalars, the isotropic cosmology with no ffiffiffiffiffiffiffiffi with G ≡ G a; ν; λ;ma ; ν ; λ ;m , and the metric deter- scalar momenta is a special case, poised on the edge of a 0 0 0 0 minant on superspaceð j is now madeÞ explicit. Again, we can qualitatively different (and perhaps ill-defined) phase. On now go to Fourier space on HM × RK introducing momenta the positive side, this finding may turn out to be a selection i i principle, telling us that anisotropic or kinetic-dominated ζ and k ; the Feynman propagator in Fourier space satisfies singularities should be excluded from the theory whereas ∂2 M K ∂ c isotropic universes with zero scalar momenta are allowed. 2 2 þ − 2 m G a; ζ;k;ma0; ζ;k;m0 If so, this would imply that black hole singularities, which ∂a þ a ∂a a þ  ð j Þ − M K locally resemble strongly anisotropic cosmological singu- −2ima ð þ Þδ a − a0 δ m − m0 95 larities, do not correspond to a bounce (contradicting the ¼ ð Þ ð ÞðÞ interpretation given by Ref. [4], for example); there would with c as in Eq. (84). Since we have already identified be no “born again” universe on the other side of the black the positive- and negative-frequency solutions (88) of the hole singularity. On the negative side, one may wonder corresponding homogeneous equation, it is immediate to whether the inclusion of inhomogeneities could lead to write down the solution to Eq. (95) with the correct problems even for the isotropic, nonkinetic cosmological boundary conditions, bounce. We emphasize that, for M 0 or M 1, any ¼ ¼ amount of classical momentum in the zero modes of the G a; m a ;m −2ima− M K δ m − m ð j 0 0Þ¼ ð þ Þ ð 0Þ anisotropy or scalar degrees of freedom takes the quantum −1 × W ψ −; ψ ψ − a0 ψ a θ a − a0 mechanics of a into the subcritical regime. Perhaps it is ð þÞ ð ð Þ þð Þ ð Þ essential to work at M>1 for the theory to make sense. ψ − a ψ a0 θ a0 − a ; 96 Clearly, we have only scratched the surface with this þ ð Þ þð Þ ð ÞÞ ð Þ discussion, and there is a great deal to explore further. where the Wronskian is

103510-18 QUANTUM PROPAGATION ACROSS COSMOLOGICAL … PHYSICAL REVIEW D 95, 103510 (2017) − M K ⃗ ik·x 4a ð þ Þ v η; x kvk η e , with v−k η vk η Ã, with the W ψ −; ψ 97 ð Þ¼ ð Þ ð Þ¼ ð Þ ð þÞ¼ iπ ð Þ coefficientsP decomposed into irreducible representations of the little group of rotations about k. Now consider the and no longer constant in a, as is consistent with the action for the perturbations. At leading order, it is quadratic appearance of a first derivative in Eq. (83). The Wronskian and it is diagonalized by the above mode decomposition. takes care of the factors of a appearing in the elimination of After a suitable time-dependent rescaling of the perturba- the first derivative, Eq. (85), and cancels the determinant tions, the kinetic terms can always be brought to canonical − M K factor a ð þ Þ. The final result is form in which the action reads [see, e.g., Ref. [31], page 269, Eq. (10.59)] G a; m a ;m ð j 0 0Þ πm − M K−1 =2 2 δ m − m0 aa0 ð þ Þ S 2 _ a 2 − ;a a 2 ¼ 2 ð Þð Þ ð Þ dη vk wk η vk ; 99 ¼ k Z ðj j ð Þj j Þ ð Þ 1 2 X;a × Hðν Þ ma Hðν Þ ma θ a − a a ↔ a 98 ð ð 0Þ ð Þ ð 0Þþð 0ÞÞ ð Þ where the index a labels the independent modes (here, with ν ≡ 1 p1 4c , which is consistent with the results of 2 þ 0 scalar and tensor), and Ref. [1] (withffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK ≡ D − 2 as only the D − 2 anisotropy degrees of freedom of a D-dimensional universe were 2;a a 2 2;a considered there). One can check that in the absence w η kcs m η 100 k ð Þ¼ð Þ þ eff ð ÞðÞ of anisotropies or minimally coupled scalar fields, K 0, this result reduces to the expression obtained in where ca is the speed of sound, 1=p3 for the scalar Sec.¼ III, i.e., the propagator for a free massive particle s acoustic modes and unity for the tensor modes. In general, in (M 1)-dimensional Minkowski spacetime. By our ffiffiffi the time-dependent “effective mass” introduces a nontrivial remarksþ below Eq. (93) the same should be true for η-dependence. However, in our chosen background, the M 0 or M 1 and general K, where the superspace ¼ ¼ effective mass vanishes for both the scalar and tensor metric is conformally flat. a a modes so wk kcs in both cases. We now make¼ the assumption that the perturbations are VI. PERTURBATIONS well described by linear theory for wide intervals of In this section, we extend our analysis to inhomogeneous conformal time η well before and well after the bounce. cosmology, treated perturbatively at both linear and non- As we see later, we cannot actually take the limit of infinite linear order. We aim to solve the following problem: given positive and negative conformal time because of the effect an incoming state at large negative a consisting of a flat, of nonlinearities in the fluid. Nevertheless, in the semi- FRW, radiation-dominated classical background universe classical approximation, and for modes whose wavelength with perturbations in their local adiabatic vacuum state, is longer than the thermal wavelength of the fluid, the what is the outgoing quantum state at large positive a,as periods of incoming and outgoing conformal time during defined by our analytic continuation prescription? This which linear theory remains valid are very large. We define question can be answered, in the semiclassical limit, by our incoming and outgoing states during these intervals. using complex solutions of the classical Einstein-matter When linear theory is valid, and when the frequencies a a a 2 field equations. If one sends in any combination of wk η change adiabatically, dwk=dη = wk ≪ 1, the linearized positive- (negative-) frequency modes then, even quantumð Þ states of the system areð well describedÞ ð Þ by those after including the effects of nonlinearities in the field of a set of decoupled harmonic oscillators. Let us denote equations, it turns out that one finds only positive- the corresponding real coordinates, i.e., the real and a (negative-) frequency linearized modes coming out. As imaginary parts of the vk, by the coordinates qm, where we now explain, this is sufficient to show, semiclassically, the single index m runs over all of the real, independent that the outgoing quantum state is also the local adiabatic modes. Each of the coordinates qm contributes an action vacuum. Hence, at a semiclassical level, there is no particle S 1 dη q_ 2 − ω η 2q2 , and the adiabatic vacuum m ¼ 2 ð m mð Þ mÞ production across the bounce. state isR just the product of the corresponding harmonic Let us see this in detail. Consider a classical time- oscillator ground states, dependent background solution of the Einstein-matter equations. If the matter is a perfect fluid, the only 2 ℏ Ψ ℏ 1=4 −ωmqm= 2 propagating degrees of freedom are scalar density pertur- 0 η;q ωm= π e ð Þ: 101 ð Þ¼ ð Þ ð Þ bations and tensor gravitational wave modes. At the Ym linearized level, we can decouple the modes by exploiting the homogeneity and isotropy of the background: for a This state is uniquely defined by amΨ0 0 for all m, for flat background, every mode is a sum of plane waves the annihilation operator ¼

103510-19 STEFFEN GIELEN and NEIL TUROK PHYSICAL REVIEW D 95, 103510 (2017) 1 ≡ ℏ d large positive time. We solve the classical Einstein- am ωmqm : 102 matter equations with these two boundary conditions p2ωmℏ  dqm þ  ð Þ in linear perturbation theory. At linear order, the ffiffiffiffiffiffiffiffiffiffiffiffi c Let us assume that the incoming state of the pertur- solution satisfying the boundary conditions is qm η~ ikc η~−η ð Þ¼ bations is Ψ η ;q Ψ0 η ;q at some large negative q e s . Below, we give the complete solution for inð 0 Þ¼ ð 0 Þ m ð Þ η0, for which linear theory is valid. The quantum generic perturbation modes at linear and nonlinear order. fluctuations in the fluid density may be shown to be We find that the solution is well described by linear small compared to the background density provided the perturbation theory at large negative and large positive wavelength of the modes is longer than the thermal times, with small nonlinear corrections, and that an wavelength, a condition which is in any case required in incoming positive (negative) frequency mode evolves to order for the fluid description to hold. The outgoing an outgoing positive (negative) frequency mode which quantum state, at some large positive time η,isthen directly implies that the outgoing quantum state is the given by propagating the incoming vacuum Ψ0 to large local adiabatic vacuum. To verify this, we need only positive times η, for which linear theory is once again apply the annihilation operators a ∝ ip ω q m m þ m m ¼ valid, using the path integral, ℏ d Ψ ωmqm to out η;qm as given in Eq. (103). dqm þ ð Þ Using the Hamilton-Jacobi equation, the result is pro- i S Ψ ℏ q;η;q0;η0 Ψ out η;q ≈ N Dqe ð Þ dqm0 0 η0;q0 ; 103 portional to ipc ω qc η , which vanishes if the ð Þ Z ð Þ ð Þ m m m Ym solution is pureð negativeþ frequency.Þð Þ Hence the incoming adiabatic vacuum evolves to the outgoing adiabatic where S q; η; q ; η is the full, nonlinear Einstein-matter 0 0 vacuum, and there is no particle production across actionð taken withÞ boundary conditions q η q, ð Þ¼ the bounce. q η0 q0; Dq indicates the complete path-integral measureð Þ¼ and N is a normalization constant. We com- pute Eq. (103) in the semiclassical approximation, by A. Basic setup and conventions finding the appropriate complex classical solution qc η~ , We study perturbations about a flat (κ 0) radiation- mð Þ ¼ η0 < η~ < η, which is a stationary point of the combined dominated FRW universe in a perturbation expansion. Ψ exponent. Substituting Eq. (101) for in and varying the We go to nonlinear order but, for simplicity, restrict exponent with respect to qm0 yields, using the Hamilton- consideration to planar symmetry so that the metric Jacobi relation, the initial condition for the classical depends only on conformal time η and one spatial solution qc, coordinate x, with two orthogonal spatial directions y; z . To keep the calculations manageable, we do c c ð Þ ipm ωmqm η0 0; 104 not introduce conformally or minimally coupled scalar ð þ Þð Þ¼ ð Þ fields, so M K 0.WeworkinEinsteingauge,i.e., c c ¼ ¼ where pm q_ m is the canonical momentum. The initial in the usual formulation of general relativity coupled to ¼ c condition (104) specifies that qm is pure negative a radiation fluid. frequency at η0, a large negative time. The final The general form of the metric compatible with our boundary condition is just qc η q ,whereη is a assumed symmetry is mð Þ¼ m

−1 ϵg η;x ϵg η;x þ ηηð Þ ηxð Þ 0 ϵgηx η;x 1 ϵgxx η;x 1 ds2 a2 η ð Þ þ ð Þ : 105 ¼ ð ÞB 1 ϵgyy η;x ϵgyz η;x C ð Þ B þ ð Þ ð Þ C B ϵg η;x 1 ϵg η;x C @ yzð Þ þ zzð Þ A We can still apply coordinate transformations that leave this form of the metric invariant. A coordinate transformation η η~ ϵg x; η~ changes the metric coefficients as ¼ þ ð Þ a_ a_ δg −2 g g_ ; δg −2g ; δg δg δg 2 g 106 ηη ¼ a þ  ηx ¼ 0 xx ¼ yy ¼ zz ¼ a ð Þ

: where here and in the remainder of this section is derivative with respect to η and 0 denotes derivative with respect to x. We use this freedom to eliminate gηx and introduce a different notation for the metric perturbation functions (note that in this section ψ denotes a scalar metric perturbation, not a solution to the Wheeler-DeWitt equation as in earlier sections),

103510-20 QUANTUM PROPAGATION ACROSS COSMOLOGICAL … PHYSICAL REVIEW D 95, 103510 (2017) see, there are also six nontrivial Einstein equations relating ds2 a2 η −1 2ϵϕ η;x dη2 ¼ ð Þð þ ð ÞÞ these. To proceed, we assume that all of the perturbation functions can further be expanded as a power series in ϵ, 1 2ϵ ψ η;x γ η;x dx2 ϵh× η;x dydz þð þ ð ð Þþ ð ÞÞÞ þ ð Þ T n−1 h η;x 2 ϕ η;x ϵ ϕ η;x ; etc: 112 1 ϵ 2ψ η;x ð Þ dy ð Þ¼ nð Þ ð Þ þ  þ  ð Þþ 2  Xn≥1 T h η;x 2 1 ϵ 2ψ η;x − ð Þ dz : 107 The idea is now to solve the Einstein equations Gμν þ  þ  ð Þ 2   ð Þ ¼ 8πGTμν order by order in ϵ; the Einstein equations also imply energy-momentum conservation ∇ Tμ 0 for the The form (107) is still left invariant by a transformation μ ν ¼ fluid. First, for the background (at order ϵ0) we have the of the form equations

x~ η η~ ϵ G η~ dXf_ X; η~ ; a_ a_ 2 8πG ¼ þ  ð Þþ ð Þ ρ_0 4 ρ0 0; ρ: 113 Z þ a ¼ a2 ¼ 3 ð Þ x x~ ϵf x;~ η~ ; 108 ¼ þ ð Þ ð Þ −4 The first one tells us that ρ0 ∝ a for some constant which we use to simplify the matter variables. The M; the Friedmann equation then gives the solution energy-momentum tensor for radiation is a η ∝ η, the simplest example of a perfect bounce that weð Þ have already discussed in the introduction to this 4 1 paper. It follows that a_ 1, and that analytic continu- T ρu u ρg ; uμu −1: 109 a ¼ η μν ¼ 3 μ ν þ 3 μν μ ¼ ð Þ ation in the scale factor a (as we have discussed in previous sections) is equivalent to analytic continuation The density ρ η;x and four-velocity uμ η;x can also in the conformal time coordinate η, which we use in be written in termsð Þ of background and perturbationð Þ as this section. At order ϵn in the perturbation expansion, the six nontrivial Einstein equations are ρ η;x ρ0 η 1 ϵδ η;x ; ð Þ¼ ð Þð þ rð ÞÞ 1 μ 0 3 6 2 6 u η;x v η;x ; ϵv η;x ; 0; 0 : 110 − 2 − _ − _ ð Þ¼a η ð ð Þ ð Þ Þ ð Þ 2 δr;n 2 ϕn ψ n00 γn ψ n J1;n; 114 ð Þ η η þ η η ¼ ð Þ μ 0 The constraint u uμ −1 can be solved for v η;x . 1 ¼ ð Þ ϕ ψ_ J2 ; 115 Under a coordinate transformation (108), we have η n0 þ n0 ¼ ;n ð Þ δv −f_,sothatwecansetv 0 everywhere, i.e., ¼ ¼ adopt a coordinate system in which the radiation is at 1 2 2 _ 4 δ − ϕ ϕ ψ_ 2ψ̈ J3 ; 116 rest everywhere (comoving gauge). The remaining η2 r;n η2 n þ η n þ η n þ n ¼ ;n ð Þ gauge freedom is then under transformations 2 ̈ T _ T T h h − h J4 ; 117 η η~ ϵG η~ ;xx~ ϵf x~ ; n þ η n ð n Þ00 ¼ ;n ð Þ ¼ þ ð Þ ¼ þ ð Þ y y~ ϵ ι1y~ ι2z~ ι3 ;zz~ ϵ ι4z~ ι5y~ ι6 2 ¼ þ ð þ þ Þ ¼ þ ð þ þ Þ ̈ × _ × × h h − h J5 ; 118 111 n þ η n ð n Þ00 ¼ ;n ð Þ ð Þ 1 2 2 2 where the ιi are arbitrary constants and G and f are − δ ϕ ψ − ϕ − γ_ − ϕ_ free functions. Under such a transformation, δϕ η2 r;n þ η2 n þ n00 n00 η n η n a_ _ a_ T 2 ¼ − a G − G, δψ a G, δγ f0, δh a − b and 4 ¼ ¼ ¼ ð Þ − ψ_ − γ̈ − 2ψ̈ J6 119 δh× 2 c d , and so functions of this form in the η n n n ¼ ;n ð Þ perturbations¼ ð þ Þ are to be considered pure gauge. We solve the Einstein equations in Fourier space, where for some “source terms” Ji;n that are nonlinear combina- the gauge freedom for the functions ϕ, ψ, hT and h× tions of the lower order perturbations. is somewhat hidden as it only becomes apparent for We first note that Eqs. (117) and (118) that govern the T × k 0. tensor modes hn and hn are already decoupled from the ¼We are left with five free functions for the metric others. For the scalars, Eqs. (114) and (115) can be solved T × (ϕ; ψ; γ;h ;h ) and the density perturbation δr.Aswe for δr and ϕ directly. From Eq. (115) we get

103510-21 STEFFEN GIELEN and NEIL TUROK PHYSICAL REVIEW D 95, 103510 (2017) x For k 0, the two independent solutions are hT constant ϕn η;x −ηψ_ n η;x Fn η η dx0J2;n η;x0 ¼ ¼ ð Þ¼ ð Þþ ð Þþ Z ð Þ and hT ∼ 1=η. We can write the general solution as

120 −ikη ikη ð Þ T d2 dk ikx e e h1 d1 e b1 k b2 k where F η is a free function; then Eq. (114) implies that ¼ þ k0η þ Z 2π  ð Þ kη þ ð Þ kη  nð Þ 126 2 ð Þ 2 2 2 η δr;n η;x − η ψ n00 ηγ_n 2Fn J1;n ð Þ¼ 3 þ 3 þ þ 3 where k0 is an arbitrary momentum scale to make d2 × x dimensionless. h1 satisfies the same differential equation; 2η dx0J2;n: 121 its general solution is þ Z ð Þ −ikη ikη Substituting these relations into Eqs. (116) and (119) and × e2 dk ikx e e h1 e1 e c1 k c2 k : taking linear combinations one obtains ¼ þ k0η þ Z 2π  ð Þ kη þ ð Þ kη  127 _ η x ð Þ γ_n ηψ n00 − 6ψ_ n − 3Fn − J1;n − 3 dx0J2;n ¼ 2 Z For a real solution we need b1 k −b1Ã −k , b2 k x − − ð Þ¼ ð Þ ð Þ¼ _ 3η b2Ã k and similar for c1 k and c2 k . − 3η dx0J2;n J3;n 122 ð Þ ð Þ ð Þ Z þ 2 ð Þ We recognize d1 and e1 as gauge modes corresponding to coordinate transformations (111), whereas d2 and e2 are and physical k 0 modes. The free functions b1 k ;b2 k ; ¼ ð Þ ð Þ c1 k and c2 k are the physical gravitational degrees of ð Þ ð Þ _ ̈ _ freedom. 2 1 F F J1 J1 ψ̈ ψ_ − ψ − n − n − ;n − η ;n Now consider the general inhomogeneous equation, n þ η n 3 n00 ¼ η 2 4 12 η 11 η J 2 _ 6;n ̈ T _ T T J20 ;n J3;n J3;n h h − h J4 : 128 þ 6 þ 12 þ 4 þ 6 n þ η n ð n Þ00 ¼ ;n ð Þ x J2;n _ η ̈ − dx0 2J2;n J2;n : Again, we go to Fourier space and first consider k ≠ 0.We Z  η þ þ 2  use the Wronskian method to determine a suitable Green’s 123 ’ ð Þ function; in contrast to the Green s function that appeared as a Feynman propagator in earlier sections, here the ’ Equation (123) can now be solved for ψ n using Green s boundary conditions are that the higher order perturbations functions; Eq. (122) then gives γ by a single integration are set to 0 at some conformal time η0 in the far past, so over η, and from Eq. (120) and Eq. (121) one can obtain that only a linear (purely positive- or purely negative- explicit expressions for δr;n and ϕn at each order. At each frequency) mode is present. Given two independent sol- order in ϵ, this provides an explicit algorithm for solving utions hT and h~ T to the homogeneous equation (124), the – 1 1 the system of Einstein equations (114) (119). Green’s function for these boundary conditions is

T ~ T T ~ T B. Tensor perturbations h η h1 η − h η h1 η G η; η 1 ð 0Þ ð Þ 1 ð Þ ð 0Þ 129 ð 0Þ¼ W η ð Þ Equations (117) and (118) are easily solved. First, ð 0Þ consider the homogeneous equation solved by the first- order perturbation, for η > η0 > η0 where η0 is the initial time at which only a linear perturbation is assumed to be present, and 0 2 otherwise. The Wronskian is W η ≡ hT h~ T − hT h~ T ̈ T _ T T 0 1 1 0 1 0 1 h h1 − h 00 0: 124 ’ ð Þ ð Þ ð Þ 1 þ η ð 1 Þ ¼ ð Þ as before. Using this Green s function, we find that one particular solution to Eq. (128) is We can easily find the general solution in Fourier space, for 1 η ≠ 0 T k , hn η;k dη0η0 sin k η − η0 J4;n η0;k ; 130 ð Þ¼kη Z ð ð ÞÞ ð Þ ð Þ dk hT η;x eikxhT η;k ; while for k 0 we find 1 ð Þ¼Z 2π 1 ð Þ ¼ −ikη ikη η T e e T 1 h η;k b1 k b2 k : 125 hn η; 0 dη0η0 η − η0 J4;n η0; 0 ; 131 1 ð Þ¼ ð Þ kη þ ð Þ kη ð Þ ð Þ¼η Z ð Þ ð Þ ð Þ

103510-22 QUANTUM PROPAGATION ACROSS COSMOLOGICAL … PHYSICAL REVIEW D 95, 103510 (2017)

→ ikη which is just the limit k 0 of Eq. (130). Expressions for dk ikx − 6 × γ1 η;x a3 x e a1 k e p3 − − ip3 hn are analogous. In the integrals in Eqs. (130) and (131) as ð Þ¼ ð Þþ 2π  ð Þ  kη  Z ffiffiffi well as in the following, the initial time η0 that should ffiffi ikη 6 appear as the lower limit of integration is suppressed for a2 k ep3 − ip3 ; 136 þ ð Þ  kη þ  ð Þ simplicity; we neglect the η0-dependent contributions as we ffiffi ffiffiffi are only interested in a particular solution. n η dk −ikη _ ikx p3 Clearly, at each order ϵ one can also add a solution of ϕ1 η;x F1 η F1 η e a1 k e the homogeneous equation to this solution for hT. This can ð Þ¼ ð Þþ2 ð ÞþZ 2π  ð Þ n ffiffi T ikη however be absorbed into the linear perturbation h1 .We 1 i 1 i × a2 k ep3 − ; hence set these arbitrary solutions to the homogeneous kη þ p3 þ ð Þ kη p3 equations to 0 for n ≥ 2. ffiffi ffiffiffi ffiffiffi 137 ð Þ

C. Scalar perturbations dk −ikη 4 4i ikx p δr;1 η;x 2F1 η e a1 k e 3 For the scalar perturbation functions ϕ; ψ; γ and δ ,we ð Þ¼ ð ÞþZ 2π  ð Þ kη þ p3 r ffiffi proceed analogously. For clarity, we first derive the general ikη 4 4i ffiffiffi a2 k ep3 − ; 138 solutions for the first-order perturbations, for which there þ ð Þ kη p3 ð Þ are no sources and the general solution is obtained ffiffi straightforwardly. The equation for ψ is Eq. (123) with ffiffiffi 1 where we recognize a3 x and F1 η as encoding the the sources set to 0, i.e., ð Þ ð Þ remaining gauge freedom in comoving gauge (111). a1 k ð Þ and a2 k correspond to the scalar degrees of freedom of ð Þ 2 1 _ ̈ the radiation fluid. ̈ F1 F1 ψ 1 ψ_ 1 − ψ 100 − − 132 In order to obtain the solutions for higher order þ η 3 ¼ η 2 ð Þ perturbations, we derive the Green’s function for the ψ equation (123), which has the general form where F1 is a free function of η. Going into Fourier space, the general solution for k ≠ 0, where F1 does not 2 1 ψ̈ ψ_ − ψ J : 139 contribute, is n þ η n 3 n00 ¼ n ð Þ

Again, the boundary condition for the Green’s function is dk ikx ψ 1 η;x e ψ 1 η;k ; ð Þ¼Z 2π ð Þ to set the higher order perturbations to 0 at some initial η ≠ 0 − i kη i kη conformal time 0. We find, for k , e p3 ep3 ψ 1 η;k a1 k a2 k : 133 η ð Þ¼ ð Þ kηffiffi þ ð Þ kffiffiη ð Þ p3 k η − η0 ψ n η;k dη0η0 sin ð Þ Jn η0;k ; 140 ð Þ¼ kηffiffiffi Z  p3  ð Þ ð Þ The Fourier mode k 0 is a gauge mode [see the ffiffiffi discussion below Eq. (111)¼ ], with general solution and for k 0 the same as for the tensors, ¼ 1 η c1 F1 η ψ n η; 0 dη0η0 η − η0 Jn η0; 0 : 141 ψ 1 η; 0 − c2 − ð Þ ; 134 ð Þ¼η Z ð Þ ð Þ ð Þ ð Þ¼ k0η þ 2 ð Þ

Again, once the solution for ψ n is found, the other since F1 is arbitrary, we can set c1 c2 0 with no loss of perturbation functions γn, ϕn and δr;n can be obtained generality. Putting this together, we¼ have¼ easily. From these expressions, we can now work out the − i kη i kη nonlinear solution for all metric perturbation functions p3 p3 dk ikx e e and the density perturbation order by order in ϵ; all we need ψ 1 η;x e a1 k a2 k ð Þ¼Z 2π  ð Þ kηffiffi þ ð Þ kffiffiη  to do is to expand Einstein equations up to any given order F1 η to find the sources Ji;n and then compute the integrals − ð Þ ; 135 2 ð Þ (130), (131), (140) and (141) to find the perturbations at the next order. where for a real solution we need a1 k −a1Ã −k and ð Þ¼ ð Þ D. Nonlinear positive-frequency modes a2 k −a2Ã −k . ðAsÞ¼ said, fromð thisÞ expression we can determine the other We are now specifically interested in the nonlinear scalar functions γ, ϕ and δr. We find extension of linear positive-frequency modes at a given

103510-23 STEFFEN GIELEN and NEIL TUROK PHYSICAL REVIEW D 95, 103510 (2017) wave number k . (The following calculations and discus- γ̈ γ_ 0 Φ −ϕ ; sion can be extended to the case of an incoming negative- ¼ þ k2 þ k2η frequency mode by simply taking the complex conjugate γ_ of all expressions below, and replacing the lower- by the Ψ −ψ − : 143 k2η upper-half complex plane etc.) We choose the linear ¼ ð Þ positive-frequency modes to be We find that for the linear perturbations Φ and Ψ are equal −ik0η=p3 2 2 e and fall off as 1=k0η at large η , ψ 1 η;x A cos k0x ; j j ð Þ¼ ð Þ k0η ffiffi

−ik0η e − i k η T η 2 p3 0 3 p3 h1 ;x B cos k0x : 142 Φ Ψ πAe ik0η ð Þ¼ ð Þ k0η ð Þ 1 η;k0 1 η;k0 − ð3 3þ Þ : ð Þ¼ ð Þ¼ ffiffi k0η ffiffiffi As seen before, the expression for ψ 1 then determines γ1, 144 ϕ1 and δr;1. These scalar quantities are all gauge dependent ð Þ but one may compute the gauge-invariant Newtonian potentials first introduced by Bardeen [32] and given by The explicit form of the sources at order ϵ2 is, in terms of (in Fourier space for k ≠ 0) the linear perturbations,

12 2 3 T 2 3 × 2 2 1 T T 1 × × 4 J1 2 ϕ h h 2γ ψ 3 ψ h h h h 4γψ 8ψψ − γγ_ ; ¼ η2 þ 16 ðð Þ0Þ þ 16 ðð Þ0Þ þ 0 0 þ ð 0Þ þ 4 ð Þ00 þ 4 ð Þ00 þ 00 þ 00 η 4 4 1 1 1 1 4 12 12 ϕγ_ − ψγ_ − hTh_ T − h_ T 2 − h×h_ × − h_ × 2 − γψ_ ϕψ_ − ψψ_ 2_γ ψ_ 3ψ_ 2; 145 þ η η 2η 16 ð Þ 2η 16 ð Þ η þ η η þ þ ð Þ 2 2 2 1 1 1 1 T _ T × _ × T _ T × _ × J2 2 γϕ − ϕϕ ψϕ ψ γ_ h h h h − ϕ ψ_ 2ψ ψ_ h h h h 2γψ_ 4ψψ_ ; ; ¼ η 0 η 0 þ η 0 þ 0 þ 16 ð Þ0 þ 16 ð Þ0 0 þ 0 þ 8 ð Þ0 þ 8 ð Þ0 þ 0 þ 0 146 ð Þ 4 1 1 1 3 1 3 8 8 2 T 2 × 2 2 T _ T _ T 2 × _ × _ × 2 _ J3 2 ϕ − h − h − 2ϕ ψ ψ h h h h h h − ϕϕ − ϕψ_ ; ¼ η2 16 ðð Þ0Þ 16 ðð Þ0Þ 0 0 þð 0Þ þ 2η þ 16 ð Þ þ 2η þ 16 ð Þ η η 8 1 1 ψψ_ − 2ϕ_ ψ_ ψ_ 2 hTḧ T h×ḧ × − 4ϕψ̈ 4ψψ̈ ; 147 þ η þ þ 4 þ 4 þ ð Þ 4 4 4 T T T T T T _ T _ T _ T _ T _ T _ T J4 2 −γ h − h ϕ − 3 h ψ − 2γ h − 4ψ h − 2h ψ − ϕh ψh − γ_h − h ϕ h ψ_ h ψ_ ; ¼ 0ð Þ0 ð Þ0 0 ð Þ0 0 ð Þ00 ð Þ00 00 η þ η þ η þ − 2ϕḧ T 2ψḧ T 2hTψ̈ ; 148 þ þ ð Þ 4 4 4 × × × × × × _ × _ × _ × _ × _ × _ × J5 2 −γ h − h ϕ − 3 h ψ − 2γ h − 4ψ h − 2h ψ − ϕh ψh − γ_h − h ϕ h ψ_ h ψ_ ; ¼ 0ð Þ0 ð Þ0 0 ð Þ0 0 ð Þ00 ð Þ00 00 η þ η þ η þ − 2ϕḧ × 2ψḧ × 2h×ψ̈ ; 149 þ þ ð Þ

4 2 1 T 2 1 × 2 2 2 1 T T 1 × × J6 2 − ϕ h h − γ ϕ ϕ γ ψ 2 ψ h h h h − 2γϕ 2ϕϕ ; ¼ η2 þ 16 ðð Þ0Þ þ 16 ðð Þ0Þ 0 0 þð 0Þ þ 0 0 þ ð 0Þ þ 8 ð Þ00 þ 8 ð Þ00 00 þ 00 4 4 4 1 1 1 1 8 − 2ψϕ 2γψ 4ψψ − γγ_ ϕγ_ − ψγ_ − γ_2 − hTh_ T − h_ T 2 − h×h_ × − h_ × 2 ϕϕ_ γ_ ϕ_ 00 þ 00 þ 00 η þ η η 4η 16 ð Þ 4η 16 ð Þ þ η þ 4 8 8 1 1 − γψ_ ϕψ_ − ψψ_ − γ_ ψ_ 2ϕ_ ψ_ −ψ_ 2 − 2γγ̈ 2ϕγ̈ − 2ψγ̈ − hTḧ T − h×ḧ × − 2γψ̈ 4ϕψ̈ − 4ψψ̈ ; 150 η þ η η þ þ 8 8 þ ð Þ where we omit the subscripts 1 on the first-order perturbations on the right-hand side of these equations. For simplicity, we also set the second tensor mode h× to 0 from now on (its dynamics are analogous to those of hT).

103510-24 QUANTUM PROPAGATION ACROSS COSMOLOGICAL … PHYSICAL REVIEW D 95, 103510 (2017) We now compute the second-order perturbations from positive-frequency modes is analyticity in the lower-half Eqs. (130), (131), (140) and (141), using the sources η plane, where these modes extend to Euclidean, T computed from the chosen linear perturbations. For h2 , asymptotically decaying modes. This can be achieved we find by defining all the branch cuts to be along the positive imaginary axis. The analytic continuation of these −i 1 1 k η 3 p3 0 modes that avoids the singularity at η 0 is then T e ð þ Þ ¼ h AB 1 cos 2k0x defined by choosing any contour that remains in the 2 ¼  2k2η2 ð þ ð Þ 0 ffiffi lower-half complex η plane. 1 −i 1 k0η i 1 p3 e ð þp3Þ Asymptotically, we find that at large k0 η , − ð þ Þ cos 2k0x j j ffiffiffi2k0η ffiffi ð Þ 2 1 7iB 2 1 1 −i 1 k0η −2ik η − ik0η 2 i 3 p3 e ð þp3Þ 2 1 ψ 2 ∼−e 0 e p3 A cos 2k0x ð þ Þ Γ 0; i 1 k0η 128k0η þ 12 þ 6 ð Þ þ ffiffiffi6k0η ffiffi þ 3   þ p3  ffiffi 2 cos 2k0x 1 12 log k0η 1 13ie−2ik0ηΓ 0; i −1 1 k η ffiffiffi − i þ ð Þð þ ð ÞÞ O ; p3 0 8p3k η  þ k2η2 ð ð þ Þ Þ cos 2k0x 0 0 þ 6k0η ffiffi ð Þ ffiffiffi 13 2ik0ηΓ 0 3 1 and one can check that all terms, including all subleading ie ; i p3 k0η − ð ð þ Þ Þ cos 2k0x : 151 ones, oscillate at positive frequencies asymptotically (either 6k0η ð Þ ð Þ 2 ffiffi at ω 2k0 or at ω k0). The nonlinear modes again ¼ ¼ p3 Here Γ 0;z are incomplete gamma functions. Their decay exponentially as k η → −i∞, and indeed define ffiffi 0 asymptoticð expansionÞ for large arguments is nonlinear positive-frequency modes. From the general structure of the equations (114)–(119), one can see that 1 1 1 the same property should hold to all higher nonlinear Γ 0;z ∼ e−z − O : 152 ð Þ z z2 þ z3 ð Þ orders: the source terms, being nonlinear in lower order perturbations, always decay exponentially sufficiently → T ’ Using this expansion, we see that as k0η ∞, h2 has the fast in imaginary time that integration with a Green s asymptotic behavior Æ function that has an exponentially growing and an expo- nentially decaying part, as in Eq. (140), gives again an

−i 1 1 k η 6 5p3 − 27 16p3 cos 2k0x exponentially decaying next-order perturbation. The T p3 0 h2 ∼ ABe ð þ Þ i þ ð þ Þ ð Þ method we have described then allows a general defi-  ffiffiffi 21 11p3 ffiffiffik0η ffiffi ð þ Þ nition of positive-frequency modes in the complex η ffiffiffi 23 4p3 − 37 20p3 cos 2k0x 1 plane, to all orders in perturbation theory. þ ð þ 2 Þ2 ð Þ O 3 3 : – þ ffiffiffi2 5 2p3 k ffiffiffiη þ k0η  The other perturbations are determined by Eqs. (120) ð þ Þ 0 ffiffiffi (122). For completeness, we give their asymptotic expres- sions for large k0 η , j j The asymptotic expansion shows in particular that all the 1 − 2 ik η i cos 2k0x k0η −i 1 k0η 2 p3 0 terms in Eq. (151) oscillate as e ð þp3Þ for large k0 η , γ2 ∼ A e − ð Þ j j  p3 and decay exponentially for large negativeffiffi imaginary η. ffiffi We can obtain expressions for the scalar perturbations 5 ffiffiffi 1 − 1 − cos 2k0x 3 log k0η O ; in exactly the same way. The expressions are similar to ð Þ4 þ ð Þ þ k0η T those for h2 but involve more terms (15 in total), as 154 there can be contributions of order A2 and B2,corre- ð Þ sponding to two tensor modes or two scalar modes 1 − 2 ik η 4i cos 2k0x k0η combining to give a scalar. Just as the second-order 2 p3 0 δr;2 ∼ A e ð Þ tensors, they contain incomplete gamma functions, but 3  p3 ffiffi there is also a term involving a logarithm, ffiffiffi 1 − 7 − cos 2k0x 7 − 12 log k0η O ; ð Þð ð ÞÞ þ k0η 2 p3 2 2 − ik0η i log k0η cos k0x 2 −A e p3 : 153 7 2 1 2 2 ð Þ ð Þ −2ik η B 2 − ik0η ik0η cos k0x ffiffiffi 2k0η ð Þ ϕ2 ∼ e 0 A e p3 ð þ ð ÞÞ ffiffi 64 þ  6p3 ffiffi These terms are potentially problematic when the 1 1 ffiffiffi 1 perturbation functions are extended to the complex η − cos 2k0x − log k0η O : þ 3 ð Þ12 ð Þ þ k0η plane as the logarithms and incomplete gamma func- 155 tions have branch cuts. However, all we require for ð Þ

103510-25 STEFFEN GIELEN and NEIL TUROK PHYSICAL REVIEW D 95, 103510 (2017) To verify the validity of our solution method, we have symmetry for matter and gravity and the observed fact that checked explicitly that the second-order perturbations solve the early Universe was dominated by radiation. Classical Einstein’s equations up to order ϵ2. cosmological solutions of this model describe a bounce, We see that none of the scalar perturbation functions with a big bang/big crunch singularity, but the singularity decay at real infinity k0 η → ∞, and some even blow up, can be avoided by going into the complex plane. While this indicating a breakdownj j of perturbation theory at large “singularity avoidance” seems ad hoc in classical gravity, times. Again, to get gauge-invariant statements about this we have shown its meaning in the quantum theory where, behavior, we can compute the Newtonian potentials, and similar to quantum tunneling, the complexified solutions find that they fall off as 1=k0η, represent legitimate saddle points to the path integral. The picture that emerges for quantum cosmology is based on 2 − 2 ik η iπp3a 1 modes that are asymptotically purely positive frequency at Φ η; 2k ∼−e p3 0 O 156 2 0 2 2 early and late times when the Universe is large and ð Þ 4k0ffiffiffiη þ k0η  ð Þ ffiffi classical, corresponding to a positive expansion rate of with similar behavior for Ψ. This compares with Φ1 ∼ the Universe, as we observe. We have shown that the O 1=k2η2 in Eq. (144), which still indicates that the addition of a positive radiation density makes a crucial ð 0 Þ perturbation expansion breaks down when ϵk0 η ∼ 1. difference, as it leads to classical solutions which connect This physical behavior is due to the nonlinear evolutionj j asymptotic contracting and expanding Lorentzian regions, in the fluid, as shown by analytical and numerical studies and which are represented by the positive-frequency modes in Ref. [13]. When we go down the imaginary axis, i.e. for defined by the Feynman propagator. We do not impose any η −iτ with τ → ∞, all perturbation functions fall off boundary conditions for the wave function at a 0, and ¼ exponentially,¼ with exponential terms of the form e−ωτ accept that some modes may even diverge there: all that is dominating any polynomially growing terms. As we have required is a consistent evolution from an asymptotic argued, this behavior persists for higher orders in the ϵ contracting to an asymptotic expanding universe, through expansion, and defines these modes by regularity for large or around the bounce, as this allows a calculation of negative imaginary η; the blowup of scalar perturbations transition amplitudes and hence, ultimately, predictions along the real axis due to nonlinearities in the fluid does not for the transition of a given state in the contracting phase to prevent us from defining nonlinear asymptotic positive- a state in the expanding phase. This formalism appears frequency modes. much more natural than an imposition of a boundary condition at a 0, where quantum effects are large and ¼ E. Summary where classical notions of singularity avoidance may cease to have any relevance. In practical terms, the fact that our We have given an algorithm for solving the Einstein- wave functions and propagators admit a semiclassical matter equations order by order in perturbation theory, WKB description in which high-curvature regions near and exhibited explicit results at second order that show in a 0 can be avoided gives hope that a semiclassical detail how the positive-frequency incoming modes match ¼ only to positive-frequency outgoing modes, and similarly approach to the quantum cosmology of bouncing scenarios for negative-frequency modes (where our results trivially can be used for predictions, even in the absence of a extend by taking complex conjugates). We have argued that complete theory of quantum gravity. this behavior should extend to all orders in perturbation Some features we are exploiting are clearly restricted to theory, as the nonlinear extension of linear positive- homogeneous cosmological models such as the FRW and frequency modes leads to perturbation functions that Bianchi I universes we have studied explicitly. It is there- decay exponentially for large negative imaginary times, fore vital to check that the formalism can be extended and branch cuts can be restricted to the positive half-plane consistently to generic perturbations around homogeneity, for positive-frequency solutions, so that the nonlinear and ultimately to fully nonlinear solutions of GR. We have metric perturbation satisfies a nonlinear notion of positive developed a systematic perturbative treatment that shows frequency. We identified some subtleties, namely that the how this question can be attacked, at linear and nonlinear 1 order, and given evidence for a consistent nonlinear perturbation expansion fails at late times k0 η ∼ , where ϵ j j ϵ extension of positive-frequency modes to the complex a is the perturbation amplitude, meaning that one has to plane. Again, one is interested in the transition of incoming restrict attention to an annulus in the complex plane, 1 asymptotic positive-frequency modes to outgoing modes ϵ

103510-26 QUANTUM PROPAGATION ACROSS COSMOLOGICAL … PHYSICAL REVIEW D 95, 103510 (2017) adiabatic vacuum state is stable across the bounce and no supported in part by Perimeter Institute for Theoretical divergencies arise. Our calculations have been limited to Physics, in particular through the Mike and Ophelia pure radiation and planar symmetry, and one focus of future Lazaridis Niels Bohr Chair. Research at Perimeter work will be to extend these results to more general cases. Institute is supported by the Government of Canada The present results already indicate that a consistent through the Department of Innovation, Science and semiclassical picture exists for nonlinear perturbations Economic Development Canada and by the Province of of cosmological models, and that this picture can be used Ontario through the Ministry of Research, Innovation for calculations of the cosmological phenomenology of and Science. The work of S. G. was supported in part by bounce scenarios of the type we consider. the People Programme (Marie Curie Actions) of the Thus, our results show how classical singularities do European Union’s Seventh Framework Programme (FP7/ not necessarily prevent a consistent quantum description 2007-2013) under REA Grant No. 622339. of bouncing cosmologies. The inclusion of quantum effects into the big bounce seems a natural and simple alternative to the development of more complicated bounce APPENDIX: MASSIVE RELATIVISTIC scenarios [8–10,33]. PROPAGATOR There are many avenues for further exploration. In Sec. V, we began to explore the quantum theory on the In this appendix, we calculate the massive relativistic real a-axis around a 0. In some cases, it may be that the propagator given in Eq. (31) exactly. First, we note that ¼ attractive inverse square potential in the Wheeler-DeWitt M 1 ∞ þ operator may lead the quantum theory to fail when further m 2 −im σ τ G x x0 i dτ e 2 ðτþ Þ; (inhomogeneous) degrees of freedom are included, but ð j Þ¼ Z0 2πiτ in others the quantum theory seems to be healthy. The 2 quantum dynamics of more general Bianchi models also is a convergent integral when σ − x − x0 is positive. ¼ ð Þ deserve to be understood; for these, the invisibility of the The τ integral may be taken along the positive real axis singularity that we have observed for Bianchi I will 0 < τ < ∞. Next, we set τ pσeu, with −∞

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