Quantum propagation across cosmological singularities

Steffen Gielen Theoretical Physics, Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada and Canadian Institute for Theoretical Astrophysics (CITA), 60 St George Street, Toronto, Ontario M5S 3H8, Canada

Neil Turok Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada (Dated: May 14, 2017) The initial singularity is the most troubling feature of the standard cosmology, which quantum effects are hoped to resolve. In this paper, we study quantum cosmology with conformal (Weyl invariant) matter. We show that it is natural to extend the scale factor to negative values, allowing a large, collapsing universe to evolve across a quantum “bounce” into an expanding universe like ours. We compute the Feynman propagator for Friedmann-Robertson-Walker backgrounds exactly, identifying curious pathologies in the case of curved (open or closed) universes. We then include anisotropies, fixing the operator ordering of the quantum Hamiltonian by imposing covariance under field redefinitions and again finding exact solutions. We show how complex classical solutions allow one to circumvent the singularity while maintaining the validity of the semiclassical approximation. The simplest isotropic universes sit on a critical boundary, beyond which there is qualitatively different behavior, with potential for instability. Additional scalars improve the theory’s stability. Finally, we study the semiclassical propagation of inhomogeneous perturbations about the flat, isotropic case, at linear and nonlinear order, showing that, at least at this level, there is no particle production across the bounce. These results form the basis for a promising new approach to quantum cosmology and the resolution of the big bang singularity.

PACS numbers: 98.80.Qc, 04.60.Kz, 98.80.Cq, 04.20.Dw

I. INTRODUCTION the running of couplings, although these are no doubt important. Our focus is on the technical calculation of the causal (Feynman) propagator in some specific (and On the largest scales we can observe, our Universe quasirealistic) cosmologies. We also postpone a discus- has a remarkably simple structure. It is homogeneous, sion of the important question of the interpretation of isotropic, and spatially flat to very high accuracy. Fur- the propagator and its use to compute probabilities to thermore, the primordial curvature fluctuations which future work. Nevertheless, we believe our findings are seeded the formation of structure apparently took an instructive and form a useful starting point for such in- extremely minimal form: a statistically homogeneous, vestigations. Gaussian-distributed pattern of very small-amplitude The simplest example of conformal matter is a perfect curvature perturbations, with an almost perfectly scale- fluid of radiation. In the context of cosmology, this is invariant power spectrum. While inflationary models are extremely well motivated since the early Universe was, we capable of fitting the data, it is nonetheless tempting to believe, radiation dominated. Furthermore, if we add a look for a simpler and more fundamental explanation. single scalar field then at least at a classical level, minimal The early Universe was dominated by radiation, a form coupling is equivalent to conformal coupling under field of matter without an intrinsic scale. In fact, it is believed redefinitions. So this case too may be considered as an that any well-defined quantum field theory must possess example of conformal matter. It is then instructive to a UV fixed point, signifying conformal invariance at high extend the discussion to include an arbitrary number of energies. These lines of argument encourage us to investi- conformally and minimally coupled free scalar fields. gate a minimal early universe cosmology, namely a quan- tum universe filled with conformally invariant matter [1]. Since the matter Lagrangian of interest is, by assump- We start by studying the quantum propagation of ho- tion, conformally (Weyl) invariant at a classical level, it mogeneous background universes, uncovering a number makes sense to “lift” general relativity (GR) to a larger of surprising features. Then, we include inhomogeneous theory possessing the same symmetry. This is done by perturbations, treated semiclassically and perturbatively introducing an extra scalar field which is locally a pure at both linear and nonlinear order. We do not, in this gauge degree of freedom. The full theory is now clas- paper, propose a realistic scenario. Nor do we proceed sically Weyl invariant and it may be viewed with ad- far enough to study the effects of renormalization and vantage in various Weyl gauges. Its solutions contain 2 all solutions of GR but the theory allows for extended The theory we consider consists of Einstein gravity and more general solutions that do not possess a global plus radiation and a number of free scalar fields. All gauge fixing to GR. In particular, it turns out that while of these forms of matter satisfy the strong energy con- classical cosmological (homogeneous and isotropic) back- dition, and any cosmological solution necessarily pos- ground solutions are typically singular and geodesically sesses a big bang singularity. However, there is no singu- incomplete in Einstein gauge (the Weyl gauge in which larity in the Feynman propagator for closed, open and the gravitational action takes Einstein-Hilbert form, with flat Friedmann-Robertson-Walker (FRW) and for flat, a fixed Newton’s gravitational constant G), they are reg- anisotropic Bianchi I cosmologies; the propagator is well ular and geodesically complete in a more general class of behaved across a bounce representing a transition from Weyl gauges, where G is no longer constant and can even a large, collapsing classical universe to a large, expand- change sign. In these gauges, generic cosmological back- ing one. We also study inhomogeneous perturbations of ground solutions pass smoothly through the “big bang the isotropic, radiation-dominated universe, studying the singularity” and into new regions of field space, includ- propagation of scalar and tensor perturbations across the ing “antigravity” regions where G is negative [2]. Strictly bounce, at linear and nonlinear order. We find that, in speaking, the lifted classical theory is still incomplete the semiclassical approximation at least and with strictly because these highly symmetrical backgrounds are un- conformal matter, the incoming vacuum state evolves stable to perturbations in the collapsing phase, leading into the outgoing vacuum state, with no particle pro- to diverging anisotropies which cannot be removed via a duction. The bounce may be viewed as an example of Weyl gauge choice. It was argued nevertheless that the quantum-mechanical tunneling, and we use complex clas- classical theory possesses a natural continuation across sical solutions as saddle points to the path integral, in a singularities of this kind [3]. manner which generalizes the use of instanton solutions to describe tunneling in more familiar contexts. Some of In this paper, following [1], we take a different tack. these results were anticipated in Ref. [1]; here we present We ask whether quantum effects might rescue the theory more details on their derivation, extend them to further from its breakdown at big bang-type singularities. First, cases not discussed in Ref. [1], and provide more mathe- we show that for the simplest types of conformal matter, matical and conceptual background. An alternate inter- the Feynman propagator for cosmological backgrounds pretation of the antigravity regions, not employing com- may be computed exactly, allowing us to explore many plex solutions, has been presented in Ref. [4]. issues with precision in this symmetry-reduced (minisu- The simplest example of a “perfect bounce” is provided perspace) context. Second, we extend the discussion to by a spatially flat, homogeneous and isotropic FRW uni- include anisotropies, resolving the ordering problems in verse, filled with a perfect radiation fluid. Adopting con- the quantum (Wheeler-DeWitt) Hamiltonian by impos- formal time (denoted by η), i.e., choosing the line element ing covariance under field redefinitions. We go further to be than the treatment of [1] by defining the quantum the- ory along the real axis in superspace, and discovering ds2 = a2(η)(−dη2 + d~x2) , (1) some remarkable features. For a flat, isotropic universe, quantum effects indeed become large near the singular- one finds the scale factor a(η) ∝ η. One way to see this is ity. Therefore, strictly speaking, one should not attempt from the trace of the Einstein equations. For the line el- to employ the real classical theory there. Instead, one ement (1), R = 6(d2a/dη2)/a3 and for conformal matter, can solve the quantum Wheeler-DeWitt-type equation the stress tensor is traceless. So the Einstein equations for the propagator in complexified superspace, along a imply a(η) ∝ η. If spatial curvature is included, it is contour which avoids the singularity in taking one from subdominant at small a so that a still vanishes linearly the incoming collapsing universe to the outgoing expand- with η. Hence a(η) ∝ η is a direct consequence of con- ing one. Provided the contour stays sufficiently far from formal invariance and cosmological symmetry. While the the singularity, the semiclassical approximation remains line element (1) clearly contains a big bang/big crunch valid all along it, so one can employ complex solutions singularity at η = 0, it is regular everywhere else, not of the classical theory to follow the quantum evolution only along the real η-axis but also in the entire complex across (or more accurately, around) the singularity. In η plane. Any complex η trajectory that connects large doing so, we find that while quantum effects are large negative and large positive a while avoiding a = 0 gives a near the singularity, they take a very special form such regular, complexified metric that asymptotes to a large, that they are “invisible” in the evolution between incom- Lorentzian universe in the past and the future, while cir- ing and outgoing states. In the final section of this paper, cumventing the big bang singularity. we treat inhomogeneous perturbations, at linear and non- Such complex solutions have been discussed in quan- linear order, showing how they may be followed smoothly tum cosmology for a long time as saddle points of the and unambiguously across (or, more accurately, around) path integral, for instance in the no-boundary proposal the big bang singularity, in a similar manner. of Hartle and Hawking [5, 6]. The crucial difference in 3 our proposal is that the complex solutions connect two the fundamental quantity of interest is really the causal large, Lorentzian regions, identified with a collapsing in- (Feynman) propagator, which is naturally defined as a coming universe and an expanding outgoing one. The path integral over four-geometries [12]. Accordingly, in Weyl-invariant lift of GR provides a convenient simplifi- Sec. IV we calculate the propagator for various cases of cation of the geometry on superspace, cleanly exhibiting interest. While the Feynman propagator for FRW uni- its Lorentzian nature and the role of the scale factor a verses is actually regular at the singularity a = 0, its as a single timelike coordinate for both “gravity” regions. asymptotic behavior for large arguments displays inter- This leads us to a novel formalism for quantum cosmology esting pathologies both for closed and, in particular, open in which, rather than restricting to positive a and impos- universes, so that only flat FRW universes seem to con- ing boundary conditions at a = 0, a is extended to the sistently admit a quantum bounce. In Sec. V, we extend entire real line. The Feynman propagator turns out to the treatment to anisotropic universes of Bianchi I type, have simple behavior at large negative and positive a, de- including for generality a number of free minimally cou- scribing a contracting and reexpanding universe, respec- pled scalar fields. We resolve the ordering problem in tively, and connecting them through a quantum bounce. the quantum Hamiltonian, and we are again able to ex- The purpose of this paper is to flesh out the details of plicitly derive the Feynman propagator. A very special, this formalism, and show how it leads to a novel form singular potential arises centered on a = 0 which, in the of singularity avoidance in the context of an extremely minisuperspace context, is harmless and actually invis- simple (but not altogether unrealistic) cosmology. ible in the scattering amplitude between incoming and Some of the features of our discussion are not new. outgoing states. The coefficient of this singular poten- The possibility of solving minisuperspace quantum cos- tial turns out to take a special value for the isotropic mology models exactly by recasting their dynamics as universe with zero or one conformally coupled scalars, those of a relativistic free particle or harmonic oscillator placing it on the edge of a potential quantum instability, was pointed out before (see, e.g., Refs. [7]). However, the as we discuss. The addition of further conformally cou- crucial new feature in our work is the existence of regu- pled scalars moves the theory away from this edge, how- lar solutions (especially complex ones) that connect two ever. This is an intriguing result that deserves further large Lorentzian universes through a quantum bounce. attention, as it could be used to select between isotropic This feature relies on having a positive energy density and strongly anisotropic universes. In Sec. VI, we add in radiation, a possibility which, as far as we know, was inhomogeneities, treated linearly and nonlinearly in the overlooked in previous work. In fact, our results suggest semiclassical approximation where one employs complex that the fact that the early Universe was dominated by classical solutions to the classical Einstein equations. We radiation may be sufficient in itself for a semiclassical show how this is sufficient to determine mixing between quantum resolution of the big bang/big crunch singular- positive- and negative-frequency modes, and hence to ity, without the need for less well-motivated ingredients compute the particle production across the “quantum such as exotic forms of matter [8], modified theories of bounce.” We find no particle production, but instead gravity [9], or a proposed theory of quantum gravity [10]. verify that the perturbation expansion breaks down at To avoid any potential confusion, let us reemphasize that late times due to the formation of shocks in the fluid, the theories we consider consist of general relativity with a phenomenon which is now physically well understood a radiation fluid and a number of free scalars, and nothing [13]. Section VII concludes. more: our use of the Weyl lift of GR does not introduce any additional degrees of freedom. The plan of our paper is as follows. In Sec. II, we in- troduce the Weyl lift of GR plus radiation and scalars, and show how the degree of freedom corresponding to the metric determinant can be isolated straightforwardly, II. WEYL-INVARIANT COSMOLOGY leading us to a Weyl-invariant notion of a, the “scale factor.” We then study homogeneous, isotropic FRW universes in Sec. III, showing that in these cases, the We start by studying the cosmology of a universe filled Einstein-matter action corresponds to that of a massive with perfectly conformal radiation and a number M of relativistic particle moving in Minkowski spacetime, ei- conformally coupled scalar fields, with gravitational dy- ther freely or subject to a quadratic, Lorentz-invariant namics governed by a lift of GR to a classically Weyl- potential. We discuss the classical and quantum dynam- invariant theory that contains an additional dilaton field ics of FRW universes, using the classical Hamiltonian φ. We stress again that the field φ is locally pure gauge, analysis to define the Wheeler-DeWitt quantum Hamil- and possesses no nontrivial dynamics of its own. This tonian. We discuss the Klein-Gordon-type inner product formalism for GR was developed in Ref. [2] and works proposed by DeWitt [11], but take the point of view that in any number of dimensions D > 2 (we will set D = 4 4 shortly). The total action we consider is (φ, ~χ) to a hyperboloid HM in (M + 1)-dimensional field space at each point in spacetime. One can introduce an Z √ 1  |J|  S = dDx −g (∂φ)2 − (∂~χ)2
− ρ √ (2) explicit parametrization of this hyperboloid by M coor- 2 −g dinates νi, (D − 2)  + (φ2 − ~χ2)R − J µ ∂ ϕ˜ + β ∂ αA
. 1 M i i 1 M 8(D − 1) µ A µ φ = φ(ν , . . . , ν ) , χ = χ (ν , . . . , ν ) , (5) The independent dynamical variables are the spacetime so that in this gauge the action (2) reads metric g (assumed to be Lorentzian throughout), the Z √ 1  |J|  µν S = dDx −g− G (ν) ∂νi · ∂νj − ρ √ “dilaton” φ, M physical scalar fields ~χ = (χ1, . . . , χM ), 2 ij −g and a densitized particle number flux J µ characteriz-  1  µ A
ing the radiation fluid. The latter can be identified by + R − J ∂µϕ˜ + βA∂µα , (6) √ 16πG J µ = −g n U µ, where n is the particle number den- sity and U µ is the four-velocity vector field satisfying where Gij(ν) is a positive definite metric of con- U 2 = −1; the energy density ρ is only a function of stant negative curvature on the gauge-fixed field space D i n, concretely ρ(n) ∝ n D−1 for radiation which is the parametrized by the ν . Again, Eq. (6) shows that there case we are interested in. There is a Lagrange mul- are no physical ghosts in the theory, at least as long as 2 2 tiplierϕ ˜ which enforces particle number conservation φ − ~χ > 0. µ There are two different sectors in the space of field ∂µJ = 0, and D − 1 further Lagrange multipliers βA, µ A configurations where Einstein gauge is available, corre- with A = 1,...,D −1, enforcing constraints J ∂µα = 0 that restrict the fluid flow to be directed along flow lines sponding to “future-directed” and “past-directed” (in labeled by the fields αA which play the role of Lagrangian field space) configurations, i.e., to φ > 0 or φ < 0. There 2 2 coordinates for the fluid. are also regions where φ − ~χ becomes negative, identi- In general, the fluid energy density ρ would also depend fied with “antigravity” in Ref. [2] as they would appear on the entropy per particle. For simplicity, we henceforth to correspond to a negative G. We identify such regions assume an isentropic fluid for which this entropy per par- with imaginary values of the scale factor and show how ticle is a constant. The fluid part of our action is then the the passage of the Universe through antigravity regions one given for isentropic fluids in Eq. (6.10) of Ref. [14], is a semiclassical representation of what is really a quan- where further details on the construction of actions of tum bounce, similar to how quantum tunneling can be relativistic fluids and their corresponding Hamiltonian described by complex classical trajectories. The anti- gravity regions do contain a ghost, as now (φ, ~χ) would be dynamics can be found. M−1,1 The action (2) is invariant under a Weyl transforma- constrained to de Sitter space dS which has a time- tion that takes like direction. These regions and their ghost excitations do not appear in the physical “in” and “out” states of the 2 (2−D)/2 gµν → Ω gµν , (φ, ~χ) → Ω (φ, ~χ) , (3) theory, which are defined in asymptotic timelike regions where φ2 − ~χ2 → ∞; nevertheless, the existence of these where Ω(x) is an arbitrary function on spacetime. Such a regions can cause pathologies in the quantum theory if transformation also takes ρ → Ω−D ρ. Because of this lo- initial gravity states can propagate into the antigravity cal conformal symmetry, the field φ does not correspond regions, as we will see in Sec. IV B. to a physical degree of freedom; indeed, if φ2 − ~χ2 > 0 ev- Setting D = 4, to make this more precise it is now erywhere, one can gauge fix the conformal symmetry to useful to define a scale factor, or rather its square a2, with recover the usual Einstein-Hilbert formulation of GR. It the following properties: it should be Weyl invariant, so is then clear that there is no physical ghost in the theory that it takes the same value in any conformal gauge. It even though φ appears in Eq. (2) with the wrong-sign should respect the O(M, 1) isometry of the metric on the kinetic term. space of scalar fields (defined by the kinetic terms) and Let us make this explicit. For φ2 − ~χ2 > 0, one can go so depend only on the combination φ2 − ~χ2, the radiation to “Einstein gauge” by performing a conformal transfor- density ρ and the metric determinant g. It should have mation (3) that takes physical dimensions of an area (in the usual conventions D − 1 ~ = c = 1), and scale like the square of the scale factor φ2 − ~χ2 → constant =: (4) 2(D − 2)πG for an FRW universe in Einstein gauge in conformal time. These properties fix the “squared scale factor,” up to an where G is Newton’s constant. Note that Eq. (4) does overall constant, to be not entirely fix the gauge freedom as one can still perform 2 1 − 1 2 2
a global rescaling that takes the constant to a different a ≡ (−g) 4 φ − ~χ . (7) one; the exact value of Newton’s constant is arbitrary 2ρ and corresponds to a choice of units. Einstein gauge cor- We use Eq. (7) as a natural definition in general gauges 1 responds to constraining the (M + 1)-vector formed by (the motivation for the factor 2 becomes clear shortly). 5

Note that a2 is in general not positive; if we assume pos- vanish or change sign. However, following the dynamical itive ρ and a Lorentzian metric, as we always do in the evolution through such points is in general problematic following, then in the antigravity regions a2 < 0 and so because the effective Newton constant diverges so grav- a is imaginary. This definition of a differs from the one ity becomes strongly coupled. This is reflected, for exam- in Ref. [2] as it depends on the energy density of the ple, in the behavior of tensor (gravitational wave) pertur- radiation. In Ref. [2], there was no such dependence. In- bations, which diverge as the effective Newton constant stead, factors of Newton’s constant were used to ensure does. This leads to a diverging Weyl curvature which the correct physical dimensions. cannot 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 the orientation in field space: a is defined to have the same electroweak Higgs boson) there is generically no Mixmas- sign as φ. The Minkowskian field space parametrized by ter chaos and one expects the classical evolution to be- (φ, ~χ) is then partitioned into two regions with real a and come ultralocal and Kasner-like. There are a number one region with imaginary a; see Fig. 1. Note that the of asymptotically conserved classical quantities, includ- entire light cone corresponds to a = 0. This picture, as ing the Kasner exponents, suggesting a natural matching we have anticipated, gives physical meaning to positive, rule across the singularity [3] but the issue has not been negative and imaginary a, generalizing the case of pure conclusively settled [15]. radiation, M = 0, where there are no spacelike directions In this paper, we take a different approach. We show and a takes values along the real axis, as in the example that by extending the classical discussion to a quantum discussed in the introduction. picture one can avoid the critical surface a = 0 where the theory becomes problematic, independently of any Weyl gauge choice. We give a description of nonsingular “gravity” quantum bounces in terms of analytic continuation in a, @ where the Universe evolves from large negative a to large φ @ 6 a > 0 - positive a along a contour in the complex a-plane which @ a = 0 @ avoids a = 0. We argue that as long as the quantum ~χ @ mechanics of the a degree of freedom make sense, the - “antigravity” @ classical singularity at a = 0 can be avoided without a ∈ iR @ obstruction. a < 0 @ @ “gravity” @ III. FRW BOUNCES

As a first step, we perform the familiar symmetry re- FIG. 1. Associating positive, negative and imaginary scale duction of our theory to homogeneous and isotropic FRW factor a to different regions in field space. The light cone universes, with the metric assumed to be of the form corresponds to the singularity a = 0. 2 2 2 2 i j ds = A (t)(−N (t)dt + hijdx dx ) (8) Apart from Einstein gauge, another gauge that we will where h is a fixed metric on hypersurfaces of constant often employ, again following the framework introduced ij t, which has constant three-curvature R(3) = 6κ. Note in Ref. [2], is “Weyl gauge” where the metric determinant our use of a conformal lapse function N; the usual defi- g is fixed to a constant (typically −1). This gauge is avail- nition of the lapse would be N (t) = A(t) · N(t). We can able whenever the metric is nonsingular; in particular, it 0 now set the function A(t) to one by a conformal trans- covers the entire field space pictured in Fig. 1, encompass- formation, so that the metric becomes nondynamical and ing both gravity and antigravity regions. It is often a con- all dynamics are in the scalar fields φ and ~χ. Also, with venient gauge to work in. In Weyl gauge, the expression √ FRW symmetry J µ = hn δµ, and the action (2) reduces for the scale factor reduces to a2 = 1 φ2 − ~χ2
, where 0 2ρ to for homogeneous models by energy-momentum conserva- tion ρ is constant. a is then proportional to the (signed) Z ~χ˙ 2 − φ˙2 κ  S = V dt  + N (φ2 − ~χ2) − ρ(n) − ϕ˜n˙  , timelike O(M, 1)-invariant distance from the origin in 0 2N 2 field space, making it a natural choice of time coordinate (9) on superspace. where√ ˙ denotes derivative with respect to t and V0 = For highly symmetric solutions such as FRW universes, R d3x h is the comoving spatial volume (which, as usual conformal symmetry can be used to eliminate curvature for minisuperspace models, must be assumed to be fi- singularities in the metric by moving them into 0’s of nite). We have simplified the last term including the La- 2 2
A the quantity φ − ~χ . Since this quantity has no ge- grange multipliers which would be −n(ϕ˜˙ + βAα˙ ) since A ometric interpretation, it is a priori reasonable for it to the equations of motion involving βA and α are clearly

1 6 redundant in FRW symmetry. As before, ρ(n) ∝ n4/3, tion are and we can replace n by ρ as the independent variable. 1 d mx˙ α  + mκxα = 0 , (13) It is evident that Eq. (9) is the action for a relativis- N dt N tic massive free particle (for κ = 0) or a relativistic 1 massive particle in a harmonic potential or a harmonic α α 2 x˙ x˙ α + κx xα = −1 , (14) “upside-down” potential (for κ 6= 0) moving in (M + 1)- N 1 α N α dimensional Minkowski spacetime. To make this more m˙ = 0 , ϕ˙ = − x˙ x˙ α + (κx xα + 1) . (15) explicit, we can introduce new variables 2N 2 The general solution to these equations is m = constant, α 1 x := √ (φ, ~χ) , α = 0, . . . , M , m := 2V0ρ , (10)  t  2ρ xα √ Z xα(t) = √1 exp i κ dt0 N(t0) κ   so that Eq. (9) now takes the form 0  t  α Z Z   x2 √ 0 0 m 1 α α  +√ exp −i κ dt N(t ) (16) S = dt x˙ x˙ α − N(κ x xα + 1) − ϕm˙ (11) κ 2 N 0

1 where we have redefined the Lagrange multiplier for sim- with x1 · x2 = − 4 ; the lapse function N(t) is arbitrary plicity, ϕ :=ϕV ˜ 0(dn/dm), and the Minkowski metric on and ϕ(t) is determined from integrating Eq. (15). For the space of scalar fields ηαβ = diag(−1, 1, 1,...) is used κ = 0, the general solution is simply a general timelike to raise and lower indices. A crucial role is played by straight line in Minkowski spacetime, the mass m which corresponds to (twice) the total en- t ergy in the radiation; the limit m → 0 would correspond Z α α 0 0 α to a massless relativistic particle moving in a potential, x (t) = x1 dt N(t ) + x2 (17) which is the case well known in minisuperspace quantum 0 cosmology with scalar fields [7]. Having a positive mass, 2 and hence timelike trajectories as classical solutions, is with x1 = −1. For κ = 0, all solutions describe a bounce, one of the essential features of our model that leads to similar to the example in the introduction: the Universe a bounce. With the definition (10), the Weyl-invariant comes in from negative real infinite a, goes through a = 0 scale factor is simply a2 = −x2, which explains the factor followed in general by an “excursion” into imaginary a, 1 and crosses a = 0 again before going off to real posi- 2 in Eq. (7). The variable a is simply a time coordinate on superspace. One can introduce it explicitly by setting tive infinity. When we go quantum, since the action is quadratic, the saddle point approximation is exact and xα = a vα , v2 = −1 (12) the quantum dynamics is given purely in terms of these classical solutions. When viewed as saddle points, these so that vα is restricted to a hyperboloid HM (see Fig. 2). trajectories can be deformed in the complex a-plane so This parametrization, which isolates the physical scalar that the singularity a = 0 is avoided. The situation is more subtle for κ < 0, where there are spacelike as well as timelike solutions, and for κ > 0 where there is a turnaround in the classical solutions and the Universe must recollapse due to the spatial curvature. In Sec. IV, @ we see how the more complicated structure of solutions @ for κ 6= 0 is reflected in a pathological behavior of the @ Feynman propagator for large arguments. @ a = const. @ CO α α @ Cx = a v A. Canonical formalism @ C @ C @ C In order to pass to the Hamiltonian formalism, follow- @C ing Dirac’s algorithm [16], one computes the canonical momenta for the action (11) and finds

FIG. 2. The points of constant (real) a form a hyperboloid ∂L m parametrized by vα. p = = x˙ , p ≈ −ϕ , p ≈ 0 , p ≈ 0 . α ∂x˙ α N α m N ϕ (18) fields as the variables vα, is useful below. While the first equation can be inverted to express the α Starting from Eq. (11), the classical equations of mo- velocitiesx ˙ in terms of the momenta pα, the last three

1 7 equations are primary constraints—we use Dirac’s notion a set of coordinates νi, i = 1,...,M, on the hyperboloid of “weak equality” ≈ for equations that hold on the con- HM . The Wheeler-DeWitt equation then becomes straint surface. The second and fourth constraint would  2  be second class, meaning one has to introduce a Dirac ∂ M ∂ 1 2 2 i + − ∆HM + m (1 − κa ) Ψ(a, ν , m) = 0 bracket and “solve” them. However, in this case, one ∂a2 a ∂a a2 can use the shortcut of simply identifying −ϕ with the (25) momentum conjugate to m and removing the separate where ∆HM is the Laplace-Beltrami operator on M-dimensional hyperbolic space, i.e., on the space variable pϕ. This is equivalent to saying that the term i parametrized by the coordinates νi. For example, using −ϕm˙ in Eq. (11) is part of the symplectic form piq˙ so Beltrami coordinates νi = xi/x0 one would have that one can read off pm = −ϕ. The Hamiltonian is then 2  ij i j
i  ∆HM = (1 − ~ν ) δ − ν ν ∂i∂j − 2ν ∂i . (26)  2  p m 2 H = N + (κx + 1) + ξ pN . (19) This coordinate choice on superspace makes the role of 2m 2 the timelike coordinate a explicit. The Wheeler-DeWitt equation can then be simplified by Fourier transforming Preservation of pN ≈ 0 under time evolution gives the i secondary, Hamiltonian constraint, from the ν coordinates to their conserved momenta ζ,  ∂2 M ∂ c  p2 m + − + m2(1 − κa2) Ψ(a, ζi, m) = 0 C := + (κx2 + 1) ≈ 0 . (20) 2 2 2m 2 ∂a a ∂a a (27) N can then be treated as a Lagrange multiplier; it only with enters linearly in the Hamiltonian, and its time evolution 1 under H is N˙ = {N, H} = ξ where ξ is undetermined. c ≡ − (M − 1)2 − ζ~2 (28) 4 Removing (N, pN ) from the phase space (and setting ξ = 0 in the Hamiltonian), we are left with the canonical corresponding to the eigenvalues of the Laplacian on HM α pairs (x , pα) and (m, pm), subject to the constraint C, (for M ≥ 1). As we see in Sec. V below, the same form which trivially satisfies {C, H} = 0. C generates time of the Wheeler-DeWitt equation applies when including reparametrizations, anisotropies in a Bianchi I model or additional minimally

α coupled scalar fields, with the only change that the con- α α Np δN x = {x ,NC} = , (21) stant c receives additional contributions from conserved m anisotropy and scalar field momenta as well as from fixing δN pα = −Nmκxα , (22) ordering ambiguities.  p2 1  A natural inner product on solutions of a second-order δ p = −N − + (κx2 + 1) , (23) N m 2m2 2 equation like Eq. (25) is the Klein-Gordon-like norm Z  which correspond to the Lagrangian notion of time M M √ ∗ i ∂ i hΨ|Φi ≡ ia d ν dm g M Ψ (a, ν , m) Φ(a, ν , m) reparametrization, by the equations of motion (13)–(15). H ∂a ∂  − Ψ∗(a, νi, m)Φ(a, νi, m) , (29) ∂a B. Quantization with gHM being a constant negative curvature metric on HM , which is conserved under time evolution, i.e. inde- Having set up the canonical formalism, we can proceed pendent of a, for solutions of Eq. (25). This inner product with quantization in the standard way. The Hamiltonian was introduced by DeWitt [11] and has the well-known constraint is imposed as an operator equation restricting problem (if it is used to define probabilities) that it is the set of physical states. In the (x, m) representation for only positive on positive-frequency solutions to Eq. (25), the wave function, this is the Wheeler-DeWitt equation when they exist. For some simple cosmological models, 1 this subspace is well defined, and may be interpreted as −2 + m2(κx2 + 1)
Ψ(x, m) = 0. (24) 2m x the space of expanding quantum universes: if a is taken to be positive, a wave function describing an expanding Different m sectors simply decouple, as a consequence of universe must be an eigenstate of pa = −i∂a with neg- conservation of the total energy in the radiation, with ative eigenvalue (note that pa = −ma/N˙ and soa ˙ > 0 no transitions between different m values allowed. An means pa < 0), i.e., a positive-frequency solution. This alternative representation of wave functions is obtained notion of positive frequency breaks down for cosmologi- by separating the scale factor from the physical scalar cal models with recollapsing solutions, such as the FRW field degrees of freedom, as in Eq. (12), and introducing universe with κ > 0, where it is only well defined until 8 one reaches the turning point, and it is known that a a natural way to describe the “emergence” of spacetime decomposition into positive and negative frequencies of from quantum-mechanical first principles [20]. the type we are using here is not available in general [17]. In this section, we show how to calculate the Feynman The question of how to define meaningful probabilities in propagator for FRW universes directly from the path quantum cosmology has, of course, been a matter of long integral; in particular, the path integral may be used debate (see, e.g., Refs. [5, 18, 19]). to define the propagator without the need for an addi- We do not aim to resolve this debate here. The only tional “i” prescription and, furthermore, the propaga- use we make of the DeWitt norm (29) is to help us con- tor directly defines the positive- and negative-frequency struct the Feynman propagator from mode function so- vacuum modes. As we have stressed, with FRW sym- lutions of the Wheeler-DeWitt equation. The expansion metry the action is quadratic and the saddle-point ap- rate −pa does play the role of an energy, which leads us proximation is therefore exact for the path integration to adopt Feynman’s picture for quantum field theory in over the phase-space variables. However, we also have a which positive energy (i.e., expanding) states are prop- constraint (the Friedmann equation) which must be im- agated forward in proper time. The natural two-point posed via an additional integration over the lapse func- function we consider below in Sec. IV is hence the Feyn- tion or Lagrange multiplier N [21]. This integral is no man propagator. In what follows we alternate between longer Gaussian and has to be considered with more care. the path integral and the Feynman propagator as basic Things are considerably simpler for the flat FRW case formulations of quantum cosmology, explicitly showing κ = 0, where we have seen that the dynamics are just their equivalence in simple cases. those of a massive relativistic free particle in Minkowski So far, this is a completely standard definition of a spacetime. We therefore begin by reviewing how the minisuperspace model in Wheeler-DeWitt quantum cos- Feynman propagator for a relativistic particle is calcu- mology. However, there is one crucial difference between lated both from a path integral and as a Green’s function our approach and previous treatments, in that we do not for the differential equation satisfied by physical wave restrict the wave function to positive a, nor do we impose functions. We then extend these methods to treat gen- any boundary condition at a = 0 [such as the popular eral FRW universes for the types of matter we consider. choice Ψ(a = 0) = 0]. At fixed m, the domain of the wave function is simply RM,1. Any boundary condition at a = 0 would seem artificial from the viewpoint of clas- A. Relativistic particle sical solutions such as classical FRW “bounces” which connect negative and positive a, as we have described, Consider the action for a relativistic massive particle and is also generally inconsistent with the wave func- in (M + 1)-dimensional Minkowski spacetime, tion describing an expanding Universe, i.e., a positive- m Z x˙ αx˙ βη  frequency solution. In our proposal, the natural choice S = dt αβ − N , (30) of wave functions corresponds to positive-frequency solu- 2 N tions that asymptote to plane waves at large |a|, where α the Universe becomes semiclassical, while we allow for where m > 0, x (t) is the parametrized particle world irregular behavior in the wave function at a = 0. The line and N(t) is the “einbein.” Classically, N may examples we consider all admit a semiclassical Wentzel- be eliminated using its equation of motion, obtaining Kramers-Brillouin (WKB) expansion in which one can the manifestly√ reparametrization-invariant action S = R 2 deform the contour from the real a-axis into complex a, −m dt −x˙ . Quantum mechanically, it is more conve- avoiding a = 0 entirely. nient to fix the reparametrization invariance [see also the discussion of gauge fixing below Eq. (40)]: one can work in a gauge in which t varies over a fixed range, conve- 1 1 IV. FEYNMAN PROPAGATOR FOR FRW niently taken to be − 2 < t < 2 , and N is a t-independent UNIVERSES constant, equal to the total, reparametrization-invariant time R dt N which we call τ. The Feynman propagator The Feynman propagator is one of the most basic in- is then given by the path integral gredients in relativistic quantum theory. In quantum " 1 # Z m Z 2 x˙ 2  gravity, it plays the role of a causal Green’s function for G(x|x0) = dτ Dx exp i dt − τ 2 1 τ the Wheeler-DeWitt equation, arising from a path inte- − 2 gral in which one integrates only over positive values of Z ∞ M+1  m  2 −i m σ +τ = i dτ e 2 ( τ ) , (31) the lapse function [12]. If one considers amplitudes in 2πiτ which a changes sign, as we do, then the Feynman prop- 0 agator takes one from a contracting universe in the initial where σ ≡ −(x − x0)2 and τ should be integrated over state to an expanding one in the final state, via a singu- positive values. Evaluating the Gaussian path integrals is larity of the big bang type. Such an amplitude provides straightforward, with the only unusual factor being the 9 prefactor of i in the second line, which arises from the negative values. The corresponding saddle point for the functional integral over x0, which has the “wrong sign” τ integral moves as shown in Fig. 3, passing below the kinetic term so that the overall phase factor contributed origin in the complex τ plane and down the imaginary τ- is e+iπ/4 rather than the usual e−iπ/4. axis. At spacelike separations, σ = −R2 < 0, the saddle The second line of Eq. (31) is, of course, just the fa- point is at τ = −iR, and the propagator falls exponen- miliar Schwinger representation of the Feynman propa- tially with spacelike separation. Notice that although the 2 2 gator, in which the exponent is the classical action eval- classical constraintx ˙ = −τ remains satisfied at the sad- uated on a solution of the equations of motionx ¨ = 0, dle point, the saddle-point value for τ is imaginary, and satisfying the correct boundary conditions, i.e., x(t) = hence classically disallowed. Hence, the propagation of 1 0 1 a massive relativistic particle in spacelike directions may x(t+ 2 )+x ( 2 −t), and the prefactor is given by the usual (regularized) functional determinant. Note that this so- be viewed as a semiclassical quantum tunneling process, lution is only the saddle point for the functional integral mediated by a complex classical solution. over paths x(t), at fixed τ. The constraintx ˙ 2 = −τ 2 arises subsequently, as the condition for a saddle point in the exponent of the τ integral. In fact, once the τ saddle point is chosen, the integration contour is then fixed (up to an equivalence class of contours yielding the same result) as the complete extension of the correspond- T ing steepest descent contour. Integrating over negative proper time in Eq. (31) would reverse the notion of time ordering, whereas integrating over both positive and neg- ative τ would lead to a symmetrized two-point function in which one sums over both time orderings, i.e., the -iR Hadamard propagator. For M > 0, the τ integral in Eq. (31) has a potential divergence at τ = 0. In fact, the integral converges at all real values of σ except σ = 0. That divergence is real: FIG. 3. Integration contours in the complex τ plane, for the the Feynman propagator is singular for null-separated relativistic massive propagator, defined in Eq. (31). As σ ≡ points. For other real values of σ, given that the integral −(x − x0)2 is varied, from timelike values σ = T 2 with T real, converges for all σ in the lower-half σ plane, one may to spacelike values σ = −R2, with R real, by passing beneath define the Feynman propagator as the boundary value of the origin in the complex σ plane, the saddle point in τ, shown the function defined by the integral, which is analytic in by the black point, moves correspondingly. In each case, the the lower-half σ plane. Traditionally, the mass m is also integral is defined by the associated steepest descent contour taken to have a small negative imaginary part, in order running from 0 to ∞, also shown. to make the τ integral absolutely convergent at large pos- itive values (Feynman’s “i” prescription). Equivalently, In order to match the path-integral definition of the one may distort the τ-contour to run to infinite values Feynman propagator to its definition as a Green’s func- below the real axis. The integral (31) may be directly tion, it is convenient to use a time slicing with maximal expressed in terms of a Hankel function, whose proper- spatial symmetry. Here, the trivial time slicing defined ties confirm these general arguments (see the appendix). by x0 is spatially homogeneous, so one can Fourier trans- It is instructive, however, to evaluate the τ integral in form in the spatial coordinates and reduce the problem Eq. (31) in the saddle-point approximation. First, con- to a single timelike dimension. Defining sider timelike separations, σ = T 2 > 0. The exponent in M the τ integral (31) is stationary at τ = +T and τ = −T , Z d ~k ~ 0 G(x|x0) = eik·(~x−~x )G (x0, x00) , (32) but only the former saddle point is relevant to the contour (2π)M k we want, which is deformable into the positive τ-axis. The saddle point at τ = +T gives rise to a positive- one finds that G (x0, x00) is given by frequency result, G ∼ e−imT at large T . The integration k contour for τ may then be taken to be the correspond-  0 00 2 ω2  Z ∞ r −i m (x −x ) + k τ ing steepest descent contour, shown as the solid line in 0 00 m 2 τ m2 Gk(x , x ) = i dτ e Fig. 3. Now consider analytically continuing T through 0 2πiτ the lower right quadrant to the negative imaginary axis, √ (33) 2 2 T → −iR. It follows that G converges as G ∼ e−mR at where ωk ≡ k + m . large R. Correspondingly, this continuation implies that By considering the saddle-point approximation to σ = T 2 runs below the origin in the complex σ plane to Eq. (33), we see that the Fourier-transformed Feynman 10 propagator asymptotically satisfies (i.e., x0-independent) inner product, or Wronskian. The

0 dependence of the Feynman propagator at large posi- 0 00 −iωkx 0 00 Gk(x , x ) ∼ e , x → +∞ , x fixed, (34) tive and negative times now determines the appropriate 00 0 00 +iωkx 00 0 k 0 k 0 Gk(x , x ) ∼ e , x → −∞ , x fixed. (35) choices for ψ1 (x ) and ψ2 (x ): comparing Eqs. (34) and (35) with Eq. (39) we infer that, up to irrelevant con- 0 0 Such asymptotic expressions can be used to fix boundary k +iωkx k −iωkx stants, ψ1 = e and ψ2 = e . Inserted into conditions for the corresponding wave (Wheeler-DeWitt) Eqs. (32) and (39), these give the usual expression for the equation, as we do shortly. Feynman propagator in “time-ordered” form. This shows 0 Formally, G(x|x ) is the matrix element how, in the example of the relativistic particle, the cor- R ∞ −iHτ 0 −1 0 hx| 0 dτ e |x i = −ihx|H |x i, where we again rect boundary conditions that define the Feynman prop- assume the integral converges at infinite τ. Hence, agator as one particular solution of Eq. (36) can be ob- 0 suitably defined, G(x|x ) should obey tained from the asymptotic behavior of its path-integral definition. We will now proceed similarly to define the H G(x|x0) = −iδM+1(x − x0), (36) x Feynman propagator for general FRW universes. where Hx is the Hamiltonian in the x-representation, 1 2 Hx = (−2x + m ). We can check Eq. (36) is in- 2m B. FRW universes deed satisfied by applying Hx to the last line of Eq. (31) and using the fact that the integrand is a product of free- particle quantum-mechanical propagators, For our cosmological model, the Feynman propagator can be defined through a phase-space path integral, tak- 1 (−2 + m2)G(x|x0) ing into account the integration over the lapse N [21], 2m x ( M+1  0 2  ) Z ∞ im (x−x ) Z d  m  2 2 τ −τ 0 0 α = i dτ i e G(x, m|x , m ) = Dx DPα Dm Dpm DN 0 dτ 2πiτ  1/2  0 2  M+1 im (x−x ) Z  m  2 2 τ α = lim e , (37) exp i dt x˙ Pα +mp ˙ m (40) τ→0 2πiτ  −1/2 where the limit should be taken along the appropriate  α   PαP m α
contour in the complex τ plane. The last line of Eq. (37) −N + (κ x xα + 1) . is a representation of the (M +1)-dimensional delta func- 2m 2 tion: separating it into a product of similar terms for each As in the previous example, due to the reparametriza- α coordinate x , we determine the coefficient of the cor- tion invariance of the theory the parameter time (speci- responding delta function by Fourier transforming with fied by t) between the initial and final configurations is α respect to x , and then taking the limit τ → 0. For the arbitrary, and we choose it to run from − 1 to 1 . In or- 0 00 2 2 timelike coordinate we obtain −iδ(x − x ), whereas for der for the path integral to be well defined, the gauge M 0 the spacelike coordinates we obtain δ (~x−~x ). Together, invariance under time reparametrizations generated by these results verify Eq. (36). the Hamiltonian constraint must be broken by fixing a The point is now that the Feynman propagator can specific gauge. One simple gauge fixing, N˙ = 0, can be also be computed by directly solving Eq. (36) in terms of obtained by introducing a new field Π and adding the mode functions, again because the Fourier transform al- term ΠN˙ to the action; we refer to Ref. [22] for details. lows reduction of the problem to one dimension. Writing Integrating over the field Π then reduces the integration both the delta function and the propagator in Eq. (36) as over N to an ordinary integral over the total conformal 0 Fourier transforms, one sees that G(x|x ) clearly satisfies time between the initial and final configurations; we make Eq. (36) as long as this explicit by again writing N as τ. 2 2 0 00 0 00 The remaining path integrals may be computed ex- (∂0 + ωk)Gk(x , x ) = −2imδ(x − x ) . (38) actly. Path integration over m and pm simply gives a This equation is solved by delta function in m, as expected since m has trivial dy- namics constraining it to be constant. One can then in- 2im 0 00 k 00 k 0 0 00 tegrate over Pα which yields Gk(x , x ) = − k k ψ1 (x )ψ2 (x )θ(x − x ) W (ψ1 , ψ2 ) 0 0 0 k 0 k 00 00 0
G(x, m|x , m )= δ(m − m ) × (41) +ψ1 (x )ψ2 (x )θ(x − x ) , (39) Z Z  2  m x˙ 2 k k dτDx expi dt − τ 1 + κx
 where ψ1 and ψ2 are two independent solutions to 2 τ 2 2 the homogeneous equation (∂0 + ωk)ψ = 0, and 0 0 W (ψ1, ψ2) = ψ1ψ2 − ψ2ψ1 is the natural conserved corresponding to M + 1 decoupled harmonic oscillators. 11

0 For κ = 0, apart from the overall delta function, this which is unique for all x, √x and τ [where we exclude is exactly the expression Eq. (31). Accordingly, the path special cases for which sin( κ τ) = 0], and the prefactor integral over x is just that of a free relativistic particle is given by the usual regularized functional determinant and the τ integral can be evaluated exactly; the result is for the harmonic oscillator, so that

1 1−M (2) 0 0 0 0 0 0 M 2 G(x, m|x , m ) G (x, m|x , m ) = δ(m−m )(−im) (2πs) H M−1 (s) , 2 2 ∞ √ M+1 Z   2 (42) 0 m κ p 0 2 (2) = iδ(m − m ) dτ √ (47) with s ≡ m −(x − x ) − i, where Hα (x) is a Han- 2iπ sin( κ τ) 0 kel function of the second kind (see the appendix). The √  m √ (x2 + x02) cos( κ τ) − 2x · x0  −i in its argument indicates that the expression is the × exp i κ √ − τ . boundary value of a function which is analytic in the 2 sin( κ τ) lower half −(x − x0)2 plane. As we have emphasized, The overall factor of i arises just as it did for the free we have derived this definition from the path integral, relativistic particle, discussed in the previous subsection. and the requirement that the integral over proper time τ As there, we are left with an ordinary integral over τ, converges. and need to establish the appropriate integration con- In order to understand the more involved case of spa- tour. The resulting integral in Eq. (47) is difficult to do tial curvature κ 6= 0, it is again helpful to recall how directly but we can use its behavior at large positive a Eq. (42) can be obtained from the Wronksian method; and large negative a0 to fix the mode functions ψ and ψ for simplicity, let us set M = 0 and label x0 ≡ a which 1 2 appearing in the Wronskian representation. As a consis- is our scale factor. Then the Feynman propagator is a tency check, we compare the resulting Green’s function solution to to a numerical evaluation of the τ integral in Eq. (47)  1 d2 m along a suitable contour, finding perfect agreement in all + G0(a, m|a0, m0) = −iδ(m − m0)δ(a − a0) 2m da2 2 cases. In this numerical evaluation, we choose x = (T,~0) (43) and x0 = (−T,~0), so that any classical real solution has and can be written in the form to pass through the singularity a = 0 at least once, which ψ (a0)ψ (a) is the situation of main relevance for our study. For ease G0(a, m|a0, m0) = −2imδ(m − m0) 1 2 θ(a − a0) W (ψ , ψ ) of comparison, we plot the flat case κ = 0, with M = 0, 1 2 i.e., G = e−2imT , with m = 7, in Fig. 4 [here and in the ψ (a)ψ (a0)  + 1 2 θ(a0 − a) (44) following we are of course plotting the function multiply- W (ψ1, ψ2) ing the singular part δ(m − m0)]. 0 0 where W (ψ1, ψ2) = ψ1(a)ψ2(a)−ψ1(a)ψ2(a) is again the 1.0 (a-independent) Wronskian and ψ1(a) and ψ2(a) are two appropriate independent solutions to the homogeneous  1 d2 m  equation 2m da2 + 2 ψ(a) = 0, found by matching 0.5 Eq. (44) with the asymptotic behavior of Eq. (31) at infinity, with σ = (a−a0)2 (and there are no spacelike di- rections to be considered). As explained below Eq. (33), 0.2 0.4 0.6 0.8 1.0 1.2 (31) asymptotes to e−ima for large positive a at fixed a0, 0 and to eima for large negative a0 at fixed a; this fixes the -0.5 ima −ima modes in (44) as ψ1(a) = e and ψ2(a) = e (up 0 to a normalization that is irrelevant for G ). Thus, one -1.0 finds

0 0 0 0 −im|a−a0| FIG. 4. Feynman propagator for flat universes as a function G (a, m|a , m ) = δ(m − m )e , (45) of T , for m = 7, showing real part (blue), imaginary part (dashed) and absolute value (black) which is constant here. in agreement with Eq. (42) for M = 0. With this in mind, we can now go beyond the simplest flat case, and consider κ 6= 0, where the dynamics of the Consider the saddle points in the τ integral for the Universe corresponds to those of a relativistic oscillator curved-space propagator, and the associated steepest de- or upside-down oscillator. The path integral over x in scent contour. The condition for the exponent to be sta- Eq. (41) is easily calculated: the classical path which tionary with respect to τ is precisely the Hamiltonian −2 2 2
fixes the exponent generalizes to constraint (Friedmann equation) τ x˙ + 1 + κ x = 0, √ √ with x(t) given by Eq. (46). Real saddle points of the x sin κ τ(t + 1 )
+ x0 sin κ τ( 1 − t)
full functional integral, when they exist, are real solu- x(t) = 2 √ 2 , (46) sin( κ τ) tions of the classical equations of motion, including the 12 constraints. Given such saddle points, one defines the Two independent solutions to Eq. (49) are given by associated τ integration cycle as the complete extension √ 1/4 ψ(a) = Di √m − 1 ((1 − i) m (−κ) a) (52) of the steepest descent contour. If this cycle can be de- 2 −κ 2 formed to the real τ-axis while maintaining the conver- and its complex conjugate, which asymptotically become gence of the integral, then the saddle point contribution pure negative and positive frequency modes as a → ∞ is relevant to the final result. but are a mixture as a → −∞. We therefore set We start with the case of negative κ, where, just as in √ 1/4 the flat case, there is always a unique classical solution: ψ4(a) = D−i √m − 1 ((1 + i) m (−κ) a) (53) 2 −κ 2 for timelike separated x and x0, the saddle point in τ is located on the real axis, and the steepest descent contour and ψ3(a) = ψ4(−a). By computing their Wronskian we obtain the Green’s function is the solid curve in Fig. 5. The singular behavior of the √ integrand at τ = 0 [cf. Eq. (37)] then ensures, just as im 1 im  G(a, m|a0, m0) = √ Γ + √ δ(m − m0) in the argument leading to Eq. (37), that Eq. (47) is a π(−κ)1/4 2 2 −κ Green’s function for the Wheeler-DeWitt equation, 0 0 0 × (ψ3(a )ψ4(a)θ(a − a ) + (a ↔ a )) . (54)  2 m  − + (κx2 + 1) G = −iδ(m − m0)δM+1(x − x0) . As we have said, this result can also be obtained from 2m 2 numerical evaluation of Eq. (47). Again we choose x = (48) (T,~0) and x0 = (−T,~0) and also fix κ = −1, M = 0 and m = 7. The resulting function of T is plotted in Fig. 6. Notice that the resulting propagator resembles the flat-space (κ = 0) expression for small T , and the effects of spatial curvature become relevant only at scales |x| ∼ √1 . |κ|

1.0

0.5 FIG. 5. Integration contours in the complex τ plane for κ ≤ 0 (solid line) and for κ > 0 (dashed line), for the case where x = (T,~0) and x0 = (−T,~0). 0.2 0.4 0.6 0.8 1.0 1.2

Once more we set M = 0 for simplicity, and obtain the -0.5 Green’s function from the Wronskian method, with the modes determined from their behavior at large argument. -1.0 The Wheeler-DeWitt equation, at fixed m, is FIG. 6. Feynman propagator for open universes as a func-  d2  + m2(1 − κa2) ψ(a) = 0 , (49) tion of T , for m = 7 and κ = −1, showing real part (blue), da2 imaginary part (dashed) and absolute value (black). and is solved by parabolic cylinder functions [23], denoted For positive κ, the behavior is rather different from by Dν (z). In order to find the modes that generalize the κ ≤ 0 in that for positive κ the real, classical solutions plane waves e±ima used in the κ = 0 case, we can again are periodic in τ; for given x and x0, when one classi- study the asymptotic limits of Eq. (47) using the saddle- cal solution exists there will be an infinite number, and point approximation, finding that we must have (with in general they should all contribute to the propagator. κ < 0) Again for consistency with the κ → 0 limit, we can choose √ the τ integration contour such that it only picks out the − 1 −i m −κa2 ψ3(a) ∼ |a| 2 e 2 , a → −∞ , simplest saddle point, where the classical solution inter- √ − 1 −i m −κa2 polating between x and x0 has no turning points. The ψ4(a) ∼ |a| 2 e 2 , a → +∞ , (50) corresponding saddle point and steepest descent contour, for the mode functions ψ3 and ψ4 appearing in the analog indicated by the dashed curve in Fig. 5, goes over to the of Eq. (44). This asymptotic behavior is also consistent unique κ = 0 saddle point and steepest descent contour with the requirement that, as Eq. (47) is invariant under in Fig. 3 in the flat limit κ → 0. 0 0 0 x → −x and x → −x , ψ3 and ψ4 must satisfy Another issue is that for large timelike x and x there is no real classical solution at all; for large timelike argu- ψ3(−a) = ψ4(a) . (51) ments, the saddle point for τ becomes imaginary and, as 13

4 of T is plotted in Fig. 7. Again, it reduces to the κ = 0 expression e−2imT for small T .

3

2 The integration contour in Fig. 5 is chosen in such

1 a way that its main contribution, for small enough T , comes from the lowest positive real saddle point in τ, 0 0.2 0.4 0.6 0.8 1.0 1.2 corresponding to the real classical solution that takes the smallest amount of proper time. As the arguments√ of -1 ~ the Feynman√ propagator approach x = (1/ κ, 0) and x0 = (−1/ κ,~0), this saddle point moves towards τ = √π FIG. 7. Feynman propagator for closed universes as a function κ where it eventually merges with another saddle point of T , for m = 7 and κ = 1, showing real part (blue), imaginary √π approaching τ = κ from above, corresponding to two part (dashed) and absolute value (black). As discussed in the classical solutions that become indistinguishable in this text, for T ≥ 1 the semiclassical interpretation fails. limit. Our choice of integration contour then becomes ambiguous and is no longer defined by consistency with the κ → 0 limit. As we extend the arguments to |T | > 1, explained above, we choose the one on the negative imag- where there is no longer a real solution, these two saddle inary axis. By the same saddle-point approximation as points separate again and start moving up and down in above, one then determines the asymptotic behavior of the imaginary direction. This is akin to the situation the relevant mode functions ψ and ψ , 5 6 for spacelike separations for the relativistic particle, and √ − √m − 1 m 2 2 κ 2 + κa means that our saddle point needs to be replaced by a ψ5(a) ∼ |a| e 2 , a → −∞ , (55) √ √π − √m − 1 m 2 saddle point on the line − iR parallel to the negative 2 κ 2 + κa κ ψ6(a) ∼ |a| e 2 , a → +∞ , (56) imaginary τ-axis. For the purposes of this paper, we are mainly interested in studying the propagator with again consistent with ψ5(a) = ψ6(−a). These results, as well as the form of the propagator, can in fact be arguments for which there is a classical solution, so that √ a semiclassical picture of the propagator as given by these obtained by replacing −κ → iκ in√ the expressions for the open case. [The factor |a|−m/(2 κ) was dropped in solutions is meaningful. the expressions√ above since it was a subleading oscillatory factor |a|−im/(2 −κ).] It is straightforward to obtain a complete analytic ex- The exponential blowup of the Feynman propagator pression for the κ > 0 propagator from parabolic cylin- for large T follows from the asymptotic behavior of the der functions by solving the homogeneous equation (49). integral (47) for large timelike x and x0. The correspond- Two independent solutions with κ > 0 are ing mode functions increase exponentially for |T | > 1 √ 1/4 when there are no classical solutions, as can be verified ψ5(a) = D− √m − 1 (−i 2m κ a) , 2 κ 2 √ explicitly from the asymptotics of the parabolic cylinder 1/4 ψ6(a) = D− √m − 1 (i 2m κ a) . (57) functions (57). The Feynman propagator is hence patho- 2 κ 2 logical for large timelike separations, and does not define These are again complex conjugates of each other, and a suitable two-point function on the entire superspace, here also already satisfy ψ5(a) = ψ6(−a), unlike for the because positive curvature forces classical solutions to open case κ < 0. On the other hand, they are not asymp- recollapse. An asymptotic description in terms of well- totic positive or negative frequency modes, but blow up defined states that can be used to formulate a quantum exponentially both at positive and at negative infinity. theory of expanding universes does not exist for positive At small a, up to corrections that vanish as κ → 0 they spatial curvature, and this case does not consistently de- reduce to plane waves e±ima. From Eq. (57), the Wron- scribe the type of quantum bounce we are interested in. skian method gives the Green’s function Of course, this situation could be altered by the inclusion √ of a positive cosmological constant, as we mention later. m 1 m  G(a, m|a0, m0) = √ Γ + √ δ(m − m0) × π κ1/4 2 2 κ 0 0 0 (ψ5(a )ψ6(a)θ(a − a ) + (a ↔ a )) . (58) For κ < 0, classical solutions with pure radiation are As before, we have verified that this expression agrees well behaved, expanding to infinite volume in the fu- with the result of a numerical integration of the τ integral ture and past and leading to a well-behaved Feynman along the chosen contour. With x = (T,~0), x0 = (−T,~0), propagator, Eq. (50). However, as we mentioned in as well as κ = 1, M = 0 and m = 7, the resulting function Sec. III, when conformally coupled scalars are introduced 14

(M ≥ 1), the general solution the observed (nearly flat) Universe lives on the corre- sponding critical boundary. This would be the case, for  t  α Z example, if we could identify the correct quantum mea- α x1 √ 0 0 x (t) = √ exp  −κ dt N(t ) −κ sure on the space of closed universes, with sufficiently 0 large cosmological constant to prevent recollapse, and if  t  this measure favored the flat case. We explore this pos- α Z x2 √ 0 0 +√ exp − −κ dt N(t ) , (59) sibility in future work. For the remainder of this paper, −κ however, we ignore spatial curvature. 0

1 with x1 ·x2 = 4 , has both spacelike antigravity and time- V. ADDING ANISOTROPIES AND FREE like gravity solutions: choosing spacelike x1 and x2 that satisfy the constraint, one finds a universe that comes in SCALAR FIELDS from one antigravity direction, turns around before en- tering gravity and then disappears into a different (or the We now extend the treatment to anisotropic cosmolo- same) antigravity direction. Such spacelike solutions are gies, choosing the simplest form of anisotropies, the far enough in the antigravity region that the curvature Bianchi I model: we still require the metric to be spa- term dominates over the positive mass in the potential, tially homogeneous, with an Abelian group of isometries m2(1 + κx2) < 0, leading to their acceleration towards acting on constant time hypersurfaces, but no longer im- spacelike infinity. Even though there are no real classical pose isotropy. The most convenient parametrization of solutions that connect incoming gravity solutions to these such a metric employs Misner variables [24], far antigravity regions, and starting in a gravity region  √ 2 2 2 2 2λ1(t)+2 3λ2(t) 2 one is guaranteed to asymptote into the other gravity ds = A (t) −N (t)dt + e dx1 region, quantum mechanically one expects the spacelike √  2λ1(t)−2 3λ2(t) 2 −4λ1(t) 2 solutions to determine the behavior of the Feynman prop- +e dx2 + e dx3 . (62) agator for spacelike separations. This is indeed confirmed The Ricci tensor, and hence the Einstein tensor, for this by finding the saddle-point approximation to Eq. (47) for metric are diagonal, which by the Einstein equations for- large spacelike separations, e.g., for x2 → ∞ at fixed x0, bids any anisotropy in the fluid, manifest in a velocity ui. √ 0 0 −1/2 i m −κx2 We hence continue to assume that the fluid moves with G(x, m|x , m ) ∼ |x| e 2 . (60) µ µ the cosmological flow, J ∝ δ0 . We can then again exploit the conformal freedom to The propagator becomes oscillatory at spacelike separa- set A(t) to 1. The Ricci scalar of (62) is then tions, so that a given initial state, e.g., a wave packet cen- tered around an initial state in the gravity region a < 0, is λ˙ 2 + λ˙ 2 propagated to large spacelike distances into the antigrav- R = 6 1 2 , (63) N 2 ity region. In this sense, the quantum theory is even more pathological for open than for closed universes, where one giving the correct canonical normalization of the 2 0 finds, again for x → ∞ at fixed x , anisotropy variables λi in the symmetry-reduced action,

√ − √m − 1 m 2 Z ˙ 2 ˙2 ˙ 2 ˙ 2 0 0 2 κ 2 − κx ~χ − φ λ + λ G(x, m|x , m ) ∼ |x| e 2 , (61) S = V dt  + 1 2 (φ2 − ~χ2) − Nρ − ϕ˜n˙  . 0 2N 2N i.e., exponential falloff for large spacelike separations. (64) This is because for κ > 0, both timelike and spacelike In terms of the variables xα and m defined in Eq. (10), classical solutions are bounded due to the potential, and this action reads never reach (timelike or spacelike) infinity. Neither κ < 0 Z   m 1  2 2 2 2   nor κ > 0 can lead to a viable perfect bounce picture S = dt x˙ − x (λ˙ + λ˙ ) − N − ϕm˙ . 2 N 1 2 in terms of a transition between incoming and outgoing (65) gravity states with a → ±∞, while the κ = 0 case leads As for the flat FRW universe which the Bianchi I uni- directly to a perfect bounce. We conclude that, at least verse generalizes, this is the action of a free massive rela- for the theories considered here, only flat FRW universes tivistic particle. However, here the particle is not moving lead to a consistent quantum theory, able to account for through a flat Minkowski spacetime but through a curved an expanding universe. The inclusion of a positive cosmo- superspace, with metric logical constant could rescue positively curved universes from this conclusion, provided the curvature is too small 2 α β 2 2 2 ds = ηαβdx dx − x (dλ1 + dλ2) (66) to cause a recollapse. Nevertheless, the quantum pathol- ogy we have identified for negatively curved FRW uni- or, if we again use the parametrization xα = avα(νi) verses is intriguing, because it raises the possibility that with v2 = −1 to separate the scale factor a, introducing 15 coordinates νi on the hyperboloid HM , which is a simple generalization of Eq. (65). The K vari-

2 2 2 2 2 2 ables λi, i = 1,...,K, can now correspond to anisotropy ds = −da + a g M + a (dλ + dλ ) (67) H 1 2 variables or minimally coupled scalar fields with the un- where gHM is a constant negative curvature metric on usual normalization (71). Equation (73) is now the ac- HM , as in Sec. III. The geometry of superspace at fixed tion of a particle moving in an (M + K + 1)-dimensional a corresponds to the maximally symmetric space HM × superspace with curved metric 2 R ; in the absence of anisotropies, the last term would K 2 α β 2 X 2 vanish and one would simply use Milne coordinates for ds = ηαβdx dx − x dλi flat Minkowski spacetime. i=1 It is well known that the dynamics of anisotropies in K 2 2 2 X 2 the Bianchi I model are equivalent to those of minimally = −da + a gHM + a dλi . (74) coupled free scalar fields in this background. To see this, i=1 we momentarily switch to Einstein gauge, in which the By an elementary generalization of the procedure de- metric is given by Eq. (62) with a general A(t). The scribed in Sec. III A, Eq. (73) gives a Hamiltonian con- action of a free scalar field in this background is straint 1 Z √ Z τ˙ 2 1 m − d4x −g (∂τ)2 = V dt A2 (68) C := gµν (x, λ)p p + ≈ 0 (75) 2 0 2N 2m µ ν 2 µν (identical to the action of a free scalar in a flat FRW where g is the inverse metric on superspace, and pµ universe; a homogeneous scalar field does not feel the includes conjugate momenta for both the variables xα anisotropies). To see that this reduces to the kinetic and the new degrees of freedom λi; more concretely, terms for the anisotropies λi in Eq. (64), we note that   1 2 1 ij 1 ij m since the scale factor (7) is Weyl invariant, one can ex- C = −p + g M (ν)ζ ζ + δ k k + 2m a a2 H i j a2 i j 2 press it both in Weyl and Einstein gauge, (76) 2 2 1 2 2 3A in terms of the momentum pa conjugate to a, momenta a = (φ − ~χ ) Weyl = Einstein (69) 2ρ 8ρ0πG ζi conjugate to the conformally coupled scalar field vari- ables νi living on HM , and momenta k conjugate to the −4 i with ρ = ρ0A in Einstein gauge (in Weyl gauge, ρ = ρ0 free scalar fields and anisotropy variables. is constant). To change gauges, one hence has to replace When quantizing this Hamiltonian constraint in order to obtain the Wheeler-DeWitt equation, there is now an 2 4πG 2 2 A → (φ − ~χ ) ; (70) Einstein 3 Weyl ambiguity, the well-known quantization ambiguity for a particle moving on a curved manifold [25]: if the Ricci the factor 4πG can be absorbed in the normalization of 3 scalar for the superspace metric (74) is R, the general the scalar fields, expression for the quantum Hamiltonian is r 4πG 1 m λ := τ (71) H = (−2 + ξR) + (77) 3 2m 2 (the anisotropy variables are dimensionless while a scalar where 2 is the Laplace-Beltrami operator for the curved field has dimensions of mass), showing the equivalence. metric (corresponding to the operator ordering that en- Hence, we obtain a simple generalization of the theory sures that the Hamiltonian is independent of the choice we have discussed by also adding an arbitrary number of coordinates on superspace) and ξ is, in general, a free of minimally coupled free scalar fields, which can repre- parameter. Halliwell [22] has given the following strong sent either physical scalar fields or anisotropy degrees of argument for fixing ξ: the (classical) Hamiltonian of min- freedom of the Bianchi I model. One has to lift the free isuperspace models is really H = NC (see Sec. III) where scalar fields to a Weyl-invariant theory by the lapse function N is arbitrary, and in particular can be −2 ˜ 1 Z √ 1 Z √ redefined arbitrarily, N → Ω N, where Ω(x, λ) is any − d4x −g (∂τ)2 → − d4x −g (φ2 − ~χ2)(∂λ)2 function on minisuperspace. Under such a redefinition, 2 2 (72) the constraint (75) becomes where λ is again dimensionless and conformally invariant. 1 m˜ (x, λ) C˜ := g˜µν (x, λ)p p + ≈ 0 (78) In Einstein gauge, the right-hand side of Eq. (72) clearly 2m µ ν 2 reduces to the right-hand side of Eq. (68). Going back µν −2 µν −2 to Weyl gauge, the total action is then withg ˜ = Ω g andm ˜ (x, λ) = Ω m, leading by the same argument as above to a quantum Hamiltonian K ! ! Z m 1 X S = dt x˙ 2 − x2 λ˙ 2 − N −ϕm˙  , (73) 1   m˜ (x, λ) 2 N i H˜ = −2˜ + ξR˜ + , (79) i=1 2m 2 16

˜ 0 1 where now 2˜ and R are the Laplace-Beltrami operator where c ≡ c + 4 (M + K)(M + K − 2), which has two and Ricci scalar for the conformally rescaled metricg ˜ on independent solutions in terms of Bessel functions of the superspace. Halliwell now asks that, since redefining the first and second kind, lapse is always possible classically, the solutions Ψ and √ √ √ √ χ1 = a J 1 0 (ma) , χ2 = a Y 1 0 (ma) . (87) Ψ˜ to HΨ = 0 and H˜ Ψ˜ = 0 be related by a conformal 2 1+4c 2 1+4c ˜ γ transformation, Ψ = Ω Ψ for some γ, and finds that this A more convenient basis is given by the Hankel functions is only possible if one fixes ξ to be the conformal coupling, H(1,2) of the first and second kind (which are just linear M + K − 1 combinations of the Bessel functions), so that two linearly ξ = (80) 4(M + K) independent solutions of Eq. (83), for fixed ζ, k and m, are finally given by (recall that the dimension of our superspace manifold is (2,1) M +K +1; Ref. [22] gives an overall minus sign for ξ, pre- ψ (a) = a−(M+K−1)/2H √ (ma) . (88) +,− 1 1+4c0 sumably due to a different sign convention for the Ricci 2 curvature). Demanding covariance under field redefini- As indicated by the subscript +, −, these functions rep- tions of the lapse function hence fixes ξ uniquely. Special resent positive- and negative-frequency modes for the cases are M +K = 0 where there is no conformal coupling Wheeler-DeWitt equation. Indeed, when extended to that can restore covariance under lapse redefinitions, and negative a through the analytic continuation (see, e.g., (2) iπν (1) (1) M + K = 1 where the Laplace-Beltrami operator is con- Ref. [26]) Hν (−z) = −e Hν (z) and Hν (−z) = formally covariant and ξ = 0. −iπν (2) −e Hν (z), ψ+,− have the interesting property of cor- The Wheeler-DeWitt equation for Ψ = Ψ(a, ν, λ, m), responding to pure positive and pure negative frequency, corresponding to the classical constraint (76), then be- respectively, both at positive and negative infinite a, comes −(M+K)/2 −ima  2  ψ+(a) ∼ a e , a → ±∞ , ∂ M + K ∂ 1 2 + − ∆ M K + ξR + m Ψ = 0 −(M+K)/2 +ima ∂a2 a ∂a a2 H ×R ψ−(a) ∼ a e , a → ±∞ . (89) (81) That is, for the Wheeler-DeWitt equation (83) one finds with that an incoming positive-frequency mode simply con- K(2M + K − 1) tinues to an outgoing positive-frequency mode, with the R = . (82) a2 potential at a = 0 not even leading to a phase shift. This 2 As in Sec. III, one can simplify the Wheeler-DeWitt equa- complete invisibility of the 1/a potential is a direct con- tion by Fourier transforming on HM × K from ν and λ sequence of the symmetry of Eq. (83) under a → λa and R −1 to the momenta ζ and k. One then has m → λ m, which forbids any phase shift. These special properties of a 1/x2 potential, and its invisibility in a  ∂2 M + K ∂ c  + − + m2 Ψ(a, ζ, k, m) = 0 scattering process, are well known in quantum mechan- ∂a2 a ∂a a2 ics. In the context of our perfect bounce scenario, they (83) imply that the Universe can go through the singularity with a = 0 without any noticeable impact on its evolution, 1 2 1 ~2 ~ 2 when viewed asymptotically. This is already true clas- c = − (M−1) + δM,0−ζ −k −ξK(2M+K−1) , (84) 4 4 sically, where the classical solutions bounce without any where we have explicitly included the case M = 0 net time delay or advance: the classical Hamiltonian is through the Kronecker delta. This is precisely the same equal to the constraint (76) times a lapse function, functional form as the Wheeler-DeWitt equation for  1  1 1  m FRW universes, Eq. (27), and so the extension of our for- 2 ij ~ 2 H = N −pa + 2 gHM (ν)ζiζj + 2 k + malism from FRW symmetry to the Bianchi I model and 2m a a 2 (90) the inclusion of minimally coupled scalars are completely The terms multiplying 1/a2 are again conserved and can straightforward. The constant c now gets contributions be replaced by a constant, −c0 with c0 < 0; classically from the eigenvalues of the Laplacian on HM × K , as R the effect of anisotropies and momenta in the scalar fields well as from the curvature on superspace. always leads to an attractive potential for a, centered on We can now obtain the general solution to Eq. (83) in the singularity at a = 0. The classical solutions to the the usual way, by setting equations of motion including the constraint are then Ψ(a, ζ, k, m) = a−(M+K)/2χ(a, ζ, k, m) (85) c0 c0 a2 = + N 2(t − t )2 = + (τ − τ )2 (91) to eliminate the first derivative. χ then satisfies the dif- m2 0 m2 0 ferential equation in terms of proper time τ = Nt. These solutions are  ∂2 c0  singular at a = 0 and perform an excursion into the anti- − + m2 χ(a, ζ, k, m) = 0 (86) ∂a2 a2 gravity region of imaginary a, just as the generic flat 17

FRW solutions described in Sec. III which would be of to a = 0. In quantum mechanics, standard arguments the exact same form. The attractive potential at a = 0 then imply an infinite number of lower energy states, speeds up the trajectory as it heads toward the singular- and hence a spectrum which is unbounded below. It has ity, but this time advance is canceled by the additional been claimed that the theory is nevertheless renormaliz- time it takes to cross antigravity. Indeed, both at large able, although the renormalization group displays a limit positive and negative a we have simply a(τ) ≈ (τ − τ0). cycle [29]. (There is a large literature on inverse square In the quantum theory, ordering ambiguities in the potentials in quantum mechanics, and even some exper- Hamiltonian constraint can alter the coefficient of the imental tests. See, e.g., Ref. [30] for a recent discussion 1/a2 potential, making it repulsive in some cases. In- and further references.) deed, the relevant coefficient of the potential is the one At the minisuperspace level discussed here, negative appearing in Eq. (86), energy states are irrelevant because we are only interested 1 1 in solutions of the Wheeler-DeWitt equation with posi- c0 = − (M − 1)2 + δ − ζ~2 − ~k2 4 4 M,0 tive energy, defined by m2. However, when we include in- M 2(M − 2) + K(M 2 − M − 1) teractions with other modes, such as the inhomogeneous + (92) 0 1 4(M + K) modes of gravitons or scalars, then for c < − 4 it is possible that the negative energy states for the scale fac- if we use the value (80) for ξ, that is, we fix the or- tor a would become excited, potentially signifying strong dering ambiguities by demanding coordinate covariance backreaction as the Universe passes through the quan- on superspace and covariance under redefinitions of the tum bounce. The problem may be avoided in two ways. lapse function, giving a purely quantum contribution in For M = 0 or M = 1 one can restrict consideration the second line of Eq. (92). If we ignore the trivial case to background cosmologies for which the zero-mode mo- M = K = 0, we can rewrite Eq. (92) as menta of the anisotropy and scalar fields are strictly 0, in −ζ~2 − 1 M ≥ 1 ,K = 0 , which case the quantum mechanics for a lies on the crit-  4 −~k2 − 1 K ≥ 1 ,M = 0 , ical boundary where it (just) makes sense. Or, one can c0 = 4 (93) include additional conformally coupled scalars, taking −ζ2 − ~k2 − 1 K ≥ 1 ,M = 1 ,  4 M > 1 so that, from the last line of Eq. (93), the quan-  ~2 ~ 2 1 (M−1)K −ζ − k − 4 + 4(M+K) K ≥ 1 , M > 1 . tum mechanics of a is well defined for a range of classical This is an intriguing result. The contributions coming anisotropy and scalar field momenta. For K = 2 (i.e., from anisotropy or scalar field momenta are both neg- only anisotropies but no minimally coupled scalars) and ative. The numerical term is fixed by covariance. The M > 4, the numerical contribution can be large enough first line corresponds to the situation of Sec. III, where to make the potential repulsive at small momenta. If we no anisotropies or minimally coupled scalars are present consider classical solutions with this (order ~ squared) [from Eq. (27), removing the first derivative term changes potential, an isotropic universe with no scalar momenta c in Eq. (28) to c0 given here]. The similarity of the first would bounce off the repulsive potential and avoid the three lines is not a coincidence; for M ≤ 1, the super- singularity altogether. Quantum mechanically, however, space metric (74) is conformally flat. As we have imposed if we extend the range of a to negative values, then a tun- conformal coupling to the Ricci scalar on superspace in nels through the barrier in a process which may be de- Eqs. (79) and (80), the dynamics must be equivalent to scribed with complex classical solutions, as we explained the flat superspace case of Sec. III. in Ref. [1]. 0 1 The value c = − 4 is well known as a critical value in The conclusion is that when anisotropy and scalar field the quantum mechanics of an inverse square potential. If degrees of freedom are included, then for small numbers 0 1 c ≥ − 4 , the negative classical potential is outweighed of conformal scalars, the isotropic cosmology with no by the kinetic energy due to the Heisenberg uncertainty scalar momenta is a special case, poised on the edge of principle, rendering the energy spectrum strictly positive. a qualitatively different (and perhaps ill-defined) phase. There are various infrared regularized versions of the the- On the positive side, this finding may turn out to be a ory in which the spectrum is made discrete by including selection principle, telling us that anisotropic or kinetic- a positive harmonic potential [27], with a taken either dominated singularities should be excluded from the the- on the infinite line, the half-line a > 0, or by imposing ory whereas isotropic universes with zero scalar momenta periodicity in a, in which case the model becomes the are allowed. If so, this would imply that black hole singu- Calogero-Sutherland model (see, e.g., Ref. [28]). These larities, which locally resemble strongly anisotropic cos- are well-defined, exactly solvable models which exhibit, mological singularities, do not correspond to a bounce among other interesting phenomena, anomalous dimen- (contradicting the interpretation given by Ref. [4], for ex- sions in the operator product expansion [27]. ample); there would be no “born again” universe on the 0 1 If, however, c < − 4 , any finite energy wave func- other side of the black hole singularity. On the negative tion has an infinite number of oscillations on the way side, one may wonder whether the inclusion of inhomo- 18 geneities could lead to problems even for the isotropic, of the corresponding homogeneous equation, it is imme- nonkinetic cosmological bounce. We emphasize that, for diate to write down the solution to Eq. (95) with the M = 0 or M = 1, any amount of classical momentum in correct boundary conditions, the zero modes of the anisotropy or scalar degrees of free- 0 0 −(M+K) 0 dom takes the quantum mechanics of a into the subcrit- G(a, m|a , m ) = −2im a δ(m − m ) × −1 0 0 ical regime. Perhaps it is essential to work at M > 1 for W (ψ−, ψ+) (ψ−(a )ψ+(a)θ(a − a ) 0 0 the theory to make sense. Clearly, we have only scratched +ψ−(a)ψ+(a )θ(a − a)) , (96) the surface with this discussion, and there is a great deal to explore further. where the Wronskian is For the remainder of the paper, we assume that the 4a−(M+K) quantum mechanics for a makes sense. As explained in W (ψ−, ψ+) = (97) Ref. [1], this allows us to calculate the propagation of the iπ Universe, and all inhomogeneous modes in it, by solving and no longer constant in a, as is consistent with the ap- the theory on complex trajectories which bypass a = 0 in pearance of a first derivative in Eq. (83). The Wronskian the complex a plane. Remarkably, as was also explained takes care of the factors of a appearing in the elimination in Ref. [1], due to its scale-invariant property, the inverse of the first derivative, Eq. (85), and cancels the determi- square potential, if present, is actually invisible in our nant factor a−(M+K). The final result is final results for “in-out” amplitudes. πm G(a, m|a0, m0) = δ(m − m0)(a a0)−(M+K−1)/2 × (98) 2   H(1)(ma0)H(2)(ma)θ(a − a0) + (a ↔ a0) A. Feynman propagator ν ν √ 1 0 with ν ≡ 2 1 + 4c , which is consistent with the re- Having defined positive- and negative-frequency modes sults of Ref. [1] (with K ≡ D − 2 as only the D − 2 by their asymptotics, given in Eq. (89) (and without using anisotropy degrees of freedom of a D-dimensional uni- any boundary condition at a = 0), it is easy to obtain the verse were considered there). One can check that in the Feynman propagator for the anisotropic case as a Green’s absence of anisotropies or minimally coupled scalar fields, function for the Wheeler-DeWitt equation, by using the K = 0, this result reduces to the expression obtained in Wronskian method as before. Sec. III, i.e., the propagator for a free massive particle in With the quantum Hamiltonian given by Eq. (77), the (M + 1)-dimensional Minkowski spacetime. By our re- Feynman propagator satisfies marks below Eq. (93) the same should be true for M = 0 or M = 1 and general K, where the superspace metric is −2 + ξR + m2
G(x, λ, m|x0, λ0, m0) (94) conformally flat. − 1 M+1 0 K 0 0 = −2im(−g) 2 δ (x − x )δ (λ − λ )δ(m − m )

− 1 where we must introduce a factor (−g) 2 for the nontriv- VI. PERTURBATIONS ial metric determinant on superspace. Again switching to the scale factor coordinate a, Eq. (94) is equivalent to In this section, we extend our analysis to inhomoge- neous cosmology, treated perturbatively at both linear  2  ∂ M + K ∂ 1 2 + − ∆ M K + ξR + m G and nonlinear order. We aim to solve the following prob- 2 2 H ×R ∂a a ∂a a lem: given an incoming state at large negative a consist- 0 M+K 0 0 δ(a − a ) δ (ν − ν , λ − λ ) 0 ing of a flat, FRW, radiation-dominated classical back- = −2im M+K √ δ(m − m ) a gHM ground universe with perturbations in their local adia- batic vacuum state, what is the outgoing quantum state with G ≡ G(a, ν, λ, m|a0, ν0, λ0, m0), and the metric de- at large positive a, as defined by our analytic continua- terminant on superspace is now made explicit. Again, tion prescription? This question can be answered, in the we can now go to Fourier space on HM ×RK introducing semiclassical limit, by using complex solutions of the clas- momenta ζi and ki; the Feynman propagator in Fourier sical Einstein-matter field equations. If one sends in any space satisfies combination of linearized positive- (negative-)frequency modes then, even after including the effects of nonlineari-  ∂2 M + K ∂ c  + − + m2 G(a, ζ, k, m|a0, ζ, k, m0) ties in the field equations, it turns out that one finds only ∂a2 a ∂a a2 positive- (negative-)frequency linearized modes coming = −2im a−(M+K)δ(a − a0)δ(m − m0) (95) out. As we now explain, this is sufficient to show, semi- classically, that the outgoing quantum state is also the with c as in Eq. (84). Since we have already identi- local adiabatic vacuum. Hence, at a semiclassical level, fied the positive- and negative-frequency solutions (88) there is no particle production across the bounce. 19

Let us see this in detail. Consider a classical time- This state is uniquely defined by amΨ0 = 0 for all m, for dependent background solution of the Einstein-matter the annihilation operator equations. If the matter is a perfect fluid, the only   propagating degrees of freedom are scalar density per- 1 d am ≡ √ ~ + ωmqm . (102) turbations and tensor gravitational wave modes. At the 2ωm~ dqm linearized level, we can decouple the modes by exploit- ing the homogeneity and isotropy of the background: for Let us assume that the incoming state of the pertur- 0 0 a flat background, every mode is a sum of plane waves bations is Ψin(η , q) = Ψ0(η , q) at some large negative P ik·x ∗ η0, for which linear theory is valid. The quantum fluc- v(η, ~x) = k vk(η)e , with v−k(η) = vk(η) , with the coefficients decomposed into irreducible representations tuations in the fluid density may be shown to be small of the little group of rotations about k. Now consider compared to the background density provided the wave- the action for the perturbations. At leading order, it length of the modes is longer than the thermal wave- is quadratic and it is diagonalized by the above mode length, a condition which is in any case required in order decomposition. After a suitable time-dependent rescal- for the fluid description to hold. The outgoing quantum ing of the perturbations, the kinetic terms can always be state, at some large positive time η, is then given by prop- brought to canonical form in which the action reads [see, agating the incoming vacuum Ψ0 to large positive times e.g., Ref. [31], page 269, Eq. (10.59)] η, for which linear theory is once again valid, using the Z path integral, (2) X  a 2 2,a a 2 S = dη |v˙k| − wk (η)|vk| , (99) Z i 0 0 k,a S(q,η;q ,η ) Y 0 0 0 Ψout(η, q) ≈ N Dq e ~ dqmΨ0(η , q ), where the index a labels the independent modes (here, m (103) scalar and tensor), and where S(q, η; q0, η0) is the full, nonlinear Einstein-matter 2,a a 2 2,a 0 wk (η) = (kcs ) + meff (η) (100) action taken with boundary conditions q(η) = q, q(η ) = √ 0 a q ; Dq indicates the complete path-integral measure and where cs is the speed of sound, 1/ 3 for the scalar acous- N is a normalization constant. We compute Eq. (103) in tic modes and unity for the tensor modes. In general, the the semiclassical approximation, by finding the appropri- time-dependent “effective mass” introduces a nontrivial c 0 ate complex classical solution qm(˜η), η < η˜ < η, which η-dependence. However, in our chosen background, the is a stationary point of the combined exponent. Substi- effective mass vanishes for both the scalar and tensor a a tuting Eq. (101) for Ψin and varying the exponent with modes so wk = kcs in both cases. 0 respect to qm yields, using the Hamilton-Jacobi relation, We now make the assumption that the perturbations the initial condition for the classical solution qc, are well described by linear theory for wide intervals of conformal time η well before and well after the bounce. c c 0 (ipm + ωmqm)(η ) = 0, (104) As we see later, we cannot actually take the limit of infi- nite positive and negative conformal time because of the c c where pm =q ˙m is the canonical momentum. The initial effect of nonlinearities in the fluid. Nevertheless, in the c condition (104) specifies that qm is pure negative fre- semiclassical approximation, and for modes whose wave- quency at η0, a large negative time. The final boundary length is longer than the thermal wavelength of the fluid, c condition is just qm(η) = qm, where η is a large positive the periods of incoming and outgoing conformal time dur- time. We solve the classical Einstein-matter equations ing which linear theory remains valid are very large. We with these two boundary conditions in linear perturba- define our incoming and outgoing states during these in- tion theory. At linear order, the solution satisfying the tervals. c ikcs(˜η−η) boundary conditions is qm(˜η) = qme . Below, we When linear theory is valid, and when the frequencies give the complete solution for generic perturbation modes a a a 2 wk(η) change adiabatically, (dwk/dη)/(wk)  1, the at linear and nonlinear order. We find that the solution quantum states of the system are well described by those is well described by linear perturbation theory at large of a set of decoupled harmonic oscillators. Let us de- negative and large positive times, with small nonlinear note the corresponding real coordinates, i.e., the real and corrections, and that an incoming positive (negative) fre- a imaginary parts of the vk, by the coordinates qm, where quency mode evolves to an outgoing positive (negative) the single index m runs over all of the real, independent frequency mode which directly implies that the outgoing modes. Each of the coordinates qm contributes an action quantum state is the local adiabatic vacuum. To ver- 1 R 2 2 2 Sm = 2 dη(q ˙m − ωm(η) qm), and the adiabatic vacuum ify this, we need only apply the annihilation operators state is just the product of the corresponding harmonic d am ∝ ipm + ωmqm = + ωmqm to Ψout(η, qm) as ~ dqm oscillator ground states, given in Eq. (103). Using the Hamilton-Jacobi equation, Y 2 c c 1/4 −ωmqm/(2 ) the result is proportional to (ip +ωmq )(η), which van- Ψ0(η, q) = (ωm/~π) e ~ . (101) m m m ishes if the solution is pure negative frequency. Hence the 20 incoming adiabatic vacuum evolves to the outgoing adia- ity, restrict consideration to planar symmetry so that the batic vacuum, and there is no particle production across metric depends only on conformal time η and one spa- the bounce. tial coordinate x, with two orthogonal spatial directions (y, z). To keep the calculations manageable, we do not introduce conformally or minimally coupled scalar fields, A. Basic setup and conventions so M = K = 0. We work in Einstein gauge, i.e., in the usual formulation of general relativity coupled to a We wish to study perturbations about a flat (κ = 0) radiation fluid. radiation-dominated FRW universe in a perturbation ex- The general form of the metric compatible with our pansion. We shall go to nonlinear order but, for simplic- assumed symmetry is

  −1 + gηη(η, x) gηx(η, x) 2 2  gηx(η, x) 1 + gxx(η, x)  ds = a (η)   . (105)  1 + gyy(η, x) gyz(η, x)  gyz(η, x) 1 + gzz(η, x)

We can still apply coordinate transformations that The density ρ(η, x) and four-velocity uµ(η, x) can also be leave this form of the metric invariant. A coordinate written in terms of background and perturbation as transformation η =η ˜ +  g(x, η˜) changes the metric coef- ficients as ρ(η, x) = ρ0(η)(1 +  δr(η, x)) , µ 1 0 a˙  u (η, x) = (v (η, x), v(η, x), 0, 0) . (110) δg = −2 g +g ˙ , δg = −2g0 , a(η) ηη a ηx µ 0 a˙ The constraint u uµ = −1 can be solved for v (η, x). Un- δg = δg = δg = 2 g (106) ˙ xx yy zz a der a coordinate transformation (108), we have δv = −f, so that we can set v = 0 everywhere, i.e., adopt a coordi- where here and in the remainder of this section · is deriva- nate system in which the radiation is at rest everywhere tive with respect to η and 0 denotes derivative with re- (comoving gauge). The remaining gauge freedom is then spect to x. We use this freedom to eliminate gηx and under transformations introduce a different notation for the metric perturba- tion functions (note that in this section ψ denotes a η =η ˜ +  G(˜η) , x =x ˜ +  f(˜x) , (111) scalar metric perturbation, not a solution to the Wheeler- y =y ˜ + (ι1y˜ + ι2z˜ + ι3) , z =z ˜ + (ι4z˜ + ι5y˜ + ι6) DeWitt equation as in earlier sections), where the ιi are arbitrary constants and G and f are free 2 2  2 a˙ ˙ ds = a (η) (−1 + 2φ(η, x)) dη functions. Under such a transformation, δφ = − a G − G, δψ = a˙ G, δγ = f 0, δhT = 2(a − b) and δh× = 2(c + +(1 + 2(ψ(η, x) + γ(η, x))) dx2 +  h×(η, x) dy dz a d), and so functions of this form in the perturbations   hT (η, x) + 1 +  2ψ(η, x) + dy2 are to be considered pure gauge. We solve the Einstein 2 equations in Fourier space, where the gauge freedom for   hT (η, x) the functions φ, ψ, hT and h× is somewhat hidden as it + 1 +  2ψ(η, x) − dz2 . (107) 2 only becomes apparent for k = 0. We are left with five free functions for the metric T × The form (107) is still left invariant by a transformation (φ, ψ, γ, h , h ) and the density perturbation δr. As we of the form will see, there are also six nontrivial Einstein equations relating these. To proceed, we assume that all of the per- Z x˜ ! η =η ˜ +  G(˜η) + dX f˙(X, η˜) , turbation functions can further be expanded as a power series in , x =x ˜ +  f(˜x, η˜) , (108) X n−1 φ(η, x) =  φn(η, x) , etc. (112) which we will use to simplify the matter variables. The n≥1 energy-momentum tensor for radiation is The idea is now to solve the Einstein equations Gµν = 4 1 8πGTµν order by order in ; the Einstein equations also T = ρu u + ρg ; uµu = −1 . (109) µ µν 3 µ ν 3 µν µ imply energy-momentum conservation ∇µT ν = 0 for 21 the fluid. First, for the background (at order 0) we have and the equations ˙ ¨ ˙ ¨ 2 ˙ 1 00 Fn Fn J1,n J1,n ψn + ψn − ψn = − − − − η (123) a˙  a˙ 2 8πG η 3 η 2 4 12 ρ˙0 + 4 ρ0 = 0 , = ρ . (113) η 11 η J a a2 3 + J 0 + J + J˙ + 6,n 6 2,n 12 3,n 4 3,n 6 −4 Z x J η  The first one tells us that ρ0 ∝ a for some constant M; − dx0 2,n + 2J˙ + J¨ . the Friedmann equation then gives the solution a(η) ∝ η, η 2,n 2 2,n the simplest example of a perfect bounce that we have Equation (123) can now be solved for ψ using Green’s already discussed in the introduction to this paper. It n functions; Eq. (122) then gives γ by a single integration follows that a˙ = 1 , and that analytic continuation in the a η over η, and from Eq. (120) and Eq. (121) one can obtain scale factor a (as we have discussed in previous sections) explicit expressions for δr,n and φn at each order. At each is equivalent to analytic continuation in the conformal order in , this provides an explicit algorithm for solving time coordinate η, which we use in this section. the system of Einstein equations (114)–(119). At order n in the perturbation expansion, the six non- trivial Einstein equations are 3 6 2 6 B. Tensor perturbations δ − φ + 2ψ00 − γ˙ − ψ˙ = J , (114) η2 r,n η2 n n η n η n 1,n Equations (117) and (118) are easily solved. First, con- 1 0 ˙ 0 φ + ψ = J2,n , (115) sider the homogeneous equation solved by the first-order η n n perturbation, 1 2 2 4 δ − φ + φ˙ + ψ˙ + 2ψ¨ = J , (116) η2 r,n η2 n η n η n n 3,n 2 h¨T + h˙ T − (hT )00 = 0 . (124) 2 1 η 1 1 h¨T + h˙ T − (hT )00 = J , (117) n η n n 4,n We can easily find the general solution in Fourier space, 2 h¨× + h˙ × − (h×)00 = J , (118) for k 6= 0, n η n n 5,n Z 1 2 T dk ikx T − δ + φ + ψ00 − φ00 h1 (η, x) = e h1 (η, k) , η2 r,n η2 n n n 2π −ikη ikη 2 2 4 T e e − γ˙ − φ˙ − ψ˙ − γ¨ − 2ψ¨ = J (119) h (η, k) = b1(k) + b2(k) . (125) η n η n η n n n 6,n 1 kη kη T for some “source terms” Ji,n that are nonlinear combi- For k = 0, the two independent solutions are h = nations of the lower order perturbations. constant and hT ∼ 1/η. We can write the general so- We first note that Eqs. (117) and (118) that govern lution as the tensor modes hT and h× are already decoupled from n n d Z dk  e−ikη eikη  the others. For the scalars, Eqs. (114) and (115) can be hT = d + 2 + eikx b (k) + b (k) 1 1 k η 2π 1 kη 2 kη solved for δ and φ directly. From Eq. (115) we get 0 r (126) Z x where k0 is an arbitrary momentum scale to make d2 φ (η, x) = −ηψ˙ (η, x) + F (η) + η dx0 J (η, x0) × n n n 2,n dimensionless. h1 satisfies the same differential equation; (120) its general solution is where Fn(η) is a free function; then Eq. (114) implies Z  −ikη ikη  × e2 dk ikx e e that h1 = e1 + + e c1(k) + c2(k) . k0η 2π kη kη 2 2 2 00 2 η (127) δr,n(η, x) = − η ψn + ηγ˙ n + 2Fn + J1,n ∗ 3 3 3 For a real solution we need b1(k) = −b1(−k), b2(k) = Z x ∗ 0 −b2(−k) and similar for c1(k) and c2(k). +2η dx J2,n . (121) We recognize d1 and e1 as gauge modes correspond- ing to coordinate transformations (111), whereas d2 and Substituting these relations into Eqs. (116) and (119) and e2 are physical k = 0 modes. The free functions taking linear combinations one obtains b1(k), b2(k), c1(k) and c2(k) are the physical gravitational Z x degrees of freedom. 00 ˙ ˙ η 0 γ˙ n = ηψ − 6ψn − 3Fn − J1,n − 3 dx J2,n Now consider the general inhomogeneous equation, n 2 Z x 3η 2 −3η dx0 J˙ + J (122) h¨T + h˙ T − (hT )00 = J . (128) 2,n 2 3,n n η n n 4,n 22

Again, we go to Fourier space and first consider k 6= 0. tribute, is We use the Wronskian method to determine a suitable Z dk ikx Green’s function; in contrast to the Green’s function that ψ1(η, x) = e ψ1(η, k) , appeared as a Feynman propagator in earlier sections, 2π − √i kη √i kη here the boundary conditions are that the higher order e 3 e 3 ψ1(η, k) = a1(k) + a2(k) . (133) perturbations are set to 0 at some conformal time η0 in kη kη the far past, so that only a linear (purely positive- or purely negative-frequency) mode is present. Given two The Fourier mode k = 0 is a gauge mode [see the discus- T ˜T sion below Eq. (111)], with general solution independent solutions h1 and h1 to the homogeneous equation (124), the Green’s function for these boundary c1 F1(η) conditions is ψ1(η, 0) = − + c2 − ; (134) k0η 2 hT (η0)h˜T (η) − hT (η)h˜T (η0) G(η, η0) = 1 1 1 1 (129) since F1 is arbitrary, we can set c1 = c2 = 0 with no loss W (η0) of generality. Putting this together, we have

0 − √i kη √i kη ! for η > η > η where η is the initial time at which only Z dk e 3 e 3 0 0 ψ (η, x) = eikx a (k) + a (k) a linear perturbation is assumed to be present, and 0 1 2π 1 kη 2 kη otherwise. The Wronskian is W (η0) ≡ hT (h˜T )0 −(hT )0h˜T 1 1 1 1 F (η) as before. Using this Green’s function, we find that one − 1 , (135) particular solution to Eq. (128) is 2 ∗ where for a real solution we need a1(k) = −a1(−k) and 1 Z η ∗ hT (η, k) = dη0 η0 sin(k(η−η0)) J (η0, k) , (130) a2(k) = −a2(−k). n kη 4,n As said, from this expression we can determine the other scalar functions γ, φ and δr. We find while for k = 0 we find Z ikη  √  dk ikx − √ 6 Z η γ1(η, x) = a3(x) + e a1(k)e 3 − − i 3 T 1 0 0 0 0 2π kη hn (η, 0) = dη η (η − η ) J4,n(η , 0) , (131) η ikη  √  √ 6  +a (k)e 3 − + i 3 , (136) 2 kη which is just the limit k → 0 of Eq. (130). Expressions for Z ikη h× are analogous. In the integrals in Eqs. (130) and (131) η dk ikx − √ n φ1(η, x) = F1(η) + F˙1(η) + e a1(k)e 3 (137) as well as in the following, the initial time η0 that should 2 2π   ikη   appear as the lower limit of integration is suppressed for 1 i √ 1 i  × + √ + a2(k)e 3 − √ , simplicity; we neglect the η0-dependent contributions as kη 3 kη 3 we are only interested in a particular solution. Z ikη   dk ikx − √ 4 4i n δr,1(η, x) = 2F1(η) + e a1(k)e 3 + √ Clearly, at each order  one can also add a solution of 2π kη 3 the homogeneous equation to this solution for hT . This n ikη   T √ 4 4i  can however be absorbed into the linear perturbation h1 . +a2(k)e 3 − √ , (138) We hence set these arbitrary solutions to the homoge- kη 3 neous equations to 0 for n ≥ 2. where we recognize a3(x) and F1(η) as encoding the re- maining gauge freedom in comoving gauge (111). a1(k) and a2(k) correspond to the scalar degrees of freedom of C. Scalar perturbations the radiation fluid. In order to obtain the solutions for higher order pertur- bations, we derive the Green’s function for the ψ equation For the scalar perturbation functions φ, ψ, γ and δr, we proceed analogously. For clarity, we first derive the gen- (123), which has the general form eral solutions for the first-order perturbations, for which ¨ 2 ˙ 1 00 there are no sources and the general solution is obtained ψn + ψn − ψ = Jn . (139) η 3 n straightforwardly. The equation for ψ1 is Eq. (123) with the sources set to 0, i.e., Again, the boundary condition for the Green’s function is to set the higher order perturbations to 0 at some initial ˙ ¨ ¨ 2 ˙ 1 00 F1 F1 conformal time η0. We find, for k 6= 0, ψ1 + ψ1 − ψ = − − (132) η 3 1 η 2 √ Z η  0  3 0 0 k(η − η ) 0 ψn(η, k) = dη η sin √ Jn(η , k) , where F1 is a free function of η. Going into Fourier space, kη 3 the general solution for k 6= 0, where F1 does not con- (140) 23 and for k = 0 the same as for the tensors, upper-half complex plane etc.) We choose the linear positive-frequency modes to be Z η 1 0 0 0 0 √ ψ (η, 0) = dη η (η − η ) J (η , 0) . (141) −ik η/ 3 n η n e 0 ψ1(η, x) = A cos(k0x) , k0η Again, once the solution for ψn is found, the other pertur- −ik0η T e bation functions γ , φ and δ can be obtained easily. h (η, x) = B cos(k0x) . (142) n n r,n 1 k η From these expressions, we can now work out the non- 0 linear solution for all metric perturbation functions and As seen before, the expression for ψ1 then determines the density perturbation order by order in ; all we need γ1, φ1 and δr,1. These scalar quantities are all gauge de- to do is to expand Einstein equations up to any given pendent but one may compute the gauge-invariant New- order to find the sources Ji,n and then compute the in- tonian potentials first introduced by Bardeen [32] and tegrals (130), (131), (140) and (141) to find the pertur- given by (in Fourier space for k 6= 0) bations at the next order. γ¨ γ˙ γ˙ Φ = −φ + + , Ψ = −ψ − . (143) k2 k2η k2η D. Nonlinear positive-frequency modes We find that for the linear perturbations Φ and Ψ are 2 2 equal and fall off as 1/k0η at large |η|, We are now specifically interested in the nonlinear ex- i √ − √ k0η tension of linear positive-frequency modes at a given wave 2π A e 3 (3 + 3i k0η) Φ1(η, k0) = Ψ1(η, k0) = − 3 3 . number k0. (The following calculations and discussion k0η can be extended to the case of an incoming negative- (144) frequency mode by simply taking the complex conjugate The explicit form of the sources at order 2 is, in terms of all expressions below, and replacing the lower- by the of the linear perturbations,

12 3 3 1 1 4 J = φ2 + ((hT )0)2 + ((h×)0)2 + 2γ0ψ0 + 3(ψ0)2 + hT (hT )00 + h×(h×)00 + 4γψ00 + 8ψψ00 − γγ˙ 1,2 η2 16 16 4 4 η 4 4 1 1 1 1 4 12 12 + φγ˙ − ψγ˙ − hT h˙ T − (h˙ T )2 − h×h˙ × − (h˙ ×)2 − γψ˙ + φψ˙ − ψψ˙ + 2γ ˙ ψ˙ + 3ψ˙ 2 , (145) η η 2η 16 2η 16 η η η 2 2 2 1 1 1 1 J = γφ0 − φφ0 + ψφ0 + ψ0γ˙ + (hT )0h˙ T + (h×)0h˙ × − φ0ψ˙ + 2ψ0ψ˙ + hT (h˙ T )0 + h×(h˙ ×)0 2,2 η η η 16 16 8 8 +2γψ˙ 0 + 4ψψ˙ 0 , (146) 4 1 1 1 3 1 3 8 8 J = φ2 − ((hT )0)2 − ((h×)0)2 − 2φ0ψ0 + (ψ0)2 + hT h˙ T + (h˙ T )2 + h×h˙ × + (h˙ ×)2 − φφ˙ − φψ˙ 3,2 η2 16 16 2η 16 2η 16 η η 8 1 1 + ψψ˙ − 2φ˙ψ˙ + ψ˙ 2 + hT h¨T + h×h¨× − 4φψ¨ + 4ψψ¨ , (147) η 4 4 4 4 4 J = −γ0(hT )0 − (hT )0φ0 − 3(hT )0ψ0 − 2γ(hT )00 − 4ψ(hT )00 − 2hT ψ00 − φh˙ T + ψh˙ T − γ˙ h˙ T − h˙ T φ˙ + hT ψ˙ + h˙ T ψ˙ 4,2 η η η −2φh¨T + 2ψh¨T + 2hT ψ¨ , (148) 4 4 4 J = −γ0(h×)0 − (h×)0φ0 − 3(h×)0ψ0 − 2γ(h×)00 − 4ψ(h×)00 − 2h×ψ00 − φh˙ × + ψh˙ × − γ˙ h˙ × − h˙ ×φ˙ + h×ψ˙ + h˙ ×ψ˙ 5,2 η η η −2φh¨× + 2ψh¨× + 2h×ψ¨ , (149) 4 1 1 1 1 J = − φ2 + ((hT )0)2 + ((h×)0)2 − γ0φ0 + (φ0)2 + γ0ψ0 + 2(ψ0)2 + hT (hT )00 + h×(h×)00 − 2γφ00 + 2φφ00 6,2 η2 16 16 8 8 4 4 4 1 1 1 1 8 −2ψφ00 + 2γψ00 + 4ψψ00 − γγ˙ + φγ˙ − ψγ˙ − γ˙ 2 − hT h˙ T − (h˙ T )2 − h×h˙ × − (h˙ ×)2 + φφ˙ +γ ˙ φ˙ η η η 4η 16 4η 16 η 4 8 8 1 1 − γψ˙ + φψ˙ − ψψ˙ − γ˙ ψ˙ + 2φ˙ψ˙ − ψ˙ 2 − 2γγ¨ + 2φγ¨ − 2ψγ¨ − hT h¨T − h×h¨× − 2γψ¨ + 4φψ¨ − 4ψψ¨ , (150) η η η 8 8

where we omit the subscripts 1 on the first-order pertur- bations on the right-hand side of these equations. For 24 simplicity, we also set the second tensor mode h× to 0 analytic continuation of these modes that avoids the sin- from now on (its dynamics are analogous to those of hT ). gularity at η = 0 is then defined by choosing any contour We now compute the second-order perturbations from that remains in the lower-half complex η plane. Eqs. (130), (131), (140) and (141), using the sources com- Asymptotically, we find that at large k0|η|, T puted from the chosen linear perturbations. For h2 , we 2 2  −2ik η 7i B − √ ik0η 2 1 1 find ψ ∼ −e 0 + e 3 A + cos(2k x) 2 128k η 12 6 0 −i(1+ √1 )k η 0 3 0 T 3e    h = AB (1 + cos(2k x)) 2 + cos(2k0x)(1 + 12 log(k0η)) 1 2 2 2 0 −i √ + O , 2k0η 2 2 8 3k0η k0η √ −i(1+ √1 )k η i(1 + 3)e 3 0 − cos(2k0x) and one can check that all terms, including all subleading 2k0η ones, oscillate at positive frequencies asymptotically (ei- √ −i(1+ √1 )k η 2 3 0   ther at ω = 2k or at ω = √ k ). The nonlinear modes i(3 + 3)e 2 1 0 3 0 + + Γ 0, i(1 + √ )k0η 6k0η 3 3 again decay exponentially as k0η → −i∞, and indeed de- fine nonlinear positive-frequency modes. From the gen- 13ie−2ik0η Γ(0, i(−1 + √1 )k η) 3 0 eral structure of the equations (114)–(119), one can see + cos(2k0x) 6k0η that the same property should hold to all higher nonlin- 13ie2ik0η Γ(0, i(3 + √1 )k η) ! ear orders: the source terms, being nonlinear in lower or- 3 0 − cos(2k0x) . (151) der perturbations, always decay exponentially sufficiently 6k0η fast in imaginary time that integration with a Green’s Here Γ(0, z) are incomplete gamma functions. Their function that has an exponentially growing and an ex- asymptotic expansion for large arguments is ponentially decaying part, as in Eq. (140), gives again an exponentially decaying next-order perturbation. The 1 1  1  Γ(0, z) ∼ e−z − + O . (152) method we have described then allows a general defini- z z2 z3 tion of positive-frequency modes in the complex η plane, to all orders in perturbation theory. Using this expansion, we see that as k η → ±∞, hT has 0 2 The other perturbations are determined by Eqs. (120)– the asymptotic behavior (122). For completeness, we give their asymptotic expres- √ √ −i(1+ √1 )k η 6 + 5 3 − (27 + 16 3) cos(2k0x) sions for large k0|η|, T 3 0 h2 ∼ AB e i √ (21 + 11 3)k0η 2  2 − √ ik0η i cos(2k0x)k0η √ √ γ ∼ A e 3 − √ (154) 23 + 4 3 − (37 + 20 3) cos(2k x)  1   2 + √ 0 + O . 3 2 2 3 3     2(5 + 2 3)k0η k0η 5 1 −1 − cos(2k0x) + 3 log(k0η) + O , The asymptotic expansion shows in particular that all the 4 k0η −i(1+ √1 )k η 0 1 √2 4i cos(2k x)k η terms in Eq. (151) oscillate as e 3 for large k0|η|, 2 − ik0η 0 0 δr,2 ∼ A e 3 √ (155) and decay exponentially for large negative imaginary η. 3 3 We can obtain expressions for the scalar perturbations   1  −7 − cos(2k0x)(7 − 12 log(k0η)) + O , in exactly the same way. The expressions are similar to k0η those for hT but involve more terms (15 in total), as there 2 2 2  −2ik η 7B 2 − √ ik0η ik0η(1 + 2 cos(2k0x)) can be contributions of order A2 and B2, corresponding φ ∼ e 0 + A e 3 √ 2 64 to two tensor modes or two scalar modes combining to 6 3     give a scalar. Just as the second-order tensors, they con- 1 1 1 + − cos(2k0x) − log(k0η) + O . tain incomplete gamma functions, but there is also a term 3 12 k0η involving a logarithm, To verify the validity of our solution method, we have √ 2 checked explicitly that the second-order perturbations 2 − √ ik0η i 3 log(k0η) cos(2k0x) − A e 3 . (153) solve Einstein’s equations up to order 2. 2k0η We see that none of the scalar perturbation functions These terms are potentially problematic when the per- decay at real infinity k0|η| → ∞, and some even blow up, turbation functions are extended to the complex η plane indicating a breakdown of perturbation theory at large as the logarithms and incomplete gamma functions have times. Again, to get gauge-invariant statements about branch cuts. However, all we require for positive- this behavior, we can compute the Newtonian potentials, frequency modes is analyticity in the lower-half η plane, and find that they fall off as 1/k0η, where these modes extend to Euclidean, asymptotically √ 2   decaying modes. This can be achieved by defining all the − √2 ik η iπ 3a 1 3 0 Φ2(η, 2k0) ∼ −e + O 2 2 (156) branch cuts to be along the positive imaginary axis. The 4k0η k0η 25 with similar behavior for Ψ. This compares with Φ1 ∼ purely positive frequency at early and late times when 2 2 O(1/k0η ) in Eq. (144), which still indicates that the the Universe is large and classical, corresponding to a perturbation expansion breaks down when k0|η| ∼ 1. positive expansion rate of the Universe, as we observe. This physical behavior is due to the nonlinear evolution We have shown that the addition of a positive radiation in the fluid, as shown by analytical and numerical stud- density makes a crucial difference, as it leads to classi- ies in Ref. [13]. When we go down the imaginary axis, cal solutions which connect asymptotic contracting and i.e. for η = −iτ with τ → ∞, all perturbation functions expanding Lorentzian regions, and which are represented fall off exponentially, with exponential terms of the form by the positive-frequency modes defined by the Feynman e−ωτ dominating any polynomially growing terms. As propagator. We do not impose any boundary conditions we have argued, this behavior persists for higher orders for the wave function at a = 0, and accept that some in the  expansion, and defines these modes by regular- modes may even diverge there: all that is required is ity for large negative imaginary η; the blowup of scalar a consistent evolution from an asymptotic contracting to perturbations along the real axis due to nonlinearities an asymptotic expanding universe, through or around the in the fluid does not prevent us from defining nonlinear bounce, as this allows a calculation of transition ampli- asymptotic positive-frequency modes. tudes and hence, ultimately, predictions for the transition of a given state in the contracting phase to a state in the expanding phase. This formalism appears much more E. Summary natural than an imposition of a boundary condition at a = 0, where quantum effects are large and where clas- We have given an algorithm for solving the Einstein- sical notions of singularity avoidance may cease to have matter equations order by order in perturbation theory, any relevance. In practical terms, the fact that our wave and exhibited explicit results at second order that show in functions and propagators admit a semiclassical WKB detail how the positive-frequency incoming modes match description in which high-curvature regions near a = 0 only to positive-frequency outgoing modes, and similarly can be avoided gives hope that a semiclassical approach for negative-frequency modes (where our results trivially to the quantum cosmology of bouncing scenarios can be extend by taking complex conjugates). We have argued used for predictions, even in the absence of a complete that this behavior should extend to all orders in perturba- theory of quantum gravity. tion theory, as the nonlinear extension of linear positive- frequency modes leads to perturbation functions that de- Some features we are exploiting are clearly restricted to cay exponentially for large negative imaginary times, and homogeneous cosmological models such as the FRW and branch cuts can be restricted to the positive half-plane for Bianchi I universes we have studied explicitly. It is there- positive-frequency solutions, so that the nonlinear met- fore vital to check that the formalism can be extended ric perturbation satisfies a nonlinear notion of positive consistently to generic perturbations around homogene- frequency. We identified some subtleties, namely that ity, and ultimately to fully nonlinear solutions of GR. 1 We have developed a systematic perturbative treatment the perturbation expansion fails at late times k0|η| ∼  , where  is the perturbation amplitude, meaning that one that shows how this question can be attacked, at linear has to restrict attention to an annulus in the complex and nonlinear order, and given evidence for a consistent 1 nonlinear extension of positive-frequency modes to the plane,  < k0|η| <  , in which the  expansion can be trusted and nonlinearities are not yet dominant [1]. complex a plane. Again, one is interested in the transi- tion of incoming asymptotic positive-frequency modes to outgoing modes which are, in general, a mixture of posi- VII. CONCLUSIONS tive and negative frequency and which signal particle pro- duction (and potential divergencies) at the bounce. We This paper represents a detailed study of a very sim- have shown that an incoming positive-frequency mode ple cosmological model, based on the principle of confor- can be continued around the singularity, and unambigu- mal symmetry for matter and gravity and the observed ously matches to an outgoing positive-frequency solution. fact that the early Universe was dominated by radiation. So the incoming adiabatic vacuum state is stable across Classical cosmological solutions of this model describe the bounce and no divergencies arise. Our calculations a bounce, with a big bang/big crunch singularity, but have been limited to pure radiation and planar symme- the singularity can be avoided by going into the complex try, and one focus of future work will be to extend these plane. While this “singularity avoidance” seems ad hoc in results to more general cases. The present results already classical gravity, we have shown its meaning in the quan- indicate that a consistent semiclassical picture exists for tum theory where, similar to quantum tunneling, the nonlinear perturbations of cosmological models, and that complexified solutions represent legitimate saddle points this picture can be used for calculations of the cosmolog- to the path integral. The picture that emerges for quan- ical phenomenology of bounce scenarios of the type we tum cosmology is based on modes that are asymptotically consider. 26

Thus, our results show how classical singularities do Mike and Ophelia Lazaridis Niels Bohr Chair. Research not necessarily prevent a consistent quantum description at Perimeter Institute is supported by the Government of bouncing cosmologies. The inclusion of quantum ef- of Canada through the Department of Innovation, Sci- fects into the big bounce seems a natural and simple al- ence and Economic Development Canada and by the ternative to the development of more complicated bounce Province of Ontario through the Ministry of Research, scenarios [8–10, 33]. Innovation and Science. The work of S.G. was supported There are many avenues for further exploration. In in part by the People Programme (Marie Curie Actions) Sec. V, we began to explore the quantum theory on the of the European Union’s Seventh Framework Programme real a-axis around a = 0. In some cases, it may be that (FP7/2007-2013) under REA grant agreement no 622339. the attractive inverse square potential in the Wheeler- Appendix A: Massive Relativistic Propagator DeWitt operator may lead the quantum theory to fail when further (inhomogeneous) degrees of freedom are in- In this appendix, we calculate the massive relativistic cluded, but in others the quantum theory seems to be propagator given in Eq. (31) exactly. First, we note that healthy. The quantum dynamics of more general Bianchi models also deserve to be understood; for these, the invis- Z ∞ M+1 0  m  2 −i m σ +τ G(x|x ) = i dτ e 2 ( τ ) , ibility of the singularity that we have observed for Bianchi 2πiτ I will presumably be replaced by a nontrivial scatter- 0 ing matrix between in and out asymptotic states. The is a convergent integral when σ = −(x − x0)2 is positive. pathologies we have identified in the Feynman propaga- The τ integral may be taken along√ the positive real axis tor for curved FRW universes should be revisited with 0 < τ < ∞. Next, we set τ = σeu, with −∞ < u < ∞, the inclusion of a positive cosmological constant. More so that basic conceptual questions concerning the interpretation Z ∞ M+1 √ of the propagator and the determination of probabilities 0 √ 1−M  m  2 −im σcoshu− M−1 u need to be investigated. Ultimately, we need to find a G(x|x ) = i du( σ) 2 e 2 −∞ 2πi compelling measure on the space of quantum universes. 1 √ 1−M (2) √ M 2 There are hints that the present flat, isotropic universe = (−im) (2πm σ) H M−1 (m σ) , (A1) 2 2 lies on a critical boundary in the quantum theory, and these may point to novel resolutions of the classic flatness where we have used the standard integral representation and isotropy puzzles. of the Hankel function of the second kind,

∞ iν+1 Z ACKNOWLEDGMENTS H(2)(z) = du e−iz cosh u−νu , (A2) ν π −∞ We thank A. Ashtekar, I. Bars, J. Halliwell, J.-L. Lehn- ers, L. Smolin, P. J. Steinhardt and especially J. Feld- and for positive real argument the function is defined as brugge for helpful discussions, and L. Sberna for com- the boundary value of a function in the lower-half com- ments on the manuscript. One of us (N.T.) thanks plex z plane where the integral converges. E. Witten for encouraging remarks. Following the discussion given in Sec. IV, the result is This research was supported in part by Perimeter In- then continued to negative values of σ by analytic con- stitute for Theoretical Physics, in particular through the tinuation through the lower-half complex σ plane.

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