Quantum cosmology at the turn of Millennium A.O. Barvinsky † 1 0 0 TheoryDepartment, LebedevPhysics Institute and LebedevResearchCenterin Physics, Lenin- 2 sky Prospect 53, Moscow 117924, Russia n a Abstract J 2 Abriefreviewofthemodernstateofquantumcosmology ispresentedasatheoryof 1 quantum initial conditions for inflationary scenario. The no-boundary and tunneling 1 states of the Universe are discussed as a possible source of probability peaks in the v distribution of initial data for inflation. It is emphasized that in the tree-level approx- 6 4 imation the existence of such peaks is in irreconcilable contradiction with the slow roll 0 regime – the difficulty that is likely to be solved only on account of quantum gravi- 1 tational effects. The low-energy (typically GUT scale) mechanism of quantum origin 0 1 of the inflationary Universe with observationally justified parameters is presented for 0 closed and open inflation models with a strong non-minimal coupling. / c q †Talk given at the IXth Marcel Grossmann Meeting, Rome, July, 2000 - r g : v i X 1. Introduction r a This is a generally recognized fact that in the last two decades the cosmological theory was dominated by the discovery and rapid progress of the inflation paradigm that scored explaining the well known paradoxes of the standard big bang scenario. Interestingly, the beginning of inflation theory was marked in early eighties by the revival of interest in quan- tum cosmology. Before that it was considered merely as a toy model testing ground for quantum gravity [1]. However, practically simultaneously with the invention of the inflation scenario several suggestions were put forward for special quantum states of the Universe [2, 3, 4, 5, 6, 7] that could serve as a source of this scenario. Quantum cosmology became the theory of quantum initial conditions in inflationary Universe. Such application of this theory was from the start marred by a number of difficulties of bothconceptualandtechnicalnature. Tobeginwith, agreatcontroversy brokeoutregarding drastic difference between two major proposals for the cosmological wavefunction – the no- boundary proposal of Hartle and Hawking [2, 3] (also semiclassically implemented in [4]) and the tunneling proposal of [5]. The origin of this difference is rooted in the peculiarities of quantum gravity theory (the presence of both positive and negative frequency solutions of the Wheeler-DeWitt equation, indefiniteness of the Euclidean gravitational action, problem 1 of time, etc.) which are not properly understood up till now, so that the fundamentals of these proposals and the discrepancies of their predictions are still being disputed [8]. Another problem was that, even within a rather shaky foundation of the no-boundary and tunneling wavefunctions, in the semiclassical approximation they would not generate well defined and sufficiently sharp probability peaks that could serve as a source of initial conditions for inflation. This is a very general property of all semiclassical models, and it is very easy to see this. Indeed, a basic characteristics of the inflation scenario is an effective Hubble constant H, its value determining at later times all main cosmological parameters of the Universe δT H = Ω, ρ, ,.... (1.1) ⇒ T In its turn it is usually generated by the inflaton scalar field ϕ, H = H(ϕ), whose initial conditions are determined by a sharp probability peak in the quantum distribution for φ. Semiclassically this distribution ρ(ϕ) is given by the exponentiated Euclidean action of the model I(ϕ) – the Hamilton-Jacobi function in the classically forbidden domain ρ(ϕ) e±I(ϕ). (1.2) ∼ In the slow-roll regime the inflaton momentum is very small, which is both true for the Lorentzian and Euclidean domains (classically allowed and forbidden regions related by analytic continuation). Therefore the derivative dI(ϕ)/dϕ is very small, the graph of I(φ) is very flat, and it cannot generate any peak-like behaviour for the corresponding distribution function. Therefore, slow-roll nature of inflation is always in irreconcilable contradiction with the demand of the semiclassical probability peaks in the wave function. Finally, in recent years another major objection arose against the issue of initial condi- tions for inflation in cosmology, in general, and in quantum cosmology, in particular. With the invention of self-reproducing inflation [9, 10], it was understood that the anthropic prin- ciple starts playing an important role [11, 12]. So provided, the self-reproducing eternal inflation regime is achieved, the total probability of observing some value of the effective Hubble constant equals the fundamental quantum probability P(H) weighted by the an- thropic probability P (H) – the probability of the existence of the observer. The anthropic latter is obviously proportional to the volume of inflating Universe and, therefore, exponen- tially depending on time, P (H) exp(3Ht). Therefore, very quickly any probability anthropic ∼ peak of P(H) gets wiped out by the anthropic factor in P (H) P(H)exp(3Ht), unless total ∼ the peak P(H) has a stronger fall off behaviour in H than the exponential one. In this paper we give a brief overview of the present state of quantum cosmology and advocate that despite the intrinsic difficulties and objections of the above type this theory remains a viable scheme of description for the very early quantum Universe. In particular, we shall try to show that its imprint on the observable large scale structure can be used as a testing ground of fundamental quantum gravity theory in the range of energies where the semiclassical expansion can be trusted. We would like to emphasize here that the role of quantum initial conditions in cosmology should not be underestimated. One of the reasons is that these conditions determine the energy scale of the cosmological evolution (encoded in the characteristic value of the Hubble constant) as compared to both the Planckian scale, where semiclassical methods break, and the self-reproduction scale of entering the eternal inflation. 2 Note that the argumentation against the issue of initial conditions in cosmology usually starts with the assumption of the Planckian energy scale at the onset of inflation, resulting in the self-reproduction conditions for eternal inflation. However, such a starting point cannot be regarded reliable because of the absence of non-perturbative methods at high energies and the absence of fully consistent quantum gravity theory at this energy scale. Within the conventional perturbation framework the predictions can be justified only when the entire evolution of the system stays in the low-energy domain. If this evolution corresponds to the conventional inflation scenario followed by the standard big-bang model cosmology evolving down to lower scales, then the criterion of our semiclassical predictions boils down to verification of the initial energy scale being much below the Planckian one. This is the point where the issue of initial conditions becomes crucially important – this low initial scale should not be imposed by hands, but rather be derived from general assumptions on the cosmological quantum state. One of the goals of this review is to demonstrate that such a possibility occurs in the cosmological model with strong non-minimal curvature coupling of the inflaton – the model for which the quantum origin of the Universe turns out to be the low-energy (typically GUT) phenomenon generating the present day observable cosmological parameters. The paper is organized as follows. We start with a brief overview of the state of art in quantum cosmology in the year 2000, discuss the two fundamental proposals for the cosmo- logical quantum state – the no-boundary and tunneling wavefunctions, their implications in context of the closed and open cosmology and then show how these states can lead to the low energy phenomenon of the quantum origin of the inflationary Universe. 2. Quantum cosmology 2000 – state of art State of art in present day quantum cosmology consists of the set of rules, rendering this field a physically complete and consistent theory, and the scope of approximation meth- ods that allow one to solve quantum cosmological problems in concrete setting. Among these main ingredients one can single out three basic subjects most important for applica- tions: i) Wheeler-DeWitt equation and the path integral; ii) semiclassical approximation and beyond; iii) collective variables – homogeneous minisuperspace modes and cosmological perturbations. 2.1. Wheeler-DeWitt equation and path integral The formalism of quantum cosmology is based on the system of Wheeler-DeWitt equa- tions – the quantum Dirac first-class constraints as equations selecting the quantum state of thegravitationalandmatterfields Ψ[g (x),φ(x)]inthefunctionalcoordinaterepresentation ab of canonical commutation relations for the operators of 3-metric g (x), matter fields φ(x) ab and their conjugated momenta pab(x) = δ/iδg (x), p(x) = δ/iδφ(x). These equations look ab like Hˆ (x)Ψ[g (x),ϕ(x)] = 0, ⊥ ab Hˆ (x)Ψ[g (x),ϕ(x)] = 0. (2.1) a ab 3 Here Hˆ (x) and Hˆ (x) are the operators of the superamiltonian and supermomentum con- ⊥ a straints – the functional variation operators of the second and first order correspondingly whose actual form will not be very important for us. FromtheviewpointoflocalquantumfieldtheorysuchoperatorsinthefunctionalSchrodinger representation are not well-defined because of poorly defined equal time products of local operators, but formally this problem is overcome by writing down a formal path integral so- lution to these equations – which all the same does not control the equal-time commutators i Ψ[g (x),ϕ(x)] = Dg (x)Dφ(x) exp S[g (x),φ(x)] , (2.2) ab µν µν h¯ Z (cid:18) (cid:19)(cid:12)gaugefixing (cid:12) g ,φ = g (x),ϕ(x). (cid:12) (2.3) µν ab (cid:12) Σ (cid:12) (cid:12) Here the in(cid:12)tegration runs over 4-metrics and matter fields in spacetime domain subject to boundary values – 3-metric g (x) and matter field ϕ(x) – arguments of the wavefunction ab defined on this spacelike boundary. ThisintegralinthenaiveformlackingthegaugefixationwasfirstproposedbyH.Leutwyler [13] and later derived in [14, 15] with a proper account of the full Feynman-DeWitt-Faddeev- Popovgaugefixingprocedureandboundaryconditionsonintegrationvariables(forlaterand more detailed formulation see [16, 17]). The Euclidean version of this path integral represen- tation for the solution of Wheeler-DeWitt equations was then used by Hartle and Hawking in the formulation of the no-boundary prescription for the cosmological wavefunction [2, 3]. 2.2. Semiclassical approximation and beyond Usefulness of the path integral consists, as is well known, in the possibility of developing a regular semiclassical expansion. In quantum cosmology, however, the knowledge of the solution to Wheeler-DeWitt equations – exact or within any approximation scheme – does not guarantee the exhaustive solution of the physical problem. Oneofthereasonsisthat, incontrast withusualquantum fieldtheory, thebasicWheeler- DeWitt equation is not evolutionary – unlike the Schrodinger equation it is hyperbolic rather than parabolic in its variables, and, moreover, it is not just one equation, but the whole infinite system of equations, the consistency of which is guaranteed by the closure of the algebra of operators in the left-hand side of eqs.(2.1). For this reason, the formalism of quantum cosmology is devoid of a unique time variable labeling the evolution, which is nothing but the manifestation of the diffeomorphism invariance of the theory. This simple property at the classical level results in disastrous complications at the quantum level – sometimes called the problem of time, the remarkable review of its status given by K.Kuchar [18]. This problem has a number manifestations which, however, altogether originate from the problem of interpreting the cosmological wavefunction – the solution of eqs.(2.1). In contrast with the evolutionary Schrodinger problem of a conventional QFT, this inter- pretation is far from being obvious. To begin with, the Wheeler-DeWitt equations in view of theirhyperbolic natureimplyasaconserved object thequantity whichisnotpositivedefinite – a direct analogueof the situation with the Klein-Gordonequation in first-quantized theory. Therefore, this conserved quantity – inner product in the space of solutions of the Wheeler- DeWitt equations – cannot be used for constructing probability amplitudes. Another side 4 of this problem is the presence of solutions of both positive and negative frequencies con- tributing with opposite signs to the probabilistic quantities. By and large, this problem has not yet been solved, so we present here only the existing preliminary steps in its partial resolution. Interestingly, those steps were undertaken simultaneously with the construction of semiclassical expansion for the cosmological wavefunction, which is very often interpreted as the fact that time (and associated with it conserved probability) in quantum gravity is not a fundamental concept, but rather is the notion which arises only in semiclassical ap- proximation. We believe, though, that such a widespread opinion is erroneous – lack of our understanding should not be hidden by smearing out the fundamental and primary notions of physics. The first serious attempt to introduce time, probability, etc. in quantum cosmology originated from the semiclassical approximation for Ψ[g (x),φ(x)] in the sector of the grav- ab itational field g (x). This was first done, at the level of minisuperspace model, by DeWitt ab in his pioneering paper on canonical quantum gravity [1] and then rederived for generic gravitational system by Rubakov and Lapchinsky in [19] and also intensively discussed by Banks [20]. With the wavefunction Ψ[g (x),φ(x)] rewritten as ab i Ψ[g (x),φ(x)] = exp S[g (x)] Ψ [g (x),φ(x)], (2.4) ab ab m ab h¯ (cid:18) (cid:19) where S[g (x)] is the Einstein-Hamilton-Jacobi function of the pure gravitational field in ab vacuum, the function Ψ [g (x),φ(x)] starts playing the role of the quantum state of quan- m ab tized matter fields in the external classical gravitational background. Such an interpretation is justified by the fact that, when this function is restricted to the solution of classical Ein- stein equations in vacuum, g (x) = gclass(t,x), then it becomes the explicit function of ab ab time variable, Ψ (t)[φ(x)] Ψ [gclass(t,x),φ(x)]. Moreover, on account of the Wheeler- m ≡ m ab DeWitt equations it satisfies in the lowest order approximation in 1/m2 – the inverse of the P Planck mass squared –theSchrodinger equation withthequantum Hamiltonianof quantized matter fields in external classical gravitational field without sources. In operator notations, Ψ (t) = Ψ (t)[φ(x)], m m | i ∂ ih¯ Ψ (t) = d3x(N⊥Hˆ +NaHˆ ) Ψ (t) , (2.5) m ⊥ a m ∂t | i | i Z where the operator Hamiltonian is explicitly written as a linear combination of matter super- hamiltonianandsupermomenta with thebackground lapseandshift functionsascoefficients. Thus, when the semiclassical approximation for the gravitational field is reliable, the time variable can be introduced into the formalism of quantum cosmology by means of the classical gravitational background that i) neither takes into account the back reaction of quantized matter nor ii) has its own quantum fluctuations. The discussion of such a way of introducing time in cosmology can be found in [19, 16, 21, 22]. Majority of results in modern cosmology have been obtained within this interpretation framework – with time introduced via the classical background. An obvious limitation of this framework is that the quantum properties of the gravitational background are inaccessible and it is incapable of accounting for quantum back reaction properties – so, in essence, this is not quantum and cosmology but rather quantum field theory in curved spacetime. The origin of this difficulty is obvious – the quantum effects of the gravitational and matter fields are not treated on equal footing: 5 the Shrodinger equation of the above type takes into account quantum effects of matter exactly (to all powers in h¯), but disregards all inverse powers of the Planck mass squared 1/m2. The way around this difficulty is to develop a regular semiclassical loop expansion P in h¯ without distinguishing those powers of Planck’s constant that arise from 1/m2 and P those from the quantum loops of matter fields, and, simultaneously not to loose time and probability interpretation inherent in the Schrodinger equation above. This program was implemented in the series of papers [23, 24, 16, 25] in the lowest non- trivial order ofh¯-expansion for quantum states having the form of a single semiclassical wave packet i Ψ[g (x),ϕ(x)] = P[g (x),ϕ(x)]exp S[g (x),ϕ(x)] . (2.6) ab ab ab h¯ (cid:18) (cid:19) Here S[g (x),ϕ(x)] is the Hamilton-Jacobi system of the full system of interacting gravita- ab tional and matter fields, and P[g (x),ϕ(x)] = P [g (x),ϕ(x)]+O(h¯) (2.7) ab 1−loop ab is the preexponential factor whose expansion in h¯ begins with the one-loop term. In [23, 24, 16] it was shown that the one-loop preexponential factor can be universally obtained in terms of the Hamilton-Jacobi function of the system. This factor was obtained by both solving the Wheeler-DeWitt equations in the approximation linear in h¯ [23, 24] and calculatingthepathintegralinthegaussianapproximation[17]. Bothmethodsgivethesame result, thus confirming formal consistency of the theory. The expression for the conserved inner product of semiclassical quantum states was derived [23, 24, 25] (with a nontrivial measure in superspace of 3-metrics and matter fields). This inner product turned out to coincide with the usual inner product of states in the physical sector of the theory arising as a result of the Hamiltonian (or ADM) reduction to true physical degrees of freedom. Physical time arising in this reduction was shown to survive the transition to the one- loop approximation, which means that concept of time is not entirely semiclassical, but admits continuation to quantum domain. All these conclusions attained in the one-loop approximation can be generalized to higher orders of semiclassical expansion by the price of tecnically complicated formalism – the only principal limitation of this framework is that the starting point remains the semiclassical wave packet of the form (2.6) not mixing opposite frequencies. This is a fundamental limitation that reflects conceptual problems in quantum cosmology, that are not yet resolved. 2.3. Collective variables – minisuperspace and cosmological per- turbations The success of applications in a physical theory to an essential extent depends on a particular approximation scheme used. Together with general semiclassical expansion, con- sidered above, one should use approximation schemes that simplify the configuration space of the theory leaving aside those degrees of freedom which are not very important in the problem in question. The extremal manifestation of this approach in cosmology consists in the so called minisuperspace reduction, when only a finite number of collective degrees of freedom is left – describing the spatially homogeneous cosmological model. At the quantum 6 level, such approximation is not completely consistent, because the zero-point fluctuations of the discarded degrees of freedom cannot be excluded by hands – they might give an im- portant contribution to the dynamics of those collective variables that determine the main features of the model. Thus, more fruitful is the approach when all the degrees of freedom are retained, but a finite number of them are treated exactly, while the rest are considered perturbatively in the linearized approximation and higher. Such a scheme matches well with the semiclassical expansion in which quantum fluctuations of linearized modes give contributions to perturbative loop effects. Inapplication to cosmology, this generalscheme implies a well known theory of cosmolog- ical perturbations (see, for example [27]), when the metric and matter field are decomposed into the spatially homogeneous background and inhomogeneous perturbations ds2 = N2(t)dt2 +a2(t)γ dxidxj +h (x)dxµdxν, (2.8) ij µν − ϕ(x) = ϕ(t)+δϕ(x), x = (t,x), (2.9) where a(t) is the scale factor, N(t) is the lapse function and γ is the homogeneous spa- ij tial metric (for closed cosmology this is a metric of the 3-sphere of unit radius). The full set of fields consists of the minisuperspace sector of spatially homogeneous variables (a(t), ϕ(t), N(t)) and inhomogeneous fields f(x) essentially depending on spatial coordi- nates xi=x f(x) = δϕ(t,x), h (t,x), χ(t,x), ψ(t,x), A (t,x),... (2.10) µν µ The role of classically non-vanishing scalar field ϕ (or the field with non-vanishing expecta- tion value) is usually played by inflaton that drives the quasi-exponential expansion at the inflationary stage of the evolution. On the other hand, the cosmological perturbations of metric, the inflaton field itself and other matter fields – initially quantum and later semiclas- sically coherent –describe theformationofthelargescale structure ontheRobertson-Walker cosmologicalbackground(including microwave backgroundradiationandothermatterinthe observable Universe). The description of cosmological perturbations requires the formalism that deals with its gauge invariance properties [28, 27]– not all the variables above are dynamically independent and physically significant, because part of the variables represent just purely coordinate degrees of freedom or those degrees of freedom that can be excluded in virtue of constraints in terms of physical variables. The dynamical content of the latter is basically the following. The spatially homogeneous sector of inflaton ϕ, scale factor a and lapse N gives rise to only one dynamical degree of freedom – it can be without loss of generality identified with the inflaton ϕ, while for the inhomogeneous perturbations, basically it is transverse, transverse- traceless, etc., components that form the physical sector. We shall collectively denote them by fT. With such a decomposition, the effective dynamics of the main collective variable in cosmology – the inflaton field ϕ is determined by the reduced density matrix obtained by tracing out the inhomogeneous fields fT. If in the physical sector the full cosmological state is denoted by Ψ(ϕ,fT), then this density matrix is given by ρˆ ρ(ϕ,ϕ′) = dfTΨ(ϕ,fT)Ψ∗(ϕ′,fT). (2.11) ≡ Z 7 The diagonal element of this density matrix is the distribution function of the inflaton ϕ, ρ(ϕ) = ρ(ϕ,ϕ), (2.12) which might yield initial conditions for inflation, provided it has a peak-like behaviour for some suitable mean values of ϕ. This goes as follows. In the chaotic inflation model the stage of inflation is generated within the slow roll approximation, when the inflaton field slowly rolls down the the potential V(ϕ) – some monotonically growing function of ϕ – which, in its turn, determines a big value of the effective Hubble constant, a˙/a H(φ), ≃ 8πV(ϕ) H2(φ) = . (2.13) 3m2 P Theinitialvalueofϕactuallydetermines allmaincosmologicalparameters, including thedu- ration of inflation in terms of the e-folding number N – the logarithmic expansion coefficient for the cosmological scale factor a during the inflation stage, tF N = dtH (2.14) Z0 (with t = 0 and t denoting the beginning and the end of inflation epoch) and the present F day value of Ω [29], 1 Ω , (2.15) ≃ 1 Bexp( 2N) ∓ − The signs here are related respectively to the closed and open models, and B is the ∓ parameter incorporating the details of the reheating and radiation-to-matter transitions in the early Universe. Depending on the model for these transitions, its order of magnitude can vary from 1025 to 1050 (when the reheating temperature varies from the electroweak to GUT scale). In what follows we shall assume the latter as the most probable value of this parameter. Eq. (2.15) clearly demonstrates rather stringent bounds on N. For the closed model the e-folding number should satisfy the lower bound N lnB/2 60 in order to generate the ≥ ≃ observable Universe of its present size, while for the open model N should be very close to this bound N 60 in order to have the present value of Ω not very close to zero or one, ≃ 0 < Ω < 1, the fact intensively discussed on the ground of the recent observational data. In the chaotic inflation model the effective Hubble constant is generated by the potential of the inflaton scalar field and all the parameters of the inflationary epoch, including its duration in units of N, can be found as functions of the initial value of the inflaton field ϕ at the onset of inflation t = 0. If this initial condition belongs to the quantum domain then it has to be considered subject to the quantum distribution (2.11)-(2.12) following from the cosmological wavefunction. If this distribution function has a sharp probability peak at certain ϕ, then, at least within the semiclassical expansion, this value of ϕ serves as the initial condition for the inflationary dynamics. 8 3. No-boundary and tunneling wave functions Quantum states that give initial conditions for inflation were suggested in early eight- ies. Basically, these are two states having in the semiclassical approximation qualitatively different behaviours. One of these states – the so called no-boundary one – was suggested by Hartle and Hawking [2, 3] in the form of the Euclidean path integral prescription. This prescription reduces to the Euclidean version of the integral (2.3) where the integration goes over Euclidean compact 4-geometries and 4-dimensional histories of matter fields ”bounded” by the 3-geometry and matter field – the arguments of the cosmological wavefunction. The underlying spacetime of Euclidean signature has the topology of a 4-dimensional ball. An- otherquantumstate–thetunnelingone–wassuggestedasaparticularsemiclassical solution of the minisuperspace Wheeler-DeWitt equation in the Robertson-Walker model with the inflaton potential, generating the effective Hubble (or cosmological constant) [5, 6, 7]. The both states were analyzed in much detail for the model of minimally coupled inflaton field ϕ having a generic potential V(ϕ) m2 1 S[g ,ϕ] = d4xg1/2 P R(g ) ( ϕ)2 V(ϕ) . (3.1) µν µν 16π − 2 ∇ − ! Z As semiclassical solutions of the minisuperspace Wheeler-DeWitt equation in this model, they can be written down in the slow roll approximation (when the derivatives with respect to ϕ are much smaller than the derivatives with respect to a). They read [6, 7] 1 π Ψ (ϕ,a) = C (a2H2(ϕ) 1)−1/4exp I(ϕ) cos S(a,ϕ)+ , (3.2) NB NB − −2 4 (cid:20) (cid:21) (cid:20) (cid:21) 1 iπ Ψ (ϕ,a) = C (a2H2(ϕ) 1)−1/4exp + I(ϕ)+iS(a,ϕ)+ (3.3) T T − 2 4 (cid:20) (cid:21) and describe two types of the nucleation of the Lorentzian quasi-DeSitter spacetime (de- scribed by the Hamilton-Jacobi function S(ϕ,a)) from the gravitational semi-instanton – the Euclidean signature hemisphere bearing the Euclidean gravitational action I(ϕ)/2 πm2 I(ϕ) = P , −H2(ϕ) πm2 S(ϕ,a) = P (a2H2(ϕ) 1)3/2. (3.4) −2H2(ϕ) − Thesizeofthishemisphere –itsinverse radius–aswell asthecurvatureofthequasi-DeSitter spacetime are determined by the effective Hubble constant (2.13) In the tree-level approximation the quantum distributions of universes with different values of the inflaton field φ (2.11)-(2.12) are, thus, given by two algorithms – the real amplitudes of (3.2)-(3.3): ρ (ϕ) e−I(ϕ) (3.5) NB ∼ for the no-boundary quantum state [2, 3, 4] and ρ (φ) eI(ϕ), (3.6) T ∼ 9 for the tunneling one [5]. For the minimally coupled inflaton, I(ϕ) – the action of the gravitational instanton – reads, up to inflationary slow roll corrections, as 3m4 I(φ) P , (3.7) ≃ −8V(ϕ) where V(ϕ) is the inflaton potential. From these equations it immeadiately follows that the no-boundary and tunneling wavefunctions lead to opposite predictions: most probable universes with a minimum of the inflaton potential in the no-boundary case and with a maximum – for the tunneling situation [6]. However, for reasons discussed in Introduction these extrema in the probability distribu- tion cannot generate inflationary scenario. The main obstacle is the irreconcilable contradic- tion between the slow-roll nature of inflation and the requirement of sharp probability peaks – for slow-roll regime the ϕ-derivatives of the distribution function are very small, because by order of magnitude they coincide with the momentum conjugated to ϕ (or rate of change of ϕ), and, therefore, ρ (ϕ) cannot have sharp enough peaks in the inflationary domain. NB,T The only means possible for overcoming this difficulty is, to the best of our knowledge, to go beyond the tree-level approximation. The loop part of the distribution function, depending on the cosmological model, can qualitatively change the predictions of the tree-level theory. We shall show this below for the model with non-minimally coupled instanton accounting for one-loop effects in the quantum ensemble of tunneling Universes. 3.1. Hawking-Turok wavefunction and open inflation Another interesting application of quantum cosmology was the attempt to generate open inflation from Euclidean quantum gravity similarly to the case of the closed Universe. Hawking and Turok have recently suggested the mechanism of quantum creation of an open Universe from the no-boundary cosmological state [29]. Motivated by the observational ev- idence for the potential possibility of Ω < 1 and the necessity to avoid a rather contrived nature of inflaton potentials in the early suggestions for open inflation [30, 31] (see also [32]) theyconstructed aspecialgravitationalinstanton. Withintheframeworkoftheno-boundary cosmological wavefunction this instanton is capable of generating expanding open homoge- neous universes without assuming any special form for the potential. The prior quantum probability of such universes weighted by the anthropic probability of galaxy formation was shown to be peaked at Ω 0.01. ∼ Very briefly, the construction of the Hawking-Turok instanton generating the open in- flation, as compared to the closed one, is as follows. The inflating Lorentzian spacetime originates in both closed and open models by the nucleation from a 3-dimensional section of the gravitational instanton. In the closed model this is the equatorial section – the boundary of the 4-dimensional quasi-hemishpere labelled by the constant value of the latitude anglular coordinate. The analytic continuation of this coordinate into the complex plane gives rise to the Lorentzian quasi-DeSitter spacetime modelling the open inflation. In the open case the Hawking-Turok suggestion was to continue the Euclidean solution beyond the equatorial sec- tion up to the point where the Euclidean scale factor again vanishes at the point antipoidal to the regular pole on the first hemisphere. The nucleation surface then has to be chosen as the longitudinal section of this quasi-spherical manifold passing through the regular pole 10