Tunneling cosmological state revisited: Origin of inflation with a non-minimally coupled Standard Model Higgs inflaton

# Tunneling cosmological state revisited: Origin of inflation with a non-minimally coupled Standard Model Higgs inflaton

Andrei O. Barvinsky Alexander Yu. Kamenshchik Claus Kiefer Christian F. Steinwachs Theory Department, Lebedev Physics Institute, Leninsky Prospect 53, Moscow 119991, Russia Dipartimento di Fisica and INFN, via Irnerio 46, 40126 Bologna, Italy L.D. Landau Institute for Theoretical Physics, Moscow 119334, Russia Institut für Theoretische Physik, Universität zu Köln, Zülpicher Strasse 77, 50937 Köln, Germany
###### Abstract

We suggest a path integral formulation for the tunneling cosmological state, which admits a consistent renormalization and renormalization group (RG) improvement in particle physics applications of quantum cosmology. We apply this formulation to the inflationary cosmology driven by the Standard Model (SM) Higgs boson playing the role of an inflaton with a strong non-minimal coupling to gravity. In this way a complete cosmological scenario is obtained, which embraces the formation of initial conditions for the inflationary background in the form of a sharp probability peak in the distribution of the inflaton field and the ongoing generation of the Cosmic Microwave Background (CMB) spectrum on this background. Formation of this probability peak is based on the same RG mechanism which underlies the generation of the CMB spectrum which was recently shown to be compatible with the WMAP data in the Higgs mass range . This brings to life a convincing unification of quantum cosmology with the particle phenomenology of the SM, inflation theory, and CMB observations.

###### keywords:
Quantum cosmology, Inflation, Higgs boson, Standard Model, Renormalization group
###### Pacs:
98.80.Cq, 14.80.Bn, 11.10.Hi

## 1 Introduction

At the dawn of inflation theory two prescriptions for the quantum state of the Universe were seriously considered as a source of initial conditions for inflation. These are the so-called no-boundary noboundary () and tunneling tunnel (); Vilenkin84 () cosmological wavefunctions (see also OUP () for a general review), whose semiclassical amplitudes are roughly inversely proportional to one another. In the model of chaotic inflation driven in the slow-roll approximation by the inflaton field with the potential these amplitudes read as , where label, respectively, the no-boundary/tunneling wavefunctions. Here, is the Einstein action of the Euclidean de Sitter instanton with the effective cosmological constant given by the value of the inflaton field ,

 SE(φ)≃−24π2M4PV(φ), (1)

in units of the reduced Planck mass (). The no-boundary state was originally formulated as a path integral over Euclidean four-geometries; the tunneling state in the form of a path integral over Lorentzian metrics was presented in Vilenkin84 (); Vilenkin94 (), and both wavefunctions were also obtained as solutions of the minisuperspace Wheeler–DeWitt equation.

The no-boundary and tunneling states lead to opposite physical conclusions. In particular, in view of the negative value of the Euclidean de Sitter action the no-boundary state strongly enhances the contribution of empty universes with in the full quantum state and, thus, leads to the very counterintuitive conclusion that infinitely large universes are infinitely more probable than those of a finite size – a property which underlies the once very popular but now nearly forgotten big-fix mechanism of S. Coleman bigfix (). On the other hand, the tunneling state favors big values of capable of generating inflationary scenarios. Thus, it would seem that the tunneling prescription is physically more preferable than the no-boundary one. However, the status of the tunneling prescription turns out to be not so simple and even controversial.

Naive attempts to go beyond the minisuperspace approximation lead to unnormalizable states in the sector of spatially inhomogeneous degrees of freedom for matter and metric and invalidate, in particular, the usual Wick rotation from the Lorentzian to the Euclidean spacetime. This problem was partly overcome by imposing the normalizability condition on the matter part of the solution of the Wheeler–DeWitt equation Vilenkin88 (), but the situation remained controversial for the following reason.

Modulo the issue of quantum interference between the “contracting” and “expanding” branches of the cosmological wavefunction discussed, for example, in Vilenkin88 (); OUP (); CK92 (); debate (), the amplitudes of the no-boundary and tunneling branches of such a semiclassical solution take the form

 (2)

where is a set of matter fields separate from the spatially homogeneous inflaton, and is their normalizable (quasi-Gaussian) part in the full wavefunction – in essence representing the Euclidean de Sitter invariant vacuum of linearized fields on the quasi-de Sitter background with . Quantum averaging over then leads to the following quantum distribution of the inflaton field

 ρ1−loop±(φ)=∫d[Φ(x)]∣∣Ψ±(φ,Φ(x))∣∣2=exp(∓SE(φ)−S1−loopE(φ)) , (3)

where is the contribution of the UV divergent one-loop effective action norm (); PhysRep (); reduc (). With the aid of this algorithm a sharp probability peak was obtained in the tunneling distribution for the model with a strong non-minimal coupling of the inflaton to gravity norm (); we-scale (); BK (). This peak was interpreted as generating the quantum scale of inflation – the initial condition for its inflationary scenario. Quite remarkably, for accidental reasons this result was free from the usual UV renormalization ambiguity. It did not require application of the renormalization scheme of absorbing the UV divergences into the redefinition of the coupling constants in the tree-level action .

However, beyond the one-loop approximation and for other physical correlators the situation changes, and one has to implement a UV renormalization in full. But with the ambiguity in (3) this renormalization would be different for the tunneling and no-boundary states. For instance, an asymptotically free theory in the no-boundary case (associated with the usual Wick rotation to the Euclidean spacetime) will not be asymptotically free in the tunneling case. The tunneling versus no-boundary gravitational modification of the theory will contradict basic field-theoretical results in flat spacetime. This strongly invalidates a naive construction of the tunneling state of the above type. In particular, it does not allow one to go beyond the one-loop approximation in the model of non-minimally coupled inflaton and perform its renormalization group (RG) improvement.

Here we suggest a solution of this problem by formulating a new path integral prescription for the tunneling state of the Universe. This formulation is based on a recently suggested construction of the cosmological density matrix slih () which describes a microcanonical ensemble of cosmological models why (). The statistical sum of this ensemble was calculated in a spatially closed model with a generic set of scalar, spinor, and vector fields conformally coupled to gravity. It was obtained in the saddle-point approximation dominated by the contribution of the thermal cosmological instantons of topology . These instantons also include the vacuum topology treated as a limiting case of the compactified time dimension in being ripped in the transition from to . This limiting case exactly recovers the Hartle–Hawking state of noboundary (), so that the whole construction of slih (); why () can be considered as a generalization of the vacuum no-boundary state to the quasi-thermal no-boundary ensemble. The basic physical conclusion for this ensemble was that it exists in a bounded range of values of the effective cosmological constant, that it is capable of generating a big-boost scenario of the cosmological acceleration bigboost () and that its vacuum Hartle–Hawking member does not really contribute because it is suppressed by the infinite positive value of its action. This is a genuine effect of the conformal anomaly of quantum fields FHH (); Starobinsky (), which qualitatively changes the tree-level action (1).

Below we shall show that the above path integral actually has another saddle point corresponding to the negative value of the lapse function , which is gauge-inequivalent to . In the main, this leads to the inversion of the sign of the action in the exponential of the statistical sum and, therefore, deserves the label “tunneling”. In this tunneling state the thermal part vanishes and its instanton turns out to be a purely vacuum one. Finally, this construction no longer suffers from the above mentioned controversy with the renormalization. A full quantum effective action is supposed to be calculated and renormalized by the usual set of counterterms on the background of a generic metric and then the result should be analytically continued to and taken at the tunneling saddle point of the path integral over the lapse function .

Below we shall apply this construction to a cosmological model for which the Lagrangian of the graviton-inflaton sector reads

 \boldmathL(gμν,Φ)=12(M2P+ξ|Φ|2)R−12|∇Φ|2−V(|Φ|), (4) V(|Φ|)=λ4(|Φ|2−v2)2,|Φ|2=Φ†Φ, (5)

where is the Standard Model (SM) Higgs boson, whose expectation value plays the role of an inflaton and which is assumed here to possess a strong non-minimal curvature coupling with . Here, as above, is a reduced Planck mass, is a quartic self-coupling of , and is an electroweak (EW) symmetry breaking scale.

The early motivation for this model with a GUT type boson non-min (); KomatsuFutamase () was to avoid an exceedingly small quartic coupling by invoking a non-minimal coupling with a large . This was later substantiated by the hope to generate the no-boundary/tunneling initial conditions for inflation we-scale (); BK (). This theory but with the SM Higgs boson instead of the abstract GUT setup of we-scale (); BK () was suggested in BezShap (), extended in we () to the one-loop level and considered regarding its reheating mechanism in GB08 (). The RG improvement in this model has predicted CMB parameters – the amplitude of the power spectrum and its spectral index – compatible with WMAP observations in a finite range of values of the Higgs mass, which is close to the widely accepted range dictated by the EW vacuum stability and perturbation theory bounds Wil (); BezShap1 (); BezShap3 (); RGH (); PLBRGH (); Clarcketal ().

The purpose of our paper is to extend the results of RGH (); PLBRGH () by suggesting that this model does not only have WMAP-compatible CMB perturbations, but can also generate the initial conditions for the inflationary background upon which these perturbations propagate. These initial conditions are realized in the form of a sharp probability peak in the tunneling distribution function of the inflaton.

Our paper is organized as follows. In Sect. 2 we present the path-integral formulation for the tunneling state and derive the relevant distribution in the space of values of the cosmological constant. In Sect. 3 we apply this distribution to the gravitating SM model with the graviton-inflaton sector (4) and obtain the probability peak in the distribution of the initial value of the Higgs-inflaton. Sect. 4 contains a short discussion.

## 2 Tunneling cosmological wavefunction within the path integral formulation

The path integral for the microcanonical statistical sum in cosmology why () can be cast into the form of an integral over a minisuperspace lapse function and scale factor of a spatially closed Euclidean FRW metric ,

 e−Γ=∫D[a,N]e−Seff[a,N], (6) e−Seff[a,N]=∫DΦ(x)e−S[a,N;Φ(x)] . (7)

Here, is the Euclidean effective action of all inhomogeneous “matter” fields (which include also metric perturbations) on the minisuperspace background of the FRW metric, is the classical Euclidean action, and the integration runs over periodic fields on the Euclidean spacetime with a compactified time (of topology).

It is important that the integration over the lapse function runs along the imaginary axis from to because this Euclidean path integral represents, in fact, the transformed version of the integral over metrics with Lorentzian signature. This transformation is the usual Wick rotation which can be incorporated by the transition from the Lorentzian lapse function to the Euclidean one by the relation why (). The Lorentzian path integral, in turn, fundamentally follows from the definition of the microcanonical ensemble in quantum cosmology which includes all true physical configurations satisfying the quantum first-class constraints – the Wheeler–DeWitt equations. The projector onto these configurations is realized in the integrand of the path integral by the delta functions of the Hamiltonian (and momentum) constraints. The Fourier representation of these delta functions in terms of the integral over the conjugated Lagrange multipliers – the lapse (and shift) functions – implies an integration with limits at infinity, , which explains the range of integration over the Euclidean .

It should be mentioned that a full non-perturbative evaluation of the path integral would require a careful inspection of the infinite contours in the complex -plane that render the integral convergent, see, for example, pathintegrals (). However, such an inspection is not needed here because we are dealing here with a semiclassical approximation in which only the vicinity of the saddle point enters.

The convenience of writing the path integral (6) in the Euclidean form follows from the needs of the semiclassical approximation. In this approximation, it is dominated by the contribution of a saddle point, , where and solve the equation of motion for and satisfy periodicity conditions dictated by the definition of the statistical sum. Such periodic solutions exist in the Euclidean domain with real rather than in the Lorentzian one with the imaginary lapse. This means that the contour of integration over along the imaginary axis should be deformed into the complex plane to traverse the real axis at some corresponding to the Euclidean solution of the equations of motion for the minisuperspace action.

The residual one-dimensional diffeomorphism invariance of this action (which is gauged out by the gauge-fixing procedure implicit in the integration measure ) allows one to fix the ambiguity in the choice of . There remains only a double-fold freedom in this choice actually inherited from the sign indefiniteness of the integration range for . This freedom is exhausted by either positive, , or negative, , values of the lapse, because, on the one hand, all values in each of these equivalence classes are gauge equivalent and, on the other hand, no continuous family of nondegenerate diffeomorphisms exists relating these classes to one another. Without loss of generality one can choose as representatives of these classes and label the relevant solutions and on-shell actions, respectively, as and

 Γ±=Seff[a±(τ),±1] . (8)

Gauge inequivalence of these two cases, , is obvious because, for example, all local contributions to the effective action are odd functionals of , . Thus we can heuristically identify the statistical sums correspondingly with the “no-boundary” and “tunneling” prescriptions for the quantum state of the Universe,

 exp(−Γno−boundary/tunnel)=e−Γ±. (9)

In other words, we use this equation to define “no-boundary” and “tunneling” in the first place. This result shows that for both prescriptions a full quantum effective action as a whole sits in the exponential of the partition function without any splitting into the minisuperspace and matter contributions weighted by different sign factors like in (3). This means that the usual renormalization scheme is applicable to the calculation of (8) – generally covariant UV counterterms should be calculated on the background of a generic metric and afterwards evaluated at the FRW metric with , depending on the choice of either the no-boundary or tunneling prescription. Below we demonstrate how this procedure works for the system dominated by quantum fields with heavy masses, whose effective action admits a local expansion in powers of the spacetime curvature and matter fields gradients.

For such a system the Euclidean effective action takes the form

 Seff[gμν]=∫d4xg1/2(M2PΛ−M2P2R(gμν)+...), (10)

where we disregard the terms of higher orders in the curvature and derivatives of the mean values of matter fields. Here the cosmological term and the (reduced) Planck mass squared can be considered as functions of these mean values and treated as constants in the approximation of slowly varying fields. This effective action does not contain the thermal part characteristic of the statistical ensemble slih () because for heavy quanta the radiation bath is not excited. This is justified by the fact that the effective temperature of this bath turns out to be vanishing.

In fact, the minisuperspace action functional for (10) reads in units of as

 Seff[a,N]=m2P∫dτN(−aa′2−a+H2a3), (11)

where , and we use the notation for the cosmological constant in terms of the effective Hubble factor . Then the saddle point for the path integral (6) – the stationary configuration with respect to variations of the lapse function, , – satisfies the Euclidean Friedmann equation

 a′2=1−H2a2. (12)

It has one turning point at below which the real solution interpolates between and . In the gauge for both no-boundary/tunneling cases this solution describes the Euclidean de Sitter metric, that is, one hemisphere of ,

 a±(τ)=1Hsin(Hτ). (13)

After the bounce from the equatorial section of the maximal scale factor , this solution spans at the contraction phase the rest of the full four-sphere111The formal analytic extension from to should not, of course, be applied to to give a negative instead of (13), because in contrast to the sign-indefinite Lagrange multiplier the path integration over in (6) semiclassically always runs in the vicinity of its positive geometrically meaningful value. For this reason, never brings sign factors into the on-shell action even though it enters the action with odd powers.. Thus, this solution is not periodic and in the terminology of slih () describes a purely vacuum contribution to the statistical sum (6). As shown in slih (), the effective temperature of this state is determined by the inverse of the full period of the instanton solution measured in units of the conformal time . Therefore, for (13) it vanishes because this period between the poles of this spherical instanton is divergent,

 η=2∫π/2H0dτa(τ)→∞ . (14)

This justifies the absence of the thermal part in (10).

Thus, with the no-boundary and tunneling on-shell actions (8) read

 Γ±=∓8π2M2PH2 (15)

and the object of major interest here – the tunneling partition function in the space of positive values of – is given by

 ρtunnel(Λ)=exp(−24π2M2PΛ),Λ>0. (16)

It coincides with the semiclassical tunneling wavefunction of the Universe tunnel (), , derived from the Wheeler–DeWitt equation in the tree-level approximation.

At the turning point , the solution (13) can be analytically continued to the Lorentzian regime, . The scale factor then expands eternally as

 aL(t)=1Hcosh(Ht) , (17)

which can be interpreted as representing the distributions of scale factors in the quantum ensemble (after decoherence) of de Sitter models distributed according to (16). Note that the attempt to extend this ensemble to negative fails, because the equation (12) with does not have turning points with nucleating real Lorentzian geometries. Moreover, virtual cosmological models with Euclidean signature are also forbidden in the tunneling state because their positive Euclidean action diverges to infinity, so that for .

## 3 Quantum origin of the Universe with the SM Higgs-inflaton non-minimally coupled to curvature

The partition function of the above type can serve as a source of initial conditions for inflation only when the cosmological constant becomes a composite field capable of a decay at the exit from inflation. Usually this is a scalar inflaton field whose quantum mean value is nearly constant in the slow roll regime, and its effective potential plays the role of the cosmological constant driving the inflation. When the contribution of the inflaton gradients is small, the above formalism remains applicable also with the inclusion of this field whose ultimate effect reduces to the generation of the effective cosmological constant and the effective Planck mass.

These constants are the coefficients of the zeroth and first order terms of the effective action expanded in powers of the curvature, and they incorporate radiative corrections due to all quantum fields in the path integral (7). Now there is no mismatch between the signs of the tree-level and loop parts of the partition function. Therefore, one can apply the usual renormalization and, if necessary, the renormalization group (RG) improvement to obtain the full effective action and then repeat the procedure of the previous section. In the slow roll approximation the effective action has the general form

 Seff[gμν,φ]=∫d4xg1/2(V(φ)−U(φ)R(gμν)+12G(φ)(∇φ)2+...), (18)

where , and are the coefficients of the derivative expansion, and we disregard the contribution of higher-derivative operators. With the slowly varying inflaton the coefficients and play the role of the effective cosmological and Planck mass constants, so that one can identify in (10) and (11) the effective and , respectively, with and . Therefore, the tunneling partition function (16) becomes the following distribution of the field

 ρtunnel(φ)=exp(−24π2M4P^V(φ)), (19) ^V(φ)=(M2P2)2V(φ)U2(φ), (20)

where in fact coincides with the potential in the Einstein frame of the action (18) RGH (); PLBRGH ().

Now we apply this formalism to the model (4) of inflation driven by the SM Higgs inflaton . As shown in RGH (); PLBRGH (), the one-loop RG improved action in this model has for large the form (18) with the coefficient functions

 V(φ)=λ(t)4Z4(t)φ4, (21) U(φ)=12(M2P+ξ(t)Z2(t)φ2), (22) G(φ)=Z2(t), (23)

determined in terms of the running couplings and , and the field renormalization . They incorporate a summation of powers of logarithms and belong to the solution of the RG equations which at the inflationary stage with a large and large read as (see RGH (); PLBRGH () for details)

 dλdt=\boldmathA16π2λ−4γλ, (24) dξdt=6λ16π2ξ−2γξ (25)

and . Here, is the anomalous dimension of the Higgs field, the running scale is normalized at the top quark mass , and is the running parameter of the anomalous scaling. This quantity was introduced in norm () as the pre-logarithm coefficient of the overall effective potential of all SM physical particles and Goldstone modes. Due to their quartic, gauge and Yukawa couplings with , they acquire masses and for large give rise to the asymptotic behavior of the Coleman-Weinberg potential,

 V1−loop(φ)=∑particles(±1)m4(φ)64π2lnm2(φ)μ2≃λ\boldmathA128π2φ4lnφ2μ2, (26)

which can serve as a definition of .

The importance of this quantity and its modification due to the RG running of the non-minimal coupling ,

 \boldmathAI=\boldmathA−12λ (27)

( gives the running of the ratio , , is that for mainly these parameters determine the quantum inflationary dynamics BK (); efeqmy () and yield the parameters of the CMB generated during inflation we (). In particular, the value of at the beginning of the inflationary stage of duration in units of the e-folding number turns out to be we ()

 φ2=−64π2M2Pξ\boldmathAI(tend)(1−ex), (28) x≡N\boldmathAI(tend)48π2, (29)

where a parameter has been introduced which directly involves taken at the end of inflation, , . This parameter also enters simple algorithms for the CMB power spectrum and its spectral index . As shown in RGH (); PLBRGH (), the application of these algorithms under the observational constraints and (the combined WMAP+BAO+SN data at the pivot point Mpc corresponding to WMAP ()) gives the CMB-compatible range of the Higgs mass , both bounds being determined by the lower bound on the CMB spectral index.

Now we want to show that, in addition to the good agreement of the spectrum of cosmological perturbations with the CMB data, this model can also describe the mechanism of generating the cosmological background itself upon which these perturbations exist. This mechanism consists in the formation of the initial conditions for inflation in the form of a sharp probability peak in the distribution function (19) at some appropriate value of the inflaton field with which the Universe as a whole starts its evolution. The shape and the magnitude of the potential (20) depicted in Fig.1 for several values of the Higgs mass clearly indicates the existence of such a peak.

Indeed, the negative of the inverse potential damps to zero after exponentiation the probability of those values of at which and, vice versa, enhances the probability at the positive maxima of the potential. The pattern of this behavior with growing Higgs mass is as follows.

As is known, for low the SM has a domain of unstable EW vacuum, characterized by negative values of running at certain energy scales. Thus we begin with the EW vacuum instability threshold espinosa (); Sher () which exists in this gravitating SM at GeV RGH (); PLBRGH () and which is slightly lower than the CMB compatible range of the Higgs mass ( is chosen in Fig. 2 and for the lowest curve in Fig. 1). The potential drops to zero at , , and forms a false vacuum RGH (); PLBRGH () separated from the EW vacuum by a large peak at . Correspondingly, the probability of creation of the Universe with the initial value of the inflaton field at the EW scale and at the instability scale is damped to zero, while the most probable value belongs to this peak. The inflationary stage of the formation of the pivotal CMB perturbation (from the moment of the first horizon crossing until the end of inflation ), which is marked by dashed lines in Fig.2, lies to the left of this peak. This conforms to the requirement of the chronological succession of the initial conditions for inflation and the formation of the CMB spectra.

The above case is, however, below the CMB-compatible range of and was presented here only for illustrative purposes222Another interesting range of is below the instability threshold where becomes negative in the “true” high energy vacuum. As mentioned in the previous section, the tunneling state rules out aperiodic solutions of effective equations with , which cannot contribute to the quantum ensemble of expanding Lorentzian signature models. Therefore, this range is semiclassically ruled out not only by the instability arguments, but also contradicts the tunneling prescription.. An important situation occurs at higher Higgs masses from the lower CMB bound on GeV until about 160 GeV. Here we get a family of a metastable vacua with . An example is the plot for the lower CMB bound GeV depicted in Fig. 3. Despite the shallowness of this vacuum its small maximum generates via (19) a sharp probability peak for the initial inflaton field. This follows from an extremely small value of , the reciprocal of which generates a rapidly changing exponential of (19). The location of the peak again precedes the inflationary stage for a pivotal CMB perturbation (also marked by dashed lines in Fig. 3).

For even larger these metastable vacua get replaced by a negative slope of the potential which interminably decreases to zero at large (at least within the perturbation theory range of the model), see Fig. 1. Therefore, for large close to the upper CMB bound 185 GeV, the probability peak of (19) gets separated from the non-perturbative domain of large over-Planckian scales due to a fast drop of to zero. This, in turn, follows from the fact that grows much faster than when they both start approaching their Landau pole RGH ().

The location of the probability peak and its quantum width can be found in analytical form, and their derivation shows the crucial role of the running for the formation of initial conditions for inflation. Indeed, the exponential of the tunneling distribution (19) for reads as

 Γ−(φ)=24π2M4P^V(φ)≃96π2ξ2λ(1+2M2PξZ2φ2), (30)

and in view of the RG equations (24)–(25) has an extremum satisfying the equation

 φdΓdφ=dΓdt=−6ξ2λ(\boldmathAI+64π2M2PξZ2φ2)=0, (31)

where we again neglect higher order terms in and (extending beyond the one-loop order). Here, is the anomalous scaling introduced in (26) and (27) – the quantity that should be negative for the existence of the solution for the probability peak,

 φ20=−64π2M2Pξ% \boldmathAIZ2∣∣ ∣∣t=t0. (32)

As shown in RGH (); PLBRGH (), this quantity is indeed negative. In the CMB-compatible range of its running starts from the range at the EW scale and reaches small but still negative values in the range at the inflation scale. Also, the running of and is very slow – the quantities belonging to the two-loop order – and the duration of inflation is very short RGH (); PLBRGH (). Therefore, , and these estimates apply also to . As a result, the second derivative of the tunneling on-shell action is positive and very large,

 d2Γ−dt2≃−12ξ2λ\boldmathAI≫1, (33)

which gives an extremely small value of the quantum width of the probability peak,

 Δφ2φ20=−λ12ξ21\boldmathAI∣∣∣t=t0∼10−10. (34)

This width is about times – one order of magnitude – higher than the CMB perturbation at the pivotal wavelength Mpc (which we choose to correspond to ). The point of the horizon crossing of this perturbation (and other CMB waves with different ’s) easily follows from equation (28) which in view of takes the form

 φ2inφ20=1−exp(−N|\boldmathAI(tend)|48π2). (35)

It indicates that for wavelengths longer than the pivotal one the instant of horizon crossing approaches the moment of “creation” of the Universe, but always stays chronologically succeeding it, as it must.

## 4 Conclusions and discussion

In this paper we have constructed the tunneling quantum state of the Universe based on the path integral for the microcanonical ensemble in cosmology. The corresponding apparent ensemble from the quantum state exists in the unbounded positive range of the effective cosmological constant, unlike the no-boundary state discussed in slih (); why () whose apparent ensemble is bounded by the reciprocated coefficient of the topological term in the overall conformal anomaly. Also, in contrast to the no-boundary case, the tunneling state turns out to be a radiation-free vacuum one.

The status of the tunneling versus no-boundary states is rather involved. In fact, the formal Euclidean path integral (6) is a transformed version of the microcanonical path integral over Lorentzian metrics, so that its lapse function integration runs along the imaginary axis from to why ()333This might seem being equivalent to the tunneling path integral of Vilenkin84 (); Vilenkin94 (), but the class of metrics integrated over is very different. We do not impose by hands as the boundary condition, but derive it from the saddle-point approximation for the integral over formally periodic configurations. The fact that periodicity gets violated by the boundary condition implies that the a priori postulated tunneling statistical ensemble is exhausted at the dynamical level by the contribution of a pure vacuum state slih (); why ().. The absence of periodic solutions for stationary points of (6) with the Lorentzian signature makes one to distort the contour of integration over into a complex plane, so that it traverses the real axis at the points or which give rise to no-boundary or tunneling states. One can show that the no-boundary thermal part of the statistical sum of slih () is not analytic in the full complex plane of . The domains are separated by the infinite sequence of its poles densely filling the imaginary axes of . Therefore, the contour of integration passing through both points is impossible, and the no-boundary and tunneling states cannot be obtained by analytic continuation from one another444In the case of the vacuum no-boundary state when the vanishing thermal part of the effective action cannot present an obstacle to analytic continuation in the complex plane of the situation stays the same. Indeed, any integration contour from to crosses the real axes an odd number of times, so that the contribution of only one such crossing survives, because any two (gauge-equivalent) saddle points traversed in opposite directions give contributions canceling one another.. They represent alternative solutions (quantum states) of the Wheeler-DeWitt equation.

The path-integral formulation of the tunneling state admits a consistent renormalization scheme and a RG resummation which is very efficient in cosmology according to a series of recent papers Wil (); BezShap1 (); BezShap3 (); RGH (); PLBRGH (); Clarcketal (). For this reason we have applied the obtained tunneling distribution to a recently considered model of inflation driven by the SM Higgs boson non-minimally coupled to curvature. In this way a complete cosmological scenario was obtained, embracing the formation of initial conditions for the inflationary background (in the form of a sharp probability peak in the inflaton field distribution) and the ongoing generation of the CMB perturbations on this background. As was shown in RGH (); PLBRGH (), the comparison of the CMB amplitude and the spectral index with the WMAP observations impose bounds on the allowed range of the Higgs mass. These bounds turn out to be remarkably consistent with the widely accepted EW vacuum stability and perturbation theory restrictions. Interestingly, the behavior of the running anomalous scaling , being crucially important for these bounds, also guarantees the existence of the obtained probability peak. The quantum width of this peak is one order of magnitude higher than the amplitude of the CMB spectrum at the pivotal wavelength, which could entail interesting observational consequences. Unfortunately, this quantum width is hardly measurable directly because it corresponds to an infinite wavelength perturbation (a formal limit of in (35)), but indirect effects of this quantum trembling of the cosmological background deserve further study.

We have entertained here the idea that we can obtain sensible predictions from peaks in the cosmological wavefunction. This is, of course, different from approaches based on the anthropic principle. We find it intriguing, however, that a consistent scenario based on our more traditional approach may be possible and even falsifiable.

To summarize, the obtained results bring to life a convincing unification of quantum cosmology with the particle phenomenology of the SM, inflation theory, and CMB observations. They support the hypothesis that an appropriately extended Standard Model nuMSM (); dark () can be a consistent quantum field theory all the way up to quantum gravity and perhaps explain the fundamentals of all major phenomena in early and late cosmology.

## Acknowledgements

The authors express their gratitude to A.A.Starobinsky for fruitful and thought provoking discussions. A.B. and A.K. acknowledge support by the grant 436 RUS 17/3/07 of the German Science Foundation (DFG) for their visit to the University of Cologne. The work of A.B. was also supported by the RFBR grant 08-02-00725 and the grant LSS-1615.2008.2. A.K. was partially supported by the RFBR grant 08-02-00923, the grant LSS-4899.2008.2 and by the Research Programme “Elementary Particles” of the Russian Academy of Sciences. The work of C.F.S. was supported by the Villigst Foundation. A.B. acknowledges the hospitality of LMPT at the University of Tours.

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