Cosmic eggs to relax the cosmological constant
CCosmic eggs to relax the cosmological constant
Thomas Hertog ∗ and Rob Tielemans † Institute for Theoretical Physics, KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium.
Thomas Van Riet ‡ Institute for Theoretical Physics, KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium & Institutionen för Fysik och Astronomi, Box 803, SE-751 08 Uppsala, Sweden
In theories with extra dimensions, the cosmological hierarchy problem can be thought of as theunnaturally large radius of the observable universe in Kaluza-Klein units. We sketch a dynamicalmechanism that relaxes this. In the early universe scenario we propose, three large spatial dimensionsarise through tunneling from a ‘cosmic egg’, an effectively one-dimensional configuration with allspatial dimensions compact and of comparable, small size. If the string landscape is dominated bylow-dimensional compactifications, cosmic eggs would be natural initial conditions for cosmology.A quantum cosmological treatment of a toy model egg predicts that, in a variant of the Hartle-Hawking state, cosmic eggs break to form higher dimensional universes with a small, but positivecosmological constant or quintessence energy. Hence cosmic egg cosmology yields a scenario in whichthe seemingly unnaturally small observed value of the vacuum energy can arise from natural initialconditions. ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ h e p - t h ] J a n I. INTRODUCTION
The cosmological constant (cc) problem is one of the most vexing and longstanding conceptual problems in theo-retical physics –for a review see e.g. [1–4]. It is therefore reasonable to anticipate that a better understanding ofthe origin of the observed small value of the vacuum energy will yield new insights in the fundamental physical lawsand their status in a cosmological context. The cc problem arises when one couples quantum field theory (QFT)to gravity. In modern parlance, the problem follows directly from the fact that the vacuum energy is a relevantoperator of dimension four. This means it is highly sensitive to the exact details of the UV physics which leave thelow-energy effective field theory (EFT) otherwise unaffected. In particular any particle species with mass m in theUV contributes, at one-loop, a term of the form Λ M p ⊃ m log (cid:18) m µ (cid:19) , (I.1)where µ is an RG scale, the precise meaning of which in this context being a matter of debate –see e.g. [5–11]. Thedominant contribution to Λ comes from the heaviest particles with masses near the cut-off of the theory. Henceany fine-tuning of the bare cc to cancel contributions from particles in loops depends sensitively on the UV.In the context of the Standard Model EFT coupled to classical gravity, the natural value of the cosmologicalconstant would therefore be Λ ≈ M , where M NP is the cut-off scale at which new physics (‘NP’) beyond theStandard Model enters. Conservatively one can take M NP = M p but in general M NP can of course be lower.A straightforward formulation of the cc problem is that the observed dark energy density M p Λ ≈ (10 − eV) isextremely –unnaturally– small in units of M NP : Λ M (cid:28) . (I.2)In terms of the length scales L Λ = | Λ | − / and L NP = M − the cc problem amounts to the observation that L Λ L NP (cid:29) . (I.3)Put differently, the observed Hubble scale L Λ is very much larger than its natural size ∼ L NP ? The supremedifficulty of the cc problem stems precisely from the fact that, at least within our current theoretical framework, itintertwines the largest and the smallest scales in physics.This strongly indicates that the cc problem can only be properly analysed –and hopefully ultimately resolved– inthe context of a UV-complete description of gravity such as string theory. It is a striking –yet underappreciated–property of string theory indeed that the theory enables a precise calculation of the vacuum energy in certaincontrolled corners, at weak coupling and low energies. Celebrated examples include holographic backgrounds such We note, however, that the calculation of the quantum contributions to the cc in QFT coupled to classical gravity isn’t entirelyunambiguous due to radiative instability. Naturalness arguments are therefore inherently qualitative. The observed dark energy can be dynamical as e.g. in quintessence [12, 13], but this hardly changes the crux of the problem. as AdS × S in IIB string theory, or AdS × S in 11-dimensional supergravity. At weak coupling and smallcurvature these are trusted backgrounds of string/M-theory, with a negative vacuum energy that does not receivesignificant quantum corrections, despite the presence of an infinite tower of light and heavy modes .Hence string theory brings to the table genuinely novel ingredients which bear directly on the cc problem. Here weconcentrate on two such elements, namely the possibility to decompactify dimensions and the shear abundance ofstring theory vacua. We then use quantum cosmology to combine these into a new toy model cosmological scenariowhich, we argue, offers an interesting new take on the cc problem.First, consider the number of dimensions. In string phenomenology one usually starts with ten dimensions of whichsix are compactified to arrive at low-energy solutions (vacua) of the form,d s = d s + d s , (I.4)with three non-compact spatial dimensions. From a cosmological viewpoint, an ansatz of this form amounts to ahighly contrived starting point. After all how did the universe get there? Specifically, a genuinely four-dimensionalcompactification requires the following scale separation, L Λ L KK (cid:29) , (I.5)where L KK is the Kaluza-Klein (KK) scale associated to d s and L Λ is the, possibly time-dependent, Hubble scaleon d s . However this scale separation already implies a cc problem because L KK is a ‘new physics’ scale and hence(I.3) is identical to (I.5) [14]. Four-dimensional compactifications obeying (I.5), therefore, have a cc problem builtin. Relatedly, equation (I.5) means that standard four-dimensional QFT arguments apply only when the vacuumalready has an unnaturally small cc. Hence to address the cc problem properly one must go beyond four-dimensionalQFT and probe the origin of such scale separated configurations.This hierarchy problem is equally clear from a ten-dimensional viewpoint. The ten-dimensional geometry (I.4)carries two vastly different scales –the two most extreme scales in nature indeed. The corresponding fine-tuning isreflected in the difficulties to construct (meta)stable vacua exhibiting scale separation or, for that matter, gentlyrolling quintessence backgrounds [27]. However the latter can at least evade the swampland constraints [28].These difficulties motivate exploring different ‘lines of attack’ on the cc problem. In this spirit, we consider a scenarioin which all spatial dimensions are initially compactified and of comparable, small size. Such configurations canbe thought of as effectively one-dimensional vacua or ‘cosmic eggs’. The absence of any scale separation meansthat cosmic eggs do not suffer from a cc problem or hierarchy problem. These eggs can be metastable and decay At first sight one might think that these examples rely on supersymmetry to cancel quantum corrections. However, the same reasoningapplies to non-supersymmetric vacua. In any controlled vacuum with weak curvature and small string coupling, all corrections mustbe sub-leading to the classical (low-energy) contributions to Λ , contrary to what one would expect on the basis of standard QFT-basedreasoning. In the limit of ten-dimensional supergravity, a no-go theorem for obeying (I.5) [14] extends the no-go theorem in [15]. In ten-dimensional supergravity with orientifolds, some constructions do yield precise and fully moduli-stabilised vacua with scale separation(see e.g. [16, 17]). Most compactifications with orientifolds, however, are of the no-scale type [18, 19], meaning they feature Minkowksivacua with some modes not stabilised. Stabilising these using quantum corrections, as in KKLT [20] or LVS [21], can give solutionswith scale separation (I.5), but never to arbitrary precision. Indeed the consistency of KKLT vacua is still being debated [22–26]. via tunneling to a configuration with large expanding dimensions . Hence they form a natural starting pointfor cosmological considerations. We put forward a toy model cosmic egg cosmology in which an effectively four-dimensional universe emerges from the breaking and subsequent decompactification of a cosmic egg. Our modelinvolves besides a cosmological constant also an axion which sets the overall size of the egg.It should be noted that the shear abundance of string vacua also means that there is not a unique cosmic egg. Onthe contrary, even the most ardent advocates of the swampland program would agree, we believe, that string theoryencompasses a ‘landscape’ of possible low-energy laws, albeit one with a set of interesting theoretical patterns thatconstrain its phenomenology. On general grounds one expects that vacua with a lower number of large dimensionsare statistically hugely favored since there are more compact manifolds, more cycles to wrap branes and fluxes, etc.The ‘landscape’ of higher dimensional vacua may well be a set of measure zero in the landscape of all vacua.This proliferation of cosmic eggs means that a cosmic egg cosmology is neither complete nor predictive withoutspecifying a state on its configuration space that defines a notion of typicality. We therefore embed cosmic eggcosmology in quantum cosmology, where the wave function of the universe provides a relative weighting of differentegg-born cosmological histories from which a measure for observations can be obtained (see e.g. [35–37]). Specificallywe consider a toy model landscape consisting of a collection of cosmic eggs with different values of axion flux and Λ in a variant of the Hartle-Hawking state. In this context we show that the model predicts the decay of a cosmic egginto an expanding, higher dimensional universe with a small positive cosmological constant. The latter is thereforeseen to arise from natural initial conditions. II. COSMIC EGGS
To implement our scenario we are led to consider compactifications of string theory down to one dimension –time–that serve as possible initial conditions for cosmic egg cosmology. The landscape of one-dimensional string theoryvacua is relatively unexplored. Some preliminary work has been done in [33, 34, 38, 39]. We will not attemptto find new compactifications to one dimension (1D) but simply work with a toy model landscape that admitsconfigurations that enable us to implement a cosmic egg cosmological scenario.Obviously we are interested in models that have 1D backgrounds that are metastable. This implies they probablyshould break supersymmetry. However, there are no 1D vacua in a strict sense once supersymmetry is broken[33, 40]. This means one should not expect cosmic eggs to be perfectly static configurations. Second, we areinterested in 1D backgrounds that can potentially tunnel to 4D universes . A convenient way to proceed is to Earlier cosmologies involving dynamical decompactification include string gas cosmology [29] (see also [30–32]) and other scenarioslike [33, 34]. For simplicity we work with an effective four-dimensional cosmological constant but our results can be readily generalized toquintessence models. We comment below on cosmic egg cosmologies with a different number of large dimensions. dimensionally reduce 4D EFTs. Consider the following action S = (cid:90) d x √− g (cid:18) R − Λ − H (cid:19) , (II.1)where H is an axion three-form field strength and Λ is a four-dimensional cc, not necessarily small. We also notethat adding more axions or two-form fluxes or even a rolling scalar potential does not significantly change thequalitatively analysis we wish to pursue here.A dimensional reduction of this system over the three spatial dimensions amounts effectively to adopting a FLRW-like ansatz, d s = − N ( t ) x ( t ) d t + x ( t ) d s , (II.2)where N is a lapse function, x acting as the volume modulus of a compact three-dimensional space with lineelement ds , which we take to be Einstein. The spatial curvature can be normalised as usual, in FLRW language k = − , , +1 with all three values consistent with compactness. The effective action for the volume-modulus is ofthe form S = − Vol (cid:90) d t N (cid:18) N ˙ x − U ( x ) (cid:19) , (II.3)with U ( x ) = − q x + 2 k −
23 Λ x. (II.4)where q is the axion-flux. Gravity in one dimension is non-dynamical and governed by a Hamiltonian constraint –the first Friedmann equation–that is enforced by the equation of motion for the lapse function N . Hence a 1D, time independent, (meta)stablevacuum only exists when U has a minimum where it exactly vanishes. It is hard, if not impossible, to achieve thiswith broken supersymmetry since quantum corrections are unlikely to sum exactly to zero. We do not want to tradeone fine-tuning problem for another. However, for our purposes, it suffices for the potential U to have a ‘pocket’region which can trap the scale factor. This would (in a weak sense) stabilize the volume modulus and create thepossibility of an effective one-dimensional cosmological phase.If the spatial curvature is positive the potential has a maximum at zero energy when q = 8 k Λ . (II.5)This corresponds to the unstable Einstein static Universe (cf. Fig 1) and is not a candidate cosmic egg . The eggs,if they exist, are to be found at yet smaller values of the volume-modulus, where the potential becomes negative. From here onward, we work in Planck units M p = 1 . To restore units, the length dimensions of the parameters are [ N ] = L, [ x ] = L , [ Q ] = L α , [ q ] = L and [Λ] = L − . Moreover since q is quantised, the required tuning for this solution to exist may not be possible. Figure 1: The effective potential U for the volume modulus x for a given axion-flux q , positive spatial curvature k = +1 and different values of the 4D cosmological constant Λ . The solid (dashed) lines represent the potential with (without)putative repulsive quantum corrections at small x added. Now, one expects quantum corrections to modify the effective potential (II.4) at small x , adding terms involvinghigher inverse powers of x . For example, higher derivative terms come with extra inverse metrics leading to termsof the form ∼ x − or higher. An F higher derivative correction in the Maxwell field, for instance, would introducea term proportional to p x − / , with p the magnetic flux. To compute whether U will ultimately be repulsive orattractive at very small volume requires full control over the quantum gravity theory which is not within reach.Interestingly, however, models of loop quantum gravity suggest a repulsive behavior [41]. Figure 1 shows a fewexamples of an effective potential with a repulsive term at small x added, for different values of the 4d cc Λ , andwith k = +1 .The essential ingredient of our toy model that makes a cosmic egg scenario possible is the existence of a potentialwell at small volume in which the scale factor can be trapped and oscillate until it quantum mechanically tunnelsthrough the barrier. As anticipated any cosmic egg is not expected to be static but rather exhibit a ‘breathing’volume modulus [40].Whether in our model an appropriate potential well exists, depends on three properties. First, the spatial curvatureshould be positive ( k = +1 ). Second, Λ should be bounded from above by the value that roughly corresponds tothe Einstein static universe (II.5), Λ (cid:46) Λ cr ≡ √ q . (II.6)Third, the cc must be positive in order for the cosmic egg to be metastable. Thus within the range < Λ < Λ cr . (II.7)breathing cosmic eggs can emerge as novel, potentially natural early universe configurations, with all spatial dimen-sions of similar size.We next turn to the quantum cosmological dynamics of this model to identify the breathing, classical cosmic eggconfigurations and their evolution . III. BREAKING COSMIC EGGS
Within the above range of values of Λ , cosmic eggs are metastable configurations. The eggs eventually ‘break’ bytunneling through the potential barrier, causing the three spatial dimensions to decompactify and grow exponen-tially. This transition corresponds to the birth of a four-dimensional expanding universe.Quantum cosmology provides a unified treatment of cosmic egg cosmology, from the quantum formation of theclassical oscillatory egg phase, to its breaking and the subsequent expansion of the universe. Furthermore in alandscape context, the wave function of the universe yields a relative weighting of different egg-born cosmologicalhistories. Here we concentrate on a mini-landscape that is a collection of 1D vacua (eggs) with different values ofthe 4D cc Λ and axion-flux.The compactified egg phase is ideally suited to be treated in a minisuperspace approximation, with the overallvolume of the three compact spatial dimensions as the only remaining light degree of freedom. Of course newdegrees of freedom will become relevant when the expansion gets underway. However our toy model does notinclude these, and hence does not describe a realistic universe. We are primarily interested here in modeling theearliest stages of evolution, the egg and tunneling phases, and to determine the relative weighting of expandinguniverses emerging from it. For these purposes it is reasonable to assume the minisuperspace approximation shouldapply. Upon quantization, (II.3) leads to the minisuperspace Wheeler-DeWitt equation, (cid:20) − d d x + U ( x ) (cid:21) Ψ( x ) = 0 , (III.1)where U is the potential (II.4) plus some quantum corrections modifying its behaviour at small x as argued before.It remains to specify the minisuperspace wave function. In compactifications where U ( x ) is repulsive as x → ,the wave function must definitely decay and eventually tend to zero in the small volume limit. This is because fora repulsive potential, the problem resembles that of a particle with an infinite wall in quantum mechanics. Thischoice of boundary condition is akin to the original Hartle-Hawking proposal [43]. It embodies its motivation thatthe big bang is a genuine beginning in a physical sense.In compactifications where U ( x ) is attractive as x → , it would seem that one has a choice of boundary conditionson the wave function at x = 0 . One can either require the wave function to vanish at the origin, or one can impose A quantum cosmology analysis of a model that shares similarities with ours was presented in [42]. The context and motivation are,however, somewhat different. x Ψ x Ψ Figure 2: Numerical solution to the Wheeler-DeWitt equation (III.1) with the Hartle-Hawking boundary condition
Ψ(0) = 0 for q = 29 . and Λ = 0 . . The wave function is everywhere real and shown in blue, the potential is representedby the gray line. The left (right) panel shows the solution with an attractive (repulsive) potential in the small volume limit.Both models clearly exhibit a classical oscillatory egg state connected through quantum tunneling to an expanding fourdimensional de Sitter-like universe. boundary conditions for which the wave function continues to rise towards the origin. The former case is again akinto the original Hartle-Hawking proposal. We will see below that the no-boundary condition on the wave functionacts as an effective quantum barrier at small x , giving rise to an effective pocket in which the wave function oscillatesand describes a classical breathing egg just as in the repulsive case above. In the latter case the wave function woulddiverge as x → and it is doubtful that it describes a physically meaningful, let alone approximately classical, eggphase.In both sets of models, we therefore require the wave function to decay to zero as x → . Figure 2 shows an exampleof a numerical solution Ψ( x ) with ‘no-boundary’ condition Ψ(0) = 0 , first in a model with an attractive small x potential (such as (II.4)) and then with a repulsive ‘quantum-corrected’ potential of the form, U ( x ) = Q x α − q x + 2 k −
23 Λ x, (III.2)with α > and k = +1 as argued before (from here onwards, we will set k = +1 ). In both cases one can clearlyidentify the classical oscillatory egg state, the exponential behavior of the wave function under the barrier, andfinally the classical expansion at large volume . The existence of a classical egg described by an oscillatory wavefunction requires q > M p (cf. Appendix A). Also, if U is repulsive at small x , then Q must be sufficiently small: Q < α (cid:20) (cid:18) − α (cid:19)(cid:21) α − q α , (III.3)in order to have a region where the potential is negative.The steepness of the potential in the small volume limit causes the WKB approximation to break down there. Thisis rather different from the standard applications of Hartle-Hawking initial conditions in the context of inflation,involving gentle and finite, positive potentials in the small volume regime, in which the minisuperspace wave See [44, 45] for an in depth discussion of the emergence of classical spacetime in quantum cosmology. As q only has a physical meaning, we write q instead of | q | to simplify notation. function can be expressed in terms of (the action of) regular semiclassical saddle points. In Appendix A we presentan approximate solution of the small x wave function with Hartle-Hawking boundary conditions using a differentapproximation scheme.On the other hand the WKB approximation becomes accurate for larger x . The general behavior of the WKBsolution in the large volume x (cid:29) x regime reads, Ψ( x ) = 1 | U ( x ) | / (cid:16) Ae iS ( x,x ) + Be − iS ( x,x ) (cid:17) , (III.4)where S ( x, x i ) is defined as S ( x, x i ) = (cid:90) xx i d x (cid:48) (cid:112) | U ( x (cid:48) ) | . (III.5)and x is the large x endpoint of the potential barrier (cf. Fig. 3).The standard WKB connection formulae determine the general form of the coefficients A and B , A = e S − i π H + e − S + i π V (III.6a) B = e S + i π H + e − S − i π V (III.6b)where S ≡ S ( x , x ) and the coefficients H ( Q, α, q, Λ) and V ( Q, α, q, Λ) depend on the choice of boundary conditionon Ψ . They are obtained by matching the small x wave function to the WKB form (III.4). We refer to AppendixA for the details of this procedure.We have numerically verified and analytically substantiated in Appendix A that both H and V are genericallynon-zero for Hartle-Hawking boundary conditions at x = 0 . Hence the intermediate tunneling dynamics in cosmicegg cosmology naturally produces a wave function in the large volume limit that involves a superposition of thetypical Hartle-Hawking behavior ∼ e S , with a Vilenkin tail ∼ e − S . Substituting (III.6) in (III.4) yields the large x wave function Ψ( x ) = 2 | U ( x ) | / (cid:104) e S H cos (cid:16) S ( x, x ) − π (cid:17) + e − S V cos (cid:16) S ( x, x ) + π (cid:17)(cid:105) ( x (cid:29) x ) . (III.7)The leading dependence on q and Λ is encoded in S . To understand this, note that S is given by the surfacearea of the function √ U between the two turning points x and x . With H the maximum height of this and ∆ x = | x − x | the distance between the turning points x and x , the area can be shown to be somewhere between H ∆ x < S < H ∆ x , (III.8)since U ( x ) is concave. One then finds H ≈ (cid:113) − q Λ ) / . (III.9)An approximate expression for ∆ x , valid when Λ is significantly lower than its critical value (II.6), is given by ∆ x ≈ − q √ . (III.10)0This yields the following estimate, S ≈ ζ (cid:18) − q √ (cid:19) (cid:113) − q Λ ) / , (III.11)with / < ζ < . In the limit Λ (cid:28) Λ cr this becomes S ≈ ζ Λ (cid:34) − (cid:18) ΛΛ cr (cid:19) / + O (cid:18) ΛΛ cr (cid:19)(cid:35) . (III.12)Hence the wave function in this limit is approximately given by Ψ( x ) ≈ H e S | U ( x ) | / cos (cid:16) S ( x, x ) − π (cid:17) ( x (cid:29) x ) . (III.13)Thus the wave function of the universe in the large volume regime in a cosmic egg scenario consists of a leadingHartle-Hawking term, given here, together with an exponentially suppressed contribution characteristic of thetunneling wave function. The mixing of both wave functions arises because the quantum dynamics of the eggcosmology involves a combination of Hartle-Hawking initial conditions to create the egg, and quantum tunnelingfrom the egg to a large expanding universe. IV. DISCUSSION
We have argued that the cc problem motivates the study of cosmological scenarios in which all spatial dimensionsare initially of comparable, small size. That is, cosmology encourages one to consider decompactification ratherthan compactification, as in string gas cosmology [29].We have proposed a new toy model in this spirit in which a universe with three large spatial dimensions is seen toemerge in a two-step process. First, we conceive of a metastable, classical, fully compactified configuration in theHartle-Hawking state. These effectively zero-dimensional cosmic egg configurations are neither de Sitter nor anti-deSitter but breathing, with an oscillating volume modulus. Their size is set by a combination of the particle physicsingredients, including an axion, and the Hartle-Hawking initial conditions. The absence of scale separation meansthat cosmic eggs do not suffer from a cc problem and are thus plausibly natural early universe configurations When the cosmological constant term is positive but below the Einstein static value in the model, we have identifieda decompactification channel in which the egg decays through tunneling into an expanding universe with three largespatial dimensions. In a landscape consisting of eggs with different parameters the characteristic Hartle-Hawkingweighting then favors the nucleation of large universes with a low value of the four dimensional cc. This cast thecc problem in a new light: a seemingly unnaturally small cc is seen to arise from natural initial conditions. As a historical comment, we note that the cosmic egg configurations we consider, differ from Lemaîtres conception of ‘primeval atom’[46] which he thought of as purely quantum, i.e. exhibiting no (classical) notion of space and time. Lemaître’s primeval atom seemsmore akin to the Euclidean region of the conventional no-boundary saddle points. Needless to say, the cosmic egg cosmologies we have considered aren’t realistic. The sole purpose of the toy modelswe have proposed, is to exhibit a concrete mechanism to relax the cc. It will require much further work to embedthis scenario in somewhat realistic models of the early universe. One possible route would appear to consider thebreathing eggs as a pre-inflationary phase, with the decay process leading to a period of inflation that generatesa large universe filled with matter and primordial perturbations. It is plausible that the general tendency of theHartle-Hawking state towards a low vacuum energy survives in such more realistic egg-based cosmologies [54].
ACKNOWLEDGMENTS
This work is supported by the C16/16/005 grant of the KU Leuven, the COST Action GWverse CA16104, and bythe FWO Grant No. G092617N. TVR would like to thank the FWO-Vlaanderen and the KU Leuven for supportinghis sabbatical research. We refer to [47–53] for a more detailed discussion of this. Figure 3: The different regions of the potential in which the Wheeler–DeWitt equation is solved. Solutions in region I, IIand III are matched within the overlap (shaded and striped region) and connected to region IV with the WKB connectionformula.
A. COSMIC EGG WAVE FUNCTION
In this Appendix we derive an approximate analytical solution of the wave function in the cosmic egg model beyondthe WKB approximation which fails in the small volume regime. Specifically we first solve the Wheeler–DeWittequation approximately in three different potential regions and then match the solutions across the regions andeventually with the WKB form at large x . Figure 3 shows the subdivision in regions, for a potential of the form U ( x ) = Q x α − q x + 2 −
23 Λ x. (A.1)It should be noted that the approximation solution derived below is valid for configurations that have Λ (cid:28) Λ cr and x (cid:28) x (cid:48)(cid:48) (cid:28) x (cid:48) (cid:28) x or, equivalently, + α − (cid:28) x x , (A.2)or Q (cid:28) (cid:18) q √ (cid:19) α . (A.3)Note that (A.3) is stronger than the aforementioned bound on Q in eq (III.3), for all values of α > .Region I is specified by x (cid:28) x (cid:48) with x (cid:48) = q √ ≈ x √ . (A.4)In this region, the first two terms of the potential (A.1) dominate over the curvature and cc term, i.e. U ( x ) ≈ Q x α − q x (region I) . (A.5)3Consequently, the Wheeler–DeWitt equation is approximately solved by Ψ I ( x ) = a √ xK ν (cid:32) ˜ Qx α − (cid:33) + b √ xI ν (cid:32) ˜ Qx α − (cid:33) , (A.6)where I ν and K ν are the modified Bessel functions of the first and second order ν and ν = 1 α − (cid:114) − q , ˜ Q = 2 √ α − Q. (A.7)Hence Ψ I only displays oscillations characteristic for a classical egg phase when q > . (A.8)In what follows we will assume this lower bound on q . Also, in the limit x → , only the first term in (A.6) remainsregular and –in particular– vanishes. The Hartle-Hawking wave function in region I is therefore given by eq. (A.6)with a = 1 and b = 0 .In region II, any quantum effects ∼ Q /x α in the potential (III.2) are damped out and the curvature term becomesrelevant while the cc term remains negligible. This region roughly corresponds to x (cid:48)(cid:48) (cid:28) x (cid:28) x (cid:48)(cid:48)(cid:48) with x (cid:48)(cid:48) = (cid:18) Q q (cid:19) α − = 2 α − x (A.9a) x (cid:48)(cid:48)(cid:48) = 32Λ = x (A.9b)The potential can be approximated by U ( x ) ≈ − q x + 2 (region II) (A.10)and the wave function is then approximately given by Ψ II ( x ) = c √ xK µ (2 x ) + d √ xM µ (2 x ) , (A.11)where M µ ( x ) is defined to be M µ ( x ) ≡ I µ ( x ) + I − µ ( x ) . (A.12)The choice of writing K µ and M µ in (A.11) simplifies the interpretation: if c and d are real, so is Ψ II . The order µ of the modified Bessel functions is again purely imaginary under the constraint (A.8) and given by µ = (cid:16) α − (cid:17) ν . (A.13)The integration constants c and d are determined by the behaviour of the wave function in region I. This is doneby a matching procedure in the overlap region between x (cid:48)(cid:48) and x (cid:48) , the shaded area in Figure 3. For x (cid:48)(cid:48) < x < x (cid:48) ,the behaviour of (A.6) is Ψ I ( x ) ∼ ˜ Q ν Γ( − ν )2 ν x − ( α − ) ν + + ˜ Q − ν − ν Γ( ν ) x ( α − ) ν + + O (cid:18)(cid:16) x x (cid:17) α − ± ( α − ) ν (cid:19) , (A.14)4while for x (cid:48)(cid:48) < x < x (cid:48) (cid:28) x , the wave function (A.11) takes the form Ψ II ( x ) ∼ x − µ + (cid:18)
12 Γ( µ ) c + d Γ(1 − µ ) (cid:19) + x µ + (cid:18)
12 Γ( − µ ) c + d Γ(1 + µ ) (cid:19) + O (cid:32)(cid:18) xx (cid:19) ± µ (cid:33) (A.15)Hence the matching condition is c = (cid:32) ˜ Q (cid:33) ν Γ(1 − µ )Γ( − ν )Γ(1 − µ )Γ( µ ) − Γ( − µ )Γ(1 + µ ) + c . c . (A.16a) d = 12 (cid:32) ˜ Q (cid:33) ν Γ(1 − µ )Γ(1 + µ )Γ( − µ )Γ( − ν )Γ( − µ )Γ(1 + µ ) − Γ(1 − µ )Γ( µ ) + c . c . (A.16b)where c.c. denotes complex conjugate. Hence it is obvious that c and d are real and, consequently, Ψ II is real.Within region II it is to be expected that the WKB approximation becomes increasingly accurate for larger x . Wewill therefore aim at giving WKB solutions in region III (overlapping with the right portion of region II) and IV.Region III is the underbarrier region between the turning points x < x < x in which the wave function takes theWKB form Ψ III ( x ) = 1 | U ( x ) | / (cid:16) Ce − S ( x,x ) + De S ( x,x ) (cid:17) , (A.17)where S ( x, x i ) is defined as S ( x, x i ) = (cid:90) xx i d x (cid:48) (cid:112) | U ( x (cid:48) ) | . (A.18)The amplitudes C and D are to be matched with c and d that specify the behaviour of the wave function in thefirst part of the underbarrier regime (the striped area in Fig 3) where the cc term can still be neglected. In thislimit, x (cid:28) x < x (cid:48)(cid:48)(cid:48) , the asymptotic form of (A.11) is Ψ II ( x ) ∼ c √ πe − x + d √ π e x + O (cid:18) e ± x (cid:16) x x (cid:17) / (cid:19) (A.19)while the WKB form (A.17) is approximately Ψ III ( x ) ∼ C √ e π √ q e − x + D √ e − π √ q e x (A.20)This is because in this limit x (cid:28) x < x (cid:48)(cid:48)(cid:48) S ( x, x ) ≈ x (cid:114) − (cid:16) x x (cid:17) − x arctan (cid:115)(cid:18) xx (cid:19) − ∼ x − π √ q + O (cid:16) x x (cid:17) (A.21)where the potential (A.10) has been inserted. Hence the amplitudes C and D are given by C = (cid:114) π e − π √ q c (A.22a) D = (cid:114) π e π √ q d (A.22b)IBLIOGRAPHY 15To make the transition to region IV, x < x , one simply uses the standard WKB connection formulae. The wavefunction in this large x regime is given by Ψ IV ( x ) = 1 | U ( x ) | / (cid:16) Ae iS ( x,x ) + Be − iS ( x,x ) (cid:17) , (A.23)where A = De S − i π + 12 Ce − S + i π (A.24a) B = De S + i π + 12 Ce − S − i π (A.24b)where S ≡ S ( x , x ) . As stated in the text, the connection formulae for this specific cosmic egg model are of thegeneral form A = e S − i π H ( Q, α, q ) + e − S + i π V ( Q, α, q ) (A.25a) B = e S + i π H ( Q, α, q ) + e − S − i π V ( Q, α, q ) (A.25b)where H and V depend on the quantum contributions to the potential as well as the scalar flux. In particular, usingequations (A.16), (A.22) and (A.24), H and V are explicitly given by H = (cid:114) π e π √ q Re (cid:34)(cid:32) ˜ Q (cid:33) ν Γ(1 − µ )Γ(1 + µ )Γ( − µ )Γ( − ν )Γ( − µ )Γ(1 + µ ) − Γ(1 − µ )Γ( µ ) (cid:35) (A.26a) V = (cid:114) π e − π √ q Re (cid:34)(cid:32) ˜ Q (cid:33) ν Γ(1 − µ )Γ( − ν )Γ(1 − µ )Γ( µ ) − Γ( − µ )Γ(1 + µ ) (cid:35) (A.26b) BIBLIOGRAPHY [1] S. Weinberg, “The Cosmological Constant Problem,”
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