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A Quantum Gravity Extension of the Inflationary Scenario Ivan Agullo,∗ Abhay Ashtekar,† and William Nelson‡ Institute for Gravitation and the Cosmos & Physics Department, Penn State, University Park, PA 16802, U.S.A. Since the standard inflationary paradigm is based on quantum field theory on classical space- times, it excludes the Planck era. Using techniques from loop quantum gravity, the paradigm is extendedtoaself-consistenttheoryfromthePlanckscaletotheonsetofslowrollinflation,covering some 11 orders of magnitude in energy density and curvature. This pre-inflationary dynamics also opens a small window for novel effects, e.g. a source for non-Gaussianities, which could extend the reach of cosmological observations to the deep Planck regime of theearly universe. PACSnumbers: 98.80.Qc,04.60.Pp,04.60.Kz 2 1 The inflationaryparadigmhas hadremarkablesuccess φ. In numerical simulations we will use the quadratic 0 in accounting for the inhomogeneities in the cosmic mi- potential V =(1/2)m2φ2 with m=1.21×10−6m , the 2 Pl crowave background (CMB) that serve as seeds for the value that comes from the 7 year WMAP data [5, 6]. p large scale structure of the universe. However it has Throughout we use natural Planck units. e S certain conceptual limitations from particle physics as The truncated phase space: We have ΓTrun = Γo×Γ1 well as quantum gravity perspectives. For example: i) where Γo is the 4-dimensional phase space of homoge- 7 The physical origin of the inflaton and its properties re- neous fields, and Γ , of the first order, purely inhomo- 1 mains unclear; ii) Since the background geometry and geneous perturbations thereon. Thus Γ is a normal ] Trun c matter satisfy Einstein’s equations, the big bang singu- bundle over Γ . The base space Γ is conveniently coor- o o q larity persists [1]; iii) One ignores pre-inflationary dy- dinatized by the scale factor a, the inflaton φ and their r- namics and simply requires that perturbations be in the conjugatemomenta. DynamicsonΓo isgeneratedbythe g BunchDavies(BD) vacuumatthe onsetofthe slowroll; single, homogeneous, Hamiltonian constraint, C = 0. o [ and, iv) When evolved back in time these perturbative On Γ1 the first order constraints can be solved and one 1 modes acquire trans-Planckian frequencies and the un- canreadilypass to the reduced phase spaceΓ˜1 which we v derlying framework of quantum field theory on classical coordinatize by two tensor modes, collectively denoted 9 space-timesbecomesunreliable. Herewewillnotaddress by T in what follows, and the Mukhanov variable Q 0 any of the particle physics issues. Rather, we focus on repre~ksenting the scalar mode. This passage to the re~k- 6 the incompleteness relatedto quantumgravityandshow duced phase space is carried out entirely in the phase 1 that this limitation can be overcome. In addition, we space frameworkwithout using any equations ofmotion. . 9 find that pre-inflationary dynamics can produce certain Finally,aconceptuallyimportantpointisthatdynam- 0 deviationsfromthe BDvacuumatthe onsetofinflation, ics on Γ , or on Γ˜ =Γ ×Γ˜ , is not generated by 2 Trun Trun o 1 leading to noveleffects which couldbe seen, e.g.,in non- 1 a constraint, or, indeed by any Hamiltonian. Rather, : Gaussianities through future measurements of the halo the dynamical vector field Xα on Γ˜ has the form v bias and the ‘µ-type distortions’ in the CMB [2]. Xα = Ωαβ∂ C + Ω˜αβ∂ C′ where ΩTruannd Ω˜ are the i o β o 1 β 2 o 1 X Loop quantum gravity (LQG) offers a natural frame- symplectic structures on Γ and Γ˜ , and C′ is the part o 1 2 r work to address these issues because effects of its under- ofthesecondorderHamiltonianconstraintinwhichonly a lying quantum geometry dominate at the Planck scale, terms that are quadratic in the first order perturbations leadingtosingularityresolutioninavarietyofcosmolog- are kept. Xα fails to be Hamiltonian on Γ˜ because Trun ical models, including some that admit anisotropies and C′ depends not only on perturbations but also on back- 2 inhomogeneities [3]. Even though LQG is still incom- ground quantities. However, given a dynamical trajec- plete, notable advances have occurred —e.g., in cosmol- tory γ (t) onΓ and a perturbationat a point on it, Xα o o ogy,analysisofblackholes,andaderivationofthegravi- providesacanonicalliftofγ (t)tothe totalspaceΓ˜ , o Trun ton propagator— by using the following strategy: First describingtheevolutionofthatperturbationalongγ (t). o carryoutatruncationoftheclassicaltheorygearedtothe Quantum Kinematics: Since Γ˜ = Γ ×Γ˜ , the to- given physical problem and then use LQG techniques to Trun o 1 tal Hilbert space is given by H=H ⊗H . The Hilbert construct the quantum theory [4]. For inflation, then, we o 1 space H of backgroundfields consists of wave functions are led to focus just on first order perturbations off the o Ψ (a,φ) and its structure is well understood from loop spatially flat Friedman backgrounds with a scalar field o quantumcosmology(LQC)[3]. Forperturbations,wein- troduceaninfraredcutoffsothatλ ≥λ ,thesizeof cutoff o the observable universe. Physically,this amounts to ‘ab- sorbingmodeswithλ>λ inthebackground’. Then ∗Electronicaddress: [email protected] cutoff †Electronicaddress: [email protected] there is a natural Hilbert space H1 on which perturba- ‡Electronicaddress: [email protected] tions Qˆ~k and Tˆ~k act. It admits an infinite dimensional 2 sub-space of 4th order adiabatic states which are invari- of wave functions of interest again follow the effective ant under spatial translations, often called ‘vacua’. H trajectories as expected. We restrict ourselves to back- 1 is generated by excitations on any one of them. (For an ground quantum geometries Ψ (a,φ) with this property. o alternatecharacterizationsee[7].) Notehoweverthat,in Each of them provides a probability amplitude for var- contrast to quantum field theory on strictly stationary ious classical space-time geometries to occur. They are space-times, H does not havea preferredvacuum state, peaked not on classical Friedmann solutions but rather 1 or a canonical notion of particles. on quantum corrected bouncing solutions. Furthermore, Thekeydifferencefromstandardinflationisthatquan- therearefluctuationsaroundthese peaks. The challenge tum fields Qˆ ,Tˆ now propagateon a quantum geometry is to capture the effects of this background quantum ge- k k represented by Ψo(a,φ) rather than on a classical Fried- ometry Ψo(a,φ) on the dynamics of perturbations. mann solution (a(t),φ(t)). These quantum geometries To meet it, we use the conceptual framework of quan- are all regular, free of singularities. Thus, by construc- tum field theory on cosmological quantum geometries, tion, the framework encompasses the Planck regime. introducedin[10]. Anextensionofthatframeworktoin- corporateaninfinite number ofmodes, with appropriate Now, the quantum geometry underlying LQG is sub- regularizationand renormalization,provides the dynam- tle [4]. For example, while there is a minimum non-zero icalequationforstatesψ(Q ,T )ofperturbationsonthe eigenvalue of the area operator, there is no such mini- ~k ~k background quantum geometry Ψ . A key result is that mum for the volume operator, although its eigenvalues o this evolution is equivalent to that of test perturbations are also discrete. In the present truncated theory, per- propagating on a dressed, effective, smooth metric turbative modes with arbitrarily high frequencies are al- lowed even though there is a quantum geometry Ψ in g˜ dxadxb ≡ ds˜2 = a˜2(φ)(−dη˜2+d~x2) o ab the background. By itself, this is not a problem. In where the dressed scale factor a˜ and the dressed confor- our homogeneous sector, for example, the inflaton mo- mal time ηãre given by mentum p can be arbitrarilylarge but still the energy (φ) density ρ is bounded above by ρmax ∼ 0.41ρPl [3]. The a˜4 =hHˆ−12 aˆ4(φ)Hˆ−21ihHˆ−1i−1; dη˜=a˜2(φ)hHˆ−1idφ. real trans-Planckian issue for us is whether the energy o o o o density inperturbationsremains(notonlyboundedbut) This result is exact within our truncation scheme. It small compared to the background all the way back to shows that the propagation of perturbations is sensitive thebounce. Onlythenwouldwebeassuredofaselfcon- to properties of the state Ψ even beyond the quantum o sistent solution, justifying our truncation which ignores corrected effective geometry followed by its peak; it also the back reaction. Otherwise one would have to await a senses quantumfluctuations aroundthis peak. However, full quantum gravity theory. interestingly,this dependence is neatly codedinjust two Quantum dynamics: Since the classical dynamics on ‘dressed’quantities,ηãnda˜. Thisisanalogoustothefact Γ˜Trun is not generated by a constraint, contrary to what that although light propagating in a medium interacts is often done, one cannot recover quantum dynamics for with its atoms, the net effect can be captured in just a the total system by imposing a quantum constraint. As few parameters such as the refractive index. in the classical theory, we can do this only in the homo- Thisresultgreatlysimplifiesourtaskconceptuallyand geneoussectorHo andwethenhaveto‘lift’theresulting enables us to use the technical tools of mode by mode quantum trajectory to the full H. On Ho one can follow regularization and renormalization from the well devel- thestandardprocedureinLQC.Itagainleadsustorein- opedadiabaticschemeofquantumfield theoryonclassi- terpret the quantum Hamiltonian constraint CôΨo = 0 cal cosmological space-times [11]. For tensor modes, for as an ‘evolution’ equation, −i~∂φΨo(a,φ)=HôΨo(a,φ), example, one obtains the following evolution equation with respect to the relational or emergent time variable φ generated by a time dependent Hamiltonian Hô [3, 8]. i~∂η˜ψ(T~k,η˜)=Hˆ1ψ(T~k,η˜) However,relativetothesimplermasslesscase[3],oneen- ≡ 1 d3k 4κ|ˆp |2+ k2a˜2 |Tˆ |2 −C (η˜) ψ(T ,η˜) counters certain technical complications because of the 2R ha˜2 ~k 4κ ~k k i ~k presence of a potential. Their origin and resolution is analogous to that in the case where φ is massless but where ˆp is the momentum conjugate to Tˆ , κ = 8πG ~k ~k there is a positive cosmologicalconstant [9]. andC (η˜)arec-numbers,derivedfromthe4thorderadi- k In the V(φ) = 0 case, detailed investigations have abatic regularizationthat depend only on k =|~k|. shown that wave functions Ψ (a,φ) of physical interest Initial conditions: Since the big bang is replaced by o remain sharply peaked even in the Planck era and fol- the big bounce, it is natural to specify initial conditions lowquantumcorrectedeffectivetrajectories. ForV(φ)= at the bounce. The initial state can be taken to be of (1/2)m2φ2, solutions to effective equations continue to the form Ψ ⊗ ψ because perturbations are treated as o undergoa bounce when ρ=ρ and to agreewith gen- test fields. This tensor product form prevails so long as max eral relativity for ρ . 10−3ρ . However, because of thethebackreactionremainsnegligibleduringevolution. max computationallimitations,sofarthequantumwavefunc- To specify the initial condition for Ψ let us first recall o tionsΨ (a,φ)havebeencalculatedonlywhenthebounce that, in effective LQC, all dynamical trajectories enter a o is kinetic energy dominated [8]. In this case, the peaks slow roll phase compatible with the 7 year WMAP data 3 6 Thepowerspectrumattheendofinflationwascomputed 1.4 in each case for both scalar and tensor modes. Results 5 1.3 1.2 are all very similar. FIG.1 shows how the LQC scalar 1.1 4 1 power spectrum relates to the prediction of standard in- PLQCPStandard23 0000....6789 6 8 10 12 14 flsvtaaacttuieounψmf.oisrWttheheefo‘ocuabnsvdeiowtuhhsa’etorerthφ‘esBtpanl=odta1ir.s1d5l’am4rtgPhel,loyarnidndesretnahsdeiitaiinvbieattitaiocl choices of initial conditions within our class. 1 Recall,however,thatthe7yearWMAPdata[5]covers onlyawindow(k ≈k⋆/8.58, k ≈2000k )inthe 0 min max min 2 4 6 8 10 12 14 16 18 20 co-movingkspace. Herethereferencemodek⋆ istheone k (co-moving) that exits the Hubble radius at time η˜k⋆ when the Hub- ble parameter is given by H(η˜k⋆) = 7.83×10−6mPl. In FIG. 1: Ratio of our LQG power spectrum for scalar per- FIG.1, numerical values of the co-moving k were calcu- turbations to the standard inflationary power spectrum. For latedusingthescalefactorconventiona =1,ratherthan small k, the ratio oscillates rapidly with k. The solid curve B a = 1. (The physical wave numbers are of course shows averages over bins of width ∆k=0.5ℓPl−1. The inset today convention independent). In each simulation, we first shows a blow-up of the interesting region around k=9. locate the scale factor a˜(η˜k⋆) by setting H = H(η˜k⋆), and then determine k⋆ via k⋆ =a˜(η˜k⋆)H(η˜k⋆). Since we have a˜B=1, values of a˜(η˜k⋆) and k⋆ depend on the pre- unlessφ ,thevalueoftheinflatonatthebounce,liesina inflationary backgrounddynamics which turns out to be B very small regionRoftheconstraintsurface[6]. Wewill governed entirely by φB. Therefore, in FIG.1 the obser- assume that, at the bounce, the background quantum vationally relevant window depends on the value of φB, state Ψo is sharply peaked at a point on the constraint moving steadily to right as φB increases. surface outside this R. In this sense the initial data for The plot has two interesting features. First, the LQG Ψ is generic. For perturbations, we assume that the power spectrum is virtually indistinguishable from that of o initial ψ is a 4th order adiabatic ‘vacuum’ such that the standard inflation if k & 9m . This occurs when min Pl expectation value of the renormalized energy density in φ & 1.2m . Second, for smaller values of k , the B Pl min ψ isnegligiblecomparedtothatinthebackground. This observational window admits modes for which the two is a large class of initial data for test fields, selected by power spectra are noticeably different. For concreteness, general symmetry requirements. letussetφ =1.15m . Thenk ≃1.07m andthese B Pl min Pl Physically, we are assuming ‘initial quantum homo- modes correspond to ℓ . 30 in the WMAP angular de- geneity’ i.e., requiring that the region which expands to compositionfor which observationalerrorbars are large. become the observable universe is homogeneous at the ThereforetheLQGpowerspectrumisalsoviablebutthe bounce except for ‘vacuum fluctuations’. While this is a predicted quantum state of perturbations at the onset of strong restriction, it may be naturally realized in LQG inflation is not the BD vacuum for φB <1.2mPl. because: i) In solutions of interest, the observable uni- Selfconsistency: Whetherthetestfieldapproximation verse has a radius . 10ℓ at the bounce; and, ii) The continues to hold in the Planck regime is an intricate Pl strong repulsive force due to quantum geometry that issue and had not been explored before. FIG.2 shows causes the bounce has a ‘diluting effect’. It could make that there do exist solutions ψ in which the renormal- this ‘quantum homogeneity’ generic, ‘washing out’ the ized energy density in perturbations remains low com- memoryofthepre-bouncedynamicsatthescale.10ℓ . paredtothebackgroundthroughouttheevolution. (Here, Pl Our remainingtaskis twofolds: i)starting fromthese we have set kcutoff =ko =30mPl, which corresponds to initialconditions,calculatethepowerspectrumforscalar φB ≈ 1.23mPl.) Furthermore, there is an analytical ar- and tensor modes at the end of the slow roll inflation; gument showing that every such ψ admits a well-defined and, ii) verify if the back reaction continues to remain neighborhood(intheinfinitedimensionalspaceofthe4th negligibleallthe wayto the onsetofthe slowrollsothat orderadiabatic‘vacua’)withthesameproperty. Thises- ourinitialtruncationisa selfconsistentapproximation. tablishes that if φB &1.23mPl, the truncated theory ad- Power spectrum: As noted above, in bounces with ki- mits a rich set of self consistent solutions. Furthermore, netic energy domination on which we focus, the quan- in each of them ψ has extremely small excitations over tum state Ψ (a,φ) is known to remain sharply peaked theBDvacuumattheonsetofslowroll. These solutions o on effective trajectories. Therefore, in numerical simula- provideviableextensionsofthestandardinflationarysce- tionsa‘meanfield’approximationwasmadebyreplacing nario all the way to the Planck scale. a˜(φ),η˜(φ)bythemeanvaluesoftheseoperators. Forthe What about the small but interesting window φ < B background,severalsimulationswerecarriedoutwithφ 1.2m ? Sofarweonlyhaveupperbounds forthe renor- B Pl in (0.93m , 1.5m ), which, as we will see below, is the malizedenergydensityinperturbationsandthesearefar Pl Pl most interesting range. Three sets of initial conditions frombeingoptimal. Thereforewedo notyethaveanex- were chosen for the quantum state ψ of perturbations. plicit solution establishing the validity of the test field 4 1 effect can be explained in terms of the new scale k de- M 0.01 fined by the universal value of the scalar curvature at 0.0001 the bounce. Excitations with k . k are created in M 1e-06 the Planck regime near the bounce. It turns out that if 1e-08 the number N ofe-foldings ina˜ betweenthe bounce and ρ 1e-10 η˜=η˜k⋆ is less than 15, then kmin <kM, whence some of 1e-12 these modes would be in the window accessible through 1e-14 CMB. N < 15 corresponds to φB . 1.2mPl, precisely theregimeinwhichtheLQCpowerspectrumisdifferent 1e-16 from the BD vacuum. Future measurements should be 1e-18 0.0001 0.01 1 100 10000 sensitive to such deviations [2]. If they are observed at tPl the scale k = k , the parameter space of initial condi- M tions for Ψ would be tightly squeezed, making much o FIG.2: Energy densityinthebackground(uppercurve)and more detailed predictions feasible. In this sense, the an upper bound on the renormalized energy density in per- framework expands the reach of observational cosmol- turbations (lower curve) are plotted against time from the ogy all the way to the deep Planck regime. This general bouncetotheonsetofslowroll, usingPlanckunits. Thetest argument also shows that the pre-inflationary dynamics fieldapproximationholdsacrossachangeofover11ordersof hasnegligibleeffectformodeswithk ≫k becausetheir magnitude in both quantities. M physicalwavelengthturnouttobesmallerthanthecur- vature scale throughout the evolution. This explains the verycloseagreementbetweenthe LQCandthestandard approximation in this window. power spectrum at high k. Summary and Discussion: UsingLQGideasandtech- Finally, interesting and complementary investigations niques, we have extended the inflationary paradigm all of LQG dynamics between the bounce and the onset of the way to the deep Planck regime. At the big bounce, slowrollhaveappearedintheliteraturerecently(see,es- one can specify natural initial conditions for the quan- pecially,[12]). Thedistinguishingfeaturesofouranalysis tumstateΨ thatencodesthe backgroundhomogeneous o are: i)Itisbasedonthegeneraltruncationstrategythat quantum geometry, as well as for ψ that describes the has proven to be successful in other problems; ii) It pro- quantum state of perturbations. There is a precise sense videsasystematicapproachtoquantumdynamics,made in which generic initial conditions for the background necessary by the fact that the classical evolution is not lead to a slow roll phase compatible with the 7 year generated by a constraint on Γ ; iii) The treatment Trun WMAPdata[6]. Wehaveshownthatthereisalargeset of initial states has been stream-lined; and, most impor- ofinitialdataforψsuchthat: i)attheonsetofslowroll, tantly, iv) While issues of regularization of the Hamil- ψ is extremely close to the BD vacuum, and, ii) the test tonian operator Hˆ , adiabatic renormalization of energy fieldapproximationbehindthetruncationstrategyholds. 1 density, and consistency of the test field approximation Eachofthesesolutionsprovidesaviablequantumgravity were ignoredso far, they have now been addressedusing completion of the standard inflationary paradigm. How- quantum field theory on quantum geometries. Details ever particle physics issues still remain. and subtleties which could not be included here will be In addition, there exists a narrow window in φ for B discussed in two forthcoming articles. which the quantum state ψ at the onset of inflation has an appreciable number of ‘BD particles’ (but within the Acknowledgments: ThisworkissupportedbytheNSF current observationallimits). The physical origin of this grant PHY-1205388. [1] A.Borde,A.GuthandA.Vilenkin,Phys.Rev.Lett.90, hinho, Phys. Rev.D83025002 (2011). 151301 (2003). [8] A.Ashtekar,T.PawlowskiandP.Singh(inpreparation). [2] I. Agullo and L. Parker, Phys. Rev. D83 063526 (2011); [9] T. Pawlowski and A. Ashtekar, Phys. Rev.D85, 064001 I.AgulloandS.Shandera,JCAP09, 007(2012); J.Ganc (2012). and E. Komatsu, Phys. Rev.D86 023518 (2012). [10] A. Ashtekar, W. Kaminski and J. Lewandowski, Phys. [3] For a review, see, e.g., A. Ashtekar and P. Singh, Class. Rev. D79064030 (2009). Quant.Grav. 28, 213001 (2011). [11] L. Parker and S. A. 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