PoS(Rio de Janeiro 2012)002 http://pos.sissa.it/ † ∗ [email protected] Speaker. A footnote may follow. This brief overview is addresseddistinguishing features to of beginning loop researchers. quantumfaced gravity, by It then any begins illustrates quantum with the gravity theory, type aissues and of discussion in concludes issues loop of with that quantum a the must brief gravity.introductory be summary For of as concreteness the well the status as emphasis of these literature, is detailed the on reviews focus cosmology. of is Since on loop several explainingintroduce quantum the the overall gravity interested reader viewpoint are to and literature. already providing a available short in bibliography the to ∗ † Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. c

7th Conference Mathematical Methods in Physics16 - to Londrina 20 2012, April 2012 Rio de Janeiro, Brazil Abhay Ashtekar Institute for Gravitational Physics and Geometry,Park, Physics PA 16802, Department, U.S.A. Penn State, University E-mail: Loop Quantum Gravity and the Very Early Universe PoS(Rio de Janeiro 2012)002 1 ]. 2 ]. For 9 , 8 , start with a 7 Abhay Ashtekar does not 2 ] and much more detailed treatments in 3 ] and references therein. These difficulties are now 2 ] and a detailed discussion of conceptual problems in [ 2 1 ] that will appear in the proceedings of the the third Zakopane school on quantum 13 , 12 , The absence of a background geometry implies that classical dynamics is generated by con- ]. 6 11 For more detailed treatments, the following reviews should be helpful to the beginning researchers. A short sum- There is a significant body of literature on this issue; see e.g., [ Keeping this limitation in mind, let us begin with a list of some of the long standing conceptual Loop quantum gravity (LQG) is a non-perturbative approach aimed at unifying general rela- In this brief overview I will use loop quantum cosmology (LQC) to illustrate the main ideas. , 1 2 • , 5 , 10 4 being discussed also in the string theory literature in the context of the AdS/CFT conjecture. mary of the history of quantum gravity can be found in [ [ geometry and quantum gravity. and technical issues which mustgravity be is faced by encoded any in quantumrelativity the theory can of very gravity. be geometry In traced of generalbig-bang, back space-time. formation relativity, of to black The this holes most dualcurvature. and emergence Already dramatic role in of features of the gravitational of classical waves geometry: as theory,dynamical general it nature ripples the took of of physicists geometry expansion space-time several and of decades toIn to get the quantum truly used gravity, universe, appreciate to this the the paradigm the absence shift of leads a to kinematic a background new geometry. level of difficulties. the most recent overviews of[ the key ideas in these and other topics, see review articles in Ref. A brief mathematical introduction to loop quantum gravity can be found in [ tivity and quantum physics. ItsUnderlying unity distinguishing of features the are: mathematical framework i)forces used Background of to independence, nature. describe gravity and, It and is ii) the backgroundeven other independent a three in space-time basic the metric. sense that Allthat there fields mediate are —whether fundamental no they forces background represent between fields, them, matter not same or constituents, footing. space- or time As gauge geometry in fields itself— general are relativity, they treated on are the all dynamical. Moreover, one Loop Quantum Gravity classical space-time metric and thenthem add as quantum corrections perturbations. to Rather, geometry all‘from in fields, birth’. small including steps, Finally, space-time treating while geometryfoundation one are of quantum maintains general relativity, mechanical the the basic deep dynamicalrics. duality variables are The between spin metric geometry connections arises as rather and a than composite gravity met- conjugate field at to constructed the from the the spin ‘electric connection. fields’by that Since are connections, the canonically one electroweak now has and an strongforces uniform interactions of mathematical are Nature. framework also to But describe mediated the all primarydeep four focus influence fundamental is on on dynamics gravity: of While alltheory, the other one quantum fields does nature especially not of in attempt space-time the toto has Planck unify the regime, all one in forces used contrast of in to Nature. QCD string In rather this than respect, the the one strategy that is drove analogous grandThe unified theories. main advantage of this strategygoals, challenges, is key that ideas, students advances and and limitations other inis a beginning that rather researchers simple this setting. can approach The grasp main the leaveshorizons drawback out through some quantum geometry, of the the issuerecent key calculations of advances of information the such loss, graviton as and propagators in the spin the foams, understanding non-perturbative especially setting of of the quantum LQG [ PoS(Rio de Janeiro 2012)002 ]. 17 , Abhay Ashtekar 16 , absolute boundary 15 3 comes to a halt. with respect to which the other arguments ‘evolve’? If not, how does all of physics emergent time ]. Consequently, in much of the literature on the Wheeler-DeWitt (WDW) approach to 14 How close to the big-bangnarios, does for a example, smooth are space-time based ofprinciples that on GR this a make is sense? space-time a continuum. safe Inflationary approximation sce- already Can at oneIs the show onset the from of Big-Bang inflation? some singularity first naturallythe resolved development by of quantum the gravity? field of This quantum possibility cosmology led in to the late 1960s. The basic idea can quantum cosmology, the scale factor isimplicitly. assumed to However, in play the closed role models of therecollapse time, scale and although cannot factor often serve fails only as to a be globalbetter time monotonic alternatives variable due at already to in least the classical in classical the theory. simple Are setting there of quantumCan cosmology? one construct a frameworksical that theory cures near the singularities, short-distance whiletheir very difficulties maintaining construction, faced an perturbative and by agreement effective the with descriptionsond have clas- it no requirement. at problem with large the However, scales? sec- physicallyscale their and By implications mathematically they can generally notputative fail be singularity. to trusted provide a In at deterministic LQG thebackground evolution independence the Planck across and situation the non-perturbative is methods, atheory just also priori has the it a opposite. is rich semi-classical not sector. clearthe Since Do whether underlying the the the novel quantum dynamical emphasis geometry corrections unleashed naturally ishave by fade on unforeseen away implications at that macroscopic prevent distanceslarge the scales? or theory Some do from of they reproducing these unforeseen general problems relativity are at discussed in, e.g., [ straint equations. In the quantum theory, physicalAll states of are physics, solutions including to the quantum constraints. dynamicalsolutions. content But of there the is theory, no has external timeled to to to be phrase ask: extracted questions from Can about these we evolution. extract, Thereforeserve we from as are the arguments of the waveone function, interpret one the framework? variable which What are can thework, physical DeWitt (i.e., proposed Dirac) that observables? the In determinant a[ of pioneering the 3-metric can be used as an ‘internal’ time Now, it is widely believed that the prediction of a singularity —such as the big-bang— in Next, the dual role of the space-time metric also implies that space-time itself ends when the • • • classical general relativity is primarily aof signal its that the validity theory and has cannot beentrue therefore pushed physics. be beyond the trusted. This domain expectation One immediatelyfor needs several leads a decades to quantum now: a theory host of of gravity questions to which analyze have been with us gravitational field becomes infinite. Thissingularity is of in one striking specific contrast field withphysics. has Minkowskian In physics, no general where bearing relativity, singularities at of all the gravitational on field the represent space-time an structure or on the rest of of space-time where Loop Quantum Gravity PoS(Rio de Janeiro 2012)002 Abhay Ashtekar 6ev. The hope was . 13 ≈ − . But these very tiny fluc- 5 2 ¯ h 2 / 4 me ]. Or, do quantum Einstein equations − 18 ] and ekpyrotic and cyclic models inspired 19 4 ]. However, in such perturbative treatments there is always a 21 , 20 Thus the origin of the observed cosmic structure lies in the tiny CMB that a similar mechanism wouldcosmological resolve the models. big-bang Can and the big quantumto crunch nature such singularities of a in resolution? geometry simple underlying LQG naturally lead Is a new principle/ boundary conditiona at deterministic the evolution? big bang The orthe most the ‘no big well boundary crunch known proposal’ essential example of to of Hartlesuffice provide and by such Hawking themselves a [ even boundary at the condition classical is singularities? Do quantum dynamical equationsdo remain they well-behaved continue even to at providebang these a branch singularities? deterministic to evolution? If ourthe so, universe The pre-big-bang has idea scenario been that in there advocated string was by a theory several pre-big- approaches, [ mostsmooth continuum notably in by the background andsingularity. Consequently, hence the the pre-big-bang dynamical branch equations is breakbranch not down joined by at to a the the deterministic current evolution. post-big-bang remedy The this hope has situation been in thatgeometry the non-perturbative in effects future. the would background and LQG, thethe treatment on LQC is the non-perturbative. quantum So, Einstein’s other one equation hand, is provideand led a the does to deterministic big-crunch? ask: not evolution Can have across the a big continuum bang If there is afoam deterministic from evolution, which what the current istative post-big-bang-branch on big-bang’? is the born, Or, ‘other say was a there side’?scenarios, ‘Planck another joined time classical to Is after universe ours there the as by pu- just deterministic in equations? the a pre-big-bang quantum and cyclic by brane world ideas [ be illustrated using an analogynamics to the the ground theory state of energyan the of accelerating hydrogen this electron atom. radiates system continuously, is Inextremely it unbounded classical unstable. must below. electrody- fall Quantum into Furthermore, the physics because stant, proton the intervenes ground making and, state the energy thanks atom is to lifted to a a non-zero finite value, Planck’s con- These are some of the central, foundational questions that must be addressed by any theory • • • of quantum gravity. In addition,offers there us are important the phenomenological best issues. chancedecades The have to early witnessed confront a universe transformation quantum ofular gravity cosmology success to theories of precision with space science, missions, observations. thanks COBE tosnapshot and the The WMAP. of These spectac- past the missions have two universe provided at aextremely rather the homogeneous, detailed surface with of tiny last inhomogeneitiestuations scattering of are which one responsible shows part for that in theinhomogeneities the 10 have observed universe been large taken was as scale then initial cosmic conditionssical to structure. carry gravity out and numerical More standard simulations (astro)physics. precisely, using They the clas- structure reproduce CMB of qualitative features the of universe. the observed large inhomogeneities. Loop Quantum Gravity PoS(Rio de Janeiro 2012)002 Abhay Ashtekar to the onset of the slow roll Pl ρ 114 . However, the inflationary scenario − Pl ρ 10 11 ∼ − 5 ].) I will conclude with a brief summary of the results and their 22 If the singularity canof be resolved, structure, can even one further,conditions extend all there? our Can the one current construct way a understanding theoretical tothrough of framework the to the the Planck evolve these era true origin initial where conditions general beginning? relativityof is space-time replaced Can by geometry? an one appropriate This set quantumcurved theory space-times would natural to require that initial on an quantum extension geometry.the If of initial one the can, conditions could quantum one from field systematicallycurvature the evolve theory to Planck on the era epoch over whenof the inflation inflation 11 began? agree orders with of Ifthe that magnitude so, reach predicted in would of by the density standard observational power and confrontation cosmology inflation? spectrum of all at quantum Such gravity the the theories an end way with extension observations. to will the move Planck scale, thereby enabling a (For a detailed review, see [ First, in the cosmological setting, one can ‘deparameterize’ the quantum Hamiltonian con- In the details of deparametrization, however, quantum geometry underlying LQG plays a key For theories of the early universe perhaps the most interesting consequence of the improved over the past 6 years or so, these and related questions have been addressed in detail through • straint of general relativity. Attions a which fundamental are level, to the be constraintsense regraded simply as that restricts physical one the quantum can wave states. func- clock— single But and out coherently it one interpret can of the be quantuminternal the ‘deparameterized’ time. constraint in dynamical This equation the is variable as necessary as the and natural evolutionamplitude the because, equation for since time in various the parameter this quantum space-time state geometries —or described totime a internal probability occur, that the are familiar commonly notions used ofuse in proper a cosmology or matter no conformal variable, longer typically haveties a a of scalar clear-cut interest field meaning. –such as as Instead, thecorresponds curvature, one ‘clock’ to shears, can with adopting anisotropies, respect a matter to ‘relational density which time’ and a physical pressure– la quanti- evolve. Leibniz. This role. It introduces newmological and singularities even totally when unforeseen one quantum introduces matter effects fields which that naturally satisfy resolve all the the standard cos- energy implications to the issues raised above. observations to date is thatmation. they Inflation have has narrowed now emerged downinhomogeneities as the and the theoretical therefore leading of scenarios candidate the for toespecially cosmic by structure successfully the structure. account for- general for relativity Therefore, community, the against althoughadvocate the CMB the an motivations that inflationary criticisms were phase raised, originally are given valid, to onejust can on now choose the to success forego thosethis of motivations standpoint, and the the focus inflationary inflationary scenario paradigm pushesfrom in back the the explaining issue the CMB of the CMB era origin inhomogeneities. when of the the From cosmic matter structure density was only Loop Quantum Gravity phase of inflation when the mater density was about 10 LQC. assumes that space-time is described byone general is relativity, led with to its ask: big bang singularity. Therefore, PoS(Rio de Janeiro 2012)002 a cannot a Abhay Ashtekar 0 and therefore one can is the Hubble parameter, = ˙ a H . In general relativity, ˙ B Pl ρ ) ρ B = ρ 41 ρ . ρ , where 0 − ρ 1 ≈ ) ( 3 B / ρ ρ G G π 3 π 8 8 6 = ( = 2 2 ) ) a a / / ˙ ˙ a a = ( = ( : : 2 2 0 we are in a contracting branch where the universe ends in a . Thus the big bang singularity is naturally resolved by quantum B H H < ρ a = ρ 0 we are in the expanding branch; the universe starts out from a big bang . Furthermore, if one uses a standard inflationary scenario and goes back in > Pl This resolution have been shown to persist if we add a cosmological constant ` a 3 and its numerical value turns out to be 0. If ˙ B ρ > ρ Now, there are also different types of singularity theorems in classical general relativity, due to Borde, Guth and This analysis provides a detailed theory of quantum geometry for homogeneous models in the 3 Vilenkin, which apply evenbecause when they all assume eternal energy inflation conditions whereas in are LQC not there is met, a as contracting phase in prior inflationary to the models. bounce. These are bypassed Loop Quantum Gravity conditions. In the cosmological setting underditions consideration, lead in to general the relativity powerful these singularityorems energy theorems are con- due bypassed because to the Penrose,geometry left Hawking effects. and hand others. side However, to of These bring Einstein’s the- equations outels is one new can physics modified recast in by these familiar the quantum terms, quantum corrections modified in are equations the shifted simply homogeneous to by mod- algebraic the manipulations rightsimplest so hand of that side, these, the i.e., the matter Friedmann,equation side. Lemaître, in For general Robertson definiteness, relativity let Walker models. is us consider The the standard Friedmann where the term in theformula for last parenthesis represents the quantum correction. There is an analytical the scale factor and thewave function ‘dot’ that refers satisfies the to Hamiltonian derivative constraint in follows proper an ‘effective’ time. trajectory given Now, by the peak of the quantum now pass smoothly fromwith a a contracting quantum branch bounce at ingravity the corrections. past to an expanding branch in the future, and/or an inflationary potential, considermodels) spatially closed or universes, the include simplest anisotropiesthis (Bianchi type sense of the inhomogeneities result andcomplicated. is gravitational robust. in waves addition (Gowdy to Butanytime models). the a in bounce curvature more In in scalar general thefurther enters models, density, growth. the the there Planck are nature The regime, alsoquantum physical of a other nature mechanism the quantum ‘bounces’. of lies bounce correction geometry. in is Roughly, kicksPlanck When a more in, scale, the new the preventing space-time repulsive repulsive curvature force forceattraction (or becomes whose a matter so origin la density) strong lies general approaches that relativity in the ita and the key overwhelms dilutes characteristic the these of classical quantities, this gravitational therebyregime, new preventing force so singularities. is that that But it generalfeature, decays relativity rapidly the quickly as potential one becomes ultraviolet-infrared moves a awaydoes tension not from good require metls the any approximation. new away. Planck boundary conditions: Finally,in the Because quantum the this version of cosmological of singularity context, Einstein’s this equations, resolution suffices. derived key Planck era. Quantum field theorywas on applied this to quantum extend the geometry inflationary has scenariothat been all the developed. we way Very just to recently, the discussed it Planck has regime.of The the approximately repulsive diluting 10 force effect which homogenizes curvature and matter of a scale big crunch. But because of the quantum correction, when and expands out forever. If ˙ vanish if PoS(Rio de Janeiro 2012)002 ] and for , Class. 23 , 22 Abhay Ashtekar radius. Therefore, it Pl ` , Encyclopedia of , Birkhäuser, Boston, 1988. (2005) 200–232 [gr-qc/0410054]. 7 7 , New Journal of Physics Quantum geometry and its applications Background independent quantum gravity: A status report , CUP, Cambridge, 2004. Conceptual problems of Quantum Gravity There are well over a thousand papers in loop quantum cosmology and I . Equations of quantum field theory on quantum space-times enables on to (2004) R53-R152 [gr-qc/0404018]. ]. This work was supported in part by the NSF grant PHY-1205388, the Eberly 21 Gravity and the Quantum 22 , Quantum Gravity 24 Mathematical Physics, Elsevier (2006) 230-236; alsowww.gravity.psu.edu/poparticles. available at http: Quant. Grav. Acknowledgments: To summarize, LQC has faced many of the long standing difficult problems that any theory I hope this overview will serve as an invitation to young researchers to go deeper into LQC [1] A. Ashtekar, [2] A. Ashtekar and J. Stachel, [3] A. Ashtekar and J. Lewandowski, [4] A. Ashtekar and J. Lewandowski, [5] C. Rovelli research funds of Penn state. References have profited a great deal fromcould discussions not with the do authors justice of to a alla large the flavor number work of of in the them. this results Of limited to courseparticularly space; I date. [ I There hope is I inevitable have overlap managed with to some convey of at the least reviews I have written, of quantum gravity mustmany address respects. in The the new cosmological effectsare context. of strong origin enough Results in to to the resolveagreement date Planck the with scale are singularity general physics surprising of relativity are at general in strongadvanced low relativity sufficiently yet curvature. and to subtle. make yet Furthermore, direct subtle the They contact with detailed enoughconceptual phenomenology framework to side, and has ensure observations. these now Finally, advances on in the gravity LQC also provide outside cosmology. hints and tools to further develop loop quantum and LQG. Loop Quantum Gravity time all the way to theobservable (density) universe arose bounce due using to effective equations expansion of of LQC a one tiny region finds of that less the than entire 10 details see the recent and forthcoming papers by Agullo, Ashtekar and Nelson.) suffices to assume initial conditions only forthat these such small regions regions were and it homogeneous is atthat then the cannot reasonable be bounce to assume gotten except rid for of thequantum even unceasing homogeneity in quantum principle. fluctuations The preciseevolve these form initial of conditions these all initial the conditions wayshown is to that called the the end resulting of power inflation. spectrumwith Detailed is numerical the very solutions close 7 have to year that WMAPinflationary of data. epoch standard all inflation, Thus, the and way compatible the todensity genesis the and of curvature. Big Finally, the Bounce, there covering cosmic is some also structurepredicts a 11 new small is orders effects window pushed of that in could the magnitude back be parameter in from space observed matter where in the the future theory missions. (For a summary see [ PoS(Rio de Janeiro 2012)002 97 , (2011) , Phys. , (2007) , Phys. , 373 28 122 Abhay Ashtekar (1983) 2960. (1967) 1113-1148. 28 , Phys. Rev. Lett. , Phys. Rept. 160 , CUP, Cambridge, at , Nuovo Cimeto , Class. Quantum Grav. , Phys. Rev. , Phys. Rev. D From Big Crunch to Big Bang The Ekpyrotic Universe: Colliding Branes and Graviton propagator in loop quantum gravity (2001) 123522 [hep-th/0103239]. 8 64 (2006) 124038 [gr-qc/0604013]. Quantum nature of the big bang: An analytical and Quantum nature of the big bang: Improved dynamics 73 A quantum gravity extension of the inflationary scenario Wave Function Of The Universe The pre-big bang scenario in string cosmology , Phys.Rev. D Group Integral Techniques for the Spinfoam Graviton Propagator (2004) 103502 [gr-qc/04-0074]. Loop quantum cosmology: A status report (2006) 6989-7028 [gr-qc/0604044]. 70 Difficulties with recollapsing models in closed isotropic loop quantum , Phys. Rev. D 23 (2002) 086007 [hep-th/0108187]. Introduction to Modern Canonical Quantum General Relativity Quantum Theory of Gravity I. The Canonical Theory An introduction to loop quantum gravity through cosmology 65 Graviton propagator from background-independent quantum gravity , Phys. Rev. D (2006) 084003 [gr-qc/0604013]. 74 cosmology numerical investigation I (2006) 151301 [gr-qc/0508124]. Class. Quantum Grav. gr-qc/0608131. Rev. D (2003) 1 [hep-th/0207130]. the Origin of the Hot Big Bang Phys.Rev. D 213001 [gr-qc/1108.0893]. Rev. Lett. (at press), gr-qc/1209.1609. 1-20 [gr-qc/0702030]. press. [6] T. Thiemann, [7] C. Rovelli, [8] E. Bianchi, L. Modesto, C. Rovelli and S. Speziale, [9] E. R. Livine and S. Speziale, Loop Quantum Gravity [16] A. Ashtekar, T. Pawlowski and P. Singh, [10] A. Ashtekar, arXiv:1201.4598. [11] H. Sahlmann and K. Giesel, arXiv:1203.2733. [12] F. Barbero, J. Lewandowski and E. Villasenor,[13] arXiv:1203.0174 C. Rovelli, arXiv:1102.3660. [14] B. S. DeWitt, [15] D. Green and W. Unruh, [17] A. Ashtekar, T. Pawlowski and P. Singh, [18] J. B. Hartle and S. W.[19] Hawking, M. Gasperini and G. Veneziano, [20] J. Khoury, B. A. Ovrut, P. J. Steinhardt and N. Turok, [21] J. Khoury, B. Ovrut, N. Seiberg, P. J. Steinhardt and N. Turok, [23] I. Agullo, A. Ashtekar and W. Nelson, [24] A. Ashtekar, [22] A. Ashtekar and P. Singh,