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Big Bang and Loop Quantum Cosmology

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Big Bang and Loop Quantum Cosmology Big Bang and Loop Quantum Cosmology Parampreet Singh Institute for Gravitational Physics and Geometry, Penn State Loop Quantum Gravity World Seminar LQG Seminar – p.1 Standard model of cosmology provides an excellent agreement with the observational data. Age of precision cosmology. Though we have a `working model' (= hot big bang + inflation + dark energy + dark matter + ...) in cosmology, important questions remain unanswered: The backward evolution of our Universe leads to a Big Bang singularity (result of powerful singularity theorems). General Relativity inadequate to provide correct physics at ¡ high curvature scales. Occurrence of singularity limit of validity of the theory Justification of various assumptions in models of very early Universe ? Primordial perturbations ? Dark energy ? Need for inputs from quantum gravity. What does Quantum Gravity tell us about the physics near the Big Bang ? (BASED ON WORK WITH ABHAY ASHTEKAR AND TOMASZ PAWLOWSKI) LQG Seminar – p.2 Some age-old questions Does the theory provides a non-singular evolution through the big bang ? What is on the other side of the big bang ? Quantum foam or a classical spacetime ? What is the scale at which the spacetime ceases to be classical ? Does the spacetime continuum exists at all scales, especially when primordial fluctuations were generated ? What are the modifications to the Friedmann dynamics at scales of high curvature and at what scales these become important ? Does the theory make testable predictions ? (complex set of questions difficult to answer without detailed analytical, phenomenological and numerical investigations) LQG Seminar – p.3 Various Innovative Ideas Pre big bang scenarios: Based on the scale factor duality of the string dilaton action. Big bang not the beginning. The post big £ ¦ ¢¡ £¥¤ ¨ © ¨ § bang phase with ¤ preceded by a pre big £ ¨ © ¨ § bang phase with § . Major Problem: No non-singular evolution between the pre big bang and the post big bang phase. Curvature ? time Pre−Big Bang Post−Big Bang Similar problems with Ekpyrotic/Cyclic models LQG Seminar – p.4 Various models propose modifications to Friedmann dynamics at high energies. Classical general relativity is envisioned as a low energy effective theory. It is modified at high energies. Popular Ideas: Modifications may arise due to higher order terms in the gravity action, large extra dimensions, ... Example: Randall-Sundrum model. Our Universe exists on a 3-brane embedded in 5 dimensional anti-de Sitter bulk. ¤¦¥ ¡ £ § £ ¡ ¨ © ¢ ¦ £ ¨ ¡ ¡ ¤ ¤ £ ¤ ¤ £ ¡ , . As , © . Singular evolution. Problems related to big bang as in standard cosmology. Extensions yield limited success (unnatural conditions to get a bounce). LQG Seminar – p.5 Hope has been that higher order perturbative corrections on the continuum spacetime might be enough to capture the details of singularity resolution – limited success Ideas assume the existence of a spacetime continuum at all scales Non-perturbative quantum gravitational modifications not included Quantum nature of dynamical spacetime not captured in these approaches Do non-perturbative quantum gravitational effects resolve the big bang singularity ? LQG Seminar – p.6 Background independent non-perturbative quantum gravity – LQG is leading approach. Our Strategy: Construct a quantum cosmological model based on ¡ LQG LQC Caveats: Homogeneous and Isotropic setting Relationship with full theory unclear Hopes: Glimpses of singularity resolution and physics of very early Universe Learning/Testing ground for model building, phenomenological applications, making testable predictions Valuable insights/lessons to complete the program in LQG and other non-perturbative approaches LQG Seminar – p.7 .8 p – Seminar = LQG 9 8:9<; and ¢¤£ ¡ . ¡ 7 6 almost . ¢ theory ¦ 4 triad of setting ¢£ itt ¡ GR). © 4 ¥ of triad) -DeW 5 fiducial algebra ¡ ; ' ¨ a © the ¦ § © © § fix for , 2 Isotropic * Wheeler solutions ¨ &% ) ¡ ¢ £ generate ¨ 3 opy: ¦ ( § – the © ¨ 2 ¡ £ © ¢ ¦ and isotr ¥ § 1 ¡ om "$# satisfying Holonomies: 0 physical ¨ fr © § orientations * 4 – and ) § of ent ¡ ¡ / ( £ ¢ ¦ ¨ ! and § factor: fer +-, . ¦ § space dif basis: possible Symmetries form variables . £ ¢ scale the ¥ of to functions (two space: Space homogeneity (on ¤ ¤ £ variables: ¡ Homogeneous ¡ © § ¡ periodic co-triad Spatial Basic Elementary Elements Hilbert Orthonormal Hilbert Relation LQC: .9 p – ) Seminar " of LQG ea ar their 6 # function # ¨ the a and (' be by & " . can ¢ 9 , ¦ # $ £ ¨ " variables divided ¨ strategy % ! states. ¢ ¡ % ¨ © new loop ( o. : a a f-Inv £ ¢ 9 zer Dif elementary to 2003) £ need ound 9 on of e 4 ar W . 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LQG ¦ ¦ ¤ . the ¥ ¥¤ ¤ of steps operator ¢ ¡ § £ had triad for equation: states. the on ¥ § eigenvalues ence of all ¨¡ dering in fer or for § ¨ based ¨ dif § ¡ to ¡ factor constant definite. ' eigenvalues e & oss § leads ¡ ar constraint © in £ § acr ) natural ¥ £ £ negative § ( which Earlier § with ¨ ¨¡ . © ¥ constraint ¥ and ¥ ¤ © steps ¡ the ¡ £ ¡ ¤ in non-singular constant Constraint ¢¡ of ¡ e operator ¡ ar ¡ ¥ ¤ self-adjoint £ ¡ ¤ is e volume Evolution Evolution which ¥ ¥ ¤ ¤ ¡ £ ¡ £ ¤ ¤ Quantization wher Quantum What is the physics of singularity resolution ? What are the physical predictions in LQC ? Algorithm to extract Physics in LQC Seek a dynamical variable that can play the role of internal time (physical interpretations more transparent). Introduce Inner Product, find Physical Hilbert Space. Construct Dirac Observables, raise them to operators. Construct physical semi-classical states representing a large classical Universe. Evolve these states backward towards Big Bang using quantum constraint. Analyze the behavior of expectation values and fluctuations of Dirac observables. Compare with classical dynamics, extract predictions. LQG Seminar – p.13 .14 p – Seminar LQG ¨ 4 5*10 ¡ § © ¡ © 4 ¢ 4*10 £ ¢ § singular 4 e © 3*10 ar ¥ v © Model ¡ § # 4 © ¡ 2*10 ¥ © # solutions ield ¡ ¦ 4 F ¦ , ¤ All ¡ 1*10 © ¡ © ¡ ¤ ¤ © § © © 0 ¥ ¤ 0 -1 § © -0.2 -0.4 -0.6 -0.8 -1.2 φ Scalar £ ¢¡ ¤ space: Phase Massless .15 p – Seminar LQG time. § geometry ¥ 7 internal way of § ole the § r – . © ¥ the get: § ' ¦ & ¤ and play © § ¥ dynamics evolves). ¢ # can trick § – ¡ ¢ § as ¦ ¡ ¡ ¥ ¨¨§ , ¡ ¢¤£ 7 ¨ ¡ © ¢ (or elational © r © ¤ © function to ¡ ¢ # Thiemann's ¡ `time' § eigenvalue efers , use r ¥ , © ¦ with ¦ § ¤ Observables: monotonic ¤ ¤ © a is Dirac For For changes Evolution .16 p – Seminar LQG in © ¢ equation § © § Laplacian-type don £ of £ ¡ £ © £ ¥ § § ¡ ¢ time, ¡ £ © ¨ Klein-Gor of § ole definite. ¡ r ¡ © £ ¡ the massless ¢ the positive § plays ¡ to ¡ and ¡ ¡ constraint: ¡ similar © . spacetime. ¡ self-adjoint is £ operator Gravitational with Constraint static .17 p – Seminar No LQG the ¨ ¡ of natural ¡ © ¡ the states. solutions. © of ¡ dering © © yields or orientation in space ¥ symmetric factor e the ¥ ¡ . of itt. on change © constraint: equir r ¡ ¨ constraint: automatically the ¤£¢ -DeW choice ¥ the ¡ £ ¡ classical detect for © B natural ¢ Wheeler constraint . can transformation ¡ limit operator: ¡ for considerations itt theory which , odynamics gauge for parity ¥ dering -DeW ge of or Physical Hamiltonian lar ¡ geometr quantum ¢ ¤ triad. observable Wheeler factor LQC For In In Laplacian Action .18 p – Seminar to LQG , + © Dirac © ¨ © £ to ¡ insensitive ¥§¦ © +¤£ ¡ ¡ ¡ ¡ esponding * ¡ ¢ edictions £ ¡ © corr Pr sectors: . £ ¡ § ¦ into © equency: fr operators ¥ ¡ ¡ of ¤ divided as: ¡ © on 7 positive act self-adjoint ¡ e action space is ar * 5 sectors. that support © of averaging ¡ Hilbert . ¨ oduct: have © ¡ ¤ observables oup Pr ¨ © choice Demand Gr observables ' the States Inner Result: Dirac Physical .19 p – Seminar a LQG Fix . © ¡ its ge Initial equations. 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