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New Developments and Perspectives in General Relativity and Cosmology Thesis Dennis Smoot University of Illinois Chicago, IL

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Table of Contents Chapter 1 The Classical Einstein-Hilbert Action 1.1 Structure of the Einstein-Hilbert Lagrangian 1.2 The Relation between the Bulk and Surface Terms 1.3 Thermodynamic Derivation of the Einstein-Hilbert Action 1.4 Thermodynamic Interpretation of the Einstein Field Equations 1.5 Curvature Chapter 2 Statistical Mechanics of Gravitating Systems Chapter 3 The Quantum Mechanical Perspective 3.1 Hawking Radiation 3.2 Unruh Effect Chapter 4 Relativistic Formalism 4.1 Generalized Actions and Entropy Functionals 4.2 Gravitational Energy Densities in the Universe

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Chapter 6 A History of the Early Universe 5.1 Introduction Newtonian Cosmology 5.2 The Radiation Era Relativistic Cosmology The Planck Epoch The String Epoch The GUT Epoch The Inflation Epoch The Electroweak Epoch The Parton Epoch The Hadron Epoch The Lepton Epoch The Nuclear Epoch 5.3 The Matter Era The Atomic Epoch 5.4 The Vacuum Era 5.5 Further Thermodynamics of the Early Universe 5.6 Free Energy in the Universe Chapter 7 Discussion and Conclusion 6.1 The first part: Gravitation 6.2 The second part: History of the Universe

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Louis Kauffman, Professor Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago A. Lewis Licht, Professor Emeritus Department of Physics University of Illinois at Chicago Advisors

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Special Relativity Einstein 1905 General Relativity Einstein 1915 Goal: Quantum Gravity Quantum Mechanics Quantum Field Theory Loop Gravity Ashtekar 1988 Strings History Modern Theories Standard Model of Particles Quantum Field Theory in Curved SpaceTime Quantum Gravity nonrenormalizable First Part. Gravitation and General Relativity Schrodinger, Heisenberg, Born 1925 Dirac, Feynman

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Can Gravity be united, combined with the other Forces What are the (basic) Degrees of Freedom What is the underlying Mechanism What are the relevant Observations, Experiments

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Thanu Padamanabhan Professor and Dean of Core Academic Programs of Inter-University Center for Astronomy and Astrophysics (IUCAA) at Pune, India books More than 200 research papers General Relativity videos on Web

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Relation between Bulk and Surface terms

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Conventional Action Surface term is ignored, subtracted off in modified Lagrangian. Lagrangian is covariant (under diffeomorphisms), but neither the bulk nor the surface are. As just shown there is a differential relation between the bulk and surface Lagrangians. In inertial coördinates and. The conclusion is that the degrees of freedom (dof) of Gravity are located on the boundary of a Region and not in the Bulk.

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Thermodynamic Derivation of the Einstein-Hilbert Action Suppose the only known is the relation between the bulk and surface terms that was quadratic in the first derivatives of the metric. Then it is possible to determine. Can be done in some simpler more tractable cases, e.g. static or stationary SpaceTimes. Require assumptions that the Entropy S is the 2-surface area and that Then it can be shown that. where is the 2-surface area.

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Thermodynamic Interpretation of Einsteins Equations Again this is done in tractable cases, but turns out in all cases that can be calculated. Static spherically symmetric metric The Einstein Equations (EE) reduce to The equations are solved using the boundary conditions of the horizon giving whereand is the horizon. Multiplying by is the Stress-Energy tensor

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Statistical Mechanics of Gravitating Systems g is Density of States Z(β) is the Partition Function Σ is a Path Integral Sum is Euclidean extension of R M is the microcanonical ensemble E-H Action is FE of spacetime

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Hawking Radiation: a BH radiates in a thermal spectrum Quantum Mechanics 1. Schwarzschild metric 2. Hamilton-Jacobi equation 3. Wave Function 4. Fourier decompose 5. Solve for Power Spectrum of 6. The latter is a Planckian thermal spectrum with

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Unruh Effect: a vacuum wrst an accelerated observer will contain a thermal spectrum of particles. Quantum Mechanics contd Vacuum state a massless scalar field, φ(u,w)= φ(û,ŵ) in light cone coördinates (u,w) Flat Minkowski space with coördinates (t, x), accelerated observer coördinates (τ, ξ) Action eom Solutions in both coördinates Fourier Transform and compare coëfficients: Bogoliubov transformations relating the creation and annihilation operators Observer particle number operator is Power spectrum is a thermal spectrum with temperature Particular solution in both coördinates

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Relativistic Formalism Generalized Actions and Entropy Functionals Dark Energy Fluid, unclustered, nonclumping, negative pressure Equation of state Problem: Cosmological Constant Neither the EH Action nor the EE are invariant under a shift by a constant in the Lagrangians But the matter Lagrangian and the matter eom are invariant under a shift by a constant Padmanabhan: Neither the DE nor the CC problem will be solved until these discrepancies are resolved. Related facts: separation of bulk and surface terms, differential relation between bulk and surface terms, noncovariance of terms, neglect of surface term, bulk term = 0 in inertial frame, light cone (sphere) structure, the horizons of accelerated observers, horizons of black holes, entanglement across horizons or tunneling through horizons.

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Relativistic Formalism, contd Modified eom For all null vectors 2 proofs:1.Using conserved Noether current 2.Using Elastic Entropy functional Now the Action and the eom are invariant under a shift by a constant On shell (eom are satisfied) Entropy Functional above is entropy of horizon agreeing with Wald Lagrangian special case: Lanczos-Lovelock lagrangians But the eom contain an arbitrary constant integration Λ It can be shown LL Lagrangians are the unique lagrangians constructed from the metric and curvature tensors, satisfying the PE, and containing no derivatives of the metric above 2nd order.

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Gravitational Energy Densities in the Universe Relativistic formalism, contd CC decouples from dof in Actions Set gauge = 0 Then DE gravity is surface not bulk Early Universe phase transitions Casimir effect too small Conclusion: Gravity ignores Bulk Energy. Detectors respond to fluctuations Unruh-Dewitt: Gravity: Inflation: energy fluctuations couple to gravity Analysis of length scales shows surface dimensions have correct energy values Conclusion: CC or DE is quantum fluctuations of Universes boundary.

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Analysis of some Padmanabhan Results The differential relation between the bulk and surface terms The Göckeler and Schücker Gauge Formulation of General Relativity D is the exterior covariant derivative, d the differential, ω the connection (potential) gl 4 1-form, the wedge product, R the curvature gl 4 2-form, and the s are an oriented basis of the cotangent space, they are 4 valued 1-forms, is the Hodge duality operator, a,b = 0…3, the metric is g ab raises and lowers indices and has signature +---, and the integration domain. Also

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To reduce the S GS [, ] to S EH [dx, ] assumptions of GR s are holonomic, a = dx a T = 0, vanishing torsion R ab cd R ab ab = R, scalar the gauge is fixed, i.e. particular coördinates are chosen where and

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Padmanabhans differential Bulk-Surface relation Substituting Splits into 2 terms Transforming to exact differential Substituting

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Padmanabhans differential Bulk-Surface relation, contd Stokes Theorem Last term above is a Surface term Now if there is a differential relation between the bulk and surface

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Padmanabhans differential Bulk-Surface relation, contd This is a condition on d This is true irregardless of whether assumptions are made for d or d -g. These results follow only from the GS gauge formulation of GR. Again if there is such a differential bulk-surface relation then the bulk and surface terms contain the same information, one being the differential of the other. Check if bulk and surface terms are generally covariant; check conditions when bulk and surface terms=0

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Reduction of above GS differential bulk-surface relation to Padmanabhans terms It has already been shown above that S GS [, ] reduces to S EH [dx, ] Reiterating differential geometric GR Substituting This is at the level of GR except it has not been contracted.

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Reduction of above GS differential bulk-surface relation to Padmanabhans terms, contd Applying Stokes Theorem and setting Claim: The respective terms will reduce to Padmanabhans bulk and surface terms bulk: surface:

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Reduction of above GS differential bulk-surface relation to Padmanabhans terms, contd bulk:surface: Pads bulk: 2nd term in Pads bulk: except there is 2 of them cancels one of the previous terms Pad surface: 1st term:

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Implications of Bulk Surface Relations Then This expresses the curvature R in terms of d. C a bc antisymmetric Case i. Within GR

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Implications of Bulk Surface Relations, contd Case ii. Outside GR Now T = 0 If the bulk-surface relation holds Places a condition on the torsion, T Some observations indicate the Torsion in order of magnitude

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Padmanabhan and Paranjapes Entropy Functional based on Elasticity This becomes Writing out in full Interchanging indices basis vectors

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Goal: History of Matter Radiation Era Matter Era Strings Goal: Degrees of freedom, mechanisms; Products, Relics Eras and Epochs Second Part. A History of the Early Universe Vacuum Era Planck Epoch String Epoch GUT Epoch Inflation Epoch Electroweak Epoch Parton Epoch Hadron Epoch Lepton Epoch Nuclear Epoch Atomic Epoch Newtonian Gravity, General Relativity

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Hubble expansion, 1925 homogeneous and isotropic Universe, 100 Mpc Robertson-Walker metric, 1935 big bang nucleosynthesis (BBN), Gamov & Alpher, 1948 cosmic microwave background radiation (CMBR), Penzias & Wilson, 1965 general relativity (GR), Einstein, 1915 standard model of particles (SMP), 1970 Theory and Observations

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Basic Equations Robertson-Walker(RW) metric Einstein equations Stress-Energy tensor Friedmann-Lemaitre(FL) equations expansion parametercurvature parameter RW metric conformal to Minkowski metric when

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Radiation Era Planck Epoch Theory of Everything(TOE) or Superunification Epoch Planck time Planck length Planck mass Fundamental Constants Planck temperature Gravitation, General Relativity Statistical Mechanics, Thermodynamics Special RelativityQuantum Mechanics Time-Temperature relation is dof

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reaction rate of gravitons at this time, Binetruy common theme Planck Epoch, contd Reaction rates Expansion rate Products, Relicsannihilation, bound, decoupled, go out of equilibrium, no longer created so gravitons decouple, go out of equilibrium, and form presumably CGBR; not detected examples

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GUT Epoch String Epoch pre-Big BangFutureD-branes, D dimensional objects dof The first use of group theory, enlarged groups Extends gauge theory to very high energies Lagrangian formulation SUSY prediction: LSP good candidate for DM; CERN Hierarchy of couplings explained by screening and antiscreening (asymptotic freedom) of unified couplings Gauge hierarchy

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Inflation Epoch Resolves BB problems: Flatness, why the Universe is so flat or so close to the critical density Horizon, why is the Universe so homogeneous when the regions are too far apart to be in causal contact. Monopoles, unobserved prediction of GUTs; diluted by expansion Tiny scale-invariant fluctuations, perturbations for later gravitational collapse Old Inflation; new Inflation reheating; inflaton decay Lasted for withand

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Electroweak Epoch SMP, at low end SMP complete for ordinary physics, but incomplete Separate couplings,, no Higgs, masses = 0, EW corrections, Higgs mass also divergent, fine tuning of Higgs mass Or Supersymmetric Epoch Strong force separates at beginning, Weak force separates at end New Physics SUSY graded Lie algebra of bosons and fermions in [, ], {, } SUSY only nontrivial extension of spacetime symmetries, largest spacetime symmetry of S-Matrix Symmetry breaking: kinetic: like Higgs OR dynamic: like SUSY, bosons, fermions; or form bound systems (recent CERN )

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Electroweak Epoch, contd Haag-Lopuszanski-Sohnius(HLS) Theorem. The largest symmetry of an interacting, unitary field theory is the direct product of a (possibly very large) gauge symmetry, a Lorentz invariance, and a (possibly extended) supersymmetry SUSY is also incomplete SUSY evades CM Theorem being a Lie SuperAlgebra Local SUSY implies graviton, mSUGRA; fewer parameters, more predictive, dynamical SB SUSY resolves Higgs mass problems, preserves hierarchy, MSSM coupling, Desert, DM SUSY not observed: degenerate masses, LSP, CERN; is a broken symmetry soft symmetry breaking at TeV scale Coleman-Mandula(CM) Thm. All possible Lie Algebra symmetries of the S-matrix under general assumptions can only be a direct product of the Poincare algebra and an internal symmetry algebra. the pattern of fermion masses and mixings, the replication of generations, the origin of CP violation. Causes new problems: baryon and lepton number, supersymmetric parameter μ M Weak, origin of SUSY breaking, why are SUSY breaking parameters < L P, CP violation of SUSY so small.

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Parton Epoch thermal distribution function for the Bose-Einstein(B-E) or Fermi-Dirac(F-D) equilibrium distributions 4 known forces of today all fundamental particles acquire masses via Higgs mechanism dof = , the SMP dof heavy and light particles initially in equilibrium later heaviest particles condense out at dof: next slide again: particle condensation (creation); structure, order decomposition (annihilation) much better understood physics:, partons; however no cosmological evidence

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Parton Epoch, contd thick blue: flat gray: linear thin blue: exponential Matter-Antimatter asymmetry Very weakly understood particle physics and no cosmological observational evidence: Baryogenesis Leptogenesis

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Hadron Epoch Universe: hot; high entropy; ratio of photons to baryons Parton confinement Leptogenesis Moments Special cases bosons, fermions number density energy density pressure highly relativistic, T m, nondegenerate, T nonrelativistic, T m Coupled Decoupled highly relativistic, T D m nonrelativistic, T D m

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Hadron Epoch, contd Boltzmann Equation derivation i. Louisville and Collision Operator ii. geodesic equation iii. Γs from RW metric v. Boltzmann Equationiv. number density GR numerical data simplifying assumptions Boltzmann equation Classical and SR

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Lepton Epoch & 3 species of ν heavier particles condensing out lepton-antilepton annihilation;--lepton residue neutrinos decouple initially at same T as γ but as T T of γ dof argument, assuming constant S, entropy CBNR not yet detectable, like the gravitons, CBGR can also consider μ, τ, the other ν

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Nuclear Epoch binding energies, MeV DTHe-3He Synthesis 0.1MeV Neutron decay, exponential Earliest, most rigorous, best understood High entropy Weak reactions cannot maintain equilibrium Nucleosynthesis starts: T ~ MeV, and neutron fraction/total nucleons = 1/6.

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Matter Era Atomic Epoch Matter expansion overtakes radiation expansion Formation of neutral H, He For T < 0.1MeV the main constituents H and He-4 nuclei, and the decoupled ν Recombination Combination Photons no longer EM interacting, decouple, form CMBR Once again the latter is delayed due to high Entropy of Universe; mechanism known, phenomena can be calculated

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Vacuum Era Dark Energy, ~ 70% Free Energy Today, 13.9 Gyr after the Big Bang Further Thermodynamics of the Early Universe Dark Matter, 24-25%Concerns: Kinetics: e.g. Boltzmann Eqn, reaction rates vs expansion rates Dynamics: e.g. boson fermion cancellation in corrections to Higgs mass, not put in by hand, energetic and preferable. primordial vacuum-inflaton fluctuations seed gravitational collapse to form nebula,stars,galaxies Symmetry and Order Order and Control parameters Equilibrium; Phase transitions today: nucleons past: + in Planck epoch, - all other times

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Conclusions First part: Gravitation and GR PE geometric kinetic curvature QG: dof of gravity: gravitons; CGBR Rindler observers accelerated frames horizons Relativity, Thermodynamics, QM temperature, entropy of horizons E-H Action and EE derived in variety of ways; independent of metric g αβ (neither is T αβ ), metric not varied Relation between bulk surface terms in Action Thermodynamic derivation of E-H Action Thermodynamic interpretation of EE E-H Action is FE of spacetime Hawking radiation: BH radiate in thermal spectrum Unruh effect: accelerated observers find vacuum is thermal particle spectrum

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Conclusions, contd Second part: Early history of the Universe The SMP & Lagrangian gauge theory generalizations form the basis of the SMC 3 Eras: Radiation, Matter, and Vacuum (Near) equilibrium and nonequilibrium (phase) transitions due to expansion The Universe is also expanding and its composition depends on a comparison between the rates of expansion and reaction Gauge hierarchy explained by screening and antiscreening Symmetry, symmetry breaking, (phase) transitions, order (bound states)

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