Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 016, 73 pages      arXiv:1109.6801      https://doi.org/10.3842/SIGMA.2012.016
Contribution to the Special Issue “Loop Quantum Gravity and Cosmology”

Introduction to Loop Quantum Cosmology

Kinjal Banerjee a, Gianluca Calcagni b and Mercedes Martín-Benito b
a) Department of Physics, Beijing Normal University, Beijing 100875, China
b) Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, D-14476 Golm, Germany

Received September 30, 2011, in final form March 13, 2012; Published online March 25, 2012

Abstract
This is an introduction to loop quantum cosmology (LQC) reviewing mini- and midisuperspace models as well as homogeneous and inhomogeneous effective dynamics.

Key words: loop quantum cosmology; loop quantum gravity.

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References

  1. Arnowitt R., Deser S., Misner C.W., The dynamics of general relativity, in Gravitation: an Introduction to Current Research, Wiley, New York, 1962, 227-265.
  2. Ashtekar A., An introduction to loop quantum gravity through cosmology, Nuovo Cim. 122 (2007), 135-155, gr-qc/0702030.
  3. Ashtekar A., Lectures on nonperturbative canonical gravity, Advanced Series in Astrophysics and Cosmology, Vol. 6, World Scientific Publishing Co. Inc., River Edge, NJ, 1991.
  4. Ashtekar A., Loop quantum cosmology: an overview, Gen. Relativity Gravitation 41 (2009), 707-741, arXiv:0812.0177.
  5. Ashtekar A., New Hamiltonian formulation of general relativity, Phys. Rev. D 36 (1987), 1587-1602.
  6. Ashtekar A., New variables for classical and quantum gravity, Phys. Rev. Lett. 57 (1986), 2244-2247.
  7. Ashtekar A., Baez J., Corichi A., Krasnov K., Quantum geometry and black hole entropy, Phys. Rev. Lett. 80 (1998), 904-907, gr-qc/9710007.
  8. Ashtekar A., Baez J.C., Krasnov K., Quantum geometry of isolated horizons and black hole entropy, Adv. Theor. Math. Phys. 4 (2000), 1-94, gr-qc/0005126.
  9. Ashtekar A., Bojowald M., Lewandowski J., Mathematical structure of loop quantum cosmology, Adv. Theor. Math. Phys. 7 (2003), 233-268, gr-qc/0304074.
  10. Ashtekar A., Campiglia M., Henderson A., Casting loop quantum cosmology in the spin foam paradigm, Classical Quantum Gravity 27 (2010), 135020, 32 pages, arXiv:1001.5147.
  11. Ashtekar A., Campiglia M., Henderson A., Loop quantum cosmology and spin foams, Phys. Lett. B 681 (2009), 347-352, arXiv:0909.4221.
  12. Ashtekar A., Corichi A., Singh P., Robustness of key features of loop quantum cosmology, Phys. Rev. D 77 (2008), 024046, 17 pages, arXiv:0710.3565.
  13. Ashtekar A., Isham C.J., Representations of the holonomy algebras of gravity and nonabelian gauge theories, Classical Quantum Gravity 9 (1992), 1433-1467.
  14. Ashtekar A., Lewandowski J., Background independent quantum gravity: a status report, Classical Quantum Gravity 21 (2004), R53-R152, gr-qc/0404018.
  15. Ashtekar A., Lewandowski J., Projective techniques and functional integration for gauge theories, J. Math. Phys. 36 (1995), 2170-2191, gr-qc/9411046.
  16. Ashtekar A., Lewandowski J., Quantum theory of geometry. I. Area operators, Classical Quantum Gravity 14 (1997), A55-A81, gr-qc/9602046.
  17. Ashtekar A., Lewandowski J., Representation theory of analytic holonomy C*-algebras, in Knots and quantum gravity (Riverside, CA, 1993), Oxford Lecture Ser. Math. Appl., Vol. 1, Oxford Univ. Press, New York, 1994, 21-61, gr-qc/9311010.
  18. Ashtekar A., Lewandowski J., Marolf D., Mourão J., Thiemann T., Quantization of diffeomorphism invariant theories of connections with local degrees of freedom, J. Math. Phys. 36 (1995), 6456-6493, gr-qc/9504018.
  19. Ashtekar A., Pawlowski T., Singh P., Quantum nature of the big bang, Phys. Rev. Lett. 96 (2006), 141301, 4 pages, gr-qc/0602086.
  20. Ashtekar A., Pawlowski T., Singh P., Quantum nature of the big bang: an analytical and numerical investigation, Phys. Rev. D 73 (2006), 124038, 33 pages, gr-qc/0604013.
  21. Ashtekar A., Pawlowski T., Singh P., Quantum nature of the big bang: improved dynamics, Phys. Rev. D 74 (2006), 084003, 23 pages, gr-qc/0607039.
  22. Ashtekar A., Pawlowski T., Singh P., Vandersloot K., Loop quantum cosmology of k=1 FRW models, Phys. Rev. D 75 (2007), 024035, 26 pages, gr-qc/0612104.
  23. Ashtekar A., Pullin J., Bianchi cosmologies: a new description, in Developments in General Relativity, Astrophysics and Quantum Theory (Jerusalem and Haifa, 1989), Ann. Israel Phys. Soc., Vol. 9, IOP, Bristol, 1990, 65-76.
  24. Ashtekar A., Singh P., Loop quantum cosmology: a status report, Classical Quantum Gravity 28 (2011), 213001, 122 pages, arXiv:1108.0893.
  25. Ashtekar A., Wilson-Ewing E., Loop quantum cosmology of Bianchi type I models, Phys. Rev. D 79 (2009), 083535, 21 pages, arXiv:0903.3397.
  26. Ashtekar A., Wilson-Ewing E., Loop quantum cosmology of Bianchi type II models, Phys. Rev. D 80 (2009), 123532, 16 pages, arXiv:0910.1278.
  27. Baez J.C., Generalized measures in gauge theory, Lett. Math. Phys. 31 (1994), 213-223, hep-th/9310201.
  28. Banerjee K., Date G., Loop quantization of the polarized Gowdy model on T3: classical theory, Classical Quantum Gravity 25 (2008), 105014, 15 pages, arXiv:0712.0683.
  29. Banerjee K., Date G., Loop quantization of the polarized Gowdy model on T3: kinematical states and constraint operators, Classical Quantum Gravity 25 (2008), 145004, 18 pages, arXiv:0712.0687.
  30. Barbero G. J.F., Real Ashtekar variables for Lorentzian signature space-times, Phys. Rev. D 51 (1995), 5507-5510, gr-qc/9410014.
  31. Barbero G. J.F., Villaseñor E.J.S., Quantization of midisuperspace models, Living Rev. Relativ. 13 (2010), 6, 55 pages, arXiv:1010.1637.
  32. Bentivegna E., Pawlowski T., Anti-de Sitter universe dynamics in loop quantum cosmology, Phys. Rev. D 77 (2008), 124025, 17 pages, arXiv:0803.4446.
  33. Berger B.K., Quantumgravitoncreation in a model universe, Ann. Physics 83 (1974), 458-490.
  34. Berger B.K., Quantum cosmology: exact solution for the Gowdy T3 model, Phys. Rev. D 11 (1975), 2770-2780.
  35. Berger B.K., Quantum effects in the Gowdy T3 cosmology, Ann. Physics 156 (1984), 155-193.
  36. Bianchi E., Krajewski T., Rovelli C., Vidotto F., Cosmological constant in spinfoam cosmology, Phys. Rev. D 83 (2011), 104015, 4 pages, arXiv:1101.4049.
  37. Bianchi E., Rovelli C., Vidotto F., Towards spinfoam cosmology, Phys. Rev. D 82 (2010), 084035, 8 pages, arXiv:1003.3483.
  38. Bojowald M., Absence of a singularity in loop quantum cosmology, Phys. Rev. Lett. 86 (2001), 5227-5230, gr-qc/0102069.
  39. Bojowald M., Consistent loop quantum cosmology, Classical Quantum Gravity 26 (2009), 075020, 10 pages, arXiv:0811.4129.
  40. Bojowald M., Homogeneous loop quantum cosmology, Classical Quantum Gravity 20 (2003), 2595-2615, gr-qc/0303073.
  41. Bojowald M., Inverse scale factor in isotropic quantum geometry, Phys. Rev. D 64 (2001), 084018, 8 pages, gr-qc/0105067.
  42. Bojowald M., Isotropic loop quantum cosmology, Classical Quantum Gravity 19 (2002), 2717-2741, gr-qc/0202077.
  43. Bojowald M., Large scale effective theory for cosmological bounces, Phys. Rev. D 75 (2007), 081301, 5 pages, gr-qc/0608100.
  44. Bojowald M., Loop quantum cosmology, Living Rev. Relativ. 11 (2008), 4, 131 pages.
  45. Bojowald M., Loop quantum cosmology and inhomogeneities, Gen. Relativity Gravitation 38 (2006), 1771-1795, gr-qc/0609034.
  46. Bojowald M., Loop quantum cosmology. I. Kinematics, Classical Quantum Gravity 17 (2000), 1489-1508, gr-qc/9910103.
  47. Bojowald M., Loop quantum cosmology. II. Volume operators, Classical Quantum Gravity 17 (2000), 1509-1526, gr-qc/9910104.
  48. Bojowald M., Loop quantum cosmology. III. Wheeler-DeWitt operators, Classical Quantum Gravity 18 (2001), 1055-1069, gr-qc/0008052.
  49. Bojowald M., Loop quantum cosmology. IV. Discrete time evolution, Classical Quantum Gravity 18 (2001), 1071-1087, gr-qc/0008053.
  50. Bojowald M., Loop quantum cosmology: recent progress, Pramana 63 (2004), 765-776, gr-qc/0402053.
  51. Bojowald M., Nonsingular black holes and degrees of freedom in quantum gravity, Phys. Rev. Lett. 95 (2005), 061301, 4 pages, gr-qc/0506128.
  52. Bojowald M., Quantization ambiguities in isotropic quantum geometry, Classical Quantum Gravity 19 (2002), 5113-5129, gr-qc/0206053.
  53. Bojowald M., Quantum nature of cosmological bounces, Gen. Relativity Gravitation 40 (2008), 2659-2683, arXiv:0801.4001.
  54. Bojowald M., Spherically symmetric quantum geometry: states and basic operators, Classical Quantum Gravity 21 (2004), 3733-3753, gr-qc/0407017.
  55. Bojowald M., Calcagni G., Inflationary observables in loop quantum cosmology, J. Cosmol. Astropart. Phys. 2011 (2011), no. 3, 032, 35 pages, arXiv:1011.2779.
  56. Bojowald M., Calcagni G., Tsujikawa S., Observational constraints on loop quantum cosmology, Phys. Rev. Lett. 107 (2011), 21130, 5 pages, arXiv:1101.5391.
  57. Bojowald M., Calcagni G., Tsujikawa S., Observational test of inflation in loop quantum cosmology, J. Cosmol. Astropart. Phys. 2011 (2011), no. 11, 046, 32 pages, arXiv:1107.1540.
  58. Bojowald M., Cartin D., Khanna G., Lattice refining loop quantum cosmology, anisotropic models, and stability, Phys. Rev. D 76 (2007), 064018, 13 pages, arXiv:0704.1137.
  59. Bojowald M., Date G., Vandersloot K., Homogeneous loop quantum cosmology: the role of the spin connection, Classical Quantum Gravity 21 (2004), 1253-1278, gr-qc/0311004.
  60. Bojowald M., Harada T., Tibrewala R., Lemaitre-Tolman-Bondi collapse from the perspective of loop quantum gravity, Phys. Rev. D 78 (2008), 064057, 30 pages, arXiv:0806.2593.
  61. Bojowald M., Hernández H., Skirzewski A., Effective equations for isotropic quantum cosmology including matter, Phys. Rev. D 76 (2007), 063511, 24 pages, arXiv:0706.1057.
  62. Bojowald M., Hossain G.M., Cosmological vector modes and quantum gravity effects, Classical Quantum Gravity 24 (2007), 4801-4816, arXiv:0709.0872.
  63. Bojowald M., Hossain G.M., Loop quantum gravity corrections to gravitational wave dispersion, Phys. Rev. D 77 (2008), 023508, 14 pages, arXiv:0709.2365.
  64. Bojowald M., Hossain G.M., Kagan M., Shankaranarayanan S., Anomaly freedom in perturbative loop quantum gravity, Phys. Rev. D 78 (2008), 063547, 31 pages, arXiv:0806.3929.
  65. Bojowald M., Hossain G.M., Kagan M., Shankaranarayanan S., Gauge invariant cosmological perturbation equations with corrections from loop quantum gravity, Phys. Rev. D 79 (2009), 043505, 21 pages, Erratum, Phys. Rev. D 82 (2010), 109903, arXiv:0811.1572.
  66. Bojowald M., Kagan M., Hernández H.H., Skirzewski A., Effective constraints of loop quantum gravity, Phys. Rev. D 75 (2007), 064022, 25 pages, gr-qc/0611112.
  67. Bojowald M., Lidsey J.E., Mulryne D.J., Singh P., Tavakol R., Inflationary cosmology and quantization ambiguities in semiclassical loop quantum gravity, Phys. Rev. D 70 (2004), 043530, 14 pages, gr-qc/0403106.
  68. Bojowald M., Swiderski R., Spherically symmetric quantum geometry: Hamiltonian constraint, Classical Quantum Gravity 23 (2006), 2129-2154, gr-qc/0511108.
  69. Bojowald M., Swiderski R., The volume operator in spherically symmetric quantum geometry, Classical Quantum Gravity 21 (2004), 4881-4900, gr-qc/0407018.
  70. Bojowald M., Tavakol R., Recollapsing quantum cosmologies and the question of entropy, Phys. Rev. D 78 (2008), 023515, 12 pages, arXiv:0803.4484.
  71. Bojowald M., Vandersloot K., Loop quantum cosmology, boundary proposals, and inflation, Phys. Rev. D 67 (2003), 124023, 10 pages, gr-qc/0303072.
  72. Bonzom V., Laddha A., Lessons from toy-models for the dynamics of loop quantum gravity, SIGMA 8 (2012), 009, 50 pages, arXiv:1110.2157.
  73. Brizuela D., Mena Marugán G.A., Pawlowski T., Big bounce and inhomogeneities, Classical Quantum Gravity 27 (2010), 052001, 8 pages, arXiv:0902.0697.
  74. Brizuela D., Mena Marugán G.A., Pawlowski T., Effective dynamics of the hybrid quantization of the Gowdy T3 universe, Phys. Rev. D 84 (2011), 124017, 21 pages, arXiv:1106.3793.
  75. Cailleteau T., Barrau A., Gauge invariance in loop quantum cosmology: Hamilton-Jacobi and Mukhanov-Sasaki equations for scalar perturbations, arXiv:1111.7192.
  76. Cailleteau T., Mielczarek J., Barrau A., Grain J., Anomaly-free scalar perturbations with holonomy corrections in loop quantum cosmology, arXiv:1111.3535.
  77. Calcagni G., Classical and quantum cosmology, unpublished.
  78. Calcagni G., Cortês M., Inflationary scalar spectrum in loop quantum cosmology, Classical Quantum Gravity 24 (2007), 829-853, gr-qc/0607059.
  79. Calcagni G., Gielen S., Oriti D., Group field cosmology: a cosmological field theory of quantum geometry, arXiv:1201.4151.
  80. Calcagni G., Gielen S., Oriti D., Two-point functions in (loop) quantum cosmology, Classical Quantum Gravity 28 (2011), 125014, 25 pages, arXiv:1011.4290.
  81. Calcagni G., Hossain G.M., Loop quantum cosmology and tensor perturbations in the early universe, Adv. Sci. Lett. 2 (2009), 184-193, arXiv:0810.4330.
  82. Campiglia M., Gambini R., Pullin J., Loop quantization of spherically symmetric midi-superspaces, Classical Quantum Gravity 24 (2007), 3649-3672, gr-qc/0703135.
  83. Campiglia M., Gambini R., Pullin J., Loop quantization of spherically symmetric midi-superspaces: the interior problem, AIP Conf. Proc. 977 (2008), 52-63, arXiv:0712.0817.
  84. Chiou D.W., Effective dynamics, big bounces, and scaling symmetry in Bianchi type I loop quantum cosmology, Phys. Rev. D 76 (2007), 124037, 19 pages, arXiv:0710.0416.
  85. Chiou D.W., Loop quantum cosmology in Bianchi type I models: analytical investigation, Phys. Rev. D 75 (2007), 024029, 33 pages, gr-qc/0609029.
  86. Copeland E.J., Mulryne D.J., Nunes N.J., Shaeri M., Gravitational wave background from superinflation in loop quantum cosmology, Phys. Rev. D 79 (2009), 023508, 8 pages, arXiv:0810.0104.
  87. Copeland E.J., Mulryne D.J., Nunes N.J., Shaeri M., Superinflation in loop quantum cosmology, Phys. Rev. D 77 (2008), 023510, 11 pages, arXiv:0708.1261.
  88. Corichi A., Cortez J., Mena Marugán G.A., Quantum Gowdy T3 model: a unitary description, Phys. Rev. D 73 (2006), 084020, 17 pages, gr-qc/0603006.
  89. Corichi A., Cortez J., Mena Marugán G.A., Unitary evolution in Gowdy cosmology, Phys. Rev. D 73 (2006), 041502, 5 pages, gr-qc/0510109.
  90. Corichi A., Cortez J., Mena Marugán G.A., Velhinho J.M., Quantum Gowdy T3 model: a uniqueness result, Classical Quantum Gravity 23 (2006), 6301-6319, gr-qc/0607136.
  91. Corichi A., Cortez J., Quevedo H., On unitary time evolution in Gowdy T3 cosmologies,Internat. J. Modern Phys. D 11 (2002), 1451-1468, gr-qc/0204053.
  92. Corichi A., Singh P., Is loop quantization in cosmology unique?, Phys. Rev. D 78 (2008), 024034, 13 pages, arXiv:0805.0136.
  93. Corichi A., Singh P., Quantum bounce and cosmic recall, Phys. Rev. Lett. 100 (2008), 161302, 4 pages, arXiv:0710.4543.
  94. Cortez J., Mena Marugán G.A., Feasibility of a unitary quantum dynamics in the Gowdy T3 cosmological model, Phys. Rev. D 72 (2005), 064020, 14 pages, gr-qc/0507139.
  95. Cortez J., Mena Marugán G.A., Olmedo J., Velhinho J.M., A unique Fock quantization for fields in non-stationary spacetimes, J. Cosmol. Astropart. Phys. 2010 (2010), no. 10, 030, 11 pages, arXiv:1004.5320.
  96. Cortez J., Mena Marugán G.A., Olmedo J., Velhinho J.M., Uniqueness of the Fock quantization of fields with unitary dynamics in nonstationary spacetimes, Phys. Rev. D 83 (2011), 025002, 13 pages, arXiv:1101.2397.
  97. Cortez J., Mena Marugán G.A., Velhinho J.M., Fock quantization of a scalar field with time dependent mass on the three-sphere: unitarity and uniqueness, Phys. Rev. D 81 (2010), 044037, 13 pages, arXiv:1001.0946.
  98. Cortez J., Mena Marugán G.A., Velhinho J.M., Uniqueness of the Fock quantization of the Gowdy T3 model, Phys. Rev. D 75 (2007), 084027, 14 pages, gr-qc/0702117.
  99. DeWitt B.S., Quantum theory of gravity. I. The canonical theory, Phys. Rev. 160 (1967), 1113-1148.
  100. Ding Y., Ma Y., Yang J., Effective scenario of loop quantum cosmology, Phys. Rev. Lett. 102 (2009), 051301, 4 pages, arXiv:0808.0990.
  101. Dittrich B., Partial and complete observables for canonical general relativity, Classical Quantum Gravity 23 (2006), 6155-6184, gr-qc/0507106.
  102. Dittrich B., Partial and complete observables for Hamiltonian constrained systems, Gen. Relativity Gravitation 39 (2007), 1891-1927, gr-qc/0411013.
  103. Gambini R., Pullin J., Diffeomorphism invariance in spherically symmetric loop quantum gravity, Adv. Sci. Lett. 2 (2009), 251-254, arXiv:0807.4748.
  104. Garay L.J., Martín-Benito M., Mena Marugán G.A., Inhomogeneous loop quantum cosmology: hybrid quantization of the Gowdy model, Phys. Rev. D 82 (2010), 044048, 17 pages, arXiv:1005.5654.
  105. Geroch R., A method for generating solutions of Einstein's equations, J. Math. Phys. 12 (1971), 918-924.
  106. Gielen S., Oriti D., Discrete and continuum third quantization of gravity, arXiv:1102.2226.
  107. Gowdy R.H., Gravitational waves in closed universes, Phys. Rev. Lett. 27 (1971), 826-829.
  108. Gowdy R.H., Vacuum spacetimes with two-parameter spacelike isometry groups and compact invariant hypersurfaces: topologies and boundary conditions, Ann. Physics 83 (1974), 203-241.
  109. Grain J., Barrau A., Cosmological footprints of loop quantum gravity, Phys. Rev. Lett. 102 (2009), 081301, 4 pages, arXiv:0902.0145.
  110. Grain J., Barrau A., Cailleteau T., Mielczarek J., Observing the big bounce with tensor modes in the cosmic microwave background: phenomenology and fundamental loop quantum cosmology parameters, Phys. Rev. D 82 (2010), 123520, 12 pages, arXiv:1011.1811.
  111. Han M., Ma Y., Huang W., Fundamental structure of loop quantum gravity, Internat. J. Modern Phys. D 16 (2007), 1397-1474, gr-qc/0509064.
  112. Hartle J.B., Hawking S.W., Wave function of the universe, Phys. Rev. D 28 (1983), 2960-2975.
  113. Hawking S.W., The boundary conditions of the universe, Pont. Acad. Sci. Scr. Varia 48 (1982), 563-572.
  114. Hawking S.W., The quantum state of the universe, Nuclear Phys. B 239 (1984), 257-276.
  115. Hellmann F., Expansions in spin foam cosmology, Phys. Rev. D 84 (2011), 103516, 9 pages, arXiv:1105.1334.
  116. Henderson A., Rovelli C., Vidotto F., Wilson-Ewing E., Local spinfoam expansion in loop quantum cosmology, Classical Quantum Gravity 28 (2011), 025003, 10 pages, arXiv:1010.0502.
  117. Hinterleitner F., Major S., Plane gravitational waves in real connection variables, Phys. Rev. D 83 (2011), 044034, 13 pages, arXiv:1006.4146.
  118. Hinterleitner F., Major S., Towards loop quantization of plane gravitational waves, arXiv:1106.1448.
  119. Hossain G.M., Primordial density perturbation in effective loop quantum cosmology, Classical Quantum Gravity 22 (2005), 2511-2532, gr-qc/0411012.
  120. Huang H., Ma Y., Qin L., Path integral and effective Hamiltonian in loop quantum cosmology, arXiv:1102.4755.
  121. Husain V., Pullin J., Quantum theory of space-times with one Killing field, Modern Phys. Lett. A 5 (1990), 733-741.
  122. Husain V., Smolin L., Exactly solvable quantum cosmologies from two Killing field reductions of general relativity, Nuclear Phys. B 327 (1989), 205-238.
  123. Immirzi G., Quantum gravity and Regge calculus, Nuclear Phys. B Proc. Suppl. 57 (1997), 65-72, gr-qc/9701052.
  124. Immirzi G., Real and complex connections for canonical gravity, Classical Quantum Gravity 14 (1997), L177-L181, gr-qc/9612030.
  125. Isenberg J., Moncrief V., Asymptotic behavior of the gravitational field and the nature of singularities in Gowdy spacetimes, Ann. Physics 199 (1990), 84-122.
  126. Kaminski W., Lewandowski J., The flat FRW model in LQC: self-adjointness, Classical Quantum Gravity 25 (2008), 035001, 11 pages, arXiv:0709.3120.
  127. Kaminski W., Lewandowski J., Pawlowski T., Physical time and other conceptual issues of quantum gravity on the example of loop quantum cosmology, Classical Quantum Gravity 26 (2009), 035012, 20 pages, arXiv:0809.2590.
  128. Kaminski W., Pawlowski T., Cosmic recall and the scattering picture of loop quantum cosmology, Phys. Rev. D 81 (2010), 084027, 19 pages, arXiv:1001.2663.
  129. Kaminski W., Pawlowski T., Loop quantum cosmology evolution operator of an FRW universe with a positive cosmological constant, Phys. Rev. D 81 (2010), 024014, 9 pages, arXiv:0912.0162.
  130. Kasner E., Geometrical Theorems on Einstein's Cosmological Equations, Amer. J. Math. 43 (1921), 217-221.
  131. Kato T., Perturbation theory for linear operators, Grundlehren der mathematischen Wissenschaften, Vol. 132, Springer-Verlag, Berlin, 1980.
  132. Kuchar K., Canonical quantization of cylindrical gravitational waves, Phys. Rev. D 4 (1971), 955-986.
  133. Lidsey J.E., Mulryne D.J., Nunes N.J., Tavakol R., Oscillatory universes in loop quantum cosmology and initial conditions for inflation, Phys. Rev. D 70 (2004), 063521, 6 pages, gr-qc/0406042.
  134. Livine E.R., Martín-Benito M., Classical setting and effective dynamics for spinfoam cosmology, arXiv:1111.2867.
  135. Manojlovic N., Mena Marugán G.A., Nonperturbative canonical quantization of minisuperspace models: Bianchi types I and II, Phys. Rev. D 48 (1993), 3704-3719, gr-qc/9304041.
  136. Manojlovic N., Mikovic A., Canonical analysis of the Bianchi models in the Ashtekar formulation, Classical Quantum Gravity 10 (1993), 559-573.
  137. Marolf D., Almost ideal clocks in quantum cosmology: a brief derivation of time, Classical Quantum Gravity 12 (1995), 2469-2486, gr-qc/9412016.
  138. Marolf D., Observables and a Hilbert space for Bianchi IX, Classical Quantum Gravity 12 (1995), 1441-1454, gr-qc/9409049.
  139. Marolf D., Quantum observables and recollapsing dynamics, Classical Quantum Gravity 12 (1995), 1199-1220, gr-qc/9404053.
  140. Marolf D., Refined algebraic quantization: systems with a single constraint, in Symplectic Singularities and Geometry of Gauge Fields (Warsaw, 1995), Banach Center Publ., Vol. 39, Polish Acad. Sci. Inst. Math., Warsaw, 1997, 331-344, gr-qc/9508015.
  141. Martín-Benito M., Garay L.J., Mena Marugán G.A., Hybrid quantum Gowdy cosmology: combining loop and Fock quantizations, Phys. Rev. D 78 (2008), 083516, 5 pages, arXiv:0804.1098.
  142. Martín-Benito M., Martín-de Blas D., Mena Marugán G.A., From the hybrid Gowdy model towards inhomogeneous Friedmann-Robertson-Walker quantum cosmologies, in preparation.
  143. Martín-Benito M., Martín-de Blas D., Mena Marugán G.A., Matter in inhomogeneous loop quantum cosmology: the Gowdy T3 model, Phys. Rev. D 83 (2011), 084050, 7 pages, arXiv:1012.2324.
  144. Martín-Benito M., Mena Marugán G.A., Olmedo J., Further improvements in the understanding of isotropic loop quantum cosmology, Phys. Rev. D 80 (2009), 104015, 11 pages, arXiv:0909.2829.
  145. Martín-Benito M., Mena Marugán G.A., Pawlowski T., Loop quantization of vacuum Bianchi I cosmology, Phys. Rev. D 78 (2008), 064008, 11 pages, arXiv:0804.3157.
  146. Martín-Benito M., Mena Marugán G.A., Pawlowski T., Physical evolution in loop quantum cosmology: the example of the vacuum Bianchi I model, Phys. Rev. D 80 (2009), 084038, 23 pages, arXiv:0906.3751.
  147. Martín-Benito M., Mena Marugán G.A., Wilson-Ewing E., Hybrid quantization: from Bianchi I to the Gowdy model, Phys. Rev. D 82 (2010), 084012, 11 pages, arXiv:1006.2369.
  148. Meissner K.A., Black-hole entropy in loop quantum gravity, Classical Quantum Gravity 21 (2004), 5245-5251, gr-qc/0407052.
  149. Mena Marugán G.A., A brief introduction to loop quantum cosmology, AIP Conf. Proc. 1130 (2009), 89-100, arXiv:0907.5160.
  150. Mena Marugán G.A., Canonical quantization of the Gowdy model, Phys. Rev. D 56 (1997), 908-919, gr-qc/9704041.
  151. Mena Marugán G.A., Martín-Benito M., Hybrid quantum cosmology: combining loop and Fock quantizations, Internat. J. Modern Phys. A 24 (2009), 2820-2838, arXiv:0907.3797.
  152. Mena Marugán G.A., Montejo M., Quantization of pure gravitational plane waves, Phys. Rev. D 58 (1998), 104017, 11 pages, gr-qc/9806105.
  153. Mena Marugán G.A., Olmedo J., Inhomogeneities and inflation in loop quantum cosmology: a hybrid approach, in preparation.
  154. Mena Marugán G.A., Olmedo J., Pawlowski T., Prescriptions in loop quantum cosmology: a comparative analysis, Phys. Rev. D 840 (2011), 064012, 18 pages, arXiv:1108.0829.
  155. Mielczarek J., Gravitational waves from the big bounce, J. Cosmol. Astropart. Phys. 2008 (2008), no. 11, 011, 17 pages, arXiv:0807.0712.
  156. Mielczarek J., Cailleteau T., Barrau A., Grain J., Anomaly-free vector perturbations with holonomy corrections in loop quantum cosmology, arXiv:1106.3744.
  157. Mielczarek J., Cailleteau T., Grain J., Barrau A., Inflation in loop quantum cosmology: dynamics and spectrum of gravitational waves, Phys. Rev. D 81 (2010), 104049, 11 pages, arXiv:1003.4660.
  158. Mielczarek J., Hrycyna O., Szydlowski M., Effective dynamics of the closed loop quantum cosmology, J. Cosmol. Astropart. Phys. 2009 (2009), no. 11, 014, 16 pages, arXiv:0906.2503.
  159. Misner C.W., A minisuperspace example: the Gowdy T3 cosmology, Phys. Rev. D 8 (1973), 3271-3285.
  160. Misner C.W., Thorne K.S., Wheeler J.A., Gravitation, W. H. Freeman and Co., San Francisco, Calif., 1973.
  161. Moncrief V., Global properties of Gowdy spacetimes with T3×R topology, Ann. Physics 132 (1981), 87-107.
  162. Moncrief V., Infinite-dimensional family of vacuum cosmological models with Taub-NUT (Newman-Unti-Tamburino)-type extensions, Phys. Rev. D 23 (1981), 312-315.
  163. Nelson W., Sakellariadou M., Lattice refining loop quantum cosmology and the matter Hamiltonian, Phys. Rev. D 76 (2007), 104003, 9 pages, arXiv:0707.0588.
  164. Pierri M., Probing quantum general relativity through exactly soluble midi-superspaces. II. Polarized Gowdy models, Internat. J. Modern Phys. D 11 (2002), 135-153, gr-qc/0101013.
  165. Reed M., Simon B., Methods of modern mathematical physics. I. Functional analysis, 2nd ed., Academic Press Inc., New York, 1980.
  166. Rendall A.D., Adjointness relations as a criterion for choosing an inner product, in Canonical Gravity: from Classical to Quantum (Bad Honnef, 1993), Lecture Notes in Phys., Vol. 434, Springer, Berlin, 1994, 319-326, gr-qc/9403001.
  167. Rendall A.D., Unique determination of an inner product by adjointness relations in the algebra of quantum observables, Classical Quantum Gravity 10 (1993), 2261-2269, gr-qc/9303026.
  168. Röken C., First-order quantum-gravitational correction from covariant, holomorphic spinfoam cosmology, arXiv:1011.3335.
  169. Rovelli C., Partial observables, Phys. Rev. D 65 (2002), 124013, 8 pages, gr-qc/0110035.
  170. Rovelli C., Quantum gravity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2004.
  171. Rovelli C., Smolin L., Discreteness of area and volume in quantum gravity, Nuclear Phys. B 442 (1995), 593-619, Erratum, Nuclear Phys. B 456 (1995), 753-754, gr-qc/9411005.
  172. Rovelli C., Smolin L., Knot theory and quantum gravity, Phys. Rev. Lett. 61 (1988), 1155-1158.
  173. Rovelli C., Smolin L., Loop space representation of quantum general relativity, Nuclear Phys. B 331 (1990), 80-152.
  174. Rovelli C., Smolin L., The physical Hamiltonian in nonperturbative quantum gravity, Phys. Rev. Lett. 72 (1994), 446-449, gr-qc/9308002.
  175. Shimano M., Harada T., Observational constraints of a power spectrum from superinflation in loop quantum cosmology, Phys. Rev. D 80 (2009), 063538, 11 pages, arXiv:0909.0334.
  176. Singh P., Loop cosmological dynamics and dualities with Randall-Sundrum braneworlds, Phys. Rev. D 73 (2006), 063508, 9 pages, gr-qc/0603043.
  177. Singh P., Toporensky A., Big crunch avoidance in k=1 semiclassical loop quantum cosmology, Phys. Rev. D 69 (2004), 104008, 5 pages, gr-qc/0312110.
  178. Singh P., Vandersloot K., Vereshchagin G.V., Nonsingular bouncing universes in loop quantum cosmology, Phys. Rev. D 74 (2006), 043510, 12 pages, gr-qc/0606032.
  179. Singh P., Vidotto F., Exotic singularities and spatially curved loop quantum cosmology, Phys. Rev. D 83 (2011), 064027, 13 pages, arXiv:1012.1307.
  180. Stone M.H., Linear transformations in Hilbert space. III. Operational methods and group theory, Proc. Natl. Acad. Sci. USA 16 (1930), 172-175.
  181. Szulc L., An open FRW model in loop quantum cosmology, Classical Quantum Gravity 24 (2007), 6191-6200, arXiv:0707.1816.
  182. Szulc L., Loop quantum cosmology of diagonal Bianchi type I model: simplifications and scaling problems, Phys. Rev. D 78 (2008), 064035, 12 pages, arXiv:0803.3559.
  183. Szulc L., Kaminski W., Lewandowski J., Closed Friedmann-Robertson-Walker model in loop quantum cosmology, Classical Quantum Gravity 24 (2007), 2621-2635, gr-qc/0612101.
  184. Taveras V., Corrections to the Friedmann equations from loop quantum gravity for a universe with a free scalar field, Phys. Rev. D 78 (2008), 064072, 9 pages, arXiv:0807.3325.
  185. Thiemann T., Anomaly-free formulation of non-perturbative, four-dimensional Lorentzian quantum gravity, Phys. Lett. B 380 (1996), 257-264, gr-qc/9606088.
  186. Thiemann T., Modern canonical quantum general relativity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2007.
  187. Thiemann T., Quantum spin dynamics (QSD), Classical Quantum Gravity 15 (1998), 839-873, gr-qc/9606089.
  188. Thiemann T., Quantum spin dynamics (QSD). V. Quantum gravity as the natural regulator of the Hamiltonian constraint of matter quantum field theories, Classical Quantum Gravity 15 (1998), 1281-1314, gr-qc/9705019.
  189. Torre C.G., Quantum dynamics of the polarized Gowdy T3 model, Phys. Rev. D 66 (2002), 084017, 12 pages, gr-qc/0206083.
  190. Tsujikawa S., Singh P., Maartens R., Loop quantum gravity effects on inflation and the CMB, Classical Quantum Gravity 21 (2004), 5767-5775, astro-ph/0311015.
  191. Vandersloot K., Loop quantum cosmology and the k=−1 Robertson-Walker model, Phys. Rev. D 75 (2007), 023523, 13 pages, gr-qc/0612070.
  192. Velhinho J.M., The quantum configuration space of loop quantum cosmology, Classical Quantum Gravity 24 (2007), 3745-3758, arXiv:0704.2397.
  193. Vidotto F., Many-node/many-link spinfoam: the homogeneous and isotropic case, Classical Quantum Gravity 28 (2011), 245005, 11 pages, arXiv:1107.2633.
  194. Vidotto F., Spinfoam cosmology, J. Phys. Conf. Ser. 314 (2011), 012049, 4 pages, arXiv:1011.4705.
  195. von Neumann J., Die Eindeutigkeit der Schrödingerschen Operatoren, Math. Ann. 104 (1931), 570-578.
  196. Wald R.M., General relativity, University of Chicago Press, Chicago, IL, 1984.
  197. Wald R.M., Quantum field theory in curved spacetime and black hole thermodynamics, Chicago Lectures in Physics, University of Chicago Press, Chicago, IL, 1994.
  198. Wands D., Malik K.A., Lyth D.H., Liddle A.R., New approach to the evolution of cosmological perturbations on large scales, Phys. Rev. D 62 (2000), 043527, 8 pages, astro-ph/0003278.
  199. Wheeler J.A., Superspace and the nature of quantum geometrodynamics, in Batelle Rencontres, Editors C.M. DeWitt, J.A. Wheeler, W.A. Benjamin, New York, 1968, 242-307.
  200. Wilson-Ewing E., Holonomy corrections in the effective equations for scalar mode perturbations in loop quantum cosmology, arXiv:1108.6265.
  201. Wilson-Ewing E., Loop quantum cosmology of Bianchi type IX models, Phys. Rev. D 82 (2010), 043508, 13 pages, arXiv:1005.5565.
  202. Yang J., Ding Y., Ma Y., Alternative quantization of the Hamiltonian in isotropic loop quantum cosmology, arXiv:0902.1913.
  203. Yang J., Ding Y., Ma Y., Alternative quantization of the Hamiltonian in loop quantum cosmology, Phys. Lett. B 682 (2009), 1-7, arXiv:0904.4379.

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