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

SIGMA 8 (2012), 001, 26 pages      arXiv:1110.0646      https://doi.org/10.3842/SIGMA.2012.001
Contribution to the Special Issue “Loop Quantum Gravity and Cosmology”

Numerical Techniques in Loop Quantum Cosmology

David Brizuela a, Daniel Cartin b and Gaurav Khanna c
a) Institute for Gravitation and the Cosmos, The Pennsylvania State University, 104 Davey Lab, University Park, Pennsylvania 16802, USA
b) Naval Academy Preparatory School, 197 Elliot Avenue, Newport, Rhode Island 02841, USA
c) Physics Department, University of Massachusetts at Dartmouth, North Dartmouth, Massachusetts 02747, USA

Received October 01, 2011, in final form December 20, 2011; Published online January 02, 2012

Abstract
In this article, we review the use of numerical techniques to obtain solutions for the quantum Hamiltonian constraint in loop quantum cosmology (LQC). First, we summarize the basic features of LQC, and describe features of the constraint equations to solve - generically, these are difference (rather than differential) equations. Important issues such as differing quantization methods, stability of the solutions, the semi-classical limit, and the relevance of lattice refinement in the difference equations are discussed. Finally, the cosmological models already considered in the literature are listed, along with typical features in these models and open issues.

Key words: quantum gravity; numerical techniques; loop quantum cosmology.

pdf (605 kb)   tex (113 kb)

References

  1. Ashtekar A., Loop quantum cosmology: an overview, Gen. Relativity Gravitation 41 (2009), 707-741, arXiv:0812.177.
  2. Ashtekar A., Bojowald M., Quantum geometry and the Schwarzschild singularity, Classical Quantum Gravity 23 (2006), 391-411, gr-qc/0509075.
  3. 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.
  4. Ashtekar A., Pawlowski T., Positive cosmological constant in loop quantum cosmology, arXiv:1112.0360.
  5. Ashtekar A., Pawlowski T., Singh P., Quantum nature of the big bang, Phys. Rev. Lett. 96 (2006), 141301, 4 pages, gr-qc/0602086.
    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.
  6. 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.
  7. 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.
  8. Ashtekar A., Singh P., Loop quantum cosmology: a status report, Classical Quantum Gravity 28 (2011), 213001, 138 pages, arXiv:1108.0893.
  9. Ashtekar A., Wilson-Ewing E., Loop quantum cosmology of Bianchi type I models, Phys. Rev. D 79 (2009), 083535, 21 pages, arXiv:0903.3397.
  10. Ashtekar A., Wilson-Ewing E., Loop quantum cosmology of Bianchi type II models, Phys. Rev. D 80 (2009), 123532, 16 pages, arXiv:0910.1278.
  11. Banerjee K., Date G., Discreteness corrections to the effective Hamiltonian of isotropic loop quantum cosmology, Classical Quantum Gravity 22 (2005), 2017-2033, gr-qc/0501102.
  12. Bentivegna E., Pawlowski T., Anti-de Sitter universe dynamics in LQC, Phys. Rev. D 77 (2008), 124025, 17 pages, arXiv:0803:4446.
  13. Böhmer C.G., Vandersloot K., Loop quantum dynamics of the Schwarzschild interior, Phys. Rev. D 76 (2007), 104030, 11 pages, arXiv:0709.2129.
  14. Bojowald M., Absence of singularity in loop quantum cosmology, Phys. Rev. Lett. 86 (2001), 5227-5230, gr-qc/0102069.
  15. Bojowald M., Inflation from quantum geometry, Phys. Rev. Lett. 89 (2002), 261301, 4 pages, gr-qc/206054.
  16. Bojowald M., Isotropic loop quantum cosmology, Classical Quantum Gravity 19 (2002), 2717-2741, gr-qc/0207038.
  17. Bojowald M., Loop quantum cosmology: recent progress, Pramana 63 (2004), 765-776, gr-qc/0402053.
  18. Bojowald M., Homogeneous loop quantum cosmology, Classical Quantum Gravity 20 (2003), 2595-2615, gr-qc/0303073.
    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.
  19. Bojowald M., Brizuela D., Hernández H.H., Koop M.J., Morales-Técotl H.A., High-order quantum back-reaction and quantum cosmology with a positive cosmological constant, Phys. Rev. D 84 (2011), 043514, 21 pages, arXiv:1011.3022.
  20. Bojowald M., Calcagni G., Tsujikawa S., Observational constraints on loop quantum cosmology, Phys. Rev. Lett. 107 (2011), 211302, 5 pages, arXiv:1101.5391.
  21. 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.
  22. Bojowald M., Date G., Consistency conditions for fundamentally discrete theories, Classical Quantum Gravity 21 (2004), 121-143, gr-qc/0307083.
  23. Bojowald M., Date G., Quantum suppression of the general chaotic behavior close to cosmological singularities, Phys. Rev. Lett. 92 (2004), 071302, 4 pages, gr-qc/0311003.
  24. Bojowald M., Hernández H.H., Skirzewski A., Effective equations for isotropic quantum cosmology including matter, Phys. Rev. D 76 (2007), 063511, 24 pages, arXiv:0706.1057.
  25. Bojowald M., Sandhöfer B., Skirzewski A., Tsobanjan A., Effective constraints for quantum systems, Rev. Math. Phys. 21 (2009), 111-154, arXiv:0804.3365.
  26. Bojowald M., Skirzewski A., Effective equations of motion for quantum systems, Rev. Math. Phys. 18 (2006), 713-746, math-ph/0511043.
  27. Bojowald M., Tsobanjan A., Effective constraints and physical coherent states in quantum cosmology: a numerical comparison, Classical Quantum Gravity 27 (2010), 145004, 22 pages, arXiv:0911.4950.
  28. Bojowald M., Vandersloot K., Loop quantum cosmology, boundary proposals, and inflation, Phys. Rev. D 67 (2003), 124023, 10 pages, gr-qc/0303072.
  29. Brizuela D., Mena Marugán G.A., Pawlowski T., Big bounce and inhomogeneities, Classical Quantum Gravity 27 (2010), 052001, 8 pages, arXiv:0902.0697.
    Brizuela D., Mena Marugán G.A., Pawlowski T., Effective dynamics of the hybrid quantization of the Gowdy T3 universe, arXiv:1106.3793.
  30. Cartin D., Khanna G., Bojowald M., Generating function techniques for loop quantum cosmology, Classical Quantum Gravity 21 (2004), 4495-4509, gr-qc/0405126.
  31. Cartin D., Khanna G., Absence of pre-classical solutions in Bianchi I loop quantum cosmology, Phys. Rev. Lett. 94 (2005), 111302, 4 pages, gr-qc/0501016.
  32. Cartin D., Khanna G., Separable sequences in Bianchi I loop quantum cosmology, Phys. Rev. D 72 (2005), 084008, 6 pages, gr-qc/0506024.
  33. Cartin D., Khanna G., Wave functions for the Schwarzschild black hole interior, Phys. Rev. D 73 (2006), 104009, 13 pages, gr-qc/0602025.
  34. Chiou D.-W., Loop quantum cosmology in Bianchi type I models: analytical investigation, Phys. Rev. D 75 (2006), 024029, 33 pages, gr-qc/0609029.
  35. Chiou D.-W., Effective dynamics, big bounces and scaling symmetry in Bianchi I loop quantum cosmology, Phys. Rev. D 76 (2007), 124037, 19 pages, arXiv:0710.0416.
  36. Chiou D.-W., Vandersloot K., The behavior of non-linear anisotropies in bouncing Bianchi I models of loop quantum cosmology, Phys. Rev. D 76 (2007), 084015, 15 pages, arXiv:0707.2548.
  37. Chiou D.-W., Phenomenological loop quantum geometry of the Schwarzschild black hole, Phys. Rev. D 78 (2008), 064040, 21 pages, arXiv:0807.0665.
  38. Chiou D.-W., Fi L.-F., Loop quantum cosmology with higher order holonomy corrections, Phys. Rev. D. 80 (2009), 043512, 19 pages, arXiv:0907.0640.
  39. Chiou D.-W., Geiller M., Unimodular loop quantum cosmology, Phys. Rev. D 82 (2010), 064012, 16 pages, arXiv:1007.0735.
  40. Corichi A., Singh P., Is loop quantization in cosmology unique?, Phys. Rev. D 78 (2008) 023034, 13 pages, arXiv:0805.0136.
  41. Date G., Absence of the Kasner singularity in the effective dynamics from loop quantum cosmology, Phys. Rev. D 71 (2005), 127502, 4 pages, gr-qc/0505002.
  42. Elaydi S.N., An introduction to difference equations, 3rd ed., Undergraduate Texts in Mathematics, Springer, New York, 2005.
  43. Grain J., Barrau A., Cosmological footprints of loop quantum gravity, Phys. Rev. Lett. 102 (2009), 081301, 4 pages, arXiv:0902.0145.
  44. Green D., Unruh W.G., Difficulties with recollapsing models in closed isotropic loop quantum cosmology, Phys. Rev. D 70 (2004), 103502, 7 pages, gr-qc/0408074.
  45. Hossain G.M., Primordial density perturbations in effective loop quantum cosmology, Classical Quantum Gravity 22 (2005), 2511-2532, gr-qc/0411012.
  46. Joe A., Khanna G., An efficient numerical technique for solving lattice-refined models in loop quantum cosmology, in preparation.
  47. Kaminski W., Pawlowski T., The LQC evolution operator of FRW universe with positive cosmological constant, Phys. Rev. D 81 (2010), 024014, 9 pages, arXiv:0912.0162.
  48. Kiefer C., Wave packets in minisuperspace, Phys. Rev. D 38 (1988), 1761-1772.
  49. Laguna P., Numerical analysis of the big bounce in loop quantum cosmology, Phys. Rev. D 75 (2007), 024033, 5 pages, gr-qc/0608117.
  50. 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.
    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.
  51. 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.
  52. 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.
  53. 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.
  54. 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.
  55. Meissner K.A., Black hole entropy in loop quantum gravity, Classical Quantum Gravity 21 (2004), 5245-5251, gr-qc/0407052.
  56. Mena Marugán G.A., Olmedo J., Pawlowski T., Prescriptions in loop quantum cosmology: a comparative analysis, Phys. Rev. D 84 (2011), 064012, 18 pages, arXiv:1108.0829.
  57. Mielczarek J., Possible observational effects of loop quantum cosmology, Phys. Rev. D 81 (2010), 063503, 12 pages, arXiv:0908.4329.
  58. Modesto L., The Kantowski-Sachs space-time in loop quantum gravity, Internat. J. Theoret. Phys. 45 (2006), 2235-2246, gr-qc/0411032.
  59. Nelson W., Sakellariadou M., Numerical techniques for solving the quantum constraint equation of generic lattice-refined models in loop quantum cosmology, Phys. Rev. D 78 (2008), 024030, 10 pages, arXiv:0803.4483.
  60. Nelson W., Sakellariadou M., Unique factor ordering in the continuum limit of loop quantum cosmology, Phys. Rev. D 78 (2008), 024006, 6 pages, arXiv:0806.0595.
  61. Nelson W., Sakellariadou M., Unstable anisotropic loop quantum cosmology, Phys. Rev. D 80 (2009), 063521, 9 pages, arXiv:0907.4057.
  62. Nelson W., Sakellariadou M., Lattice refining loop quantum cosmology from an isotropic embedding of anisotropic cosmology, Classical Quantum Gravity 27 (2010), 145014, 16 pages, arXiv:0906.0292.
  63. Noui K., Perez A., Vandersloot K., On the physical Hilbert space of loop quantum cosmology, Phys. Rev. D 71 (2005), 044025, 14 pages, gr-qc/0411039.
  64. Rosen J., Jung J.-H., Khanna G., Instabilities in numerical loop quantum cosmology, Classical Quantum Gravity 23 (2006), 7075-7084, gr-qc/0607044.
  65. Sabharwal S., Khanna G., Numerical solutions in lattice-refined models in loop quantum cosmology, Classical Quantum Gravity 25 (2008), 085009, 12 pages, arXiv:0711.2086.
  66. Shimano M., Harada T., Observational constraints on a power spectrum from superinflation in loop quantum cosmology, Phys. Rev. D 80 (2009), 063538, 11 pages, arXiv:0909.0334.
  67. Singh P., Loop cosmological dynamics and dualities with Randall-Sundrum braneworlds, Phys. Rev. D 73 (2006), 063508, 9 pages, gr-qc/0603043.
  68. Singh P., Are loop quantum cosmologies never singular?, Classical Quantum Gravity 26 (2009), 125005, 17 pages, arXiv:0901.2750.
    Singh P., Vidotto F., Exotic singularities and spatially curved loop quantum cosmology, Phys. Rev. D 83 (2011), 064027, 13 pages, arXiv:1012.1307.
  69. Szulc L., Open FRW model in loop quantum cosmology, Classical Quantum Gravity 24 (2007), 6191-6200, arXiv:0709.4225.
  70. Szulc L., Kaminski W., Lewandowski J., Friedmann-Robertson-Walker model in loop quantum cosmology, Classical Quantum Gravity 24 (2007), 2621-2635, gr-qc/0612101.
  71. 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.
  72. 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.
  73. 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.
  74. Vandersloot K., Loop quantum cosmology and the k=−1 Robertson-Walker model, Phys. Rev. D 75 (2007), 023523, 13 pages, gr-qc/0612070.
  75. Wilson-Ewing E., Loop quantum cosmology of Bianchi type IX models, Phys. Rev. D 82 (2010), 043508, 13 pages, arXiv:1005.5565.

Next article   Contents of Volume 8 (2012)