Table of Contents Author Guidelines Submit a Manuscript
Advances in Astronomy
Volume 2010, Article ID 576273, 26 pages
http://dx.doi.org/10.1155/2010/576273
Review Article

Second-Order Gauge-Invariant Cosmological Perturbation Theory: Current Status

Optical and Infrared Astronomy Division, National Astronomical Observatory of Japan, Osawa, Mitaka, Tokyo 181-8588, Japan

Received 19 April 2010; Accepted 12 July 2010

Academic Editor: Eiichiro Komatsu

Copyright © 2010 Kouji Nakamura. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. J. M. Bardeen, “Gauge-invariant cosmological perturbations,” Physical Review D, vol. 22, no. 8, pp. 1882–1905, 1980. View at Publisher · View at Google Scholar
  2. H. Kodama and M. Sasaki, “Cosmological perturbation theory,” Progress of Theoretical Physics Supplement, no. 78, p. 1, 1984. View at Google Scholar
  3. V. F. Mukhanov, H. A. Feldman, and R. H. Brandenberger, “Theory of cosmological perturbations,” Physics Report, vol. 215, no. 5-6, pp. 203–333, 1992. View at Google Scholar
  4. C. L. Bennett, M. Halpern, G. Hinshaw et al., “First-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: preliminary maps and basic results,” Astrophysical Journal, Supplement Series, vol. 148, no. 1, pp. 1–27, 2003. View at Publisher · View at Google Scholar
  5. E. Komatsu, J. Dunkley, M. R. Nolta et al., “Five-year wilkinson microwave anisotropy probe observations: cosmological interpretation,” Astrophysical Journal, Supplement Series, vol. 180, no. 2, pp. 330–376, 2009. View at Publisher · View at Google Scholar
  6. V. Acquaviva, N. Bartolo, S. Matarrese, and A. Riotto, “Gauge-invariant second-order perturbations and non-Gaussianity from inflation,” Nuclear Physics B, vol. 667, no. 1-2, pp. 119–148, 2003. View at Publisher · View at Google Scholar
  7. J. Maldacena, “Non-gaussian features of primordial fluctuations in single field inflationary models,” Journal of High Energy Physics, vol. 2003, no. 5, article 013, 2003. View at Google Scholar
  8. K. A. Malik and D. Wands, “Evolution of second-order cosmological perturbations,” Classical and Quantum Gravity, vol. 21, no. 11, pp. L65–L71, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  9. N. Bartolo, S. Matarrese, and A. Riotto, “Non-Gaussianity in the curvaton scenario,” Physical Review D, vol. 69, no. 4, Article ID 043503, 2004. View at Publisher · View at Google Scholar
  10. N. Bartolo, S. Matarrese, and A. Riotto, “Enhancement of non-gaussianity after inflation,” Journal of High Energy Physics, vol. 8, no. 4, article 006, pp. 147–159, 2004. View at Google Scholar
  11. D. H. Lyth and Y. Rodríguez, “Non-Gaussianity from the second-order cosmological perturbation,” Physical Review D, vol. 71, no. 12, Article ID 123508, 2005. View at Publisher · View at Google Scholar
  12. F. Vernizzi, “Conservation of second-order cosmological perturbations in a scalar field dominated universe,” Physical Review D, vol. 71, no. 6, Article ID 061301, 5 pages, 2005. View at Publisher · View at Google Scholar
  13. N. Bartolo, S. Matarrese, and A. Riotto, “Evolution of second-order cosmological perturbations and non-Gaussianity,” Journal of Cosmology and Astroparticle Physics, vol. 2004, no. 1, article 003, pp. 47–70, 2004. View at Publisher · View at Google Scholar
  14. N. Bartolo, S. Matarrese, and A. Riotto, “Gauge-invariant temperature anisotropies and primordial non-Gaussianity,” Physical Review Letters, vol. 93, no. 23, Article ID 231301, 2004. View at Publisher · View at Google Scholar
  15. N. Bartolo, E. Komatsu, S. Matarrese, and A. Riotto, “Non-Gaussianity from inflation: theory and observations,” Physics Reports, vol. 402, no. 3-4, pp. 103–266, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  16. N. Bartolo, S. Matarrese, and A. Riotto, “The full second-order radiation transfer function for large-scale CMB anisotropies,” Journal of Cosmology and Astroparticle Physics, vol. 2006, no. 5, article 010, 2006. View at Publisher · View at Google Scholar
  17. N. Bartolo, S. Matarrese, and A. Riotto, “Cosmic microwave background anisotropies at second order: I,” Journal of Cosmology and Astroparticle Physics, vol. 2006, no. 6, article 024, 2006. View at Publisher · View at Google Scholar
  18. N. Bartolo, S. Matarrese, and A. Riotto, “CMB anisotropies at second-order II: analytical approach,” Journal of Cosmology and Astroparticle Physics, vol. 2007, no. 1, article 019, 2007. View at Publisher · View at Google Scholar
  19. D. Nitta, E. Komatsu, N. Bartolo, S. Matarrese, and A. Riotto, “CMB anisotropies at second order III: bispectrum from products of the first-order perturbations,” Journal of Cosmology and Astroparticle Physics, vol. 2009, no. 5, article 014, 2009. View at Publisher · View at Google Scholar
  20. C. Pitrou, J.-P. Uzan, and F. Bernardeau, “Cosmic microwave background bispectrum on small angular scales,” Physical Review D, vol. 78, no. 6, Article ID 063526, 2008. View at Publisher · View at Google Scholar
  21. L. Senatore, S. Tassev, and M. Zaldarriaga, “Cosmological perturbations at second order and recombination perturbed,” Journal of Cosmology and Astroparticle Physics, vol. 2009, no. 8, article 031, 2009. View at Publisher · View at Google Scholar
  22. K. Nakamura, “Gauge invariant variables in two-parameter nonlinear perturbations,” Progress of Theoretical Physics, vol. 110, no. 4, pp. 723–756, 2003. View at Google Scholar
  23. K. Nakamura, “Second-order gauge invariant perturbation theory—perturbative curvatures in the two-parameter case,” Progress of Theoretical Physics, vol. 113, no. 3, pp. 481–511, 2005. View at Publisher · View at Google Scholar
  24. M. Bruni, S. Matarrese, S. Mollerach, and S. Sonego, “Perturbations of spacetime: gauge transformations and gauge invariance at second order and beyond,” Classical and Quantum Gravity, vol. 14, no. 9, pp. 2585–2606, 1997. View at Google Scholar
  25. K. Nakamura, “Gauge-invariant formulation of second-order cosmological perturbations,” Physical Review D, vol. 74, no. 10, Article ID 101301, 2006. View at Publisher · View at Google Scholar
  26. K. Nakamura, “Second-order gauge invariant cosmological perturbation theory—Einstein equations in terms of gauge invariant variables,” Progress of Theoretical Physics, vol. 117, no. 1, pp. 17–74, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  27. K. Nakamura, “Perturbations of matter fields in the second-order gauge-invariant cosmological perturbation theory,” Physical Review D, vol. 80, no. 12, Article ID 124021, 2009. View at Publisher · View at Google Scholar
  28. K. Nakamura, “Consistency of equations in the second-order gauge-invariant cosmological perturbation theory,” Progress of Theoretical Physics, vol. 121, no. 6, pp. 1321–1360, 2009. View at Publisher · View at Google Scholar
  29. C. Pitrou, “Gauge-invariant Boltzmann equation and the fluid limit,” Classical and Quantum Gravity, vol. 24, no. 24, pp. 6127–6158, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  30. C. Pitrou, “The radiative transfer at second order: a full treatment of the Boltzmann equation with polarization,” Classical and Quantum Gravity, vol. 26, no. 6, Article ID 065006, 2009. View at Publisher · View at Google Scholar
  31. K. Nakamura, ““Gauge” in general relativity: second-order general relativistic gauge-invariant perturbation theory,” in Lie Theory and Its Applications in Physics VII, V. K. Dobrev et al., Ed., Heron Press, Sofia, Bulgaria, 2008. View at Google Scholar
  32. R. K. Sachs, “Gravitational radiation,” in Relativity, Groups and Topology, C. DeWitt and B. DeWitt, Eds., Gordon and Breach, New York, NY, USA, 1964. View at Google Scholar
  33. J. M. Stewart and M. Walker, “Perturbations of space-times in general relativity,” Proceedings of the Royal Society of London A, vol. 341, pp. 49–74, 1974. View at Google Scholar
  34. J. M. Stewart, “Perturbations of Friedmann-Robertson-Walker cosmological models,” Classical and Quantum Gravity, vol. 7, no. 7, pp. 1169–1180, 1990. View at Publisher · View at Google Scholar
  35. J. M. Stewart, Advanced General Relativity, Cambridge University Press, Cambridge, UK, 1991.
  36. S. Sonego and M. Bruni, “Gauge dependence in the theory of non-linear spacetime perturbations,” Communications in Mathematical Physics, vol. 193, no. 1, pp. 209–218, 1998. View at Google Scholar
  37. S. Matarrese, S. Mollerach, and M. Bruni, “Relativistic second-order perturbations of the Einstein-de Sitter universe,” Physical Review D, vol. 58, no. 4, Article ID 043504, 22 pages, 1998. View at Google Scholar
  38. M. Bruni, L. Gualtieri, and C. F. Sopuerta, “Two-parameter nonlinear spacetime perturbations: Gauge transformations and gauge invariance,” Classical and Quantum Gravity, vol. 20, no. 3, pp. 535–556, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  39. C. F. Sopuerta, M. Bruni, and L. Gualtieri, “Nonlinear N-parameter spacetime perturbations: Gauge transformations,” Physical Review D, vol. 70, no. 6, Article ID 064002, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  40. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Volume I, John Wiley & Sons, New York, NY, USA, 1996.
  41. M. Bruni and S. Sonego, “Observables and gauge invariance in the theory of nonlinear spacetime perturbations,” Classical and Quantum Gravity, vol. 16, no. 7, pp. L29–L36, 1999. View at Google Scholar
  42. R. M. Wald, General Relativity, University of Chicago Press, Chicago, Ill, USA, 1984.
  43. S. Weinberg, Gravitation and Cosmology, John Wiley & Sons, New York, NY, USA, 1972.
  44. F. C. Mena, D. J. Mulryne, and R. Tavakol, “Nonlinear vector perturbations in a contracting universe,” Classical and Quantum Gravity, vol. 24, no. 10, p. 2721, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  45. T. H.-C. Lu, K. Ananda, and C. Clarkson, “Vector modes generated by primordial density fluctuations,” Physical Review D, vol. 77, no. 4, Article ID 043523, 2008. View at Publisher · View at Google Scholar
  46. T. H.-C. Lu, K. Ananda, C. Clarkson, and R. Maartens, “The cosmological background of vector modes,” Journal of Cosmology and Astroparticle Physics, vol. 2009, no. 2, article 023, 2009. View at Publisher · View at Google Scholar
  47. A. J. Christopherson, K. A. Malik, and D. R. Matravers, “Vorticity generation at second order in cosmological perturbation theory,” Physical Review D, vol. 79, no. 12, Article ID 123523, 2009. View at Publisher · View at Google Scholar
  48. K. N. Ananda, C. Clarkson, and D. Wands, “Cosmological gravitational wave background from primordial density perturbations,” Physical Review D, vol. 75, no. 12, Article ID 123518, 2007. View at Publisher · View at Google Scholar
  49. B. Osano, C. Pitrou, P. Dunsby, J.-P. Uzan, and C. Clarkson, “Gravitational waves generated by second order effects during inflation,” Journal of Cosmology and Astroparticle Physics, vol. 2007, no. 4, article 003, 2007. View at Publisher · View at Google Scholar
  50. D. Baumann, P. Steinhardt, K. Takahashi, and K. Ichiki, “Gravitational wave spectrum induced by primordial scalar perturbations,” Physical Review D, vol. 76, no. 8, Article ID 084019, 2007. View at Publisher · View at Google Scholar
  51. N. Bartolo, S. Matarrese, A. Riotto, and A. Väihkönen, “Maximal amount of gravitational waves in the curvaton scenario,” Physical Review D, vol. 76, no. 6, Article ID 061302, 2007. View at Publisher · View at Google Scholar
  52. P. Martineau and R. Brandenberger, “A back-reaction induced lower bound on the tensor-to-scalar ratio,” Modern Physics Letters A, vol. 23, no. 10, pp. 727–735, 2008. View at Publisher · View at Google Scholar
  53. R. Saito and J. Yokoyama, “Gravitational-wave background as a probe of the primordial black-hole abundance,” Physical Review Letters, vol. 102, no. 16, Article ID 161101, 2009. View at Publisher · View at Google Scholar
  54. F. Arroja, H. Assadullahi, K. Koyama, and D. Wands, “Cosmological matching conditions for gravitational waves at second order,” Physical Review D, vol. 80, no. 12, Article ID 123526, 2009. View at Publisher · View at Google Scholar
  55. K. Nakamura, A. Ishibashi, and H. Ishihara, “Dynamics of a string coupled to gravitational waves: gravitational wave scattering by a Nambu-Goto straight string,” Physical Review D, vol. 62, no. 10, Article ID 101502, 5 pages, 2000. View at Google Scholar
  56. K. Nakamura and H. Ishihara, “Dynamics of a string coupled to gravitational waves. II. Perturbations propagate along an infinite Nambu-Goto string,” Physical Review D, vol. 63, no. 12, Article ID 127501, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  57. K. Nakamura, “Initial condition of a gravitating thick loop cosmic string and linear perturbations,” Classical and Quantum Gravity, vol. 19, no. 4, pp. 783–797, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  58. K. Nakamura, “Does a Nambu-Goto wall emit gravitational waves? Cylindrical Nambu-Goto wall as an example of gravitating nonspherical walls,” Physical Review D, vol. 66, no. 8, Article ID 084005, 14 pages, 2002. View at Google Scholar
  59. K. Nakamura, “Comparison of the oscillatory behavior of a gravitating Nambu-Goto string and a test string,” Progress of Theoretical Physics, vol. 110, no. 2, pp. 201–232, 2003. View at Google Scholar