• Shohei Saga
Part of the Springer Theses book series (Springer Theses)


Recent development of various cosmological observations plays an important role in establishing the standard cosmology. Here, the standard cosmology means the theory that the universe begins with the extreme high temperature, called the hot big-bang model, and with the initial conditions seeded in the inflationary era. The universe based on the standard cosmology contains a dark energy, dark matters, and baryons. In this section, we introduce the history and development of the cosmological perturbation theory and clarify the standpoint of this thesis.


Cosmological perturbation theory Cosmic microwave background radiation Non-gaussianity 


  1. 1.
    E. Hubble, A relation between distance and radial velocity among extra-galactic nebulae. Proc. Natl. Acad. Sci. 15, 168–173 (1929)ADSCrossRefzbMATHGoogle Scholar
  2. 2.
    M. Livio, Lost in translation: mystery of the missing text solved. Nature 479, 171 (2011)ADSCrossRefGoogle Scholar
  3. 3.
    A.A. Penzias, R.W. Wilson, A measurement of excess antenna temperature at 4080 Mc/s. APJ 142, 419–421 (1965)ADSCrossRefGoogle Scholar
  4. 4.
    E. Lifshitz, On the gravitational stability of the expanding universe. J. Phys. (USSR) 10, 116 (1946)MathSciNetzbMATHGoogle Scholar
  5. 5.
    E. Lifshitz, I. Khalatnikov, Investigations in relativistic cosmology. Adv. Phys. 12, 185–249 (1963)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    E.R. Harrison, Normal modes of vibrations of the universe. Rev. Mod. Phys. 39, 862–882 (1967)ADSCrossRefGoogle Scholar
  7. 7.
    J.M. Bardeen, Gauge-invariant cosmological perturbations. Phys. Rev. 22, 1882–1905 (1980)ADSMathSciNetGoogle Scholar
  8. 8.
    H. Kodama, M. Sasaki, Cosmological perturbation theory. Progress Theor. Phys. Suppl. 78, 1 (1984)ADSCrossRefGoogle Scholar
  9. 9.
    P.J.E. Peebles, J.T. Yu, Primeval adiabatic perturbation in an expanding universe. APJ 162, 815 (1970)ADSCrossRefGoogle Scholar
  10. 10.
    G.F. Smoot, C.L. Bennett, A. Kogut, E.L. Wright, J. Aymon, N.W. Boggess, E.S. Cheng, G. de Amici, S. Gulkis, M.G. Hauser, G. Hinshaw, P.D. Jackson, M. Janssen, E. Kaita, T. Kelsall, P. Keegstra, C. Lineweaver, K. Loewenstein, P. Lubin, J. Mather, S.S. Meyer, S.H. Moseley, T. Murdock, L. Rokke, R.F. Silverberg, L. Tenorio, R. Weiss, D.T. Wilkinson, Structure in the COBE differential microwave radiometer first-year maps. APJ 396, L1–L5 (1992)ADSCrossRefGoogle Scholar
  11. 11.
    V.C. Rubin, W.K. Ford Jr., Rotation of the andromeda nebula from a spectroscopic survey of emission regions. APJ 159, 379 (1970)ADSCrossRefGoogle Scholar
  12. 12.
    J.R. Bond, G. Efstathiou, Cosmic background radiation anisotropies in universes dominated by nonbaryonic dark matter. APJ 285, L45–L48 (1984)ADSCrossRefGoogle Scholar
  13. 13.
    N. Vittorio, J. Silk, Fine-scale anisotropy of the cosmic microwave background in a universe dominated by cold dark matter. APJ 285, L39–L43 (1984)ADSCrossRefGoogle Scholar
  14. 14.
    R.K. Sachs, A.M. Wolfe, Perturbations of a cosmological model and angular variations of the microwave background. APJ 147, 73 (1967)ADSCrossRefGoogle Scholar
  15. 15.
    J. Silk, Cosmic black-body radiation and galaxy formation. APJ 151, 459 (1968)ADSCrossRefGoogle Scholar
  16. 16.
    W. Hu, N. Sugiyama, Anisotropies in the cosmic microwave background: an analytic approach. Astrophys. J. 444, 489–506 (1995), arXiv:astro-ph/9407093
  17. 17.
    S.D.S.S. Collaboration, M. Tegmark et al., Cosmological constraints from the SDSS luminous red galaxies. Phys. Rev. D 74, 123507 (2006), arXiv:astro-ph/0608632
  18. 18.
    W.M.A.P. Collaboration, G. Hinshaw et al., Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) observations: cosmological parameter results. Astrophys. J. Suppl. 208, 19 (2013), arXiv:1212.5226
  19. 19.
    A.G. Sanchez et al., The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: cosmological implications of the full shape of the clustering wedges in the data release 10 and 11 galaxy samples. Mon. Not. Roy. Astron. Soc. 440(3), 2692–2713 (2014), arXiv:1312.4854
  20. 20.
    P.A.R. Ade, Planck Collaboration et al., Planck 2015 results. XIII. Cosmological parameters. Astron. Astrophys. 594, A13 (2016), arXiv:1502.01589
  21. 21.
    C.M.S. Collaboration, S. Chatrchyan et al., Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC. Phys. Lett. B 716, 30–61 (2012), arXiv:1207.7235
  22. 22.
    ATLAS Collaboration, G. Aad et al., Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC. Phys. Lett. B 716, 1–29 (2012), arXiv:1207.7214
  23. 23.
    R.A. Hulse, J.H. Taylor, Discovery of a pulsar in a binary system. APJ 195, L51–L53 (1975)ADSCrossRefGoogle Scholar
  24. 24.
    J.H. Taylor, J.M. Weisberg, A new test of general relativity-ravitational radiation and the binary pulsar PSR 1913+16. APJ 253, 908–920 (1982)ADSCrossRefGoogle Scholar
  25. 25.
    J.M. Weisberg, J.H. Taylor, Relativistic binary pulsar B1913+16: thirty years of observations and analysis. ASP Conference Series, vol. 328 (2005), p. 25, arXiv:astro-ph/0407149
  26. 26.
    Virgo, LIGO Scientific Collaboration, B.P. Abbott et al., Observation of gravitational waves from a binary black hole merger. Phys. Rev. Lett. 116(6), 061102 (2016), arXiv:1602.03837
  27. 27.
    Virgo, LIGO Scientific Collaboration, B.P. Abbott et al., Tests of general relativity with GW150914. Phys. Rev. Lett. 116(22), 221101 (2016), arXiv:1602.03841
  28. 28.
    Virgo, LIGO Scientific Collaboration, B.P. Abbott et al., Observing gravitational-wave transient GW150914 with minimal assumptions. Phys. Rev. D 93(12), 122004 (2016), arXiv:1602.03843. Addendum: Phys. Rev. D 94(6), 069903 (2016)
  29. 29.
    P. Amaro-Seoane, S. Aoudia, S. Babak, P. Binetruy, E. Berti et al., eLISA/NGO: Astrophysics and cosmology in the gravitational-wave millihertz regime. GW Notes 6, 4–110 (2013), arXiv:1201.3621
  30. 30.
    J. Crowder, N.J. Cornish, Beyond LISA: exploring future gravitational wave missions. Phys. Rev. D 72, 083005 (2005), arXiv:gr-qc/0506015
  31. 31.
    KAGRA Collaboration Collaboration, K. Somiya, Detector configuration of KAGRA: the Japanese cryogenic gravitational-wave detector. Class. Quant. Grav. 29, 124007 (2012), arXiv:1111.7185
  32. 32.
    M. Kamionkowski, A. Kosowsky, A. Stebbins, Statistics of cosmic microwave background polarization. Phys. Rev. D 55, 7368–7388 (1997), arXiv:astro-ph/9611125
  33. 33.
    M. Zaldarriaga, U. Seljak, An all sky analysis of polarization in the microwave background. Phys. Rev. D 55, 1830–1840 (1997), arXiv:astro-ph/9609170
  34. 34.
    B. Reichborn-Kjennerud, A.M. Aboobaker, P. Ade, F. Aubin, C. Baccigalupi et al., EBEX: a balloon-borne CMB polarization experiment, arXiv:1007.3672
  35. 35.
    QUIET Collaboration Collaboration, D. Samtleben, Measuring the cosmic microwave background radiation (CMBR) polarization with QUIET. Nuovo Cim. B 122, 1353–1358 (2007), arXiv:0802.2657
  36. 36.
    W.M.A.P. Collaboration, C.L. Bennett et al., Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) observations: final maps and results. Astrophys. J. Suppl. 208, 20 (2013), arXiv:1212.5225
  37. 37.
    QUaD collaboration Collaboration, M. Brown et al., Improved measurements of the temperature and polarization of the CMB from QUaD. Astrophys. J. 705, 978–999 (2009), arXiv:0906.1003
  38. 38.
    H. Chiang, P. Ade, D. Barkats, J. Battle, E. Bierman et al., Measurement of CMB polarization power spectra from two years of BICEP data. Astrophys. J. 711, 1123–1140 (2010), arXiv:0906.1181
  39. 39.
    J.P. Ostriker, E.T. Vishniac, Generation of microwave background fluctuations from nonlinear perturbations at the ERA of galaxy formation. APJ 306, L51–L54 (1986)ADSCrossRefGoogle Scholar
  40. 40.
    W. Hu, D. Scott, J. Silk, Reionization and cosmic microwave background distortions: a complete treatment of second order compton scattering. Phys. Rev. D 49, 648–670 (1994), arXiv:astro-ph/9305038
  41. 41.
    N. Bartolo, S. Matarrese, A. Riotto, The full second-order radiation transfer function for large-scale cmb anisotropies. JCAP 0605, 010 (2006), arXiv:astro-ph/0512481
  42. 42.
    N. Bartolo, S. Matarrese, A. Riotto, CMB anisotropies at second-order. 2: analytical approach. JCAP 0701, 019 (2007), arXiv:astro-ph/0610110
  43. 43.
    D. Nitta, E. Komatsu, N. Bartolo, S. Matarrese, A. Riotto, CMB anisotropies at second order III: bispectrum from products of the first-order perturbations. JCAP 0905, 014 (2009), arXiv:0903.0894
  44. 44.
    L. Senatore, S. Tassev, M. Zaldarriaga, Cosmological perturbations at second order and recombination perturbed. JCAP 0908, 031 (2009), arXiv:0812.3652
  45. 45.
    W. Hu, Weak lensing of the CMB: a harmonic approach. Phys. Rev. D 62, 043007 (2000), arXiv:astro-ph/0001303
  46. 46.
    C. Pitrou, The radiative transfer at second order: a full treatment of the boltzmann equation with polarization. Class. Quant. Grav. 26, 065006 (2009), arXiv:0809.3036
  47. 47.
    C. Pitrou, J.-P. Uzan, F. Bernardeau, The cosmic microwave background bispectrum from the non-linear evolution of the cosmological perturbations. JCAP 1007, 003 (2010), arXiv:1003.0481
  48. 48.
    C. Pitrou, The radiative transfer for polarized radiation at second order in cosmological perturbations. Gen. Rel. Grav. 41, 2587–2595 (2009), arXiv:0809.3245
  49. 49.
    M. Beneke, C. Fidler, Boltzmann hierarchy for the cosmic microwave background at second order including photon polarization. Phys. Rev. D 82, 063509 (2010), arXiv:1003.1834
  50. 50.
    U. Seljak, M. Zaldarriaga, A line of sight integration approach to cosmic microwave background anisotropies. Astrophys. J. 469, 437–444 (1996), arXiv:astro-ph/9603033
  51. 51.
    C. Fidler, K. Koyama, G.W. Pettinari, A new line-of-sight approach to the non-linear Cosmic Microwave Background. JCAP1504(04), 037 (2015), arXiv:1409.2461
  52. 52.
    R. Saito, A. Naruko, T. Hiramatsu, M. Sasaki, Geodesic curve-of-sight formulae for the Cosmic Microwave Background: a unified treatment of redshift, time delay, and lensing. JCAP 1410(10), 051 (2014), arXiv:1409.2464
  53. 53.
    A. Naruko, C. Pitrou, K. Koyama, M. Sasaki, Second-order Boltzmann equation: gauge dependence and gauge invariance. Class. Quant. Grav. 30, 165008 (2013), arXiv:1304.6929
  54. 54.
    M. Beneke, C. Fidler, K. Klingmuller, B polarization of cosmic background radiation from second-order scattering sources. JCAP 1104, 008 (2011), arXiv:1102.1524
  55. 55.
    C. Fidler, G.W. Pettinari, M. Beneke, R. Crittenden, K. Koyama, D. Wands, The intrinsic B-mode polarisation of the Cosmic Microwave Background. JCAP 1407, 011 (2014), arXiv:1401.3296
  56. 56.
    J.M. Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models. JHEP 0305, 013 (2003), arXiv:astro-ph/0210603
  57. 57.
    E. Komatsu, D.N. Spergel, Acoustic signatures in the primary microwave background bispectrum. Phys. Rev. D 63, 063002 (2001), arXiv:astro-ph/0005036
  58. 58.
    Planck Collaboration, P.A.R. Ade, N. Aghanim, M. Arnaud, F. Arroja, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini, A.J. Banday et al., Planck 2015 results. XVII. Constraints on primordial non-Gaussianity. Astron. Astrophys. 594, A17 (2016), arXiv:1502.01592
  59. 59.
    G.W. Pettinari, C. Fidler, R. Crittenden, K. Koyama, D. Wands, The intrinsic bispectrum of the Cosmic Microwave Background. JCAP 1304, 003 (2013), arXiv:1302.0832
  60. 60.
    S. Mollerach, D. Harari, S. Matarrese, CMB polarization from secondary vector and tensor modes. Phys. Rev. D 69, 063002 (2004), arXiv:astro-ph/0310711
  61. 61.
    K.N. Ananda, C. Clarkson, D. Wands, The cosmological gravitational wave background from primordial density perturbations. Phys. Rev. D 75, 123518 (2007), arXiv:gr-qc/0612013
  62. 62.
    H. Assadullahi, D. Wands, Constraints on primordial density perturbations from induced gravitational waves. Phys. Rev. D 81, 023527 (2010), arXiv:0907.4073
  63. 63.
    D. Baumann, P.J. Steinhardt, K. Takahashi, K. Ichiki, Gravitational wave spectrum induced by primordial scalar perturbations. Phys. Rev. D 76, 084019 (2007), arXiv:hep-th/0703290
  64. 64.
    S. Saga, K. Ichiki, N. Sugiyama, Impact of anisotropic stress of free-streaming particles on gravitational waves induced by cosmological density perturbations. Phys. Rev. D 91(2), 024030 (2015), arXiv:1412.1081
  65. 65.
    N. Seto, S. Kawamura, T. Nakamura, Possibility of direct measurement of the acceleration of the universe using 0.1-Hz band laser interferometer gravitational wave antenna in space. Phys. Rev. Lett. 87, 221103 (2001), arXiv:astro-ph/0108011
  66. 66.
    S. Dimopoulos, P.W. Graham, J.M. Hogan, M.A. Kasevich, S. Rajendran, Atomic gravitational wave interferometric sensor. Phys. Rev. D 78, 122002 (2008)Google Scholar
  67. 67.
    S. Saga, K. Ichiki, K. Takahashi, N. Sugiyama, Magnetic field spectrum at cosmological recombination revisited. Phys. Rev. D 91(12), 123510 (2015), arXiv:1504.03790
  68. 68.
    S. Saga, D. Yamauchi, K. Ichiki, Weak lensing induced by second-order vector mode. Phys. Rev. D 92(6), 063533 (2015), arXiv:1505.02774
  69. 69.
    S. Saga, Observable cosmological vector mode in the dark ages. Phys. Rev. D 94(6), 063523 (2016), arXiv:1607.03973

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Yukawa Institute for Theoretical PhysicsKyoto UniversityKyotoJapan

Personalised recommendations