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Power-law inflation with minimal and nonminimal coupling

  • Mithun Bairagi
  • Amitava ChoudhuriEmail author
Regular Article
  • 15 Downloads

Abstract.

Power-law inflationary scenarios based on minimal and non-minimal coupling of a scalar field to gravity with exponential- and power-law potential, respectively, are studied using the symmetry-based approach. In particular, we obtained only one parameter Lie point symmetry for both the minimal and non-minimal coupling cases and it is interesting to note that the non-minimal coupled equation is invariant under a scale transformation. We find the exact analytical group invariant solutions from invariant curve condition for both the minimal and non-minimal cases of the power-law inflationary model. The solutions obtained are consistent with the Friedmann equations subject to constraints on the inflationary potential parameter \( \lambda\) for the minimal case and on the coupling parameter \( \zeta\) for the non-minimal case. In this scenario, we find transformation relations for various inflationary parameters e.g. amplitude of the scalar power spectrum, spectral index, slow-roll parameters (SRP), tensor-to-scalar perturbation ratio, equation of state parameters, non-Gaussianity parameter as well as the form of the potential in two different frames, namely the Jordan and Einstein frames by making use of the conformal transformation. The results for various inflationary parameters for the non-minimal case are presented in the background of Planck2015 and Planck2018 and are in good agreement with the cosmological observations if the non-minimal coupling parameter is chosen properly. We treat minimally and non-minimally coupled scalar field equations by the dynamical system theory and present critical point analysis. By checking the stability of the critical points in the phase space for both cases we have shown that the solutions obtained from the Lie symmetry approach are the stable attractor solutions.

References

  1. 1.
    S. Weinberg, Cosmology (Oxford University Press Inc., New York, 2008)Google Scholar
  2. 2.
    A.H. Guth, Phys. Rev. D 23, 347 (1981)ADSCrossRefGoogle Scholar
  3. 3.
    E.W. Kolb, M.S. Turner, The Early Universe (Addison-Wesley, New York, 1990)Google Scholar
  4. 4.
    V.F. Mukhanov, Physical Foundations of Cosmology (Cambridge University Press, 2005)Google Scholar
  5. 5.
    D.H. Lyth, A. Riotto, Phys. Rep. 314, 1 (1999)ADSCrossRefGoogle Scholar
  6. 6.
    A.D. Linde, Phys. Lett. B 108, 389 (1982)ADSCrossRefGoogle Scholar
  7. 7.
    R. Kallosh, A. Linde, D. Roest, Phys. Rev. Lett. 112, 011303 (2014)ADSCrossRefGoogle Scholar
  8. 8.
    K. Nozari, S.D. Sadatian, Mod. Phys. Lett. A 23, 2933 (2008)ADSCrossRefGoogle Scholar
  9. 9.
    J.A. Stein-Schabas, Phys. Rev. D 35, 2345 (1987)ADSCrossRefGoogle Scholar
  10. 10.
    F. Lucchin, S. Matarrese, Phys. Rev. D 32, 1316 (1985)ADSCrossRefGoogle Scholar
  11. 11.
    L.F. Abbott, M.B. Wise, Nucl. Phys. B 244, 541 (1987)ADSCrossRefGoogle Scholar
  12. 12.
    E. Elizalde, S. Nojiri, S.D. Odintsov, Phys. Rev. D 70, 043539 (2004)ADSCrossRefGoogle Scholar
  13. 13.
    E. Elizalde, S. Nojiri, S.D. Odintsov, D. Saez-Gomez, V. Faraoni, Phys. Rev. D 77, 106005 (2008)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    J.J. Halliwell, Phys. Lett. B 185, 341 (1987)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    J.D. Barrow, Phys. Lett. B 187, 12 (1987)ADSCrossRefGoogle Scholar
  16. 16.
    D. Babich, P. Creminelli, M. Zaldarriaga, J. Cosmol. Astropart. Phys. 08, 009 (2004) arXiv:astro-ph/0405356ADSCrossRefGoogle Scholar
  17. 17.
    A.P.S. Yadav, B.D. Wandelt, Adv. Astron. 2010, 1 (2010) arXiv:1006.0275CrossRefGoogle Scholar
  18. 18.
    Planck Collaboration (P.A.R. Ade et al.), Astron. Astrophys. 594, A17 (2016) arXiv:1502.01592CrossRefGoogle Scholar
  19. 19.
    K. Asadi, K. Nozari, Nucl. Phys. B 934, 118 (2018)ADSCrossRefGoogle Scholar
  20. 20.
    N.A. Chernikov, E.A. Tagirov, Ann. Inst. H. Poincaré A 9, 109 (1968)ADSGoogle Scholar
  21. 21.
    C.G. Callan, S. Coleman, R. Jackiw, Ann. Phys. 59, 42 (1970)ADSCrossRefGoogle Scholar
  22. 22.
    N.D. Birrell, P.C. Davies, Quantum Fields in Curved Space (Cambridge University Press, Cambridge, UK, 1980)Google Scholar
  23. 23.
    B. Nelson, P. Panangaden, Phys. Rev. D 25, 1019 (1982)ADSCrossRefGoogle Scholar
  24. 24.
    S. Capozziello, M. De Laurentis, Phys. Rep. 509, 167 (2011)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    K. Nozari, N. Rashidi, Phys. Rev. D 86, 043505 (2012)ADSCrossRefGoogle Scholar
  26. 26.
    P. Jordan, Z. Phys. 157, 112 (1959)ADSCrossRefGoogle Scholar
  27. 27.
    C. Brans, R.H. Dicke, Phys. Rev. 124, 925 (1961)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    V. Faraoni, Phys. Rev. D 62, 023504 (2000)ADSCrossRefGoogle Scholar
  29. 29.
    V. Faraoni, E. Gunzig, P. Nardone, Fundam. Cosmic Phys. 20, 121 (1999)ADSGoogle Scholar
  30. 30.
    E. Komatsu, T. Futamase, Phys. Rev. D 59, 064029 (1999)ADSCrossRefGoogle Scholar
  31. 31.
    R. Fakir, W.G. Unruh, Astrophys. J. 394, 396 (1992)ADSCrossRefGoogle Scholar
  32. 32.
    V. Faraoni, Phys. Rev. D 53, 6813 (1996)ADSCrossRefGoogle Scholar
  33. 33.
    J.R. Morris, Class. Quantum Grav. 18, 2977 (2001)ADSCrossRefGoogle Scholar
  34. 34.
    J.P. Uzan, Phys. Rev. D 59, 123510 (1999)ADSCrossRefGoogle Scholar
  35. 35.
    A.A. Starobinsky, S. Tsujikawa, J. Yokoyama, Nucl. Phys. B 610, 383 (2001)ADSCrossRefGoogle Scholar
  36. 36.
    L. Amendola, Phys. Rev. D 60, 043501 (1999)ADSCrossRefGoogle Scholar
  37. 37.
    R. Fakir, W.G. Unruh, Phys. Rev. D 41, 1783 (1990)ADSCrossRefGoogle Scholar
  38. 38.
    K. Nozari, S. Shafizadeh, Phys. Scr. 82, 015901 (2010)ADSCrossRefGoogle Scholar
  39. 39.
    P.J. Olver, Applications of Lie Groups to Differential Equations (Springer, New York, 1993)Google Scholar
  40. 40.
    Amitava Choudhuri, Phys. Scr. 90, 055004 (2015)CrossRefGoogle Scholar
  41. 41.
    K. Andriopoulos, P.G.L. Leach, Cent. Eur. J. Phys. 6, 469 (2008)Google Scholar
  42. 42.
    Amitava Choudhuri, Nonlinear Evolution Equations: Lagrangian Approach (LAP Lambert Academic Publishing, 2011)Google Scholar
  43. 43.
    P. Kanti, Olive, Phys. Rev. D 60, 043502 (1999)ADSCrossRefGoogle Scholar
  44. 44.
    A.B. Burd, J.D. Barrow, Nucl. Phys. B 308, 929 (1988)ADSCrossRefGoogle Scholar
  45. 45.
    E. Elizalde, S. Nojiri, S.D. Odintsov, Phys. Rev. D 70, 043539 (2004)ADSCrossRefGoogle Scholar
  46. 46.
    F.M. Mahomed, P.G.L. Leach, Quaest. Math. 8, 241 (1985)CrossRefGoogle Scholar
  47. 47.
    D.S. Salopek, J.R. Bond, J.M. Bardeen, Phys. Rev. D 40, 1753 (1989)ADSCrossRefGoogle Scholar
  48. 48.
    F.L. Bezrukov, M. Shaposhnikov, Phys. Lett. B 659, 703 (2008)ADSCrossRefGoogle Scholar
  49. 49.
    N. Makino, M. Sasaki, Prog. Theor. Phys. 86, 103 (1991)ADSCrossRefGoogle Scholar
  50. 50.
    T. Futamase, T. Rothman, R. Matzner, Phys. Rev. D 39, 405 (1989)ADSCrossRefGoogle Scholar
  51. 51.
    E. Komatsu, T. Futamase, Phys. Rev. D 58, 023004 (1998) 58ADSCrossRefGoogle Scholar
  52. 52.
    E. Komatsu, T. Futamase, Phys. Rev. D 59, 064029 (1999)ADSCrossRefGoogle Scholar
  53. 53.
    T. Qiu, J. Cosmol. Astropart. Phys. 06, 041 (2012)ADSCrossRefGoogle Scholar
  54. 54.
    Planck Collaboration (P.A.R. Ade et al.), Astron. Astrophys. 594, A20 (2016) arXiv:1502.02114v1CrossRefGoogle Scholar
  55. 55.
    G. Hinshaw, D. Larson, E. Komatsu et al., Astrophys. J. Suppl. Ser. 208, 19 (2013)ADSCrossRefGoogle Scholar
  56. 56.
    Planck Collaboration (P.A.R. Ade et al.), Astron. Astrophys. 571, A22 (2013)CrossRefGoogle Scholar
  57. 57.
    P.A.R. Ade et al., Phys. Rev. Lett. 112, 241101 (2014)ADSCrossRefGoogle Scholar
  58. 58.
    F. Wu, Y. Li, Y. Lu, X. Chen, Sci. China Phys. Mech. Astron. 57, 1449 (2014)ADSCrossRefGoogle Scholar
  59. 59.
    Planck Collaboration (P.A.R. Ade et al.), Astron. Astrophys. 594, A13 (2016) arXiv:1502.01589v2CrossRefGoogle Scholar
  60. 60.
    Planck Collaboration (P.A.R. Ade), Planck2018 results. X. Constraints on inflation, arXiv:1807.06211v1Google Scholar
  61. 61.
    K.T. Alligood, T. Sauer, J.A. Yorke, Chaos: An Introduction to Dynamical Systems (Springer-Verlag, New York, 1997)Google Scholar
  62. 62.
    E.F. Bunn, A.R. Liddle, M.J. White, Phys. Rev. D 54, R5917 (1996)ADSCrossRefGoogle Scholar
  63. 63.
    A.R. Liddle, P. Parsons, J.D. Barrow, Phys. Rev. D 50, 7222 (1994)ADSCrossRefGoogle Scholar
  64. 64.
    J.M. Bardeen, Phys. Rev. D 22, 1882 (1980)ADSMathSciNetCrossRefGoogle Scholar
  65. 65.
    H. Kodama, M. Sasaki, Prog. Theor. Phys. Suppl. 78, 1 (1984)ADSCrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of PhysicsThe University of BurdwanGolapbagIndia

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