A dynamical system approach to Bianchi III cosmology for Hu–Sawicki type f(R) gravity

  • Sebika Kangsha Banik
  • Debika Kangsha Banik
  • Kalyan Bhuyan
Research Article
  • 46 Downloads

Abstract

The cosmological dynamics of spatially homogeneous but anisotropic Bianchi type-III space-time is investigated in presence of a perfect fluid within the framework of Hu–Sawicki model. We use the dynamical system approach to perform a detailed analysis of the cosmological behaviour of this model for the model parameters \(n=1, c_1=1\), determining all the fixed points, their stability and corresponding cosmological evolution. We have found stable fixed points with de Sitter solution along with unstable radiation like fixed points. We have identified a matter like point which act like an unstable spiral and when the initial conditions of a trajectory are very close to this point, it stabilizes at a stable accelerating point. Thus, in this model, the universe can naturally approach to a phase of accelerated expansion following a radiation or a matter dominated phase. It is also found that the isotropisation of this model is affected by the spatial curvature and that all the isotropic fixed points are found to be spatially flat.

Keywords

Accelerated expansion of the universe Dynamical system approach Bianchi III cosmology Hu–Sawicki f(R) gravity 

References

  1. 1.
    Sotiriou, T.P., Faraoni, V.: Rev. Mod. Phys. 82, 451 (2010)ADSCrossRefGoogle Scholar
  2. 2.
    Nojiri, S., Odintsov, S.D.: Phys. Rep. 505(2), 59 (2011)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Capozziello, S., Francaviglia, M.: Gen. Relativ. Gravit. 40, 357 (2008)ADSCrossRefGoogle Scholar
  4. 4.
    Nojiri, S., Odintsov, S.D.: Int. J. Geom. Methods Mod. Phys. 04, 115 (2007)CrossRefGoogle Scholar
  5. 5.
    Felice, A.D., Tsujikawa, S.: Living Rev. Relativ. 13, 3 (2010)ADSCrossRefGoogle Scholar
  6. 6.
    Carroll, S.M., Felice, A.D., Duvvuri, V., Easson, D.A., Trodden, M., Turner, M.S.: Phys. Rev. D 71(6), 063513 (2005)ADSCrossRefGoogle Scholar
  7. 7.
    Capozziello, S., Laurentis, M.D.: Phys. Rept. 509, 167 (2011)ADSCrossRefGoogle Scholar
  8. 8.
    Motohashi, H., Starobinsky, A.A.: Int. J. Mod. Phys. D 20, 1347 (2011)ADSCrossRefGoogle Scholar
  9. 9.
    Capozziello, S., Laurentis, M.D., Faraoni, V.: Open Astron. J. 3, 49 (2010)ADSGoogle Scholar
  10. 10.
    Lobo, F.S.N.: (2008). arXiv: 0807.1640 [gr-qc]
  11. 11.
    Hu, W., Sawicki, I.: Phys. Rev. D 76, 064004 (2007)ADSCrossRefGoogle Scholar
  12. 12.
    MacCallum, M.A.H.: Anisotropic and inhomogeneous relativistic cosmologies. In: Hawking, S., Israel, W. (eds.) General Relativity: An Einstein Centenary Survey. Cambridge University Press, Cambridge (1979)Google Scholar
  13. 13.
    Batakis, N.A., Kehagias, A.A.: Nucl. Phys. B 449, 248 (1995)ADSCrossRefGoogle Scholar
  14. 14.
    Suresh, P.K.: JCAP 04, 023 (2008)ADSCrossRefGoogle Scholar
  15. 15.
    COBE Collaboration, Smoot, G.F., et al.: Astrophys. J. L 396, 1 (1992)Google Scholar
  16. 16.
    WMAP Collaboration, Hinshaw, G., et al.: Astrophys. J. Suppl. 148, 135 (2003)Google Scholar
  17. 17.
    WMAP Collaboration, Hinshaw, G., et al.: Astrophys. J. Suppl. 208, 19 (2013)Google Scholar
  18. 18.
    WMAP Collaboration, Bennett, C.L., et al.: Astrophys. J. Suppl. 148, 1 (2003)Google Scholar
  19. 19.
    WMAP Collaboration, Bennett, C.L., et al.: Astrophys. J. Suppl. 208, 20 (2013)Google Scholar
  20. 20.
    Planck Collaboration, Ade, P.A.R., et al.: Astron. Astrophys. 571, A16 (2014)Google Scholar
  21. 21.
    Planck Collaboration, Ade, P.A.R., et al.: Astron. Astrophys. 594, A13 (2016)Google Scholar
  22. 22.
    Bonometto, S., et al.: In: Ellis, G.F.R. (ed.) Modern Cosmology. Institute of Physics Publishing, Bristol and Philadelphia (2002)CrossRefGoogle Scholar
  23. 23.
    Ellis, G.F.R., MacCallum, M.A.H.: Commun. Math. Phys. 12, 108 (1969)ADSCrossRefGoogle Scholar
  24. 24.
    Wainwright, J., Ellis, G.F.R.: Dynamical Systems in Cosmology. Cambridge University Press, Cambridge (1997)CrossRefMATHGoogle Scholar
  25. 25.
    Ellis, G.F.R., Elst, H.V.: Math. Phys. Sci. 541, 1 (1999). arXiv:gr-qc/9812046v5 (2008)Google Scholar
  26. 26.
    Ellis, G.F.R.: Gen. Relativ. Gravit. 38, 1003 (2006)ADSCrossRefGoogle Scholar
  27. 27.
    Shamir, M.F.: Astrophys. Space Sci. 330, 183 (2010)ADSCrossRefGoogle Scholar
  28. 28.
    Sharif, M., Shamir, M.F.: Class. Quant. Grav. 26, 235020 (2009)ADSCrossRefGoogle Scholar
  29. 29.
    Sharif, M., Shamir, M.F.: Gen. Relativ. Gravit. 42, 2643 (2010)ADSCrossRefGoogle Scholar
  30. 30.
    Tripathy, S.K., Mishra, B.: Eur. Phys. J. Plus 131, 273 (2016)CrossRefGoogle Scholar
  31. 31.
    Banik, D.K., Banik, S.K., Bhuyan, K.: Indian J. Phys. 91(1), 109 (2017)ADSCrossRefGoogle Scholar
  32. 32.
    Banik, D.K., Banik, S.K., Bhuyan, K.: Astrophys. Space Sci. 362, 51 (2017)ADSCrossRefGoogle Scholar
  33. 33.
    Tikekar, R., Patel, L.K.: Gen. Relativ. Gravit. 24, 397 (1992)ADSCrossRefGoogle Scholar
  34. 34.
    Yadav, M.K., Rai, A., Pradhan, A.: Int. J. Theor. Phys. 46, 2677 (2007)CrossRefGoogle Scholar
  35. 35.
    Pradhan, A., Amirhashchi, H., Zainuddin, H.: Astrophys. Space Sci. 331, 679 (2011)ADSCrossRefGoogle Scholar
  36. 36.
    Bagora, A.: Astrophys. Space Sci 319, 155 (2009)ADSCrossRefGoogle Scholar
  37. 37.
    Akarsu, O., Kilinc, C.B.: Gen. Relativ. Gravit. 42, 763 (2010)ADSCrossRefGoogle Scholar
  38. 38.
    Pradhan, A., Amirhashchi, H.: Astrophys. Space Sci. 332, 441 (2011)ADSCrossRefGoogle Scholar
  39. 39.
    Shamir, M.F., Bhatti, A.A.: Canadian J. Phys. 90(2), 193 (2012)ADSCrossRefGoogle Scholar
  40. 40.
    Sharif, M., Kausar, H.R.: Phys. Lett. B 697(1), 1 (2011)ADSCrossRefGoogle Scholar
  41. 41.
    Reddy, D.R.K., Kumar, R.S., Kumar, T.V.P.: Int. J. Theor. Phys. 52, 239 (2013)CrossRefGoogle Scholar
  42. 42.
    Chandel, S., Ram, S.: Indian J. Phys 87, 12 1283 (2013)CrossRefGoogle Scholar
  43. 43.
    Mishra, B., Sahoo, P.K., Varma, Ch B.S.: Int. J. Pure Appl. Math. 99, 1 109 (2015)CrossRefGoogle Scholar
  44. 44.
    Reddy, D.R.K., Ramesh, G., Umadevi, S.: (2016). arXiv:1601.02648
  45. 45.
    Chan, N.: Dynamical systems in cosmology. Ph.D. Thesis, University College, London (2012)Google Scholar
  46. 46.
    Perko, L.: Differential Equations and Dynamical Systems. Springer, NewYork (1996)CrossRefMATHGoogle Scholar
  47. 47.
    Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, NewYork (2003)MATHGoogle Scholar
  48. 48.
    Carloni, S., Dunsby, P.K.S., Capozziello, S., Troisi, A.: Class. Quant. Grav. 22, 4839 (2005)ADSCrossRefGoogle Scholar
  49. 49.
    Abdelwahab, M., Goswami, R., Dunsby, P.K.S.: Phys. Rev. D 85, 08351 (2012)CrossRefGoogle Scholar
  50. 50.
    Kandhai, S., Dunsby, P. K. S.: (2015). arXiv: 1511.00101 [gr-qc]
  51. 51.
    Abdelwahab, M., Carloni, S., Dunsby, P.K.S.: Class. Quant. Grav. 25, 135002 (2008)ADSCrossRefGoogle Scholar
  52. 52.
    Carloni, S., Troisi, A., Dunsby, P.K.S.: Gen. Relativ. Gravit. 41, 1757 (2009)ADSCrossRefGoogle Scholar
  53. 53.
    Leach, J.A., Carloni, S., Dunsby, P.K.S.: Class. Quant. Grav. 23, 4915 (2006)ADSCrossRefGoogle Scholar
  54. 54.
    Banik, S.K., Bhuyan, K.: Indian J. Phys 89, 11 1213 (2015)CrossRefGoogle Scholar
  55. 55.
    Leon, G., Saridakis, E.N.: Class. Quant. Grav. 28, 065008 (2011)ADSCrossRefGoogle Scholar
  56. 56.
    Leon, G.: Int. J. Mod. Phys. E 20, 19 (2014)ADSCrossRefGoogle Scholar
  57. 57.
    Leon, G., Roque, A.A.: JCAP 05, 032 (2014)ADSCrossRefGoogle Scholar
  58. 58.
    Goheer, N., Leach, J.A., Dunsby, P.K.S.: Class. Quant. Grav. 24, 5689 (2007)ADSCrossRefGoogle Scholar
  59. 59.
    Fay, S.: Gen. Relativ. Gravit. 32, 203 (2000)ADSMathSciNetCrossRefGoogle Scholar
  60. 60.
    Fay, S.: Class. Quantum Grav. 17, 1667 (2000)ADSMathSciNetCrossRefGoogle Scholar
  61. 61.
    Chen, X., Saridakis, E.N., Leon, G.: JCAP 07, 005 (2012)Google Scholar
  62. 62.
    Fadragas, C.R., Leon, G., Saridakis, E.N.: Class. Quant. Grav. 31, 075018 (2014)ADSCrossRefGoogle Scholar
  63. 63.
    Santos, B., Campista, M., Santos, J., Campista, J.S.: A. & A. 548, A31 (2008)ADSCrossRefGoogle Scholar
  64. 64.
    Schmidt, F., Vikhlinin, A., Hu, W.: Phys. Rev. D 80, 083505 (2009)ADSCrossRefGoogle Scholar
  65. 65.
    Oyaizu, H., Lima, M., Hu, W.: Phys. Rev. D 78, 123524 (2008)ADSCrossRefGoogle Scholar
  66. 66.
    Gil-Marin, H., Schmidt, F., Hu, W., Jimenez, R., Verde, L.: JCAP 11, 019 (2011)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Sebika Kangsha Banik
    • 1
  • Debika Kangsha Banik
    • 1
  • Kalyan Bhuyan
    • 1
  1. 1.Department of PhysicsDibrugarh UniversityDibrugarhIndia

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