Skip to main content
SpringerLink
Log in

Viability of Noether Symmetry of F(R) Theory of Gravity

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

Recently, we have explored vices and virtues of \(R^{\frac{3}{2}}\) term in the action which has in-built Noether symmetry and anticipated that a linear term might improve the situation (Sarkar et al., arXiv:1201.2987 [astro-ph.CO], 2012). In the absence of a conserved current it is extremely difficult to obtain an analytical solution of the said fourth order theory of gravity in the presence of a linear term. Here, we therefore enlarge the configuration space by including a scalar field in addition and also taking some of the anisotropic models (in the absence of a scalar field) into account. We observe that Noether symmetry remains obscure and it does not even reproduce the one that already exists in the literature (Sanyal, Gen. Relativ. Gravit., 37:407, 2005). However, there exists in general, a conserved current for F(R) theory of gravity in the presence of a non-minimally coupled scalar field (Sanyal, Phys. Lett. B, 624:81, 2005; Mod. Phys. Lett. A, 25:2667, 2010), which simplifies the field equations considerably. Here, we briefly expatiate the non-Noether conserved current and show that indeed the situation is modified.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Nojiri, S., Odintsov, S.D.: Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models. Phys. Rep. 505, 59 (2011). arXiv:1011.0544

    Article  MathSciNet  ADS  Google Scholar 

  2. Vollick, D.N.: 1/R curvature corrections as the source of the cosmological acceleration. Phys. Rev. D 68, 063510 (2003). arXiv:astro-ph/0306630

    Article  ADS  Google Scholar 

  3. Meng, X., Wang, P.: Modified Friedmann equations in R −1-modified gravity. Class. Quantum Gravity 20, 4949 (2003). arXiv:astro-ph/0307354

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. Meng, X., Wang, P.: Cosmological evolution in 1/R-gravity theory. Class. Quantum Gravity 21, 951 (2004). arXiv:astro-ph/0308031; ibid., arXiv:astro-ph/0308284

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Sotiriou, T.P.: Unification of inflation and cosmic acceleration in the Palatini formalism. Phys. Rev. D 73, 063515 (2006). arXiv:gr-qc/0509029

    Article  MathSciNet  ADS  Google Scholar 

  6. Lee, S.: Palatini f(R) Cosmology. Mod. Phys. Lett. A 23, 1388 (2008). arXiv:0801.4606 [gr-qc]

    Article  ADS  MATH  Google Scholar 

  7. Li, B., Barrow, J.D., Mota, D.F.: Cosmology of Ricci-tensor-squared gravity in the Palatini variational approach. Phys. Rev. D 76, 104047 (2007). arXiv:0707.2664 [gr-qc]

    Article  MathSciNet  ADS  Google Scholar 

  8. Poplawski, N.J.: The cosmic snap parameter in f(R) gravity. Class. Quantum Gravity 24, 3013 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. Capozziello, S., Carloni, S., Troisi, A.: Quintessence without scalar fields. Recent Res. Dev. Astron. Astrophys. 1, 625 (2003). arXiv:astro-ph/0303041

    Google Scholar 

  10. Li, B., Barrow, J.D.: The cosmology of f(R) gravity in the metric variational approach. Phys. Rev. D 75, 084010 (2007). arXiv:gr-qc/0701111

    Article  MathSciNet  ADS  Google Scholar 

  11. Carroll, S.M., Duvvuri, V., Trodden, M., Turner, M.S.: Is cosmic speed-up due to new gravitational physics? Phys. Rev. D 70, 043528 (2004). arXiv:astro-ph/0306438

    Article  ADS  Google Scholar 

  12. Lue, A., Scoccimarro, R., Starkman, G.: Differentiating between modified gravity and dark energy. Phys. Rev. D 69, 044005 (2004). arXiv:astro-ph/0307034

    Article  ADS  Google Scholar 

  13. Nojiri, S., Odintsov, S.D.: Where new gravitational physics comes from: M-theory? Phys. Lett. B 576, 5 (2003). arXiv:hep-th/0307071

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. Nojiri, S., Odintsov, S.D.: Modified gravity with negative and positive powers of the curvature: unification of the inflation and of the cosmic acceleration. Phys. Rev. D 68, 123512 (2003). arXiv:hep-th/0307288

    Article  MathSciNet  ADS  Google Scholar 

  15. Nojiri, S., Odintsov, S.D.: Modified gravity with lnR terms and cosmic acceleration. Gen. Relativ. Gravit. 36, 1765 (2004). arXiv:hep-th/0308176

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. Dick, R.: On the Newtonian limit in gravity models with inverse powers of R. Gen. Relativ. Gravit. 36, 217 (2004). arXiv:gr-qc/0307052

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. Song, Y.-S., Hu, W., Sawicki, I.: Large scale structure of f(R) gravity. Phys. Rev. D 75, 044004 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  18. Bamba, K., Geng, C.Q., Nojiri, S., Odintsov, S.D.: Crossing of the phantom divide in modified gravity. Phys. Rev. D 79, 083014 (2009). arXiv:0810.4296v2 [hep-th]

    Article  MathSciNet  ADS  Google Scholar 

  19. Nojiri, S., Odintsov, S.D.: Unifying inflation with ΛCDM epoch in modified f(R) gravity consistent with solar system tests. Phys. Lett. B 657, 238 (2007). arXiv:0707.1941 [hep-th]

    Article  MathSciNet  ADS  Google Scholar 

  20. Nojiri, S., Odintsov, S.D.: Modified f(R) gravity unifying R m inflation with ΛCDM epoch. Phys. Rev. D 77, 026007 (2008). arXiv:0710.1738 [hep-th]

    Article  MathSciNet  ADS  Google Scholar 

  21. Cognola, G., Elizalde, E., Nojiri, S., Odintsov, S.D., Sebastiani, L., Zerbini, S.: Class of viable modified f(R) gravities describing inflation and the onset of accelerated expansion. Phys. Rev. D 77, 046009 (2008)

    Article  ADS  Google Scholar 

  22. Dolgov, A.D., Kawasaki, M.: Can modified gravity explain accelerated cosmic expansion? Phys. Lett. B 573, 1 (2003). arXiv:astro-ph/0307285

    Article  ADS  MATH  Google Scholar 

  23. Amendola, L., Polarski, D., Tsujikawa, S.: Are f(R) dark energy models cosmologically viable? Phys. Rev. Lett. 98, 131302 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  24. Brookfield, A.W., van de Bruck, C., Hall, L.M.H.: Viability of f(R) theories with additional powers of curvature. Phys. Rev. D 74, 064028 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  25. Lee, S.: Stability of Palatini-f(R) cosmology. arXiv:0710.2395 [gr-qc]

  26. Allemandi, G., Francaviglia, M., Ruggiero, M.L., Tartaglia, A.: Post-Newtonian parameters from alternative theories of gravity. Gen. Relativ. Gravit. 37, 1891 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  27. Koivisto, T., Suonio, H.K.: Cosmological perturbations in the Palatini formulation of modified gravity. Class. Quantum Gravity 23, 2355 (2006). arXiv:astro-ph/0509422

    Article  ADS  MATH  Google Scholar 

  28. Tsujikawa, S., Uddin, K., Tavakol, R.: Density perturbations in f(R) gravity theories in metric and Palatini formalisms. Phys. Rev. D 77, 043007 (2008). arXiv:0712.0082 [astro-ph]

    Article  MathSciNet  ADS  Google Scholar 

  29. Capozziello, S., Darabi, F., Vernieri, D.: Equivalence between Palatini and metric formalisms of f(R)-gravity by divergence free current. Mod. Phys. Lett. A 26, 65 (2011). arXiv:1006.0454 [gr-qc]

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. Borunda, M., Janssen, B., Gil, M.B.: Palatini versus metric formulation in higher curvature gravity. J. Cosmol. Astropart. Phys. 0811, 008 (2008). arXiv:0804.4440 [hep-th]

    Article  ADS  Google Scholar 

  31. de Ritis, R., Marmo, G., Platania, G., Rubano, C., Scudellaro, P., Stornaiolo, C.: New approach to find exact solutions for cosmological models with a scalar field. Phys. Rev. D 42, 1091 (1990)

    Article  MathSciNet  ADS  Google Scholar 

  32. de Ritis, R., Platania, G., Rubano, C., Sabatino, R.: Scalar fields and matter cosmologies. Phys. Lett. A 161, 230 (1991)

    Article  MathSciNet  ADS  Google Scholar 

  33. Demianski, M., de Ritis, R., Marmo, G., Platania, G., Rubano, C., Scudellaro, P., Stornaiolo, C.: Scalar field, nonminimal coupling, and cosmology. Phys. Rev. D 44, 3136 (1991)

    Article  MathSciNet  ADS  Google Scholar 

  34. Demianski, M., de Ritis, R., Rubano, C., Scudellaro, P.: Scalar fields and anisotropy in cosmological models. Phys. Rev. D 46, 1391 (1992)

    Article  MathSciNet  ADS  Google Scholar 

  35. Capozziellio, S., de Ritis, R.: Relation between potential and nonminimal coupling in inflationary cosmology. Phys. Lett. A 177, 1 (1993)

    Article  MathSciNet  ADS  Google Scholar 

  36. Capozziellio, S., de Ritis, R.: Nöther’s symmetries and exact solutions in flat non-minimally coupled cosmological models. Class. Quantum Gravity 11, 107 (1994)

    Article  ADS  Google Scholar 

  37. Capozziello, S., de Ritis, R., Scudellaro, P.: Nonminimal coupling and cosmic no-hair theorem. Phys. Lett. A 188, 130 (1994)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  38. Capozziello, S., de Ritis, R., Scudellaro, P.: Nöther’s symmetries in nonflat cosmologies. Nuovo Cimento B 109, 159 (1994)

    Article  ADS  Google Scholar 

  39. Capozziello, S., Marmo, G., Rubano, C., Scudellaro, P.: Noether symmetries in Bianchi universes. J. Mod. Phys. D 6, 491 (1997). arXiv:gr-qc/9606050

    MathSciNet  ADS  MATH  Google Scholar 

  40. Capozziello, S., Piedipalumbo, E., Rubano, C., Scudellaro, P.: Noether symmetry approach in phantom quintessence cosmology. Phys. Rev. D 80, 104030 (2009). arXiv:0908.2362 [astro-ph.CO]

    Article  ADS  Google Scholar 

  41. Fay, S.: Noether symmetry of the hyperextended scalar tensor theory for the FLRW models. Class. Quantum Gravity 18, 4863 (2001). arXiv:gr-qc/0309088

    Article  MathSciNet  ADS  MATH  Google Scholar 

  42. Sanyal, A.K.: Noether and some other dynamical symmetries in Kantowski-Sachs model. Phys. Lett. B 524, 177 (2002). arXiv:gr-qc/0107053

    Article  MathSciNet  ADS  MATH  Google Scholar 

  43. Sanyal, A.K., Rubano, C., Piedipalumbo, E.: Coupling parameters and the form of the potential via Noether symmetry. Gen. Relativ. Gravit. 35, 1617 (2003). arXiv:astro-ph/0210063

    Article  MathSciNet  ADS  MATH  Google Scholar 

  44. Sanyal, A.K., Rubano, C., Piedipalumbo, E.: Noether symmetry for Gauss-Bonnet dilatonic gravity. Gen. Relativ. Gravit. 43, 2807 (2011). arXiv:1107.0560 [astro-ph]

    Article  MathSciNet  ADS  MATH  Google Scholar 

  45. Kamilya, S., Modak, B.: Noether symmetry study in general scalar tensor theory. Gen. Relativ. Gravit. 36, 673 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  46. Kamilya, S., Modak, B., Biswas, S.: Induced gravity theory from Noether symmetry. Gen. Relativ. Gravit. 36, 661 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  47. Bonanno, A., Esposito, G., Rubano, C., Scudellaro, P.: Noether symmetry approach in matter-dominated cosmology with variable G and Λ. Gen. Relativ. Gravit. 39, 189 (2007). arXiv:astro-ph/0612091

    Article  MathSciNet  ADS  MATH  Google Scholar 

  48. Zhang, Y., Gong, Y.-G., Zhu, Z.-H.: Noether symmetry approach in multiple scalar fields scenario. Phys. Lett. B 688, 13 (2010). arXiv:0912.0067 [hep-ph]

    Article  ADS  Google Scholar 

  49. de Souza, R.C., Kremer, G.M.: Constraining non-minimally coupled tachyon fields by the Noether symmetry. Class. Quantum Gravity 26, 135008 (2009)

    Article  ADS  Google Scholar 

  50. Esposito, G., Roychowdhury, R., Rubano, C., Scudellaro, P.: On the transition from complex to real scalar fields in modern cosmology. arXiv:1009.2887 [hep-th]

  51. Sanyal, A.K., Modak, B., Rubano, C., Piedipalumbo, E.: Noether symmetry in the higher order gravity theory. Gen. Relativ. Gravit. 37, 407 (2005). arXiv:astro-ph/0310610

    Article  MathSciNet  ADS  MATH  Google Scholar 

  52. Basilakos, S., Tsamparlis, M., Paliathanasis, A.: Using the Noether symmetry approach to probe the nature of dark energy. Phys. Rev. D 83, 103512 (2011). arXiv:1104.2980 [astro-ph.CO]

    Article  ADS  Google Scholar 

  53. Capozziello, S., Lambiase, G.: Higher-order corrections to the effective gravitational action from Noether symmetry approach. Gen. Relativ. Gravit. 32, 295 (2000). arXiv:gr-qc/9912084

    Article  MathSciNet  ADS  MATH  Google Scholar 

  54. Capozziello, S., Stabile, A., Troisi, A.: Spherically symmetric solutions in f(R)-gravity via Noether symmetry approach. Class. Quantum Gravity 24, 2153 (2007). arXiv:gr-qc/0703067

    Article  MathSciNet  ADS  MATH  Google Scholar 

  55. Capozziello, S., Nesseris, S., Perivolaropoulos, L.: Reconstruction of the scalar-tensor Lagrangian from a LCDM background and Noether symmetry. J. Cosmol. Astropart. Phys. 0712, 009 (2007). arXiv:0705.3586 [astro-ph]

    Article  MathSciNet  ADS  Google Scholar 

  56. Capozziello, S., M-Moruno, P., Rubano, C.: Dark energy and dust matter phases from an exact f(R)-cosmology model. Phys. Lett. B 664, 12 (2008). arXiv:0804.4340 [astro-ph]

    Article  ADS  Google Scholar 

  57. Capozziello, S., Nesseris, S., Perivolaropoulos, L.: Exact f(R)-cosmological model coming from the request of the existence of a Noether symmetry. AIP Conf. Proc. 1122, 213 (2009). arXiv:0812.2138 [gr-qc]

    Article  ADS  Google Scholar 

  58. Capozziello, S., De, A.: Felice, f(R) cosmology by Noether’s symmetry. J. Cosmol. Astropart. Phys. 0808, 016 (2008). arXiv:0804.2163 [gr-qc]

    Article  ADS  Google Scholar 

  59. Vakili, B.: Noether symmetry in f(R) cosmology. Phys. Lett. B 664, 16 (2008). arXiv:0804.3449 [gr-qc]

    Article  MathSciNet  ADS  Google Scholar 

  60. Vakili, B.: Noether symmetric f(R) quantum cosmology and its classical correlations. Phys. Lett. B 669, 206 (2008). arXiv:0809.4591 [gr-qc]

    Article  MathSciNet  ADS  Google Scholar 

  61. Paliathanasis, A., Tsamparlis, M., Basilakos, S.: Constraints and analytical solutions of f(R) theories of gravity using Noether symmetries. arXiv:1111.4547 [astro-ph.CO]

  62. Sarkar, K., Sk, N., Ruz, S., Debnath, S., Sanyal, A.K.: Why Noether symmetry of F(R) theory yields three-half power law? (2012). arXiv:1201.2987 [astro-ph.CO]

  63. Sanyal, A.K., Modak, B.: Quantum cosmology with a curvature squared action. Phys. Rev. D 63, 064021 (2001). arXiv:gr-qc/0107001

    Article  MathSciNet  ADS  Google Scholar 

  64. Sanyal, A.K., Modak, B.: Is Noether symmetric approach consistent with dynamical equation in non-minimal scalar-tensor theories? Class. Quantum Gravity 18, 3767 (2001). arXiv:gr-qc/0107052

    Article  MathSciNet  ADS  MATH  Google Scholar 

  65. Sanyal, A.K.: Quantum mechanical probability interpretation in the mini-superspace model of higher order gravity theory. Phys. Lett. B 542, 147 (2002). arXiv:gr-qc/0205053

    Article  MathSciNet  ADS  MATH  Google Scholar 

  66. Sanyal, A.K.: Quantum mechanical formulation of quantum cosmology for brane-world effective action. In: Ross, L.V. (ed.) Focus on Astrophysics Research, p. 109. Nova Publ., New York (2003). arXiv:gr-qc/0305042

    Google Scholar 

  67. Sanyal, A.K.: Hamiltonian formulation of curvature squared action. Gen. Relativ. Gravit. 37, 1957 (2005). arXiv:hep-th/0407141

    Article  MathSciNet  ADS  MATH  Google Scholar 

  68. Sanyal, A.K., Debnath, S., Ruz, S.: Canonical formulation of curvature squared action in the presence of Lapse function. Class. Quantum Gravity 29, 215007 (2012). arXiv:1108.5869 [gr-qc]

    Article  ADS  Google Scholar 

  69. Candelas, P., Horowitz, G.T., Strominger, A., Witten, E.: Vacuum configurations for superstrings. Nucl. Phys. B 258, 46 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  70. Metsaev, R.R., Tseytlin, A.A.: Curvature cubed terms in string theory effective actions. Phys. Lett. B 185, 52 (1987)

    Article  MathSciNet  ADS  Google Scholar 

  71. Green, M.B., Schwarz, J.H., Witten, E.: Superstring Theory, Vols. 1 & 2,. Cambridge University Press, Cambridge (1987)

    Google Scholar 

  72. Deser, S.: Gravity from strings. Phys. Scr. T 15, 138 (1987)

    Article  MathSciNet  ADS  Google Scholar 

  73. Ketov, S.V.: Tree string generated corrections to Einstein gravity from the sigma model approach. Gen. Relativ. Gravit. 22, 193 (1990)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  74. Horava, P., Witten, E.: Heterotic and type I string dynamics from eleven dimensions. Nucl. Phys. B 460, 506 (1996). Also in Nucl. Phys. B 475, 94 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  75. Kanno, S., Soda, J.: Brane world effective action at low energies and AdS/CFT correspondence. Phys. Rev. D 66, 043526 (2002). arXiv:hep-th/0205188

    Article  MathSciNet  ADS  Google Scholar 

  76. Gregory, J.P., Padilla, A.: Braneworld holography in Gauss–Bonnet gravity. Class. Quantum Gravity 20, 4221 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  77. Cecotti, S., Ferrara, S., Girardello, L., Porrati, M., Pasquinucci, A.: Matter coupling in higher-derivative supergravity. Phys. Rev. D 33, R2504 (1986)

    Article  ADS  Google Scholar 

  78. Chamseddine, A.H.: Topological gauge theory of gravity in five and all odd dimensions. Phys. Lett. B 233, 291 (1989)

    Article  MathSciNet  ADS  Google Scholar 

  79. M-Hoissen, F.: From Chern–Simons to Gauss–Bonnet. Nucl. Phys. B 346, 235 (1990)

    Article  ADS  Google Scholar 

  80. Allemandi, G., Francaviglia, M., Raiteri, M.: Charges and energy in Chern–Simons theories and Lovelock gravity. Class. Quantum Gravity 20, 5103 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  81. Horowitz, G.T.: Quantum cosmology with a positive-definite action. Phys. Rev. D 31, 1169 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  82. Pollock, M.D.: On the semi-classical approximation to the wave function of the universe and its stochastic interpretation. Nucl. Phys. B 306, 931 (1988)

    Article  ADS  Google Scholar 

  83. Capozziello, S., De Laurentis, M., Odintsov, S.D.: Hamiltonian dynamics and Noether symmetries in extended gravity cosmology. arXiv:1206.4842 [gr-qc]

  84. Sanyal, A.K.: Scalar tensor theory of gravity carrying a conserved current. Phys. Lett. B 624, 81 (2005). arXiv:hep-th/0504021

    Article  MathSciNet  ADS  MATH  Google Scholar 

  85. Sanyal, A.K.: Study of symmetry in F(R) theory of gravity. Mod. Phys. Lett. A 25, 2667 (2010). arXiv:0910.2385 [astro-ph.CO]

    Article  MathSciNet  ADS  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Physics, University of Kalyani, Nadia, 741235, India

    Kaushik Sarkar

  2. Dept. of Physics, Ranitala High School, Murshidabad, 742135, India

    Nayem Sk

  3. Dept. of Physics, Jangipur College, Murshidabad, 742213, India

    Subhra Debnath & Abhik Kumar Sanyal

Authors
  1. Kaushik Sarkar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nayem Sk
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Subhra Debnath
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Abhik Kumar Sanyal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Abhik Kumar Sanyal.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sarkar, K., Sk, N., Debnath, S. et al. Viability of Noether Symmetry of F(R) Theory of Gravity. Int J Theor Phys 52, 1194–1213 (2013). https://doi.org/10.1007/s10773-012-1436-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-012-1436-8

Keywords

Search

Navigation