Isaac Scientific Publishing

Advances in Astrophysics

Noether Symmetries and Conserved Quantities of f(R) Cosmology Models Containing Time Transformation

Download PDF (393.9 KB) PP. 93 - 105 Pub. Date: August 1, 2016

DOI: 10.22606/adap.2016.12004

Author(s)

  • Jing-Li Fu*
    Institute of Mathematical Physics, Zhejiang Sci-Tech University, Hangzhou 310018, China
  • Yong-Xin Guo
    Department of Physics, Liaoning University, Shenyang 110036, China

Abstract

According to the invariance of Hamilton action of the generic f(R) cosmological model under the infinitesimal transformations with respect to the time t, the scale factor a and the Ricci scalar R, the generalized variation principle of the f(R) cosmological model is presented. The definition and the criterion of Noether symmetry are given for this model. The Noether identical equation and the Noether theorem of generic f(R) cosmological models are obtained. The Lie group of transformation, the conserved quantity and the form of generic f(R) gravity are derived via the method of Noether symmetry. It is also shown that the resulting form of f(R) yields as conservative law of the cosmological scale factor. Finally, an example is discussed to illustrate these results.

Keywords

f(R) cosmological model, Noether symmetry, conserved quantity, time transformation.

References

[1] S. Capozziello, M. De Laurentis, S. Nojiri, and S. D. Odintsov. “Classifying and avoiding singularities in the alternative gravity dark energy models.” Physical. review. D, vol.79, no.12, pp. 320-330, 2009

[2] S. Nojiri and S.D. Odintsov, “Introduction to modified gravity and gravitational alternative for dark energy” Int. J. Geom. Methods Mod. Phys. Vol.4,no.1,pp.115-145, 2007

[3] S. Nojiri and S. D. Odintsov. “Effective equation of state and energy conditions in phantom/tachyon inflationary cosmology perturbed by quantum effects,” Physics Letters B,vol.571,no.1-2,pp. 1-10,

[4] S. Nojiri and S.D. Odintsov,” Unified cosmic history in modified gravity: from F(R) theory to Lorentz noninvariant models, Physics Reports, vol. 505,no.2-4,pp. 59–144, 2010

[5] N.V. Dan “1/R Curvature Corrections as the Source of the Cosmological Acceleration”,Phys. Rev. D,vol. 68 ,no.6, pp. 484-504, 2003 .

[6] E.E. Flanagan, “Palatini Form of 1/R Gravity”Phys. Rev. Lett. vol. 92, no.7,pp. 2835-2842, 2004

[7] T.P. Sotiriou, S. Liberati, “Metric-affine f(R) theories of gravity,” Ann. Phys. vol.322, no.4,pp.935-966,2007

[8] P.J. Olver, “Applications of Lie Groups to Differential Equations,”Springer, New York, 1993.

[9] L.V. Ovisiannikov, “Group Analysis of Difference Equations”, Academic, New York,1982

[10] N.H. Ibragimov, “Transformation Groups Applied to Mathematical Physics”, Reidel, Boston, 1985.

[11] G.W. Bluman, S. Kumei, “Symmetries of Differential Equations”, Springer, Berlin, 1989.

[12] F.X. Mei, “Applications of Lie Group and Lie Algebra to Constraint Mechanical Systems,” Science Press, Beijing, 1999 (in Chinese)

[13] J.L. Fu, L.Q. Chen, “Non-Noether symmetries and conserved quantities of nonconservative dynamical systems,”Phys. Lett. A , vol.317 ,no. 3-4, 255-259, 2003

[14] J.L. Fu, L.Q. Chen, J. Salvador, Y.F. T ang, “Non-Noether symmetries and Lutzky conserved quantities for mechanico-electrical systems,”Phys. Lett. A,vol. 358, no.1,pp. 5-10, 2006

[15] J.L. Fu, L.Q. Chen and B.Y. Chen. “Noether-type theorem for discrete nonconservative dynamical systems with nonregular lattices,” Sci. China: Phys. Mech. & Astron. Vol. 53,no.3, pp.545-554, 2010

[16] U. Camci and Y. Kucukakca.” Noether symmetries of Bianchi I, Bianchi III, and Kantowski-Sachs spacetimes in scalar-coupled gravity theories,” Phys. Rev. D, vol 76, no.76, pp. 366-367

[17] A. K. Sanyal,B. Modak, C. Rubano , E. Piedipalumbo. “Noether symmetry in the higher order gravity theory,”Gen. Relativ. Gravit. Vol. 37 ,no.2, pp. :407-417 , 2005

[18] S. Capozziello, A. Stabile and A. Troisi. “Spherically symmetric solutions in f(R) gravity via the Noether symmetry approach.” Class. Quantum Grav. vol.24,no.8,pp. 2153-2166 ,2007

[19] B. Vakili. “Noether symmetry in f(R) cosmology,”Phys. Lett. B, vol. 664,no. 1-2,pp. 16–20, 2008

[20] B. Vakili. “Noether symmetric f(R) quantum cosmology and its classical correlations,” Phys. Lett. B, vol.669 ,no.3-4,pp. 206-211, 2008

[21] S. Capozziello, G. Lambiase, “Selection Rules in Minisuperspace Quantum Cosmology,”Gen. Relativ. Gravit, vol. 32,no.4,pp. 673-696,2000

[22] M. Rosha and Shojai F., “Palatini [formula omitted] cosmology and Noether symmetry,”Phys. Lett. B, vol. 668,no.3, pp. 238-240, 2008