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Effects of CDTT model on the stability of spherical collapse in Palatini f(R) gravity

Abstract

The objective of this paper is to study the stability of an adiabatic anisotropic collapsing sphere in the context of Palatini f(R) gravity. In this framework, we construct the collapse equation with the help of the contracted Bianchi identities of the effective as well as the usual energy-momentum tensor. The perturbation scheme is applied on the fluid variables which accordingly cause a perturbation on the Ricci scalar. We explore the instability ranges in the Newtonian and post-Newtonian regimes. It is concluded that the stability of the star is governed by adiabatic index Γ 1, which depends on the energy density profile, anisotropic pressure and dark source terms of the chosen f(R) model. We also explore our results when f(R)→R.

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References

  1. S. Perlmutter et al., Astrophys. J. 483, 565 (1997)

    ADS  Article  Google Scholar 

  2. S. Perlmutter et al., Nature 391, 51 (1998)

    ADS  Article  Google Scholar 

  3. S.W. Allen, R.W. Schmidt, H. Ebeling, A.C. Fabian, L.V. Speybroeck, Mon. Not. R. Astron. Soc. 353, 457 (2004)

    ADS  Article  Google Scholar 

  4. B. Jain, A. Taylor, Phys. Rev. Lett. 91, 141302 (2003)

    ADS  Article  Google Scholar 

  5. C.L. Bennett et al., Astrophys. J. Suppl. Ser. 148, 1 (2003)

    ADS  Article  Google Scholar 

  6. B. Famaey, S. McGaugh, Living Rev. Relativ. 15, 10 (2012)

    ADS  Google Scholar 

  7. S. Capozziello, M.D. Laurentis, Phys. Rep. 509, 167 (2011)

    MathSciNet  ADS  Article  Google Scholar 

  8. G.J. Olmo, Int. J. Mod. Phys. D 20, 413 (2011)

    MathSciNet  ADS  Article  MATH  Google Scholar 

  9. L. Amendola, R. Gannouji, D. Polarski, S. Tsujikawa, Phys. Rev. D 75, 083504 (2007)

    ADS  Article  Google Scholar 

  10. L. Amendola, D. Polarski, S. Tsujikawa, Phys. Rev. Lett. 98, 131302 (2007)

    MathSciNet  ADS  Article  Google Scholar 

  11. S. Tsujikawa, K. Uddin, R. Tavakol, Phys. Rev. D 77, 043007 (2008)

    MathSciNet  ADS  Article  Google Scholar 

  12. A.A. Starobinsky, JETP Lett. 86, 157 (2007)

    ADS  Article  Google Scholar 

  13. W. Hu, I. Sawicki, Phys. Rev. D 76, 064004 (2007)

    ADS  Article  Google Scholar 

  14. V. Miranda, S.E. Joras, I. Waga, M. Quartin, Phys. Rev. Lett. 102, 221101 (2009)

    ADS  Article  Google Scholar 

  15. V. Reijonen, arXiv:0912.0825

  16. S. Chandrasekhar, Astrophys. J. 140, 417 (1964)

    MathSciNet  ADS  Article  MATH  Google Scholar 

  17. M. Gleiser, Phys. Rev. D 38, 2376 (1988)

    ADS  Article  Google Scholar 

  18. L. Herrera, N.O. Santos, G. Le Denmat, Mon. Not. R. Astron. Soc. 237, 257 (1989)

    ADS  MATH  Google Scholar 

  19. R. Chan, L. Herrera, N.O. Santos, Mon. Not. R. Astron. Soc. 265, 533 (1993)

    ADS  Google Scholar 

  20. R. Chan, L. Herrera, N.O. Santos, Mon. Not. R. Astron. Soc. 267, 637 (1994)

    ADS  Google Scholar 

  21. R. Chan, Mon. Not. R. Astron. Soc. 316, 588 (2000)

    ADS  Article  Google Scholar 

  22. G.S. Bisnovatyi-Kogan, M. Marafina, R. Ruffini, E. Vesperini, Astrophys. J. 500, 217 (1998)

    ADS  Article  Google Scholar 

  23. M. Saijo, M. Shibata, T.W. Baumgarte, S.L. Shapiro, Astrophys. J 548, 919 (2001)

    ADS  Article  Google Scholar 

  24. K. Dev, M. Gleiser, Gen. Relativ. Gravit. 35, 1435 (2003)

    MathSciNet  ADS  Article  MATH  Google Scholar 

  25. C.G. Böhmer, T. Harko, Phys. Rev. D 71, 084026 (2005)

    ADS  Article  Google Scholar 

  26. D.D. Doneva, K.D. Kokkatas, I.Z. Stefanov, S.S. Yazadjiev, Phys. Rev. D 84, 084021 (2011)

    ADS  Article  Google Scholar 

  27. M.D. Seifert, Phys. Rev. D 76, 064002 (2007)

    MathSciNet  ADS  Article  Google Scholar 

  28. K. Bamba, S. Nojiri, S.D. Odintsov, Phys. Lett. B 698, 451 (2011)

    ADS  Article  Google Scholar 

  29. S. Capozziello, M.D. Laurentis, S.D. Odinstsov, A. Stabile, Phys. Rev. D 83, 064004 (2011)

    ADS  Article  Google Scholar 

  30. F. Schmidt, M. Lima, H. Oyaizu, W. Hu, Phys. Rev. D 79, 083518 (2009)

    ADS  Article  Google Scholar 

  31. M. Sharif, Z. Yousaf, Phys. Rev. D 88, 024020 (2013)

    ADS  Article  Google Scholar 

  32. M. Sharif, Z. Yousaf, Dynamical instability of spherical stars in Palatini f(R) gravity (2013, submitted)

  33. M. Sharif, G. Abbas, Eur. Phys. J. Plus 128, 102 (2013)

    Article  Google Scholar 

  34. R. Dick, Gen. Relativ. Gravit. 36, 217 (2004)

    MathSciNet  ADS  Article  MATH  Google Scholar 

  35. X.H. Meng, P. Wang, Gen. Relativ. Gravit. 36, 1947 (2004)

    MathSciNet  ADS  Article  MATH  Google Scholar 

  36. A.E. Dominguez, D.E. Barraco, Phys. Rev. D 70, 043505 (2004)

    ADS  Article  Google Scholar 

  37. G.J. Olmo, Phys. Rev. D 72, 083505 (2005)

    ADS  Article  Google Scholar 

  38. K. Kainulainen, V. Reijonen, D. Sunhede, Phys. Rev. D 76, 043503 (2007)

    MathSciNet  ADS  Article  Google Scholar 

  39. M. Ferraris, M. Francaviglia, I. Volovich, Class. Quantum Gravity 11, 1505 (1994)

    MathSciNet  ADS  Article  MATH  Google Scholar 

  40. B.J. Ahmedov, F.J. Fattoyev, Phys. Rev. D 78, 047501 (2008)

    ADS  Article  Google Scholar 

  41. M. Sharif, Z. Yousaf, Chin. Phys. Lett. 29, 050403 (2012)

    ADS  Article  Google Scholar 

  42. M. Sharif, Z. Yousaf, Int. J. Mod. Phys. D 21, 1250095 (2012)

    MathSciNet  ADS  Article  Google Scholar 

  43. A.D. Dolgov, M. Kawasaki, Phys. Lett. B 573, 1 (2003)

    ADS  Article  MATH  Google Scholar 

  44. M.E. Soussa, R.P. Woodard, Gen. Relativ. Gravit. 36, 855 (2004)

    MathSciNet  ADS  Article  MATH  Google Scholar 

  45. S.M. Carroll, V. Duvvuri, M. Trodden, M.S. Turner, Phys. Rev. D 70, 043528 (2004)

    ADS  Article  Google Scholar 

  46. K. Henttunen, T. Multamaki, I. Vilja, Phys. Rev. D 77, 024040 (2008)

    MathSciNet  ADS  Article  Google Scholar 

  47. V. Faraoni, Phys. Rev. D 74, 104017 (2006)

    MathSciNet  ADS  Article  Google Scholar 

  48. I. Navarro, K. Acoleyen Van, J. Cosmol. Astropart. Phys. 02, 022 (2007)

    ADS  Article  Google Scholar 

  49. Y.S. Song, W. Hu, I. Sawicki, Phys. Rev. D 75, 044004 (2007)

    MathSciNet  ADS  Article  Google Scholar 

  50. R. Chan, S. Kichenassamy, G.L. Denmat, N.O. Santos, Mon. Not. R. Astron. Soc. 239, 91 (1989)

    ADS  MATH  Google Scholar 

  51. B.K. Harrison, K.S. Thorne, M. Wakano, J.A. Wheeler, Gravitation Theory and Gravitational Collapse (University of Chicago Press, Chicago, 1965)

    Google Scholar 

  52. E. Barausse, T.P. Sotiriou, J.C. Miller, Class. Quantum Gravity 25, 062001 (2008)

    MathSciNet  ADS  Article  Google Scholar 

  53. G.J. Olmo, Phys. Rev. D 78, 104026 (2008)

    MathSciNet  ADS  Article  Google Scholar 

  54. M. Sharif, Z. Yousaf, Mon. Not. R. Astron. Soc. 432, 264 (2013)

    ADS  Article  Google Scholar 

  55. M. Sharif, Z. Yousaf, Mon. Not. R. Astron. Soc. 434, 2529 (2013)

    ADS  Article  Google Scholar 

  56. H.R. Kausar, J. Cosmol. Astropart. Phys. 01, 007 (2013)

    MathSciNet  Article  Google Scholar 

  57. D. Horvat, S. Ilijic, A. Marunovic, Class. Quantum Gravity 28, 025009 (2011)

    MathSciNet  ADS  Article  Google Scholar 

  58. M. Sharif, M.Z. Bhatti, J. Cosmol. Astropart. Phys. (2013, to appear)

  59. M. Sharif, M.Z. Bhatti, Stability of the charged radiating cylinder (2013, submitted)

  60. C.V. Vishveshwara, Phys. Rev. D 1, 2870 (1970)

    ADS  Article  Google Scholar 

  61. E. Winstanley, Class. Quantum Gravity 16, 1963 (1999)

    MathSciNet  ADS  Article  MATH  Google Scholar 

  62. A. Friedland, M. Giannotti, M. Wise, Phys. Rev. Lett. 110, 061101 (2013)

    ADS  Article  Google Scholar 

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Correspondence to M. Sharif.

Appendix

Appendix

We have the following:

$$\begin{aligned} D_0 =&\frac{A}{\kappa} \biggl[ \biggl\{ \frac{-1}{(HA)^2} \biggl(\dot{f'_R} -\frac{A'}{A} \dot{f_R}-\frac{\dot{H}}{H}f'_R- \frac{5}{2} \frac{\dot{f_R}f'_R}{f_R} \biggr) \biggr\} _{,1} \\ &{}-\frac{1}{A^2} \biggl\{ \frac{f-Rf_R}{2}+ \frac{f''_R}{(H)^2} +\frac{\dot{f_R}}{A^2} \biggl(\frac{\dot{H}}{H}- \frac{3}{4} \frac{\dot{f_R}}{f_R} \biggr) \\ &{}-\frac{f'_R}{H^2} \biggl(\frac{H'}{H} +\frac{1}{4}\frac{f'_R}{f_R} -\frac{2}{r} \biggr) \biggr\} _{,0} \\ &{}+\frac{\dot{f_R}}{2f_RA^2} \biggl\{ \frac{2\ddot{f_R}}{A^2} -(f-Rf_R) - 3\frac{f''_R}{H^2} \\ &{}+\frac{\dot{f_R}}{A^2} \biggl(-3\frac{ \dot{H}}{H}-2\frac{\dot{A}}{A} +\frac{5}{2}\frac{\dot{f_R}}{f_R} \biggr) \\ &{}- \frac{f'_R}{H^2} \biggl(\frac{5}{r}-3\frac{H'}{H} -3 \frac{f'_R}{f_R} \biggr) \biggr\} \\ &{} +\frac{\dot{H}}{A^2H} \biggl\{ - \frac{f''_R}{H^2}+ \frac{\ddot{f_R}}{A^2}+\frac{\dot{f_R}}{A^2} \biggl(\frac{\dot{f_R}}{2f_R}-\frac{\dot{A}}{A}- \frac{\dot{H}}{H} \biggr) \\ &{}+\frac{f'_R}{H^2} \biggl(\frac{A'}{A}- \frac{5}{2}\frac{f'_R}{f_R} -\frac{H'}{H} \biggr) \biggr\} \\ &{}- \frac{1}{(HA)^2} \biggl(\dot{f'_R} - \frac{5}{2}\frac{\dot{f_R}f'_R}{f_R}-\frac{A'}{A}\dot{f_R} - \frac{\dot{H}}{H}f'_R \biggr) \\ &{}\times\biggl( \frac{3A'}{A} +3\frac{f'_R}{f_R}+\frac{H'}{H} +\frac{2}{r} \biggr) \biggr], \end{aligned}$$
(A.1)
$$\begin{aligned} D_1 =&\frac{H}{\kappa} \biggl[ \biggl\{ \frac{-1}{(HA)^2} \biggl(\dot{f'_R} -5 \frac{\dot{f_R}f'_R}{f_R}-\frac{A'}{A}\dot{f_R}-\frac{\dot{H}}{H} f'_R \biggr) \biggr\} _{,0} \\ &{}+\frac{1}{H^2} \biggl\{ \frac{\ddot{f_R}}{A^2} +\frac{f-Rf_R}{2} +\frac{\dot{f_R}}{A^2} \biggl( \frac{-\dot{A}}{A}-\frac{1}{4}\frac{ \dot{f_R}}{f_R} \biggr) \\ &{}+ \frac{f'_R}{H^2} \biggl(-\frac{2}{r}-\frac{A'}{A} - \frac{9}{4}\frac{f'_R}{f_R} \biggr) \biggr\} _{,1} \\ &{}+\frac{A'}{H^2A} \biggl\{ -\frac{f''_R}{H^2}+ \frac{\ddot{f_R}}{ A^2}+\frac{\dot{f_R}}{A^2} \biggl(\frac{\dot{f_R}}{2f_R}- \frac{\dot{A}}{A} -\frac{\dot{H}}{H} \biggr) \\ &{}+\frac{f'_R}{H^2} \biggl(- \frac{5}{2}\frac{f'_R}{f_R} -\frac{A'}{A} -\frac{H'}{H} \biggr) \biggr\} \\ &{}+\frac{f'_R}{2H^2f_R} \biggl\{ \frac{f-Rf_R}{2}+2\frac{\ddot{f_R}}{A^2} \\ &{}+ \frac{\dot{f_R}}{A^2} \biggl(\frac{\dot{f_R}}{4f_R}-2\frac{\dot{A}}{A} -2\frac{\dot{H}}{H} \biggr) \\ &{}+\frac{f'_R}{H^2} \biggl(-\frac{3}{r}-2\frac{A'}{ A}-2\frac{H'}{H}- \frac{29}{4}\frac{f'_R}{f_R} \biggr) \biggr\} \\ &{}+\frac{1}{H^2r} \biggl\{ \frac{f''_R}{H^2} -\frac{\dot{f_R}}{A^2} \biggl(\frac{\dot{H}}{H} \biggr) \\ &{} + \frac{f'_R}{H^2} \biggl(-\frac{1}{r}-\frac{H'}{H}- \frac{5}{2}\frac{f'_R}{f_R} \biggr) \biggr\} \\ &{}-\frac{1}{(HA)^2} \biggl(- \frac{A'}{A}\dot{f_R} + \dot{f'_R}- \frac{5}{2}\frac{ \dot{f_R}f'_R}{f_R}-\frac{\dot{H}}{H}f'_R \biggr) \\ &{}\times \biggl(\frac{\dot{A}}{A}+\frac{3\dot{H}}{H}+\frac{5}{2} \frac{\dot{f_R}}{f_R} \biggr) \biggr], \end{aligned}$$
(A.2)
$$\begin{aligned} D_{1S} =&\frac{1}{H_0^2} \biggl[\frac{2\delta^4R'_0}{R_0^3H_0^2} \biggl(-\frac{2}{r}-\frac{A'_0}{A_0}-\frac{9\delta^4R'_0}{ R_0(R_0^2-\delta^4)} \biggr)+ \frac{\delta^4}{R_0} \biggr]' \\ &{} +\frac{\delta^4}{R_0} \frac{R'_0}{(R_0^2-\delta^4)H_0^2} \biggl[\frac{\delta^4}{R_0} - \frac{2\delta^4R'_0}{R_0^3} \biggl(2\frac{A'_0}{A_0} +2\frac{H'_0}{H_0} \\ &{} + \frac{3}{r}+\frac{29\delta^4}{2R_0} \frac{R'_0}{ (R_0^2-\delta^4)} \biggr) \biggr] \\ &{} + \frac{A'_0}{A_0H_0^2} \biggl[ -\frac{2\delta^4}{R_0^3H_0^2} \biggl(R''_0-3 \frac{R'^2_0}{R_0} -\frac{2\delta^4}{R_0} \biggr) \\ &{}\times\frac{R'_0}{(R_0^2-\delta^4)} \biggl(\frac{A'_0}{A_0} + \frac{H'_0}{H_0}+\frac{5\delta^5R'_0}{R_0(R_0^2-\delta^4)} \biggr) \biggr] \\ &{} +\frac{1}{rH_0^2} \biggl[ \frac{2\delta^4}{H_0^2R_0^3} \biggl(R''_0 -3 \frac{R'^2_0}{R_0} \biggr) \\ &{} -\frac{2\delta^4R'_0}{R_0^3H_0^2} \biggl(\frac{1}{r}+ \frac{H'_0}{H_0}-\frac{5\delta^4R'_0}{ R_0(R_0^2-\delta^4)} \biggr) \biggr], \end{aligned}$$
$$\begin{aligned} D_2 =&\frac{1}{\kappa} \biggl[ \biggl[ \frac{e\delta^4}{R_0^2}-\frac{2\delta^4}{H_0^2R_0^3} \biggl\{ \frac{\delta^4}{2R_0(R_0^2-\delta^4)} \biggl(e''-\frac{3e'R'_0}{R_0} \\ &{} -\frac{3eR''_0}{R_0}+12 \frac{eR'^2_0}{R_0^2} -\frac{2h}{H_0} \biggl(R''_0 -3\frac{R'^2_0}{R_0} \biggr) \biggr) \biggr\} \\ &{} -\frac{2\delta^4}{H_0^2R_0^3} \biggl(e'-\frac{3R'_0e}{R_0}- \frac{2hR'_0}{H_0} \biggr) \\ &{} \times\biggl(\frac{H'_0}{H_0}-\frac{2}{r} +\frac{\delta^4R'_0}{2R_0(R_0^2-\delta^4)} \biggr) \\ &{} -\frac{2\delta^4R'_0}{H_0^2R_0^3} \biggl\{ \frac{\delta^4}{2R_0}\frac{1}{(R_0^2-\delta^4)} \biggl(e' - \frac{3R'_0e}{R_0} \\ &{} -\frac{2e\delta^4R'_0}{R_0} \frac{1}{(R^2_0-\delta^4)} \biggr)+ \biggl(\frac{h}{H_0} \biggr)' \biggr\} \biggr] \\ &{} + \frac{e\delta^4}{ R_0(R_0^2-\delta^4)} \biggl[-\frac{6\delta^4}{R_0^3H_0^2} \biggl(R''_0 -3\frac{R'^2_0}{R_0} \biggr) \\ &{}-2\frac{\delta^4}{R_0}+2\frac{\delta^4R'_0}{R_0^3H_0^2} \biggl( \frac{5}{r} -3\frac{H'_0}{H_0}-\frac{6\delta^4R'_0}{R_0(R_0^2-\delta^4)} \biggr) \biggr] \\ &{} + \frac{h}{A_0H_0} \biggl[-\frac{2\delta^4}{H_0^2R_0^3} \biggl(R''_0-3 \frac{R'^2_0}{R_0} \biggr) \\ &{} +2\frac{\delta^4R'_0}{H_0R_0^3} \biggl(\frac{A'_0}{A_0}- \frac{H'_0}{H_0}-\frac{5\delta^4R'_0}{ R^4_0(R_0^2-\delta^4)} \biggr) \biggr] \\ &{} +\frac{2\delta^4}{R_0^3A_0H_0^2} \biggl\{ \biggl(e'-3e\frac{R'_0}{R_0} - \frac{eA'_0}{A_0}-\frac{R'_0h}{H_0} \\ &{} -\frac{5\delta^4eR'_0}{R_0(R_0^2\,{-}\,\delta^4)} \biggr) \biggl(\frac{3A'_0}{A_0}\,{+}\,\frac{H'_0}{H_0}\,{+}\,\frac{2}{r} \,{+}\,\frac{6\delta^4R'_0}{R_0(R_0^2\,{-}\,\delta^4)} \biggr) \biggr\} \\ &{} + \biggl\{ \frac{2\delta^4}{R_0^3A_0^2H_0^2} \biggl(e'-\frac{3eR'_0}{R_0} -e \frac{A'_0}{A_0}-\frac{hR'_0}{H_0} \\ &{} -\frac{5\delta^4e}{R^4_0} \frac{R'_0}{(R_0^2-\delta^4)} \biggr) \biggr\} _{,1} \biggr], \end{aligned}$$
(A.3)
$$\begin{aligned} D_3 =&\frac{2\ddot{T}\delta^4}{R_0^3A_0^2H_0} \biggl(\frac{3eR'_0}{R_0} \,{+}\, \frac{eA'_0}{A_0}\,{+}\,\frac{hR'_0}{H_0}\,{+}\,\frac{10e\delta^4R'_0}{R_0(R_0^2\,{-}\,\delta^4)} \,{-}\,e' \biggr) \\ &{}+\frac{T}{H_0} \biggl[-\frac{e\delta^4}{R_0^2} +\frac{2\delta^4R'_0}{R_0^3H_0^2} \biggl\{ - \biggl(\frac{a}{A_0} \biggr)' \\ &{}+\frac{9}{2}\frac{\delta^4}{R_0(R_0^2\,{-}\,\delta^4)} \biggl(e'\,{-}\,3 \frac{eR'_0}{R_0} \,{-}\,\frac{2e\delta^4R'_0}{R_0} \frac{1}{(R_0^2\,{-}\,\delta^4)} \biggr) \biggr\} \\ &{}- \frac{2\delta^4}{R_0^3H_0^2} \biggl(e'-\frac{3eR'_0}{R_0}-\frac{2hR'_0}{H_0} \biggr) \\ &{}\times \biggl(\frac{2}{r}+\frac{9\delta^4R'_0}{ R_0(R_0^2-\delta^4)} +\frac{A'_0}{A_0} \biggr) \biggr]_{,1}- \frac{Th}{H_0^2} \biggl[\frac{\delta^4}{R_0} \\ &{}+\frac{2\delta^4R'_0}{H_0^2R_0^3} \biggl(- \frac{2}{r}-\frac{A'_0}{A_0} -\frac{9}{2}\frac{\delta^4R'_0}{R_0(R_0^2-\delta^4)} \biggr) \biggr]_{,1} +2\delta^4 \\ &{}\times \biggl(\frac{e}{A_0^2H_0R_0^3} \biggr)\ddot{T} - \frac{T\delta^4R'_0}{R_0(R_0^2-\delta^4)H_0} \biggl[\frac{\delta^4e}{R^2_0} \\ &{}+\frac{2\delta^4R'_0}{R-0^3H_0^2} \biggl\{ 2 \biggl( \frac{a}{A_0} \biggr)' +2 \biggl(\frac{h}{h_0} \biggr)' \\ &{}+ \frac{29\delta^4}{2R_0(R_0^2-\delta^4)} \biggl(e'-\frac{3eR'_0}{R_0}-\frac{2e\delta^4R'_0}{R_0(R_0^2-\delta^4)} \biggr) \biggr\} \\ &{}+\frac{2\delta^4}{R_0^3H_0^2} \biggl(e'-\frac{3eR'_0}{R_0} - \frac{2hR'_0}{H_0} \biggr) \\ &{}\times \biggl(-2\frac{A'_0}{A_0}-2\frac{H'_0}{H_0} - \frac{3}{r}-\frac{29\delta^4R'_0}{2R_0(R_0^2-\delta^4)} \biggr) \biggr] \\ &{}+\frac{\delta^4}{H_0R_0(R_0^2-\delta^4)} \\ &{}\times\biggl(e'-3\frac{eR'_0}{R_0} - \frac{2\delta^4eR'_0}{R_0(R_0^2-\delta^4)}-\frac{hR'_0}{H_0} \biggr) \\ &{}\times\biggl\{ \frac{\delta^4}{R_0}+ \frac{2\delta^4R'_0}{H_0^2R_0^3} \biggl(-2\frac{A'_0}{A_0}-2\frac{H'_0}{H_0} \\ &{}- \frac{3}{r} -\frac{29\delta^4R'_0}{2R_0(R_0^2-\delta^4)} \biggr) \biggr\} \\ &{}+\frac{4e\delta^8\ddot{T}R'_0}{A^2_0H_0R_0^4(R_0^2-\delta^4)} +\frac{2\delta^4\ddot{T}eA'_0}{A_0^3H_0R_0^3} \\ &{}+\frac{TA'_0}{A_0H_0} \biggl[-\frac{2\delta^4}{R_0^3H_0^2} \biggl\{ e''-\frac{3e'R'_0}{R_0} -\frac{3eR''_0}{R_0} \\ &{}+ \frac{12eR'^2_0}{R_0^2}-\frac{2h}{H_0} \times \biggl(R''_0-3 \frac{R'^2_0}{R_0} \biggr) \biggr\} \\ &{}+\frac{2\delta^4R'_0}{R_0^3H_0^2} \biggl\{ - \biggl( \frac{h}{H_0} \biggr)' - \biggl(\frac{a}{A_0} \biggr)'+\frac{5\delta^4}{R_0(R_0^2-\delta^4)} \\ &{} \times\biggl(e'-3e\frac{R'_0}{R_0}- \frac{2e\delta^4R'_0}{R_0(R_0^2-\delta^4)} \biggr) \biggr\} \\ &{}-\frac{2\delta^4}{H_0R_0^3} \biggl(e'- \frac{3eR'_0}{R_0} -\frac{2hR'_0}{H_0} \biggr) \\ &{}\times \biggl(\frac{H'_0}{H_0}+\frac{A'_0}{A_0} + \frac{5\delta^4R'_0}{R_0(R_0^2-\delta^4)} \biggr) \biggr] \\ &{}+\frac{T}{H_0} \biggl[\frac{2\delta^4}{R_0^3H_0^2} \biggl\{ e''-\frac{3e'R'_0}{R_0} -3e\frac{R''_0}{R_0} \\ &{} + \frac{12eR'^2_0}{R_0^2}-2\frac{h}{H_0} \biggl(R''_0-3\frac{R'^2_0}{R_0} \biggr) \biggr\} \\ &{}+\frac{2\delta^4R'_0}{R_0^3H_0^2} \biggl\{ - \biggl(\frac{h}{H_0} \biggr)'- \frac{5\delta^4}{R_0(R_0^2-\delta^4)} \\ &{}\times\biggl(e'-\frac{3eR'_0}{R_0}- \frac{2e\delta^4R'_0}{R^2_0(R_0^2-\delta^4)} \biggr) \biggr\} \\ &{}-\frac{2\delta^4}{H_0^2R_0^3} \biggl(e'- \frac{3eR'_0}{R_0} -\frac{2hR'_0}{H_0} \biggr) \\ &{}\times \biggl(\frac{1}{r}+\frac{H'_0}{H_0} + \frac{5\delta^4R'_0}{R_0(R_0^2-\delta^4)} \biggr) \biggr] \\ &{}-\frac{Th}{rH_0^2} \biggl[\frac{2\delta^4}{R_0^3H_0^2} \biggl(R''_0-\frac{3R'^2_0}{R_0} \biggr) - \frac{2\delta^4R'_0}{R_0^3H_0^2} \\ &{} \times\biggl(\frac{H'_0}{H_0}+\frac{1}{r} + \frac{5\delta^4R'_0}{R_0(R_0^2-\delta^4)} \biggr) \biggr], \end{aligned}$$
(A.4)
$$\begin{aligned} D_4 =&\frac{2{\omega}\delta^4}{R_0^3} \biggl(\frac{3eR'_0}{R_0} +hR'_0+\frac{10e\delta^4R'_0}{R_0(R_0^2-\delta^4)} -e' \biggr) \\ &{} + \biggl[-\frac{e\delta^4}{R_0^2}\,{+}\,\frac{2\delta^4R'_0}{R_0^3} \,{+}\,\frac{2\delta^4R'_0}{R_0^3} \biggl\{ -a' \,{+}\, \frac{9}{2}\frac{\delta^4}{R_0(R_0^2-\delta^4)} \\ &{} \times\biggl(e'-3\frac{eR'_0}{R_0} -\frac{2e\delta^4R'_0}{R_0(R_0^2-\delta^4)} \biggr) \biggr\} -\frac{2\delta^4}{R_0^3} \\ &{}\times \biggl(e'-\frac{3eR'_0}{R_0}-2hR'_0 \biggr) \biggl(\frac{2}{r}+\frac{9\delta^4R'_0}{ R_0(R_0^2-\delta^4)} \biggr) \biggr]_{,1} \\ &{} -h \biggl[\frac{\delta^4}{R_0} +\frac{2\delta^4R'_0}{R_0^3} \biggl(-\frac{2}{r} -\frac{9}{2} \frac{\delta^4R'_0}{R_0(R_0^2-\delta^4)} \biggr) \biggr]_{,1} \\ &{} +2\delta^4{\omega} \biggl( \frac{e}{R_0^3} \biggr)' -\frac{\delta^4R'_0}{R_0(R_0^2-\delta^4)} \biggl[ \frac{\delta^4e}{R^2_0} \\ &{}+\frac{2\delta^4R'_0}{R_0^3} \biggl\{ 2a'+2h'+ \frac{29\delta^4}{2R_0(R_0^2-\delta^4)} \\ &{} \times \biggl(e'-\frac{3eR'_0}{R_0}-\frac{2e\delta^4R'_0}{R_0(R_0^2-\delta^4)} \biggr) \biggr\} \\ &+\frac{2\delta^4}{R_0^3} \biggl(e'\,{-}\,\frac{3eR'_0}{R_0}\,{-}\,2hR'_0 \biggr) \biggl(-\frac{3}{r}\,{-}\, \frac{29\delta^4R'_0}{2R_0(R_0^2\,{-}\,\delta^4)} \biggr) \biggr] \\ &{} +\frac{\delta^4}{R_0(R_0^2-\delta^4)}\biggl(e'-3\frac{eR'_0}{R_0} - \frac{2\delta^4eR'_0}{R_0(R_0^2-\delta^4)}-hR'_0 \biggr) \\ &{} \times\biggl\{ \frac{\delta^4}{R_0}+\frac{2\delta^4R'_0}{R_0^3} \biggl(-\frac{29\delta^4R'_0}{2R_0(R_0^2-\delta^4)} -\frac{3}{r} \biggr) \biggr\} \\ &{} + \frac{4e\delta^8{\omega}R'_0}{R_0^4(R_0^2-\delta^4)}+\frac{1}{r} \biggl[\frac{2\delta^4}{R_0^3} \biggl\{ e''-\frac{3e'R'_0}{R_0} \\ &{} -3e\frac{R''_0}{R_0}+ \frac{12eR'^2_0}{R_0^2} - 2h \biggl( R''_0-3 \frac{R'^2_0}{R_0} \biggr) \biggr\} \\ &{} +\frac{2\delta^4R'_0}{R_0^3} \biggl\{ -h'- \frac{5\delta^4}{R_0(R_0^2-\delta^4)} \\ &{} \times\biggl(e'-\frac{3eR'_0}{R_0} -\frac{2e\delta^4R'_0}{R_0(R_0^2-\delta^4)} \biggr) \biggr\} \\ &{} - \frac{2\delta^4}{R_0^3} \biggl(e'-\frac{3eR'_0}{R_0} -2hR'_0 \biggr) \biggl(\frac{1}{r} +\frac{5\delta^4R'_0}{R_0(R_0^2-\delta^4)} \biggr) \biggr] \\ &{} -\frac{h}{r} \biggl[\frac{2\delta^4}{R_0^3} \biggl(R''_0- \frac{3R'^2_0}{R_0} \biggr) \\ &{} -\frac{2\delta^4R'_0}{R_0^3} \biggl(\frac{1}{r} + \frac{5\delta^4R'_0}{R_0(R_0^2-\delta^4)} \biggr) \biggr] \\ &{} +a' \biggl[-\frac{2\delta^4}{R_0^3} \biggl(R''_0\,{-}\,3 \frac{R'^2_0}{R_0} \biggr) \\ &{} +\frac{2\delta^4R'_0}{ R_0^3} \biggl(\frac{-5\delta^4R'_0}{R_0(R_0^2\,{-}\,\delta^4)} \biggr) \biggr], \end{aligned}$$
(A.5)
$$\begin{aligned} \omega^2(r) =& e-\frac{1}{H_0^2} \biggl[ \frac{2\delta^4R'_0}{R_0(R_0^2-\delta^4)} \biggl\{ \biggl(\frac{h}{H_0} \biggr)' \\ &{} + \frac{4\delta^4}{R_0(R_0^2-\delta^4)} \biggl(e'-3e\frac{R'_0}{R_0}-\frac{2e\delta^4}{R_0} \\ &{}\times\frac{R'_0}{(R_0^2-\delta^4)} \biggr)- \frac{2\delta^4}{R_0^3} \biggl(e'-3e\frac{R'_0}{R_0} \biggr) - \biggl( \frac{a}{A_0} \biggr)' \biggr\} \\ &{} +\frac{2\delta^4}{R_0(R_0^2-\delta^4)} \biggl(e' -\frac{3eR'_0}{R_0}-\frac{2\delta^4eR'_0}{R_0^2(R_0^2-\delta^4)} \biggr) \\ &{}\times \biggl(\frac{H'_0}{H_0}+\frac{4\delta^4R'_0}{R_0(R_0^2-\delta^4)} -\frac{2}{r}-\frac{2\delta^4R'_0}{R^3_0} -\frac{A'_0}{A_0} \biggr) \\ &{}+2\frac{H'_0}{H_0} \biggl(\frac{a}{A_0} \biggr)'+ \biggl(\frac{h}{H_0} \biggr)' \biggl( \frac{2A'_0}{A_0}+\frac{4}{r} \biggr) \\ &{} +\frac{2}{r^2} \biggl\{ r^2 \biggl(\frac{a''}{A_0}-a\frac{A''_0}{A_0^2} \biggr)-2r \biggl(\frac{a}{A_0} \biggr)' \biggr\} \biggr] \\ &{} +2\frac{h}{H_0^3} \biggl[\frac{2\delta^4R'_0}{R_0 (R_0^2-\delta^4)} \biggl( \frac{H'_0}{H_0}+\frac{4\delta^4R'_0}{R_0(R_0^2-\delta^4)} \\ &{} -\frac{2}{r}-\frac{2\delta^4R'_0}{R^3_0}-\frac{A'_0}{A_0} \biggr)+\frac{H'_0}{H_0} \biggl( \frac{2A'_0}{A_0}+\frac{4}{r} \biggr) \\ &{} +\frac{2}{r^2} \biggl( r^2\frac{A''_0}{A_0}-2r\frac{A'_0}{A_0} \biggr) \biggr] \\ &{}\times \biggl[\frac{2h(1-\delta^4R_0^{-2})+2\delta^4eR_0^{-3}}{ A_0^2(1-\delta^4R_0^{-2})} \biggr]^{-1}. \end{aligned}$$
(A.6)

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Sharif, M., Yousaf, Z. Effects of CDTT model on the stability of spherical collapse in Palatini f(R) gravity. Eur. Phys. J. C 73, 2633 (2013). https://doi.org/10.1140/epjc/s10052-013-2633-1

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  • DOI: https://doi.org/10.1140/epjc/s10052-013-2633-1

Keywords

  • Black Hole
  • Dark Energy
  • Dynamical Instability
  • Ricci Scalar
  • Adiabatic Index