4 These boundary conditions impose a non-trivial condi- tion on ω [12, 13], and those that satisfy both simultane- ously are called quasinormal frequencies. For the P¨ oshl- Teller potential one can show [12, 13] that they are given by thermore, we compare these predictions against the nu- merical results by KZ, extrapolated to the extremal limit ( ρ = 1 in KZ notation). The agreement is remarkable. We conclude therefore that an instability is present ω = κ b [ − ( j + 1 2 ) i + √ V 0 κ 2 b − 1 4 ] , j = 0 , 1 , ... . (23) V. CONCLUSIONS We have shown analytically that charged black holes in de Sitter backgrounds are unstable for a wide range of charge and mass of the black hole, confirming previous numerical studies [7]. The stability properties of the ex- tremal D = 6 black hole remain unknown. Our methods and results and inconclusive at this precise point, further dedicated investigations would be necessary. Our analytical result in the near-extremal regime could be used to investigate further the nature of this instabil- ity, something we have not attempted to do here. A possible refinement concerns the large- D limit of the in- stability, where it could be possible to find an analytical expression throughout all range of parameters. We have in mind results and techniques similar to those of Kol and Sorkin [14]. It would also be interesting to investigate the stability properties, using this or other techniques, of near-extremal Kerr-dS black holes, which have recently been conjectured to have an holographic description [15]. TABLE I: The threshold of instability for near-extremal RNdS black holes (i.e., black holes for which the cosmological and event horizon almost coincide) for l = 2 modes. We show the prediction from the exact, analytic expression obtained in the near extremal limit (24), which we label Q/Q N ext and the one from criterium (14) which we label as Q/Q V ext . Both these results are compared to the numerical results by KZ. D 7 8 9 10 11 D → ∞ Q/Q N ext 0.913 0.774 0.683 0.617 0.567 p 2 /D Q/Q V ext 0.913 0.775 0.684 0.618 0.568 p 2 /D Q/Q Num ext 0.94 0.78 0.68 0.61 0.55 — whenever V 0 is negative. The threshold of stability in the near-extremal regime is therefore given by V S $_{±}$( r b ) f = 0 , (24) Acknowledgements The expression for V S $_{±}$( r b ) /f is lengthy, and we won’t present it here. The values of the charge Q/Q ext that sat- isfy the condition above are given in Table I (for l = 2), and compared to the prediction from the analysis in Sec- tion III, criterium (14). The agreement is excellent. Fur- We warmly thank Roman Konoplya and Alexander Zhidenko for useful correspondence and for sharing their numerical results with us. This work was partially funded by Funda¸c˜ ao para a Ciˆencia e Tecnologia (FCT)- Portugal through projects PTDC/FIS/64175/2006, PTDC/ FIS/098025/2008, PTDC/FIS/098032/2008 and CERN/FP/109290/2009. [1] R. Gregory and R. Laflamme, Phys. Rev. Lett. 70 , 2837 (1993); H. Kudoh, Phys. Rev. D 73 , 104034 (2006); V. Cardoso and O. J. C. Dias, Phys. Rev. Lett. 96 , 181601 (2006). [2] R. Emparan and R. C. Myers, JHEP 0309 , 025 (2003); O. J. C. Dias, P. Figueras, R. Monteiro, J. E. Santos and R. Emparan, arXiv:0907.2248 [hep-th]. [3] V. Cardoso, O. J. C. Dias, J. P. S. Lemos and S. Yoshida, Phys. Rev. D 70 , 044039 (2004) [Erratum-ibid. D 70 , 049903 (2004)]; V. Cardoso and O. J. C. Dias, Phys. Rev. D 70 , 084011 (2004); V. Cardoso and S. Yoshida, JHEP 0507 , 009 (2005); V. Cardoso, O. J. C. Dias and S. Yoshida, Phys. Rev. D 74 , 044008 (2006); V. Car- doso and J. P. S. Lemos, Phys. Lett. B 621 , 219 (2005); H. Kodama, Prog. Theor. Phys. Suppl. 172 , 11 (2008); A. N. Aliev and O. Delice, Phys. Rev. D 79 , 024013 (2009); H. Kodama, R. A. Konoplya and A. Zhidenko, Phys. Rev. D 79 , 044003 (2009); K. Murata, Prog. Theor. Phys. 121 , 1099 (2009); N. Uchikata, S. Yoshida and T. Futamase, Phys. Rev. D 80 , 084020 (2009). [4] G. Dotti and R. J. Gleiser, Phys. Rev. D 72 , 044018 (2005); R. J. Gleiser and G. Dotti, Phys. Rev. D 72 , 124002 (2005); M. Beroiz, G. Dotti and R. J. Gleiser, Phys. Rev. D 76 , 024012 (2007). [5] T. Takahashi and J. Soda, Phys. Rev. D 79 , 104025 (2009); T. Takahashi and J. Soda, arXiv:0907.0556 [gr- qc]. [6] T. Harmark, V. Niarchos and N. A. Obers, Class. Quant. Grav. 24 , R1 (2007). [7] R. A. Konoplya and A. Zhidenko, Phys. Rev. Lett. 103 , 161101 (2009). [8] H. Kodama and A. Ishibashi, Prog. Theor. Phys. 111 , 29 (2004). [9] W. F. Buell and B. A. Shadwick, Am. J. Phys. 63 , 256 (1995). [10] V. Cardoso and J. P. S. Lemos, Phys. Rev. D 67 , 084020 (2003). [11] G. P¨ oshl and E. Teller, Z. Phys. 83 , 143 (1933). [12] E. Berti, V. Cardoso and A. O. Starinets, Class. Quant. Grav. 26 , 163001 (2009).