- 
								[1]
								F. W. Dyson, A. S. Eddington, and C. Davidson, Philosophical Transactions of the Royal Society of London Series A 220, 291 (1920) 
- 
								[2]
								B. P. Abbott, LIGO Scientific Collaboration, and Virgo Collaboration, Phys. Rev. Lett. 116, 061102 (2016), arXiv: 1602.03837[gr-qc] 
- 
								[3]
								K. Akiyama et al., Astrophys. J. 875, L1 (2019), arXiv: 1906.11238[astroph.GA] 
- 
								[4]
								K. Akiyama et al., Astrophys. J. Lett 930, L12 (2022) 
- 
								[5]
								V. Bozza, S. Capozziello, G. Iovane et al., General Relativity and Gravitation 33, 1535 (2001), arXiv: grqc/0102068[gr-qc] 
- 
								[6]
								V. Bozza, Phys. Rev. D 66, 103001 (2002), arXiv: grqc/0208075[gr-qc] 
- 
								[7]
								E. F. Eiróa and D. F. Torres, Phys. Rev. D. 69, 063004 (2004) 
- 
								[8]
								V. Bozza, Phys. Rev. D 78, 103005 (2008), arXiv: 0807.3872[gr-qc] 
- 
								[9]
								K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D. 62, 084003 (2000) 
- 
								[10]
								K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D. 65, 103004 (2002) 
- 
								[11]
								S. U. Islam, R. Kumar, and S. G. Ghosh, J. Cosmol. A. P 2020, 030 (2020) 
- 
								[12]
								C.-Y. Wang, Y.-F. Shen, and Y. Xie, JCAP 2019, 022 (2019), arXiv: 1902.03789[gr-qc] 
- 
								[13]
								X. Lu and Y. Xie, Eur. Phys. J. C 79, 1016 (2019) 
- 
								[14]
								N. U. Molla, H. Chaudhary, G. Mustafa et al., Eur. Phys. J. C 84, 574 (2024), arXiv: 2310.14234[gr-qc] 
- 
								[15]
								Y.-X. Gao and Y. Xie, Phys. Rev. D 103, 043008 (2021) 
- 
								[16]
								X. Lu and Y. Xie, Eur. Phys. J. C 81, 627 (2021) 
- 
								[17]
								G. S. Bisnovatyi-Kogan and O. Y. Tsupko, Mon. Not. R. Astron. Soc. 404, 1790 (2010) 
- 
								[18]
								A. Rogers, Mon. Not. R. Astron. Soc. 451, 17 (2015) 
- 
								[19]
								V. S. Morozova, B. J. Ahmedov, and A. A. Tursunov, Astrophys Space Sci 346, 513 (2013) 
- 
								[20]
								F. Atamurotov, A. Abdujabbarov, and W.-B. Han, Phys. Rev. D 104, 084015 (2021) 
- 
								[21]
								S. Hensh, A. Abdujabbarov, J. Schee et al., European Physical Journal C 79, 533 (2019), arXiv: 1904.08776[grqc] 
- 
								[22]
								F. Atamurotov, A. Abdujabbarov, and J. Rayimbaev, Eur. Phys. J. C. 81, 118 (2021) 
- 
								[23]
								G. Z. Babar, F. Atamurotov, and A. Z. Babar, Phys. Dark Universe 32, 100798 (2021) 
- 
								[24]
								G. Z. Babar, F. Atamurotov, S. Ul Islam et al., Phys. Rev. D 103, 084057 (2021), arXiv: 2104.00714[gr-qc] 
- 
								[25]
								W. Javed, I. Hussain, and A. Övgün, Eur. Phys. J. Plus 137, 148 (2022), arXiv: 2201.09879[gr-qc] 
- 
								[26]
								A. Hakimov and F. Atamurotov, Astrophys Space Sci 361, 112 (2016) 
- 
								[27]
								F. Atamurotov, S. Shaymatov, P. Sheoran et al., J. Cosmol. A. P. 2021, 045 (2021), arXiv: 2105.02214[gr-qc] 
- 
								[28]
								F. Atamurotov, H. Alibekov, A. Abdujabbarov et al., Symmetry 15, 848 (2023) 
- 
								[29]
								F. Atamurotov, D. Ortiqboev, A. Abdujabbarov et al., Eur. Phys. J. C 82, 659 (2022) 
- 
								[30]
								S.-S. Zhao and Y. Xie, Phys. Lett. B. 774, 357 (2017) 
- 
								[31]
								F. Atamurotov and S. G. Ghosh, Eur. Phys. J. Plus 137, 662 (2022) 
- 
								[32]
								F. Atamurotov, M. Jamil, and K. Jusufi, Chin. Phys. C 47, 035106 (2023), arXiv: 2212.12949[gr-qc] 
- 
								[33]
								F. Atamurotov, I. Hussain, G. Mustafa et al., Chinese Physics C 47, 025102 (2023) 
- 
								[34]
								H. Hoshimov, O. Yunusov, F. Atamurotov et al., Physics of the Dark Universe 43, 101392 (2024), arXiv: 2312.10678[gr-qc] 
- 
								[35]
								F. Atamurotov, M. Alloqulov, A. Abdujabbarov et al., Eur. Phys. J. Plus 137, 634 (2022) 
- 
								[36]
								F. Atamurotov, F. Sarikulov, A. Abdujabbarov et al., Eur. Phys. J. Plus 137, 336 (2022) 
- 
								[37]
								M. Alloqulov, F. Atamurotov, A. Abdujabbarov et al., Chinese Physics C 47, 075103 (2023) 
- 
								[38]
								M. Alloqulov, F. Atamurotov, A. Abdujabbarov et al., Chinese Physics C 48, 025104 (2024) 
- 
								[39]
								X.-M. Kuang and A. Övgün, Annals of Physics 447, 169147 (2022) 
- 
								[40]
								A. Al-Badawi, M. Alloqulov, S. Shaymatov, et al., Chinese Physics C 48, 095105 (2024) (2024) 
- 
								[41]
								J. L. Synge, Mon. Not. Roy. Astron. Soc. 131, 463 (1966) 
- 
								[42]
								J. P. Luminet, Astron.Astrophys. 75, 228 (1979) 
- 
								[43]
								H. Falcke, F. Melia, and E. Agol, Astrophys. J. 528, L13 (2000), arXiv: astro-ph/9912263[astro-ph] 
- 
								[44]
								V. Perlick and O. Y. Tsupko, Phys. Rep. 947, 1 (2022), arXiv: 2105.07101[gr-qc] 
- 
								[45]
								K. Hioki and K.-I. Maeda, Phys. Rev. D 80, 024042 (2009) 
- 
								[46]
								C. Bambi and K. Freese, Phys. Rev. D. 79, 043002 (2009), arXiv: 0812.1328[astro-ph] 
- 
								[47]
								L. Amarilla and E. F. Eiróa, Phys. Rev. D 85, 064019 (2012) 
- 
								[48]
								A. Abdujabbarov, M. Amir, B. Ahmedov et al., Phys. Rev. D 93, 104004 (2016), arXiv: 1604.03809[gr-qc] 
- 
								[49]
								F. Atamurotov, A. Abdujabbarov, and B. Ahmedov, Phys. Rev. D 88, 064004 (2013) 
- 
								[50]
								N. Tsukamoto, Phys. Rev. D. 97, 064021 (2018), arXiv: 1708.07427[gr-qc] 
- 
								[51]
								S.-W. Wei and Y.-X. Liu, Eur. Phys. J. Plus 136, 436 (2021), arXiv: 2003.07769[gr-qc] 
- 
								[52]
								X. Hou, Z. Xu, and J. Wang, J. Cosmol. A. P 2018, 040 (2018) 
- 
								[53]
								V. Perlick, O. Y. Tsupko, and G. S. Bisnovatyi-Kogan, Phys. Rev. D. 97, 104062 (2018), arXiv: 1804.04898[gr-qc] 
- 
								[54]
								P. V. P. Cunha, N. A. Eiró, C. A. R. HerdEiró et al., J. Cosmol. A. P 2020, 035 (2020), arXiv: 1912.08833[gr-qc] 
- 
								[55]
								M. Afrin, R. Kumar, and S. G. Ghosh, Mon. Not. R. Astron. Soc. (2021), arXiv: 2103.11417[gr-qc] 
- 
								[56]
								F. Atamurotov, S. G. Ghosh, and B. Ahmedov, Eur. Phys. J. C 76, 273 (2016), arXiv: 1506.03690[gr-qc] 
- 
								[57]
								F. Atamurotov, U. Papnoi, and K. Jusufi, Classical and Quantum Gravity 39, 025014 (2022), arXiv: 2104.14898[gr-qc] 
- 
								[58]
								F. Atamurotov, I. Hussain, G. Mustafa et al., Eur. Phys. J. C 82, 831 (2022), arXiv: 2209.01652[gr-qc] 
- 
								[59]
								F. Sarikulov, F. Atamurotov, A. Abdujabbarov et al., Eur. Phys. J. C 82, 771 (2022) 
- 
								[60]
								G. Mustafa, F. Atamurotov, I. Hussain et al., Chinese Physics C 46, 125107 (2022), arXiv: 2207.07608[gr-qc] 
- 
								[61]
								T. Ohgami and N. Sakai, Phys. Rev. D 91, 124020 (2015), arXiv: 1704.07065[gr-qc] 
- 
								[62]
								X. Wang, P.-C. Li, C.-Y. Zhang et al., Phys. Lett. B 811, 135930 (2020), arXiv: 2007.03327[gr-qc] 
- 
								[63]
								M. Wielgus, J. Horák, F. Vincent et al., Phys. Rev. D 102, 084044 (2020), arXiv: 2008.10130[gr-qc] 
- 
								[64]
								K. A. Bronnikov, R. A. Konoplya, and T. D. Pappas, Phys. Rev. D 103, 124062 (2021), arXiv: 2102.10679[gr-qc] 
- 
								[65]
								K. Jusufi, S. Kumar, M. Azreg-Aïnou et al., Eur. Phys. J. C 82, 633 (2022), arXiv: 2106.08070[gr-qc] 
- 
								[66]
								A. Abdujabbarov, B. Juraev, B. Ahmedov et al., Astrophys. Space Sci. 361, 226 (2016) 
- 
								[67]
								S. Kumar, A. Uniyal, and S. Chakrabarti, arXiv: 2308.05545[gr-qc] 
- 
								[68]
								S. Vagnozzi, R. Roy, Y.-D. Tsai et al., Classical and Quantum Gravity 40, 165007 (2023) 
- 
								[69]
								G. Lambiase, R. C. Pantig, D. J. Gogoi et al., The European Physical Journal C 83, (2023) 
- 
								[70]
								V. Vertogradov and A. Övgün, Physics Letters B 854, 138758 (2024) 
- 
								[71]
								M. Zubair, M. A. Raza, F. Sarikulov et al., J. Cosmol. A. P. 2023, 058 (2023), arXiv: 2305.16888[gr-qc] 
- 
								[72]
								J. Rayimbaev, B. Majeed, M. Jamil et al., Physics of the Dark Universe 35, 100930 (2022), arXiv: 2202.11509[gr-qc] 
- 
								[73]
								V. Vertogradov and A. Övgün, Physics of the Dark Universe 45, 101541 (2024) 
- 
								[74]
								A. Övgün, R. C. Pantig, and Á. Rincón, Annals of Physics 463, 169625 (2024), arXiv: 2402.14190[gr-qc] 
- 
								[75]
								M. Okyay and A. Övgün, Journal of Cosmology and Astroparticle Physics 2022, 009 (2022) 
- 
								[76]
								A. Uniyal, S. Chakrabarti, M. Fathi et al., Annals of Physics 462, 169614 (2024) 
- 
								[77]
								B. Puliçe, R. C. Pantig, A. Övgün et al., Classical and Quantum Gravity 40, 195003 (2023) 
- 
								[78]
								A. Al-Badawi, S. Shaymatov, M. Alloqulov et al., Communications in Theoretical Physics 76, 085401 (2024) 
- 
								[79]
								J. Rayimbaev, D. Bardiev, T. Mirzaev et al., International Journal of Modern Physics D 31, 2250055 (2022) 
- 
								[80]
								Y. Kumaran and A. Övgün, Eur. Phys. J. C 83, 812 (2023), arXiv: 2306.04705[gr-qc] 
- 
								[81]
								S. U. Khan, J. Rayimbaev, F. Sarikulov et al., Chinese Journal of Physics 90, 690 (2024), arXiv: 2310.05860[gr-qc] 
- 
								[82]
								O. Ruiz and E. Tuiran, Phys. Rev. D 107, 066003 (2023) 
- 
								[83]
								S. Mandal, General Relativity and Gravitation 54, 142 (2022), arXiv: 2207.05062[gr-qc] 
- 
								[84]
								J. Rayimbaev, A. Abdujabbarov, M. Jamil et al., Phys. Rev. D 102, 084016 (2020), arXiv: 2010.15079[gr-qc] 
- 
								[85]
								F. H. Zuluaga and L. A. Sánchez, Eur. Phys. J. C 81, 840 (2021), arXiv: 2106.03140[gr-qc] 
- 
								[86]
								Y.-X. Chen, P.-H. Mou, and G.-P. Li, Symmetry 14, 1959 (2022) 
- 
								[87]
								R. A. Konoplya, A. F. Zinhailo, J. Kunz et al., JCAP 2022, 091 (2022), arXiv: 2206.14714[grqc] 
- 
								[88]
								J. M. Ladino and E. Larrañaga, International Journal of Modern Physics D 31, 2250091 (2022), arXiv: 2302.12213[gr-qc] 
- 
								[89]
								C. Bambi, The Astrophysical Journal 761, 174 (2012), arXiv: 1210.5679[gr-qc] 
- 
								[90]
								R. Konoplya, L. Rezzolla, and A. Zhidenko, Phys. Rev. D 93, 064015 (2016), arXiv: 1602.02378[gr-qc] 
- 
								[91]
								J. M. Ladino, C. A. Benavides-Gallego, E. Larranaga et al., Eur. Phys. J. C 83, 989 (2023), arXiv: 2305.15350[gr-qc] 
- 
								[92]
								J. L. Synge, Relativity: The General Theory (Amsterdam: North-Holland, 1960) 
- 
								[93]
								O. Y. Tsupko and G. S. Bisnovatyi-Kogan, Gravit. Cosmol. 15, 184 (2009) 
- 
								[94]
								L. Spitzer, Interscience Tracts on Physics and Astronomy 28, 262 (1968) 
- 
								[95]
								L. Spitzer, Physical processes in the interstellar medium (New York: John Wiley & Sons, Ltd, 1978) 
- 
								[96]
								V. Perlick, O. Y. Tsupko, and G. S. Bisnovatyi-Kogan, Phys. Rev. D. 92, 104031 (2015), arXiv: 1507.04217[gr-qc] 
- 
								[97]
								F. Atamurotov, K. Jusufi, M. Jamil et al., Phys. Rev. D 104, 064053 (2021), arXiv: 2109.08150[gr-qc]