- 
								[1]
								S. W. Hawking and G. F. R. Ellis, The large-scale structure of space-time (Cambridge: Cambridge University Press, 1973) 
- 
								[2]
								A. Borde, A. H. Guth, and A. Vilenkin, Phys. Rev. Lett. 90, 151301 (2003), arXiv: gr-qc/0110012[gr-qc] 
- 
								[3]
								A. Borde and A. Vilenkin, Phys. Rev. Lett. 72, 3305 (1994), arXiv: gr-qc/9312022[gr-qc] 
- 
								[4]
								R. J. Adler, Am. J. Phys. 78, 925 (2010), arXiv: 1001.1205[gr-qc] 
- 
								[5]
								Y. J. Ng, Mod. Phys. Lett. A 18, 1073 (2003), arXiv: grqc/0305019[gr-qc] 
- 
								[6]
								B. P. Abbott and et al. (Virgo and LIGO Scientific Collaborations), Phys. Rev. Lett. 116, 061102 (2016), arXiv: 1602.03837[gr-qc] 
- 
								[7]
								B. P. Abbott and et al. (Virgo and LIGO Scientific Collaborations), Phys. Rev. Lett. 116, 241102 (2016), arXiv: 1602.03840[gr-qc] 
- 
								[8]
								K. Akiyama and et al. (Event Horizon Telescope Collaboration), Astrophys. J. 875, L1 (2019), arXiv: 1906.11238[astro-ph.GA] 
- 
								[9]
								K. Akiyama and et al. (Event Horizon Telescope Collaboration), Astrophys. J. 875, L6 (2019), arXiv: 1906.11243[astro-ph.GA] 
- 
								[10]
								A. Ashtekar and J. Lewandowski, Class. Quantum Gravity 21, R53 (2004), arXiv: gr-qc/0404018[gr-qc] 
- 
								[11]
								P. Singh and A. Toporensky, Phys. Rev. D 69, 104008 (2004), arXiv: gr-qc/0312110[gr-qc] 
- 
								[12]
								A. Ashtekar, T. Pawlowski, and P. Singh, Phys. Rev. D 74, 084003 (2006), arXiv: gr-qc/0607039[gr-qc] 
- 
								[13]
								A. Ashtekar, T. Pawlowski, and P. Singh, Phys. Rev. Lett. 96, 141301 (2006), arXiv: gr-qc/0602086[gr-qc] 
- 
								[14]
								G. Date and G. M. Hossain, Phys. Rev. Lett. 94, 011302 (2005), arXiv: gr-qc/0407074[gr-qc] 
- 
								[15]
								A. Ashtekar and P. Singh, Class. Quantum Gravity 28, 213001 (2011), arXiv: 1108.0893[gr-qc] 
- 
								[16]
								T. Papanikolaou, Class. Quantum Gravity 40, 134001 (2023), arXiv: 2301.11439[gr-qc] 
- 
								[17]
								B.-F. Li and P. Singh, arXiv: 2304.05426 [gr-qc] 
- 
								[18]
								L. Modesto, Int. J. Theor. Phys. 49, 1649 (2010), arXiv: 0811.2196[gr-qc] 
- 
								[19]
								L. Modesto and I. Prémont-Schwarz, Phys. Rev. D 80, 064041 (2009), arXiv: 0905.3170[hep-th] 
- 
								[20]
								S. Sahu, K. Lochan, and D. Narasimha, Phys. Rev. D 91, 063001 (2015), arXiv: 1502.05619[gr-qc] 
- 
								[21]
								A. Ashtekar, J. Olmedo, and P. Singh, Phys. Rev. Lett. 121, 241301 (2018), arXiv: 1806.00648[gr-qc] 
- 
								[22]
								A. Ashtekar, J. Olmedo, and P. Singh, Phys. Rev. D 98, 126003 (2018), arXiv: 1806.02406[gr-qc] 
- 
								[23]
								M. Bojowald, S. Brahma, and D.-H. Yeom, Phys. Rev. D 98, 046015 (2018), arXiv: 1803.01119[gr-qc] 
- 
								[24]
								E. Alesci, S. Bahrami, and D. Pranzetti, Phys. Lett. B 797, 134908 (2019), arXiv: 1904.12412[gr-qc] 
- 
								[25]
								M. Assanioussi, A. Dapor, and K. Liegener, Phys. Rev. D 101, 026002 (2020), arXiv: 1908.05756[gr-qc] 
- 
								[26]
								N. Bodendorfer, F. M. Mele, and J. Münch, Phys. Lett. B 819, 136390 (2021), arXiv: 1911.12646[gr-qc] 
- 
								[27]
								A. Ashtekar, Universe 6, 21 (2020), arXiv: 2001.08833[grqc] 
- 
								[28]
								W.-C. Gan, N. O. Santos, F.-W. Shu et al., Phys. Rev. D 102, 124030 (2020), arXiv: 2008.09664[grqc] 
- 
								[29]
								A. Perez, Rep. Prog. Phys. 80, 126901 (2017), arXiv: 1703.09149[gr-qc] 
- 
								[30]
								Y.-C. Liu, J.-X. Feng, F.-W. Shu et al., Phys. Rev. D 104, 106001 (2021), arXiv: 2109.02861[gr-qc] 
- 
								[31]
								Q.-M. Fu and X. Zhang, Phys. Rev. D 105, 064020 (2022), arXiv: 2111.07223[gr-qc] 
- 
								[32]
								S. Brahma, C.-Y. Chen, and D.-H. Yeom, Phys. Rev. Lett. 126, 181301 (2021), arXiv: 2012.08785[gr-qc] 
- 
								[33]
								S. Yang, W.-D. Guo, Q. Tan et al., Phys. Rev. D 108, 024055 (2023), arXiv: 2304.06895[gr-qc] 
- 
								[34]
								J.-M. Yan, C. Liu, T. Zhu et al., Phys. Rev. D 107, 084043 (2023), arXiv: 2302.10482[gr-qc] 
- 
								[35]
								S. Albuquerque, I. P. Lobo, and V. B. Bezerra, Class. Quant. Grav. 40, 174001 (2023), arXiv: 2301.07746[grqc] 
- 
								[36]
								H. X. Jiang, C. Liu, I. K. Dihingia et al., JCAP 01, 059 (2024), arXiv: 2312.04288[gr-qc] 
- 
								[37]
								H. Jiang, M. Alloqulov, Q. Wu et al., Phys. Dark Universe 46, 101627 (2024) 
- 
								[38]
								J. Synge, Mon. Not. R. Astron. Soc. 131, 463 (1966) 
- 
								[39]
								J. Luminet, Astron. Astrophys. 75, 228 (1979) 
- 
								[40]
								L. Amarilla and E. F. Eiroa, Phys. Rev. D 87, 044057 (2013), arXiv: 1301.0532[gr-qc] 
- 
								[41]
								S.-W. Wei and Y.-X. Liu, J. Cosmol. Astropart. Phys. 11, 063 (2013), arXiv: 1311.4251[gr-qc] 
- 
								[42]
								R. A. Konoplya, Phys. Lett. B 795, 1 (2019), arXiv: 1905.00064[gr-qc] 
- 
								[43]
								S. Vagnozzi and L. Visinelli, Phys. Rev. D 100, 024020 (2019) 
- 
								[44]
								M. Afrin, R. Kumar, and S. G. Ghosh, Mon. Not. R. Astron. Soc. 504, 5927 (2021), arXiv: 2103.11417[gr-qc] 
- 
								[45]
								F. Atamurotov, S. G. Ghosh, and B. Ahmedov, Eur. Phys. J. C 76, 273 (2016), arXiv: 1506.03690[gr-qc] 
- 
								[46]
								R. A. Konoplya and A. Zhidenko, Phys. Rev. D 100, 044015 (2019) 
- 
								[47]
								G. Mustafa, F. Atamurotov, I. Hussain et al., Chin. Phys. C 46, 125107 (2022), arXiv: 2207.07608[gr-qc] 
- 
								[48]
								N. Tsukamoto, Z. Li, and C. Bambi, J. Cosmol. Astropart. Phys. 2014, 043 (2014) 
- 
								[49]
								F. Atamurotov, S. Shaymatov, P. Sheoran et al., J. Cosmol. Astropart. Phys. 2021, 045 (2021), arXiv: 2105.02214[gr-qc] 
- 
								[50]
								N. Tsukamoto, Phys. Rev. D 97, 064021 (2018) 
- 
								[51]
								J. L. Rosa, Phys. Rev. D 107, 084048 (2023) 
- 
								[52]
								J. Badía and E. F. Eiroa, Phys. Rev. D 107, 124028 (2023), arXiv: 2210.03081[gr-qc] 
- 
								[53]
								A. Al-Badawi, M. Alloqulov, S. Shaymatov et al., Chin. Phys. C 48, 095105 (2024), arXiv: 2401.04584[gr-qc] 
- 
								[54]
								G. J. Olmo, J. L. Rosa, D. Rubiera-Garcia et al., Class. Quantum Gravity 40, 174002 (2023) 
- 
								[55]
								R. A. Konoplya and A. Zhidenko, Phys. Rev. D 103, 104033 (2021) 
- 
								[56]
								S. H. Hendi, K. Jafarzade, and B. Eslam Panah, J. Cosmol. Astropart. Phys. 2023, 022 (2023), arXiv: 2206.05132[gr-qc] 
- 
								[57]
								J. W. Moffat, Eur. Phys. J. C 75, 130 (2015), arXiv: 1502.01677[gr-qc] 
- 
								[58]
								J. W. Moffat and V. T. Toth, Phys. Rev. D 101, 024014 (2020), arXiv: 1904.04142[gr-qc] 
- 
								[59]
								A. Al-Badawi, S. Shaymatov, M. Alloqulov et al., Commun. Theor. Phys. 76, 085401 (2024), arXiv: 2401.12723[gr-qc] 
- 
								[60]
								A. S. Eddington, The Observatory 42, 119 (1919) 
- 
								[61]
								G. S. Bisnovatyi-Kogan and O. Y. Tsupko, Mon. Not. R. Astron. Soc. 404, 1790 (2010) 
- 
								[62]
								O. Y. Tsupko and G. S. Bisnovatyi-Kogan, Gravit. Cosmol. 18, 117 (2012) 
- 
								[63]
								P. V. P. Cunha, N. A. Eiró, C. A. R. Herdeiro et al., J. Cosmol. A. P 2020, 035 (2020), arXiv: 1912.08833[gr-qc] 
- 
								[64]
								G. Z. Babar, F. Atamurotov, and A. Z. Babar, Phys. Dark Universe 32, 100798 (2021) 
- 
								[65]
								W. Javed, M. Atique, and A. Övgün, Gen. Relativ. Gravit. 54, 135 (2022), arXiv: 2210.17277[gr-qc] 
- 
								[66]
								K. Jafarzade, M. Kord Zangeneh, and F. S. N. Lobo, J. Cosmol. Astropart. Phys. 2021, 008 (2021), arXiv: 2010.05755[gr-qc] 
- 
								[67]
								F. Atamurotov, D. Ortiqboev, A. Abdujabbarov, and G. Mustafa, Eur. Phys. J. C 82, 659 (2022) 
- 
								[68]
								A. Övgün, İ. Sakallı, and J. Saavedra, Ann. Phys. 411, 167978 (2019), arXiv: 1806.06453[gr-qc] 
- 
								[69]
								S. Rahvar and J. W. Moffat, MNRAS 482, 4514 (2019), arXiv: 1807.07424[gr-qc] 
- 
								[70]
								R. N. Izmailov, R. K. Karimov, E. R. Zhdanov et al., MNRAS 483, 3754 (2019), arXiv: 1905.01900[gr-qc] 
- 
								[71]
								A. Al-Badawi, S. Shaymatov, S. K. Jha et al., Eur. Phys. J. C 84, 722 (2024), arXiv: 2406.17501[gr-qc] 
- 
								[72]
								A. Al-Badawi, S. Shaymatov, and İ. Sakallı, Eur. Phys. J. C 84, 825 (2024), arXiv: 2408.09228[gr-qc] 
- 
								[73]
								A. Rogers, Mon. Not. R. Astron. Soc. 451, 17 (2015) 
- 
								[74]
								G. Z. Babar, A. Z. Babar, and F. Atamurotov, Eur. Phys. J. C. 80, 761 (2020) 
- 
								[75]
								W. Javed, S. Riaz, R. C. Pantig et al., Eur. Phys. J. C 82, 1057 (2022), arXiv: 2212.00804[gr-qc] 
- 
								[76]
								W. Javed, I. Hussain, and A. Övgün, Eur. Phys. J. Plus 137, 148 (2022), arXiv: 2201.09879[gr-qc] 
- 
								[77]
								C. Benavides-Gallego, A. Abdujabbarov, and C. Bambi, Eur. Phys. J. C. 78, 694 (2018) 
- 
								[78]
								F. Atamurotov, S. Shaymatov, and B. Ahmedov, Galaxies 9, 54 (2021) 
- 
								[79]
								F. Atamurotov, M. Alloqulov, A. Abdujabbarov et al., Eur. Phys. J. Plus 137, 634 (2022) 
- 
								[80]
								J. Lewandowski, Y. Ma, J. Yang et al., Phys. Rev. Lett. 130, 101501 (2023) 
- 
								[81]
								J.-P. Ye, Z.-Q. He, A.-X. Zhou et al., Phys. Lett. B 851, 138566 (2024), arXiv: 2312.17724[gr-qc] 
- 
								[82]
								C.-Y. Shao, C. Zhang, W. Zhang et al., Phys. Rev. D 109, 064012 (2024), arXiv: 2309.04962[gr-qc] 
- 
								[83]
								C. Zhang, Y. Ma, and J. Yang, Phys. Rev. D 108, 104004 (2023), arXiv: 2302.02800[gr-qc] 
- 
								[84]
								H. Gong, S. Li, D. Zhang, G. Fu et al., Phys. Rev. D 110, 044040 (2024), arXiv: 2312.17639[gr-qc] 
- 
								[85]
								S. Yang, Y.-P. Zhang, T. Zhu et al., arXiv: 2407.00283 
- 
								[86]
								J. L. Synge, Relativity: The General Theory (Amsterdam: North-Holland Pub. Co., 1960) 
- 
								[87]
								O. Y. Tsupko and G. S. Bisnovatyi-Kogan, Gravit. Cosmol. 15, 184 (2009) 
- 
								[88]
								V. Perlick, O. Y. Tsupko, and G. S. Bisnovatyi-Kogan, Phys. Rev. D. 92, 104031 (2015), arXiv: 1507.04217[grqc] 
- 
								[89]
								K. Akiyama and et al. (Event Horizon Telescope Collaboration), Astrophys. J. 910, L12 (2021) 
- 
								[90]
								C. Bambi, K. Freese, S. Vagnozzi et al., Phys. Rev. D 100, 044057 (2019) 
- 
								[91]
								V. Bozza, Phys. Rev. D 78, 103005 (2008) 
- 
								[92]
								F. Atamurotov, A. Abdujabbarov, and J. Rayimbaev, Eur. Phys. J. C 81, 118 (2021) 
- 
								[93]
								M. Alloqulov, F. Atamurotov, A. Abdujabbarov et al., Chin. Phys. C 47, 075103 (2023) 
- 
								[94]
								M. Alloqulov, F. Atamurotov, A. Abdujabbarov et al., Chin. Phys. C 48, 025104 (2024) 
- 
								[95]
								M. Alloqulov, S. Shaymatov, A. Jawad et al., Commun. Theor. Phys. 77, 015402 (2025) 
- 
								[96]
								Y. Wang, J. Cosmol. Astropart. Phys. 2005, 005 (2005), arXiv: astro-ph/0406635[astro-ph] 
- 
								[97]
								E. J. Gonzalez, G. Foëx, J. L. Nilo Castellón et al., MNRAS 452, 2225 (2015), arXiv: 1504.03364[astro-ph.CO] 
- 
								[98]
								Z. Kalantari, S. Rahvar, and A. Ibrahim, Astrophys. J. 934, 106 (2022), arXiv: 2205.05278[astroph.HE]