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Shape of dark energy: Constraining its evolution with a general parametrization

Dong Ha Lee1,*, Weiqiang Yang2,†, Eleonora Di Valentino1,‡, Supriya Pan3,4,§, and Carsten van de Bruck1,∥

  • *Contact author: dhlee1@sheffield.ac.uk
  • †Contact author: d11102004@163.com
  • ‡Contact author: e.divalentino@sheffield.ac.uk
  • §Contact author: supriya.maths@presiuniv.ac.in
  • ∥Contact author: c.vandebruck@sheffield.ac.uk
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Phys. Rev. D 113, 063554 – Published 20 March, 2026

DOI: https://doi.org/10.1103/z7y2-yvhg

Abstract

We consider a general dark energy (DE) model parametrized by its equation of state (EOS), featuring three free parameters: w0 (the present-day value of the DE EOS), wβ (quantifying the dynamical nature of the DE EoS), and β (governing various dynamical forms of the DE EOS). The key controlling parameter β can recover several existing DE models in the literature, such as the Chevallier-Polarski-Linder (CPL) parametrization (β=1), the logarithmic parametrization (in the limit β→0), and the linear parametrization (β=−1), alongside generate a class of new DE parametrizations for other values of β. The resulting DE scenario is constrained using a suite of the latest cosmological probes, including cosmic microwave background (CMB) temperature and polarization anisotropies from three different experiments (Planck 2018 and the Atacama Cosmology Telescope combined with WMAP), CMB lensing, baryon acoustic oscillations from Dark Energy Spectroscopic Instrument Year 2, and PantheonPlus from Type Ia supernovae. Our analyses reveal that stringent constraints on the DE parameters are obtained only when all cosmological probes are combined; otherwise, some parameters remain unconstrained. The present-day value of the DE EOS remains in the quintessence regime according to our results, and no significant evidence for a dynamical DE EOS is found. However, based on the Δχ2 and Bayesian evidence analyses, we observe a mild preference for the present three-parameter DE parametrization over the CPL parametrization when all cosmological probes are taken into account. Nonetheless, the Bayesian evidence difference remains below the threshold for statistical significance according to the revised Jeffreys scale, indicating that both models are effectively equally preferred by the data.

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References (223)

  1. A. G. Riess et al. (Supernova Search Team Collaboration), Astron. J. 116, 1009 (1998).
  2. S. Perlmutter et al. (Supernova Cosmology Project Collaboration), Astrophys. J. 517, 565 (1999).
  3. P. J. E. Peebles and B. Ratra, Rev. Mod. Phys. 75, 559 (2003).
  4. E. J. Copeland, M. Sami, and S. Tsujikawa, Int. J. Mod. Phys. D 15, 1753 (2006).
  5. V. Sahni and A. Starobinsky, Int. J. Mod. Phys. D 15, 2105 (2006).
  6. J. Frieman, M. Turner, and D. Huterer, Annu. Rev. Astron. Astrophys. 46, 385 (2008).
  7. M. Li, X.-D. Li, S. Wang, and Y. Wang, Commun. Theor. Phys. 56, 525 (2011).
  8. K. Bamba, S. Capozziello, S. Nojiri, and S. D. Odintsov, Astrophys. Space Sci. 342, 155 (2012).
  9. S. Nojiri and S. D. Odintsov, eConf C 0602061, 06 (2006).
  10. T. P. Sotiriou and V. Faraoni, Rev. Mod. Phys. 82, 451 (2010).
  11. A. De Felice and S. Tsujikawa, Living Rev. Relativity 13, 3 (2010).
  12. S. Capozziello and M. De Laurentis, Phys. Rep. 509, 167 (2011).
  13. T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis, Phys. Rep. 513, 1 (2012).
  14. K. Koyama, Rep. Prog. Phys. 79, 046902 (2016).
  15. Y.-F. Cai, S. Capozziello, M. De Laurentis, and E. N. Saridakis, Rep. Prog. Phys. 79, 106901 (2016).
  16. S. Nojiri, S. D. Odintsov, and V. K. Oikonomou, Phys. Rep. 692, 1 (2017).
  17. S. Bahamonde, K. F. Dialektopoulos, C. Escamilla-Rivera, G. Farrugia, V. Gakis, M. Hendry, M. Hohmann, J. Levi Said, J. Mifsud, and E. Di Valentino, Rep. Prog. Phys. 86, 026901 (2023).
  18. N. Bilic, G. B. Tupper, and R. D. Viollier, Phys. Lett. B 535, 17 (2002).
  19. T. Padmanabhan and T. R. Choudhury, Phys. Rev. D 66, 081301 (2002).
  20. S. Capozziello, Int. J. Mod. Phys. D 11, 483 (2002).
  21. J. S. Bagla, H. K. Jassal, and T. Padmanabhan, Phys. Rev. D 67, 063504 (2003).
  22. V. Gorini, A. Kamenshchik, and U. Moschella, Phys. Rev. D 67, 063509 (2003).
  23. A. D. Dolgov and M. Kawasaki, Phys. Lett. B 573, 1 (2003).
  24. S. M. Carroll, M. Hoffman, and M. Trodden, Phys. Rev. D 68, 023509 (2003).
  25. G. R. Farrar and P. J. E. Peebles, Astrophys. J. 604, 1 (2004).
  26. S. Nojiri and S. D. Odintsov, Phys. Rev. D 68, 123512 (2003).
  27. S. M. Carroll, A. De Felice, V. Duvvuri, D. A. Easson, M. Trodden, and M. S. Turner, Phys. Rev. D 71, 063513 (2005).
  28. M. Li, Phys. Lett. B 603, 1 (2004).
  29. Z.-K. Guo, Y.-S. Piao, X.-M. Zhang, and Y.-Z. Zhang, Phys. Lett. B 608, 177 (2005).
  30. S. Nojiri, S. D. Odintsov, and M. Sasaki, Phys. Rev. D 71, 123509 (2005).
  31. D. Pavon and W. Zimdahl, Phys. Lett. B 628, 206 (2005).
  32. H. Wei, R.-G. Cai, and D.-F. Zeng, Classical Quantum Gravity 22, 3189 (2005).
  33. B. Gumjudpai, T. Naskar, M. Sami, and S. Tsujikawa, J. Cosmol. Astropart. Phys. 06 (2005) 007.
  34. E. Elizalde, S. Nojiri, S. D. Odintsov, and P. Wang, Phys. Rev. D 71, 103504 (2005).
  35. I. H. Brevik and O. Gorbunova, Gen. Relativ. Gravit. 37, 2039 (2005).
  36. G. Cognola, E. Elizalde, S. Nojiri, S. D. Odintsov, and S. Zerbini, Phys. Rev. D 73, 084007 (2006).
  37. R. Ferraro and F. Fiorini, Phys. Rev. D 75, 084031 (2007).
  38. B. Li, J. D. Barrow, and D. F. Mota, Phys. Rev. D 76, 044027 (2007).
  39. S. Nojiri and S. D. Odintsov, Phys. Rev. D 77, 026007 (2008).
  40. H. Wei and R.-G. Cai, Phys. Lett. B 660, 113 (2008).
  41. A. De Felice and S. Tsujikawa, Phys. Lett. B 675, 1 (2009).
  42. C.-J. Feng and X.-Z. Li, Phys. Lett. B 680, 355 (2009).
  43. E. N. Saridakis, Eur. Phys. J. C 67, 229 (2010).
  44. J. B. Dent, S. Dutta, and E. N. Saridakis, J. Cosmol. Astropart. Phys. 01 (2011) 009.
  45. T. Harko, F. S. N. Lobo, S. Nojiri, and S. D. Odintsov, Phys. Rev. D 84, 024020 (2011).
  46. G. J. Olmo, Int. J. Mod. Phys. D 20, 413 (2011).
  47. G. Gubitosi, F. Piazza, and F. Vernizzi, J. Cosmol. Astropart. Phys. 02 (2013) 032.
  48. J. K. Bloomfield, E. E. Flanagan, M. Park, and S. Watson, J. Cosmol. Astropart. Phys. 08 (2013) 010.
  49. M. Maggiore, Phys. Rev. D 89, 043008 (2014).
  50. W. Yang and L. Xu, Phys. Rev. D 89, 083517 (2014).
  51. M. Maggiore and M. Mancarella, Phys. Rev. D 90, 023005 (2014).
  52. M. Kamionkowski, J. Pradler, and D. G. E. Walker, Phys. Rev. Lett. 113, 251302 (2014).
  53. J. Solà and A. Gómez-Valent, Int. J. Mod. Phys. D 24, 1541003 (2015).
  54. A. V. Astashenok, S. D. Odintsov, and V. K. Oikonomou, Classical Quantum Gravity 32, 185007 (2015).
  55. T. Katsuragawa and S. Matsuzaki, Phys. Rev. D 95, 044040 (2017).
  56. C. Lin and S. Mukohyama, J. Cosmol. Astropart. Phys. 10 (2017) 033.
  57. L. Heisenberg, M. Bartelmann, R. Brandenberger, and A. Refregier, Phys. Rev. D 98, 123502 (2018).
  58. D. Langlois, Int. J. Mod. Phys. D 28, 1942006 (2019).
  59. E. Di Valentino, A. Melchiorri, O. Mena, and S. Vagnozzi, Phys. Dark Universe 30, 100666 (2020).
  60. M. Braglia, M. Ballardini, F. Finelli, and K. Koyama, Phys. Rev. D 103, 043528 (2021).
  61. V. Motta, M. A. García-Aspeitia, A. Hernández-Almada, J. Magaña, and T. Verdugo, Universe 7, 163 (2021).
  62. P. G. S. Fernandes, P. Carrilho, T. Clifton, and D. J. Mulryne, Classical Quantum Gravity 39, 063001 (2022).
  63. M. Raveri, L. Pogosian, M. Martinelli, K. Koyama, A. Silvestri, and G.-B. Zhao, J. Cosmol. Astropart. Phys. 02 (2023) 061.
  64. S. Kumar, R. C. Nunes, S. Pan, and P. Yadav, Phys. Dark Universe 42, 101281 (2023).
  65. L. Heisenberg, Phys. Rep. 1066, 1 (2024).
  66. A. Chudaykin and M. Kunz, Phys. Rev. D 110, 123524 (2024).
  67. W. Yang, S. Pan, E. Di Valentino, O. Mena, D. F. Mota, and S. Chakraborty, Phys. Rev. D 111, 103509 (2025).
  68. I. D. Gialamas, G. Hütsi, M. Raidal, J. Urrutia, M. Vasar, and H. Veermäe, Phys. Rev. D 112, 063551 (2025).
  69. S. S. Mishra, W. L. Matthewson, V. Sahni, A. Shafieloo, and Y. Shtanov, J. Cosmol. Astropart. Phys. 11 (2025) 018.
  70. W. J. Wolf, P. G. Ferreira, and C. García-García, Phys. Rev. D 111, L041303 (2025).
  71. S. Dhawan and E. Mörtsell, arXiv:2506.22599.
  72. S. Weinberg, Rev. Mod. Phys. 61, 1 (1989).
  73. I. Zlatev, L.-M. Wang, and P. J. Steinhardt, Phys. Rev. Lett. 82, 896 (1999).
  74. L. Verde, T. Treu, and A. G. Riess, Nat. Astron. 3, 891 (2019).
  75. E. Di Valentino et al., Astropart. Phys. 131, 102605 (2021).
  76. E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri, D. F. Mota, A. G. Riess, and J. Silk, Classical Quantum Gravity 38, 153001 (2021).
  77. L. Perivolaropoulos and F. Skara, New Astron. Rev. 95, 101659 (2022).
  78. N. Schöneberg, G. Franco Abellán, A. Pérez Sánchez, S. J. Witte, V. Poulin, and J. Lesgourgues, Phys. Rep. 984, 1 (2022).
  79. P. Shah, P. Lemos, and O. Lahav, Astron. Astrophys. Rev. 29, 9 (2021).
  80. E. Abdalla et al., J. High Energy Astrophys. 34, 49 (2022).
  81. E. Di Valentino, Universe 8, 399 (2022).
  82. M. Kamionkowski and A. G. Riess, Annu. Rev. Nucl. Part. Sci. 73, 153 (2023).
  83. W. Giarè, CMB anomalies and the Hubble tension”, in The Hubble Constant Tension (Springer, Singapore, 2024).
  84. J.-P. Hu and F.-Y. Wang, Universe 9, 94 (2023).
  85. L. Verde, N. Schöneberg, and H. Gil-Marín, Annu. Rev. Astron. Astrophys. 62, 287 (2024).
  86. The Hubble Constant Tension, edited by E. Di Valentino and D. Brout Springer Series in Astrophysics and Cosmology (Springer, New York, 2024).
  87. L. Perivolaropoulos, Phys. Rev. D 110, 123518 (2024).
  88. E. Di Valentino et al. (CosmoVerse Collaboration), Phys. Dark Universe 49, 101965 (2025).
  89. A. R. Cooray and D. Huterer, Astrophys. J. Lett. 513, L95 (1999).
  90. G. Efstathiou, Mon. Not. R. Astron. Soc. 310, 842 (1999).
  91. M. Chevallier and D. Polarski, Int. J. Mod. Phys. D 10, 213 (2001).
  92. E. V. Linder, Phys. Rev. Lett. 90, 091301 (2003).
  93. C. Wetterich, Phys. Lett. B 594, 17 (2004).
  94. B. Feng, M. Li, Y.-S. Piao, and X. Zhang, Phys. Lett. B 634, 101 (2006).
  95. S. Hannestad and E. Mortsell, J. Cosmol. Astropart. Phys. 09 (2004) 001.
  96. J.-Q. Xia, B. Feng, and X.-M. Zhang, Mod. Phys. Lett. A 20, 2409 (2005).
  97. Y.-g. Gong and Y.-Z. Zhang, Phys. Rev. D 72, 043518 (2005).
  98. H. K. Jassal, J. S. Bagla, and T. Padmanabhan, Phys. Rev. D 72, 103503 (2005).
  99. S. Nesseris and L. Perivolaropoulos, Phys. Rev. D 72, 123519 (2005).
  100. D.-J. Liu, X.-Z. Li, J. Hao, and X.-H. Jin, Mon. Not. R. Astron. Soc. 388, 275 (2008).
  101. E. M. Barboza, Jr. and J. S. Alcaniz, Phys. Lett. B 666, 415 (2008).
  102. E. M. Barboza, J. S. Alcaniz, Z. H. Zhu, and R. Silva, Phys. Rev. D 80, 043521 (2009).
  103. J.-Z. Ma and X. Zhang, Phys. Lett. B 699, 233 (2011).
  104. I. Sendra and R. Lazkoz, Mon. Not. R. Astron. Soc. 422, 776 (2012).
  105. L. Feng and T. Lu, J. Cosmol. Astropart. Phys. 11 (2011) 034.
  106. E. M. Barboza, Jr. and J. S. Alcaniz, J. Cosmol. Astropart. Phys. 02 (2012) 042.
  107. A. De Felice, S. Nesseris, and S. Tsujikawa, J. Cosmol. Astropart. Phys. 05 (2012) 029.
  108. C.-J. Feng, X.-Y. Shen, P. Li, and X.-Z. Li, J. Cosmol. Astropart. Phys. 09 (2012) 023.
  109. H. Wei, X.-P. Yan, and Y.-N. Zhou, J. Cosmol. Astropart. Phys. 01 (2014) 045.
  110. J. Magaña, V. H. Cárdenas, and V. Motta, J. Cosmol. Astropart. Phys. 10 (2014) 017.
  111. O. Akarsu, T. Dereli, and J. A. Vazquez, J. Cosmol. Astropart. Phys. 06 (2015) 049.
  112. S. Pan, J. de Haro, A. Paliathanasis, and R. J. Slagter, Mon. Not. R. Astron. Soc. 460, 1445 (2016).
  113. E. Di Valentino, A. Melchiorri, and J. Silk, Phys. Lett. B 761, 242 (2016).
  114. R. C. Nunes, A. Bonilla, S. Pan, and E. N. Saridakis, Eur. Phys. J. C 77, 230 (2017).
  115. R. C. Nunes, S. Pan, E. N. Saridakis, and E. M. C. Abreu, J. Cosmol. Astropart. Phys. 01 (2017) 005.
  116. J. Magana, V. Motta, V. H. Cardenas, and G. Foex, Mon. Not. R. Astron. Soc. 469, 47 (2017).
  117. W. Yang, S. Pan, and A. Paliathanasis, Mon. Not. R. Astron. Soc. 475, 2605 (2018).
  118. S. Pan, E. N. Saridakis, and W. Yang, Phys. Rev. D 98, 063510 (2018).
  119. G. Panotopoulos and A. Rincón, Phys. Rev. D 97, 103509 (2018).
  120. W. Yang, S. Pan, E. Di Valentino, E. N. Saridakis, and S. Chakraborty, Phys. Rev. D 99, 043543 (2019).
  121. L. G. Jaime, M. Jaber, and C. Escamilla-Rivera, Phys. Rev. D 98, 083530 (2018).
  122. A. Das, A. Banerjee, S. Chakraborty, and S. Pan, Pramana 90, 19 (2018).
  123. W. Yang, S. Pan, E. Di Valentino, and E. N. Saridakis, Universe 5, 219 (2019).
  124. X. Li and A. Shafieloo, Astrophys. J. Lett. 883, L3 (2019).
  125. W. Yang, S. Pan, A. Paliathanasis, S. Ghosh, and Y. Wu, Mon. Not. R. Astron. Soc. 490, 2071 (2019).
  126. S. Pan, W. Yang, E. Di Valentino, A. Shafieloo, and S. Chakraborty, J. Cosmol. Astropart. Phys. 06 (2020) 062.
  127. D. Tamayo and J. A. Vazquez, Mon. Not. R. Astron. Soc. 487, 729 (2019).
  128. S. Pan, W. Yang, and A. Paliathanasis, Eur. Phys. J. C 80, 274 (2020).
  129. E. Di Valentino, A. Mukherjee, and A. A. Sen, Entropy 23, 404 (2021).
  130. M. Rezaei, T. Naderi, M. Malekjani, and A. Mehrabi, Eur. Phys. J. C 80, 374 (2020).
  131. D. Perkovic and H. Stefancic, Eur. Phys. J. C 80, 629 (2020).
  132. A. Banihashemi, N. Khosravi, and A. Shafieloo, J. Cosmol. Astropart. Phys. 06 (2021) 003.
  133. M. Jaber-Bravo, E. Almaraz, and A. de la Macorra, Astropart. Phys. 115, 102388 (2020).
  134. H. B. Benaoum, W. Yang, S. Pan, and E. Di Valentino, Int. J. Mod. Phys. D 31, 2250015 (2022).
  135. W. Yang, E. Di Valentino, S. Pan, A. Shafieloo, and X. Li, Phys. Rev. D 104, 063521 (2021).
  136. M. Jaber, G. Arciniega, L. G. Jaime, and O. A. Rodríguez-López, Phys. Dark Universe 37, 101069 (2022).
  137. G. Alestas, D. Camarena, E. Di Valentino, L. Kazantzidis, V. Marra, S. Nesseris, and L. Perivolaropoulos, Phys. Rev. D 105, 063538 (2022).
  138. J. Yang, X.-Y. Fan, C.-J. Feng, and X.-H. Zhai, Chin. Phys. Lett. 40, 019801 (2023).
  139. H. G. Escudero, J.-L. Kuo, R. E. Keeley, and K. N. Abazajian, Phys. Rev. D 106, 103517 (2022).
  140. M. N. Castillo-Santos, A. Hernández-Almada, M. A. García-Aspeitia, and J. Magaña, Phys. Dark Universe 40, 101225 (2023).
  141. W. Yang, W. Giarè, S. Pan, E. Di Valentino, A. Melchiorri, and J. Silk, Phys. Rev. D 107, 063509 (2023).
  142. S. Dahmani, A. Bouali, I. El Bojaddaini, A. Errahmani, and T. Ouali, Phys. Dark Universe 42, 101266 (2023).
  143. L. A. Escamilla, W. Giarè, E. Di Valentino, R. C. Nunes, and S. Vagnozzi, J. Cosmol. Astropart. Phys. 05 (2024) 091.
  144. M. Rezaei, S. Pan, W. Yang, and D. F. Mota, J. Cosmol. Astropart. Phys. 01 (2024) 052.
  145. S. A. Adil, O. Akarsu, E. Di Valentino, R. C. Nunes, E. Özülker, A. A. Sen, and E. Specogna, Phys. Rev. D 109, 023527 (2024).
  146. J. A. Lozano Torres, Mon. Not. R. Astron. Soc. 533, 1865 (2024).
  147. J. K. Singh, P. Singh, E. N. Saridakis, S. Myrzakul, and H. Balhara, Universe 10, 246 (2024).
  148. M. Rezaei, Astrophys. J. 967, 2 (2024).
  149. M. Reyhani, M. Najafi, J. T. Firouzjaee, and E. Di Valentino, Phys. Dark Universe 44, 101477 (2024).
  150. A. G. Adame et al. (DESI Collaboration), J. Cosmol. Astropart. Phys. 02 (2025) 021.
  151. M. Cortês and A. R. Liddle, J. Cosmol. Astropart. Phys. 12 (2024) 007.
  152. D. Shlivko and P. J. Steinhardt, Phys. Lett. B 855, 138826 (2024).
  153. O. Luongo and M. Muccino, Astron. Astrophys. 690, A40 (2024).
  154. W. Yin, J. High Energy Phys. 05 (2024) 327.
  155. I. D. Gialamas, G. Hütsi, K. Kannike, A. Racioppi, M. Raidal, M. Vasar, and H. Veermäe, Phys. Rev. D 111, 043540 (2025).
  156. B. R. Dinda, J. Cosmol. Astropart. Phys. 09 (2024) 062.
  157. M. Najafi, S. Pan, E. Di Valentino, and J. T. Firouzjaee, Phys. Dark Universe 45, 101539 (2024).
  158. H. Wang and Y.-S. Piao, Phys. Lett. B 873, 140180 (2026).
  159. G. Ye, M. Martinelli, B. Hu, and A. Silvestri, Phys. Rev. Lett. 134, 181002 (2025).
  160. Y. Tada and T. Terada, Phys. Rev. D 109, L121305 (2024).
  161. Y. Carloni, O. Luongo, and M. Muccino, Phys. Rev. D 111, 023512 (2025).
  162. C.-G. Park, J. de Cruz Pérez, and B. Ratra, Phys. Rev. D 110, 123533 (2024).
  163. K. Lodha et al. (DESI Collaboration), Phys. Rev. D 111, 023532 (2025).
  164. O. F. Ramadan, J. Sakstein, and D. Rubin, Phys. Rev. D 110, L041303 (2024).
  165. A. Notari, M. Redi, and A. Tesi, J. Cosmol. Astropart. Phys. 11 (2024) 025.
  166. L. Orchard and V. H. Cárdenas, Phys. Dark Universe 46, 101678 (2024).
  167. A. Hernández-Almada, M. L. Mendoza-Martínez, M. A. García-Aspeitia, and V. Motta, Phys. Dark Universe 46, 101668 (2024).
  168. S. Pourojaghi, M. Malekjani, and Z. Davari, Phys. Rev. D 111, 083547 (2025).
  169. W. Giarè, M. Najafi, S. Pan, E. Di Valentino, and J. T. Firouzjaee, J. Cosmol. Astropart. Phys. 10 (2024) 035.
  170. J. a. Rebouças, D. H. F. de Souza, K. Zhong, V. Miranda, and R. Rosenfeld, J. Cosmol. Astropart. Phys. 02 (2025) 024.
  171. W. Giarè, Phys. Rev. D 112, 023508 (2025).
  172. C.-G. Park, J. de Cruz Perez, and B. Ratra, Int. J. Mod. Phys. D 34, 2550058 (2025).
  173. N. Menci, A. A. Sen, and M. Castellano, Astrophys. J. 976, 227 (2024).
  174. T.-N. Li, Y.-H. Li, G.-H. Du, P.-J. Wu, L. Feng, J.-F. Zhang, and X. Zhang, Eur. Phys. J. C 85, 608 (2025).
  175. J.-X. Li and S. Wang, J. Cosmol. Astropart. Phys. 07 (2025) 047.
  176. A. Notari, M. Redi, and A. Tesi, J. Cosmol. Astropart. Phys. 04 (2025) 048.
  177. Q. Gao, Z. Peng, S. Gao, and Y. Gong, Universe 11, 10 (2025).
  178. R. Fikri, E. ElKhateeb, E. S. Lashin, and W. El Hanafy, Ann. Phys. (Amsterdam) 481, 170190 (2025).
  179. J.-Q. Jiang, D. Pedrotti, S. S. da Costa, and S. Vagnozzi, Phys. Rev. D 110, 123519 (2024).
  180. J. Zheng, D.-C. Qiang, and Z.-Q. You, J. Cosmol. Astropart. Phys. 08 (2025) 056.
  181. A. Gómez-Valent and J. Solà Peracaula, Phys. Lett. B 864, 139391 (2025).
  182. S. Roy Choudhury and T. Okumura, Astrophys. J. Lett. 976, L11 (2024).
  183. A. Lewis and E. Chamberlain, J. Cosmol. Astropart. Phys. 05 (2025) 065.
  184. A. Paliathanasis, Phys. Dark Universe 48, 101956 (2025).
  185. W. J. Wolf, C. García-García, and P. G. Ferreira, J. Cosmol. Astropart. Phys. 05 (2025) 034.
  186. A. J. Shajib and J. A. Frieman, Phys. Rev. D 112, 063508.
  187. W. Giarè, T. Mahassen, E. Di Valentino, and S. Pan, Phys. Dark Universe 48, 101906 (2025).
  188. E. Chaussidon et al., Phys. Rev. D 112, 063548 (2025).
  189. D. A. Kessler, L. A. Escamilla, S. Pan, and E. Di Valentino, arXiv:2504.00776.
  190. Y.-H. Pang, X. Zhang, and Q.-G. Huang, Sci. China Phys. Mech. Astron. 68, 280410 (2025).
  191. S. Roy Choudhury, Astrophys. J. Lett. 986, L31 (2025).
  192. M. Scherer, M. A. Sabogal, R. C. Nunes, and A. De Felice, Phys. Rev. D 112, 043513 (2025).
  193. E. Specogna, S. A. Adil, E. Ozulker, E. Di Valentino, R. C. Nunes, O. Akarsu, and A. A. Sen, arXiv:2504.17859.
  194. H. Cheng, E. Di Valentino, L. A. Escamilla, A. A. Sen, and L. Visinelli, J. Cosmol. Astropart. Phys. 09 (2025) 031.
  195. H. Cheng, E. Di Valentino, and L. Visinelli, arXiv:2505.22066.
  196. E. Özülker, E. Di Valentino, and W. Giarè, arXiv:2506.19053.
  197. M. Abdul Karim et al. (DESI Collaboration), Phys. Rev. D 112, 083515 (2025).
  198. Z. Lu, T. Simon, and P. Zhang, arXiv:2503.04602.
  199. G.-B. Zhao et al., Nat. Astron. 1, 627 (2017).
  200. Z. Zhang, G. Gu, X. Wang, Y.-H. Li, C. G. Sabiu, H. Park, H. Miao, X. Luo, F. Fang, and X.-D. Li, Astrophys. J. 878, 137 (2019).
  201. L. A. Escamilla, S. Pan, E. Di Valentino, A. Paliathanasis, J. A. Vázquez, and W. Yang, Phys. Rev. D 111, 023531 (2025).
  202. A. N. Ormondroyd, W. J. Handley, M. P. Hobson, and A. N. Lasenby, Mon. Not. R. Astron. Soc. 541, 3388 (2025).
  203. A. N. Ormondroyd, W. J. Handley, M. P. Hobson, and A. N. Lasenby, arXiv:2503.17342.
  204. Y. Ma, J. Zhang, S. Cao, X. Zheng, T. Xu, and J. Qi, Eur. Phys. J. C 77, 891 (2017).
  205. W. Yang, N. Banerjee, and S. Pan, Phys. Rev. D 95, 123527 (2017); 96, 089903(A) (2017).
  206. C.-P. Ma and E. Bertschinger, Astrophys. J. 455, 7 (1995).
  207. N. Aghanim et al. (Planck Collaboration), Astron. Astrophys. 641, A5 (2020).
  208. T. Louis et al. (ACT Collaboration), J. Cosmol. Astropart. Phys. 11 (2025) 062.
  209. C. L. Bennett et al. (WMAP Collaboration), Astrophys. J. Suppl. Ser. 208, 20 (2013).
  210. L. Pagano, J. M. Delouis, S. Mottet, J. L. Puget, and L. Vibert, Astron. Astrophys. 635, A99 (2020).
  211. M. S. Madhavacheril et al. (ACT Collaboration), Astrophys. J. 962, 113 (2024).
  212. F. J. Qu et al. (ACT Collaboration), Astrophys. J. 962, 112 (2024).
  213. J. Carron, M. Mirmelstein, and A. Lewis, J. Cosmol. Astropart. Phys. 09 (2022) 039.
  214. D. Brout et al., Astrophys. J. 938, 110 (2022).
  215. A. Lewis, A. Challinor, and A. Lasenby, Astrophys. J. 538, 473 (2000).
  216. C. Howlett, A. Lewis, A. Hall, and A. Challinor, J. Cosmol. Astropart. Phys. 04 (2012) 027.
  217. J. Torrado and A. Lewis, J. Cosmol. Astropart. Phys. 05 (2021) 057.
  218. A. Gelman and D. B. Rubin, Stat. Sci. 7, 457 (1992).
  219. A. Heavens, Y. Fantaye, A. Mootoovaloo, H. Eggers, Z. Hosenie, S. Kroon, and E. Sellentin, arXiv:1704.03472.
  220. W. Giare, wgcosmo: Cosmological analysis tools, https://github.com/williamgiare/wgcosmoURL (2025).
  221. R. E. Kass and A. E. Raftery, J. Am. Stat. Assoc. 90, 773 (1995).
  222. N. Aghanim et al. (Planck Collaboration), Astron. Astrophys. 641, A6 (2020); 652, C4(E) (2021).
  223. W. Handley, J. Open Source Softwaare 3, 849 (2018).

Outline

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Phys. Rev. D 113, 063554– Published 20 March, 2026
  • Received 7 August 2025
  • Accepted 27 February 2026
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