- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Constraints on attractor models of inflation and reheating from Planck, BICEP/Keck, ACT DR6, and SPT-3G data

John Ellis*

Marcos A. G. Garcia†

Keith A. Olive‡

Sarunas Verner§

  • Theoretical Particle Physics and Cosmology Group, Department of Physics, King’s College London, London WC2R 2LS, United Kingdom and Theoretical Physics Department, CERN, CH-1211 Geneva 23, Switzerland

  • William I. Fine Theoretical Physics Institute, School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA

  • Kavli Institute for Cosmological Physics, University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois 60637, USA

  • *Contact author: John.Ellis@cern.ch
  • †Contact author: marcos.garcia@fisica.unam.mx
  • ‡Contact author: olive@umn.edu
  • §Contact author: verner@uchicago.edu
PDF

Phys. Rev. D 113, 063571 – Published 30 March, 2026

DOI: https://doi.org/10.1103/d35r-7bn8

Abstract

We analyze the latest cosmic microwave background (CMB) constraints on the scalar spectral index ns and tensor-to-scalar ratio r from Planck 2018, BICEP/Keck 2018, the Atacama Cosmology Telescope Data Release 6 (ACT DR6), and the South Pole Telescope (SPT-3G) data, focusing on their implications for attractor models of inflation. We compare systematically observational bounds with theoretical predictions for both E-model (α-Starobinsky) and T-model potentials. The observational constraints accommodate E-models with α≲25, with the canonical Starobinsky model (α=1) predicting ns=0.958–0.963 for reheating temperatures between 100−1010  GeV, in good agreement with Planck 2018 data and within the 95% CL region determined by the Planck-ACT-SPT combination, but below the 95% confidence region of the Planck-ACT-DESI combination. Higher reheating temperatures from near-instantaneous reheating improve the compatibility. T-models predict slightly lower ns values (0.956–0.961), in some tension with Planck 2018 data, and we find an upper limit of α≲11 in these models. We extend our analysis to generalized α-attractors with monomial potentials V(φ)∝φk near the minimum, demonstrating that models with k≥6 naturally predict ns≃0.965–0.968 for typical number of e-folds, in better agreement with the ACT DR6 data. We also consider deformed E- and T-models, which allow significantly higher values of ns for low values of α≃1.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (135)

  1. K. A. Olive, Phys. Rep. 190 (1990) 307; A. D. Linde, Particle Physics and Inflationary Cosmology (Harwood, Chur, Switzerland, 1990); D. H. Lyth and A. Riotto, Phys. Rep. 314, 1 (1999); J. Martin, C. Ringeval, and V. Vennin, Phys. Dark Universe 5–6, 75 (2014); J. Martin, C. Ringeval, R. Trotta, and V. Vennin, J. Cosmol. Astropart. Phys. 03 (2014) 039; J. Martin, Astrophys. Space Sci. Proc. 45, 41 (2016); J. Ellis and D. Wands, arXiv:2312.13238, in S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024).
  2. D. N. Spergel et al. (WMAP Collaboration), Astrophys. J. Suppl. Ser. 148, 175 (2003).
  3. N. Aghanim et al. (Planck Collaboration), Astron. Astrophys. 641, A6 (2020); 652, C4(E) (2021); Y. Akrami et al. (Planck Collaboration), 641, A10 (2020).
  4. P. A. R. Ade et al. (BICEP Collaboration and Keck Collaboration), Phys. Rev. Lett. 127, 151301 (2021).
  5. M. Tristram, A. J. Banday, K. M. Górski, R. Keskitalo, C. R. Lawrence, K. J. Andersen, R. B. Barreiro, J. Borrill, L. P. L. Colombo, H. K. Eriksen et al., Phys. Rev. D 105, 083524 (2022).
  6. A. A. Starobinsky, Phys. Lett. 91B, 99 (1980).
  7. K. S. Stelle, Gen. Relativ. Gravit. 9, 353 (1978); B. Whitt, Phys. Lett. 145B, 176 (1984).
  8. J. D. Barrow and S. Cotsakis, Phys. Lett. B 214, 515 (1988).
  9. S. Kalara, N. Kaloper, and K. A. Olive, Nucl. Phys. B341, 252 (1990); K. Maeda, Phys. Rev. D 39, 3159 (1989).
  10. T. Louis et al. (ACT Collaboration), J. Cosmol. Astropart. Phys. 11 (2025) 062.
  11. E. Calabrese et al. (ACT Collaboration), J. Cosmol. Astropart. Phys. 11 (2025) 063.
  12. E. Camphuis et al. (SPT-3G Collaboration), arXiv:2506.20707.
  13. A. G. Adame et al. (DESI Collaboration), J. Cosmol. Astropart. Phys. 04 (2025) 012.
  14. A. G. Adame et al. (DESI Collaboration), J. Cosmol. Astropart. Phys. 02 (2025) 021.
  15. M. Abdul Karim et al. (DESI Collaboration), Phys. Rev. D 112, 083514 (2025).
  16. M. Abdul Karim et al. (DESI Collaboration), Phys. Rev. D 112, 083515 (2025).
  17. E. Cremmer, S. Ferrara, C. Kounnas, and D. V. Nanopoulos, Phys. Lett. B 133, 61 (1983); J. R. Ellis, A. B. Lahanas, D. V. Nanopoulos, and K. Tamvakis, Phys. Lett. 134B, 429 (1984).
  18. A. B. Lahanas and D. V. Nanopoulos, Phys. Rep. 145, 1 (1987).
  19. J. Ellis, D. V. Nanopoulos, and K. A. Olive, Phys. Rev. Lett. 111, 111301 (2013).
  20. R. Kallosh and A. Linde, J. Cosmol. Astropart. Phys. 07 (2013) 002.
  21. J. Ellis, D. V. Nanopoulos, and K. A. Olive, J. Cosmol. Astropart. Phys. 10 (2013) 009.
  22. R. Kallosh and A. Linde, J. Cosmol. Astropart. Phys. 10 (2013) 033.
  23. R. Kallosh, A. Linde., and D. Roest, J. High Energy Phys. 11 (2013) 198.
  24. J. Ellis, D. V. Nanopoulos, K. A. Olive, and S. Verner, J. Cosmol. Astropart. Phys. 09 (2019) 040.
  25. R. Kallosh and A. Linde, C.R. Phys. 16, 914 (2015).
  26. J. J. M. Carrasco, R. Kallosh, and A. Linde, Phys. Rev. D 92, 063519 (2015).
  27. R. Kallosh, A. Linde, and D. Roest, Phys. Rev. Lett. 135, 161001 (2025).
  28. S. Aoki, H. Otsuka, and R. Yanagita, Phys. Rev. Lett. 135, 161001 (2025).
  29. A. Berera, S. Brahma, Z. Qiu, R. O. Ramos, and G. S. Rodrigues, J. Cosmol. Astropart. Phys. 11 (2025) 059.
  30. S. Brahma and J. Calderón-Figueroa, arXiv:2504.02746.
  31. I. D. Gialamas, A. Karam, A. Racioppi, and M. Raidal, Phys. Rev. D 112, 103544 (2025).
  32. I. Antoniadis, J. Ellis, W. Ke, D. V. Nanopoulos, and K. A. Olive, J. Cosmol. Astropart. Phys. 08 (2025) 090.
  33. A. Salvio, Phys. Rev. D 112, L061301 (2025).
  34. C. Dioguardi and A. Karam, Phys. Rev. D 111, 123521 (2025).
  35. Q. Gao, Y. Gong, Z. Yi, and F. Zhang, Phys. Dark Universe 50, 102106 (2025).
  36. M. He, M. Hong, and K. Mukaida, J. Cosmol. Astropart. Phys. 09 (2025) 080.
  37. M. Drees and Y. Xu, Phys. Lett. B 867, 139612 (2025).
  38. S. Maity, Phys. Lett. B 870, 139913 (2025).
  39. W. Yin, J. Cosmol. Astropart. Phys. 09 (2025) 062.
  40. C. T. Byrnes, M. Cortês., and A. R. Liddle, arXiv:2505.09682.
  41. A. Biswas, S. Ganguly, D. Nanda, and S. K. Sahoo, arXiv:2505.13624.
  42. M. R. Haque and D. maity, Phys. Lett. B 873, 140187 (2026).
  43. D. Frolovsky and S. V. Ketov, Mod. Phys. Lett. A 40, 2550182 (2025).
  44. H. Heidarian, M. Solbi, S. Heydari, and K. Karami, Phys. Lett. B 869, 139833 (2025).
  45. S. Choudhury, B. Gulnur, S. K. Singh, and K. Yerzanov, arXiv:2506.15407.
  46. C. Dioguardi, A. J. Iovino, and A. Racioppi, Phys. Lett. B 868, 139664 (2025).
  47. M. R. Haque, S. Pal, and D. Paul, arXiv:2505.01517.
  48. D. S. Zharov, O. O. Sobol, and S. I. Vilchinskii, Phys. Rev. D 112, 023544 (2025).
  49. L. Liu, Z. Yi, and Y. Gong, arXiv:2505.02407.
  50. R. Mondal, S. Mondal, and A. Chakraborty, arXiv:2505.13387.
  51. G. German and J. C. Hidalgo, Mod. Phys. Lett. A 40, 2550182 (2025).
  52. J. Kim, X. Wang, Y. l. Zhang, and Z. Ren, J. Cosmol. Astropart. Phys. 09 (2025) 011.
  53. I. D. Gialamas, T. Katsoulas, and K. Tamvakis, J. Cosmol. Astropart. Phys. 09 (2025) 060.
  54. M. R. Haque, S. Pal, and D. Paul, Phys. Lett. B 869, 139852 (2025).
  55. Yogesh, A. Mohammadi, Q. Wu, and T. Zhu, J. Cosmol. Astropart. Phys. 10 (2025) 010.
  56. A. Addazi, Y. Aldabergenov, and S. V. Ketov, Phys. Lett. B 869, 139883 (2025).
  57. W. Ahmed and M. U. Rehman, Phys. Rev. D 112, 063519 (2025).
  58. T. Modak, Phys. Rev. D 112, 115006 (2025).
  59. D. Y. Cheong, M. G. Park, and S. C. Park, Phys. Lett. B 873, 140217 (2026).
  60. Z. Yi, X. Wang, Q. Gao, and Y. Gong, Phys. Lett. B 871, 140002 (2025).
  61. R. Kallosh and A. Linde, Gen. Relativ. Gravit. 57, 135 (2025).
  62. Z. Z. Peng, Z. C. Chen, and L. Liu, arXiv:2505.12816.
  63. C. Pallis, Phys. Lett. B 868, 139739 (2025).
  64. A. Chakraborty, D. Maity, and R. Mondal, Phys. Rev. D 112, 123547 (2025).
  65. S. D. Odintsov and V. K. Oikonomou, Phys. Lett. B 868, 139779 (2025).
  66. W. J. Wolf, J. Cosmol. Astropart. Phys. 02 (2026) 088.
  67. J. Han, H. M. Lee, and J. H. Song, arXiv:2506.21189.
  68. C. Pallis, J. Cosmol. Astropart. Phys. 09 (2025) 061.
  69. N. Okada and O. Seto, Phys. Rev. D 112, 083549 (2025).
  70. M. Hazumi, P. A. R. Ade, Y. Akiba, D. Alonso, K. Arnold, J. Aumont, C. Baccigalupi, D. Barron, S. Basak, and S. Beckman et al., J. Low Temp. Phys. 194, 443 (2019).
  71. J. L. Evans, M. A. G. García, and K. A. Olive, J. Cosmol. Astropart. Phys. 03 (2014) 022.
  72. J. Ellis, D. V. Nanopoulos, K. A. Olive., and S. Verner, J. Cosmol. Astropart. Phys. 08 (2020) 037.
  73. M. Kawasaki, K. Kohri, and N. Sugiyama, Phys. Rev. D 62, 023506 (2000); P. F. de Salas, M. Lattanzi, G. Mangano, G. Miele, S. Pastor, and O. Pisanti, 92, 123534 (2015); S. Hannestad, 70, 043506 (2004); T. Hasegawa, N. Hiroshima, K. Kohri, R. S. L. Hansen, T. Tram, and S. Hannestad, J. Cosmol. Astropart. Phys. 12 (2019) 012.
  74. J. Ellis, M. A. G. Garcia, N. Nagata, D. V. Nanopoulos, and K. A. Olive, J. Cosmol. Astropart. Phys. 12 (2025) 038.
  75. G. D. Diamandis, A. B. Lahanas, and K. Tamvakis, Phys. Rev. D 92, 105023 (2015).
  76. G. A. Diamandis, B. C. Georgalas, K. Kaskavelis, A. B. Lahanas, and G. Pavlopoulos, Phys. Rev. D 96, 044033 (2017).
  77. J. Ellis, D. V. Nanopoulos, and K. A. Olive, Phys. Rev. D 97, 043530 (2018).
  78. J. Ellis, M. A. G. García, N. Nagata, D. V. Nanopoulos, K. A. Olive, and S. Verner, Int. J. Mod. Phys. D 29, 2030011 (2020).
  79. Y. Ema, M. A. G. Garcia, W. Ke, K. A. Olive, and S. Verner, Universe 10, 239 (2024).
  80. J. Ellis, D. V. Nanopoulos, K. A. Olive, and S. Verner, J. High Energy Phys. 03 (2019) 099.
  81. S. Cecotti, Phys. Lett. B 190, 86 (1987).
  82. E. Witten, Phys. Lett. 155B, 151 (1985); see also S. B. Giddings, S. Kachru, and J. Polchinski, Phys. Rev. D 66, 106006 (2002); V. Balasubramanian, P. Berglund, J. P. Conlon, and F. Quevedo, J. High Energy Phys. 03 (2005) 007.
  83. D. Roest and M. Scalisi, Phys. Rev. D 92, 043525 (2015).
  84. J. Ellis, M. A. G. Garcia, D. V. Nanopoulos, K. A. Olive, and S. Verner, Phys. Rev. D 105, 043504 (2022).
  85. L. Iacconi, M. Fasiello, J. Väliviita, and D. Wands, J. Cosmol. Astropart. Phys. 10 (2023) 015.
  86. J. A. Zebrowski et al. (SPT-3G Collaboration), Phys. Rev. D 112, 123520 (2025).
  87. F. Ge et al. (SPT-3G Collaboration), Phys. Rev. D 111, 083534 (2025).
  88. P. A. R. Ade et al. (Planck Collaboration), Astron. Astrophys. 594, A13 (2016).
  89. P. A. R. Ade et al. (BICEP2 Collaboration and Keck Array Collaboration), Phys. Rev. Lett. 121, 221301 (2018).
  90. Y. Akrami et al. (Planck Collaboration), Astron. Astrophys. 643, A42 (2020).
  91. B. Popovic, W. D. Kenworthy, M. Ginolin, A. Goobar, P. Shah, B. M. Boyd, A. Do, D. Brout, D. Scolnic, M. Vincenzi et al., arXiv:2506.05471.
  92. E. G. M. Ferreira, E. McDonough, L. Balkenhol, R. Kallosh, L. Knox, and A. Linde, Phys. Rev. D 113, 043524 (2026).
  93. J. Ellis, M. A. G. García, D. V. Nanopoulos, and K. A. Olive, J. Cosmol. Astropart. Phys. 07 (2015) 050.
  94. A. R. Liddle and S. M. Leach, Phys. Rev. D 68, 103503 (2003).
  95. J. Martin and C. Ringeval, Phys. Rev. D 82, 023511 (2010).
  96. D. J. Fixsen, Astrophys. J. 707, 916 (2009).
  97. M. A. G. Garcia, K. Kaneta, Y. Mambrini, and K. A. Olive, J. Cosmol. Astropart. Phys. 04 (2021) 012.
  98. M. Fukugita and T. Yanagida, Phys. Lett. B 174, 45 (1986).
  99. S. Davidson and S. Sarkar, J. High Energy Phys. 11 (2000) 012.
  100. K. Harigaya, K. Mukaida, and M. Yamada, J. High Energy Phys. 07 (2019) 059; K. Harigaya, M. Kawasaki, K. Mukaida, and M. Yamada, Phys. Rev. D 89, 083532 (2014); K. Harigaya and K. Mukaida, J. High Energy Phys. 05 (2014) 006.
  101. K. Mukaida and M. Yamada, J. Cosmol. Astropart. Phys. 02 (2016) 003.
  102. S. Passaglia, W. Hu, A. J. Long, and D. Zegeye, Phys. Rev. D 104, 083540 (2021).
  103. M. A. G. Garcia and M. A. Amin, Phys. Rev. D 98, 103504 (2018).
  104. M. Drees and B. Najjari, J. Cosmol. Astropart. Phys. 10 (2021) 009.
  105. M. Drees and B. Najjari, J. Cosmol. Astropart. Phys. 08 (2023) 037.
  106. K. Mukaida and M. Yamada, J. High Energy Phys. 10 (2022) 116.
  107. G. F. Giudice, E. W. Kolb, and A. Riotto, Phys. Rev. D 64, 023508 (2001).
  108. J. Ellis, M. A. G. García, D. V. Nanopoulos, K. A. Olive, and M. Peloso, J. Cosmol. Astropart. Phys. 03 (2016) 008.
  109. M. A. G. García, K. Kaneta, Y. Mambrini., and K. A. Olive, Phys. Rev. D 101, 123507 (2020).
  110. D. Gorbunov and A. Tokareva, J. Cosmol. Astropart. Phys. 02 (2013) 021.
  111. M. Endo, K. Kadota, K. A. Olive, F. Takahashi., and T. T. Yanagida, J. Cosmol. Astropart. Phys. 02 (2007) 018.
  112. J. Ellis, M. A. G. Garcia, D. V. Nanopoulos, and K. A. Olive, J. Cosmol. Astropart. Phys. 10 (2015) 003.
  113. M. Endo, F. Takahashix, and T. T. Yanagida, Phys. Rev. D 76, 083509 (2007).
  114. T. Moroi, M. Yamaguchi., and T. Yanagida, Phys. Lett. B 342, 105 (1995).
  115. J. R. Ellis, J. E. Kim, and D. V. Nanopoulos, Phys. Lett. 145B, 181 (1984).
  116. J. R. Ellis, D. V. Nanopoulos, K. A. Olive, and S. J. Rey, Astropart. Phys. 4, 371 (1996).
  117. M. Bolz, A. Brandenburg, and W. Buchmuller, Nucl. Phys. B606, 518 (2001); B790, 336(E) (2008).
  118. J. Pradler and F. D. Steffen, Phys. Rev. D 75, 023509 (2007).
  119. V. S. Rychkov and A. Strumia, Phys. Rev. D 75, 075011 (2007).
  120. H. Eberl, I. D. Gialamas, and V. C. Spanos, Phys. Rev. D 103, 075025 (2021).
  121. H. Eberl, I. D. Gialamas, and V. C. Spanos, J. Cosmol. Astropart. Phys. 01 (2025) 079.
  122. M. Drees and Y. Xu, J. Cosmol. Astropart. Phys. 09 (2021) 012.
  123. J. R. Ellis, D. V. Nanopoulos, K. A. Olive, and K. Tamvakis, Phys. Lett. 118B, 335 (1982); Nucl. Phys.B221, 524 (1983); Phys. Lett.120B, 331 (1983).
  124. J. Ellis, T. Gherghetta, K. Kaneta, W. Ke, and K. A. Olive, Phys. Rev. D 112, 123530 (2025).
  125. K. D. Lozanov and M. A. Amin, Phys. Rev. Lett. 119, 061301 (2017).
  126. K. D. Lozanov and M. A. Amin, Phys. Rev. D 97, 023533 (2018).
  127. M. A. G. Garcia and M. Pierre, J. Cosmol. Astropart. Phys. 11 (2023) 004.
  128. M. A. G. Garcia, M. Gross, Y. Mambrini, K. A. Olive, M. Pierre, and J. H. Yoon, J. Cosmol. Astropart. Phys. 12 (2023) 028.
  129. M. A. G. Garcia and M. Pierre, J. Cosmol. Astropart. Phys. 09 (2024) 054.
  130. A. D. Linde and A. Westphal, J. Cosmol. Astropart. Phys. 03 (2008) 005
  131. J. R. Ellis, C. Kounnas, and D. V. Nanopoulos, Nucl. Phys. B241, 406 (1984).
  132. M. Drewes, J. U. Kang, and U. R. Mun, J. High Energy Phys. 11 (2017) 072.
  133. G. German, Int. J. Mod. Phys. D 31, 2250081 (2022).
  134. Z. Lalak, D. Langlois, S. Pokorski., and K. Turzynski, J. Cosmol. Astropart. Phys. 07 (2007) 014.
  135. J. Ellis, M. A. G. García, D. V. Nanopoulos, and K. A. Olive, J. Cosmol. Astropart. Phys. 01 (2015) 010.

Outline

Information

Phys. Rev. D 113, 063571– Published 30 March, 2026
  • Received 1 November 2025
  • Accepted 3 February 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation