- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

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
  • Access by Xinjiang University

Slow rolling down the curvature: A reassessment of the Planck constraints on ϕ2 inflation in a closed universe

Enrico Specogna1,*, Tatevik Vardanyan2,†, William Giarè1,‡, and Eleonora Di Valentino1,§

  • 1School of Mathematical and Physical Sciences, University of Sheffield, Hounsfield Road, Sheffield S3 7RH, United Kingdom
  • 2Faculty of Mathematics and Natural Sciences, Institute for Theoretical Physics, University of Cologne, Cologne, Germany

  • *Contact author: especogna1@sheffield.ac.uk
  • †Contact author: tatevik@thp.uni-koeln.de
  • ‡Contact author: w.giare@sheffield.ac.uk
  • §Contact author: e.divalentino@sheffield.ac.uk
PDF

Phys. Rev. D 113, 123526 – Published 12 June, 2026

DOI: https://doi.org/10.1103/dy2d-z3cc

Abstract

We revisit the cosmic microwave background (CMB) constraints on the spatial curvature of the Universe, assessing how they change when the curvature parameter and the primordial inflationary scalar spectrum are treated consistently within theoretically motivated frameworks. Instead of relying on the phenomenological parametrization commonly used to capture curvature effects at the largest scales, we present a case study based on closed quadratic inflation, where the primordial spectrum is derived in full generality and in a gauge-invariant manner. Within this framework, we analyze both the plik public release 3 (PR3) and camspec PR4 Planck CMB likelihoods and find that the constraints on ΩK shift toward spatial flatness. In plik the preference for ΩK<0 decreases from ≳3.5σ to ∼2.5σ, while in camspec it reduces further to ∼2σ. At large angular scales (ℓ<10), our model explains the low-ℓ power suppression anomaly, notably improving the fit to the quadrupole. However, the reduced preference for highly negative values of ΩK only partially accounts for the lensing anomaly at high multipoles, worsening the fit to the plik spectrum at small scales. By contrast, in the camspec PR4 spectrum, where the lensing anomaly is less pronounced, the model yields an overall improvement. Our analysis highlights a key conceptual point: closed-inflation models tie the curvature parameter to the inflationary dynamics and the primordial spectrum, enforcing consistency conditions that do not necessarily allow for the large deviations from flatness seen in phenomenological parametrizations. In the case of quadratic inflation, these restrictions reduce the apparent evidence for negative curvature reported by earlier analyses, while allowing for a mildly closed geometry.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (152)

  1. E. Di Valentino et al. (CosmoVerse Network), Phys. Dark Universe 49, 101965 (2025).
  2. L. Verde, T. Treu, and A. G. Riess, Nat. Astron. 3, 891 (2019).
  3. E. Di Valentino et al., Astropart. Phys. 131, 102605 (2021).
  4. 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).
  5. L. Perivolaropoulos and F. Skara, New Astron. Rev. 95, 101659 (2022).
  6. N. Schöneberg, G. Franco Abellán, A. Pérez Sánchez, S. J. Witte, V. Poulin, and J. Lesgourgues, Phys. Rep. 984, 1 (2022).
  7. P. Shah, P. Lemos, and O. Lahav, Astron. Astrophys. Rev. 29, 9 (2021).
  8. E. Abdalla et al., J. High Energy Astrophys. 34, 49 (2022).
  9. E. Di Valentino, Universe 8, 399 (2022).
  10. M. Kamionkowski and A. G. Riess, Annu. Rev. Nucl. Part. Sci. 73, 153 (2023).
  11. W. Giarè, arXiv:2305.16919.
  12. J.-P. Hu and F.-Y. Wang, Universe 9, 94 (2023).
  13. L. Verde, N. Schöneberg, and H. Gil-Marín, Annu. Rev. Astron. Astrophys. 62, 287 (2024).
  14. E. Di Valentino and D. Brout, eds., The Hubble Constant Tension, Springer Series in Astrophysics and Cosmology (Springer, New York, 2024).
  15. L. Perivolaropoulos, Phys. Rev. D 110, 123518 (2024).
  16. A. G. Riess et al., Astrophys. J. Lett. 934, L7 (2022).
  17. L. Breuval, A. G. Riess, S. Casertano, W. Yuan, L. M. Macri, M. Romaniello, Y. S. Murakami, D. Scolnic, G. S. Anand, and I. Soszyński, Astrophys. J. 973, 30 (2024).
  18. E. Camphuis et al. (SPT-3G Collaboration), Phys. Rev. D 113, 083504 (2026).
  19. G. Efstathiou, arXiv:2007.10716.
  20. E. Mortsell, A. Goobar, J. Johansson, and S. Dhawan, Astrophys. J. 933, 212 (2022).
  21. E. Mortsell, A. Goobar, J. Johansson, and S. Dhawan, Astrophys. J. 935, 58 (2022).
  22. A. Sharon, D. Kushnir, W. Yuan, L. Macri, and A. Riess, Mon. Not. R. Astron. Soc. 528, 6861 (2024).
  23. Y. S. Murakami, A. G. Riess, B. E. Stahl, W. D. Kenworthy, D.-M. A. Pluck, A. Macoretta, D. Brout, D. O. Jones, D. M. Scolnic, and A. V. Filippenko, J. Cosmol. Astropart. Phys. 11 (2023) 046.
  24. A. G. Riess, G. S. Anand, W. Yuan, S. Casertano, A. Dolphin, L. M. Macri, L. Breuval, D. Scolnic, M. Perrin, and R. I. Anderson, Astrophys. J. Lett. 956, L18 (2023).
  25. A. Bhardwaj et al., Astrophys. J. Lett. 955, L13 (2023).
  26. D. Brout and A. Riess, arXiv:2311.08253.
  27. A. M. Dwomoh, E. R. Peterson, D. Scolnic, C. Ashall, J. M. DerKacy, A. Do, J. Johansson, D. O. Jones, A. G. Riess, and B. J. Shappee, Astrophys. J. 965, 90 (2024).
  28. S. A. Uddin et al., Astrophys. J. 970, 72 (2024).
  29. A. G. Riess, G. S. Anand, W. Yuan, S. Casertano, A. Dolphin, L. M. Macri, L. Breuval, D. Scolnic, M. Perrin, and I. R. Anderson, Astrophys. J. Lett. 962, L17 (2024).
  30. W. L. Freedman, B. F. Madore, T. J. Hoyt, I. S. Jang, A. J. Lee, and K. A. Owens, Astrophys. J. 985, 203 (2025).
  31. A. G. Riess et al., Astrophys. J. 977, 120 (2024).
  32. A. H. Guth, Phys. Rev. D 23, 347 (1981).
  33. A. D. Linde, Phys. Lett. 108B, 389 (1982).
  34. M. Abdul Karim et al. (DESI Collaboration), Phys. Rev. D 112, 083515 (2025).
  35. S. Alam et al. (eBOSS Collaboration), Phys. Rev. D 103, 083533 (2021).
  36. T. M. C. Abbott et al. (DES Collaboration), Phys. Rev. D 107, 083504 (2023).
  37. T. Tröster et al. (KiDS Collaboration), Astron. Astrophys. 649, A88 (2021).
  38. E. Calabrese et al. (ACT Collaboration), J. Cosmol. Astropart. Phys. 11 (2025) 063.
  39. N. Aghanim et al. (Planck Collaboration), Astron. Astrophys. 641, A6 (2020); 652, C4(E) (2021).
  40. E. Di Valentino, A. Melchiorri, and J. Silk, Nat. Astron. 4, 196 (2019).
  41. W. Handley, Phys. Rev. D 103, L041301 (2021).
  42. E. Rosenberg, S. Gratton, and G. Efstathiou, Mon. Not. R. Astron. Soc. 517, 4620 (2022).
  43. M. Tristram et al., Astron. Astrophys. 682, A37 (2024).
  44. C.-G. Park and B. Ratra, Astrophys. J. 882, 158 (2019).
  45. J. Ryan, Y. Chen, and B. Ratra, Mon. Not. R. Astron. Soc. 488, 3844 (2019).
  46. G. Efstathiou and S. Gratton, Mon. Not. R. Astron. Soc. 496, L91 (2020).
  47. E. Di Valentino, A. Melchiorri, and J. Silk, Astrophys. J. Lett. 908, L9 (2021).
  48. D. Benisty and D. Staicova, Astron. Astrophys. 647, A38 (2021).
  49. S. Vagnozzi, E. Di Valentino, S. Gariazzo, A. Melchiorri, O. Mena, and J. Silk, Phys. Dark Universe 33, 100851 (2021).
  50. S. Vagnozzi, A. Loeb, and M. Moresco, Astrophys. J. 908, 84 (2021).
  51. E. Di Valentino, A. Melchiorri, O. Mena, S. Pan, and W. Yang, Mon. Not. R. Astron. Soc. 502, L23 (2021).
  52. W. Yang, S. Pan, E. Di Valentino, O. Mena, and A. Melchiorri, J. Cosmol. Astropart. Phys. 10 (2021) 008.
  53. G. Bargiacchi, M. Benetti, S. Capozziello, E. Lusso, G. Risaliti, and M. Signorini, Mon. Not. R. Astron. Soc. 515, 1795 (2022).
  54. S. Cao, J. Ryan, and B. Ratra, Mon. Not. R. Astron. Soc. 504, 300 (2021).
  55. S. Dhawan, J. Alsing, and S. Vagnozzi, Mon. Not. R. Astron. Soc. 506, L1 (2021).
  56. B. R. Dinda, Phys. Rev. D 105, 063524 (2022).
  57. L. Zuckerman, E. Zuckerman, and L. A. Anchordoqui, J. High Energy Astrophys. 33, 10 (2022).
  58. J. E. Gonzalez, M. Benetti, R. von Marttens, and J. Alcaniz, J. Cosmol. Astropart. Phys. 11 (2021) 060.
  59. O. Akarsu, E. Di Valentino, S. Kumar, M. Ozyigit, and S. Sharma, Phys. Dark Universe 39, 101162 (2023).
  60. E. Di Valentino, W. Giarè, A. Melchiorri, and J. Silk, Phys. Rev. D 106, 103506 (2022).
  61. E. Di Valentino, W. Giarè, A. Melchiorri, and J. Silk, Mon. Not. R. Astron. Soc. 520, 210 (2023).
  62. S. Cao and B. Ratra, Mon. Not. R. Astron. Soc. 513, 5686 (2022).
  63. A. Glanville, C. Howlett, and T. M. Davis, Mon. Not. R. Astron. Soc. 517, 3087 (2022).
  64. J. Bel, J. Larena, R. Maartens, C. Marinoni, and L. Perenon, J. Cosmol. Astropart. Phys. 09 (2022) 076.
  65. W. Yang, W. Giarè, S. Pan, E. Di Valentino, A. Melchiorri, and J. Silk, Phys. Rev. D 107, 063509 (2023).
  66. J. Stevens, H. Khoraminezhad, and S. Saito, J. Cosmol. Astropart. Phys. 07 (2023) 046.
  67. Q. Vigneron and V. Poulin, Phys. Rev. D 108, 103518 (2023).
  68. R. Jimenez, A. R. Khalifeh, D. F. Litim, S. Matarrese, and B. D. Wandelt, J. Cosmol. Astropart. Phys. 09 (2023) 007.
  69. T. Liu, S. Cao, X. Li, H. Zheng, Y. Liu, W. Guo, and C. Zheng, Astron. Astrophys. 668, A51 (2022).
  70. N. Banerjee, P. Mukherjee, and D. Pavón, Mon. Not. R. Astron. Soc. 521, 5473 (2023).
  71. A. Favale, A. Gómez-Valent, and M. Migliaccio, Mon. Not. R. Astron. Soc. 523, 3406 (2023).
  72. K. Zhang, T. Zhou, B. Xu, Q. Huang, and Y. Yuan, Astrophys. J. 957, 5 (2023).
  73. B. Blachier, P. Auclair, C. Ringeval, and V. Vennin, Phys. Rev. D 108, 123510 (2023).
  74. J.-Z. Qi, P. Meng, J.-F. Zhang, and X. Zhang, Phys. Rev. D 108, 063522 (2023).
  75. W. Giarè, E. Di Valentino, and A. Melchiorri, Phys. Rev. D 109, 103519 (2024).
  76. H. Foidl and T. Rindler-Daller, arXiv:2412.04126.
  77. W.-N. Deng and W. Handley, Phys. Rev. D 110, 103528 (2024).
  78. E. Jensko, Classical Quantum Gravity 42, 055011 (2025).
  79. S. Gariazzo, W. Giarè, O. Mena, and E. Di Valentino, Phys. Rev. D 111, 023540 (2025).
  80. S. Sanz-Wuhl, H. Gil-Marín, A. J. Cuesta, and L. Verde, J. Cosmol. Astropart. Phys. 05 (2024) 116.
  81. L. Amendola, M. Marinucci, and M. Quartin, Phys. Rev. Lett. 134, 101004 (2025).
  82. S. Bhattacharya, G. Borghetto, A. Malhotra, S. Parameswaran, G. Tasinato, and I. Zavala, J. Cosmol. Astropart. Phys. 09 (2024) 073.
  83. M. Shimon and Y. Rephaeli, Universe 11, 143 (2025).
  84. V. E. Kuzmichev and V. V. Kuzmichev, Phys. Dark Universe 52, 102312 (2026).
  85. W.-N. Deng and W. Handley, Phys. Rev. D 113, 023546 (2026).
  86. Y.-Y. Wang, L. Lei, S.-P. Tang, and Y.-Z. Fan, J. Cosmol. Astropart. Phys. 01 (2026) 009.
  87. T. Sekiguchi and T. Takahashi, Phys. Rev. D 103, 083507 (2021).
  88. N. Schöneberg and L. Vacher, J. Cosmol. Astropart. Phys. 03 (2025) 004.
  89. Y. Toda, W. Giarè, E. Özülker, E. Di Valentino, and S. Vagnozzi, Phys. Dark Universe 46, 101676 (2024).
  90. S. D. Odintsov, S. D’Onofrio, and T. Paul, Phys. Dark Universe 48, 101920 (2025).
  91. Y. Hu and S. Dhawan, Mon. Not. R. Astron. Soc. 535, 3211 (2024).
  92. D. H. Lyth and E. D. Stewart, Phys. Lett. B 252, 336 (1990).
  93. B. Ratra and P. J. E. Peebles, Phys. Rev. D 52, 1837 (1995).
  94. G. Efstathiou, Mon. Not. R. Astron. Soc. 343, L95 (2003).
  95. J. Asorey and J. de Cruz Pérez, arXiv:2506.13537.
  96. B. Ratra, Phys. Rev. D 96, 103534 (2017).
  97. W. Handley, Phys. Rev. D 100, 123517 (2019).
  98. D.-H. Zhang and C.-Y. Sun, Chin. Phys. Lett. 21, 1865 (2004).
  99. A. Thavanesan, D. Werth, and W. Handley, Phys. Rev. D 103, 023519 (2021).
  100. Q. Vigneron and J. Larena, Classical Quantum Gravity 41, 22LT01 (2024).
  101. L. F. Abbott and R. K. Schaefer, Astrophys. J. 308, 546 (1986).
  102. U. H. Gerlach and U. K. Sengupta, Phys. Rev. D 18, 1773 (1978).
  103. C. Wolf, Phys. Scr. 38, 129 (1988).
  104. C. Wolf, Aust. J. Phys. 43, 117 (1990).
  105. J. Ponce de Leon, J. Math. Phys. (N.Y.) 31, 371 (1990).
  106. H. J. Schmidt, Gen. Relativ. Gravit. 25, 87 (1993).
  107. A. D. Linde, Phys. Lett. B 351, 99 (1995).
  108. S. A. Pavluchenko, N. Y. Savchenko, and A. V. Toporensky, Int. J. Mod. Phys. D 11, 805 (2002).
  109. G. F. R. Ellis, W. R. Stoeger, S. J., P. McEwan, and P. Dunsby, Gen. Relativ. Gravit. 34, 1445 (2002).
  110. A. D. Linde, J. Cosmol. Astropart. Phys. 05 (2003) 002.
  111. J.-P. Uzan, U. Kirchner, and G. F. R. Ellis, Mon. Not. R. Astron. Soc. 344, L65 (2003).
  112. S. A. Pavluchenko, Phys. Rev. D 67, 103518 (2003).
  113. A. Lasenby and C. Doran, Phys. Rev. D 71, 063502 (2005).
  114. D. R. Green, Phys. Rev. D 76, 103504 (2007).
  115. M. Cicoli and A. Mazumdar, J. Cosmol. Astropart. Phys. 09 (2010) 025.
  116. M. Kleban and M. Schillo, J. Cosmol. Astropart. Phys. 06 (2012) 029.
  117. J. Ooba, B. Ratra, and N. Sugiyama, Astrophys. J. 864, 80 (2018).
  118. L. Gordon, B.-F. Li, and P. Singh, Phys. Rev. D 103, 046016 (2021).
  119. T. Murata and T. Kobayashi, Phys. Rev. D 104, 083514 (2021).
  120. M. Motaharfar and P. Singh, Phys. Rev. D 104, 106006 (2021).
  121. B. Bonga, B. Gupt, and N. Yokomizo, J. Cosmol. Astropart. Phys. 05 (2017) 021.
  122. B. Bonga, B. Gupt, and N. Yokomizo, J. Cosmol. Astropart. Phys. 10 (2016) 031.
  123. R. D’Agostino, M. Califano, N. Menadeo, and D. Vernieri, Phys. Rev. D 108, 043538 (2023).
  124. D. Müller and A. Toporensky, Eur. Phys. J. Plus 139, 358 (2024).
  125. P. Peter and J.-P. Uzan, Primordial Cosmology (Oxford University Press, Oxford, 2009).
  126. V. F. Mukhanov, H. A. Feldman, and R. H. Brandenberger, Phys. Rep. 215, 203 (1992).
  127. E. Calabrese, A. Slosar, A. Melchiorri, G. F. Smoot, and O. Zahn, Phys. Rev. D 77, 123531 (2008).
  128. E. Di Valentino, A. Melchiorri, and J. Silk, Phys. Rev. D 93, 023513 (2016).
  129. E. Specogna, E. Di Valentino, J. Levi Said, and N.-M. Nguyen, Phys. Rev. D 109, 043528 (2024).
  130. E. Specogna, W. Giarè, and E. Di Valentino, Phys. Rev. D 111, 103510 (2025).
  131. W. Giarè, O. Mena, E. Specogna, and E. Di Valentino, Phys. Rev. D 112, 103520 (2025).
  132. F. Renzi, E. Di Valentino, and A. Melchiorri, Phys. Rev. D 97, 123534 (2018).
  133. G. Domènech, X. Chen, M. Kamionkowski, and A. Loeb, J. Cosmol. Astropart. Phys. 10 (2020) 005.
  134. G. E. Addison, C. L. Bennett, M. Halpern, G. Hinshaw, and J. L. Weiland, Astrophys. J. 974, 187 (2024).
  135. J. A. Kable, G. E. Addison, and C. L. Bennett, Astrophys. J. 905, 164 (2020).
  136. G. E. Addison, Y. Huang, D. J. Watts, C. L. Bennett, M. Halpern, G. Hinshaw, and J. L. Weiland, Astrophys. J. 818, 132 (2016).
  137. N. Aghanim et al. (Planck Collaboration), Astron. Astrophys. 641, A5 (2020).
  138. C. Kiefer and T. Vardanyan, Gen. Relativ. Gravit. 54, 30 (2022).
  139. T. Vardanyan and C. Kiefer, J. Phys. Conf. Ser. 2533, 012036 (2023).
  140. J. M. Bardeen, P. J. Steinhardt, and M. S. Turner, Phys. Rev. D 28, 679 (1983).
  141. D. Brizuela, C. Kiefer, and M. Krämer, Phys. Rev. D 94, 123527 (2016).
  142. A. Lewis, A. Challinor, and A. Lasenby, Astrophys. J. 538, 473 (2000).
  143. C. Howlett, A. Lewis, A. Hall, and A. Challinor, J. Cosmol. Astropart. Phys. 04 (2012) 027.
  144. J. Torrado and A. Lewis, J. Cosmol. Astropart. Phys. 05 (2021) 057.
  145. A. Lewis, J. Cosmol. Astropart. Phys. 08 (2025) 025.
  146. Y. Akrami et al. (Planck Collaboration), Astron. Astrophys. 641, A4 (2020).
  147. N. Aghanim et al. (Planck Collaboration), Astron. Astrophys. 641, A8 (2020).
  148. Y. Akrami et al. (Planck Collaboration), Astron. Astrophys. 643, A42 (2020).
  149. Y. Akrami et al. (Planck Collaboration), Astron. Astrophys. 641, A10 (2020).
  150. P. A. R. Ade et al. (BICEP Collaboration and Keck Collaboration), Phys. Rev. Lett. 127, 151301 (2021).
  151. M. Forconi, W. Giarè, E. Di Valentino, and A. Melchiorri, Phys. Rev. D 104, 103528 (2021).
  152. F. O. Franco and T. S. Pereira, J. Cosmol. Astropart. Phys. 11 (2017) 022.

Outline

Information

Phys. Rev. D 113, 123526– Published 12 June, 2026
  • Received 7 October 2025
  • Accepted 12 May 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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation