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  • Open Access

Gravitational production of massive spin-2 particles during reheating

Sarunas Verner*

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

  • *Contact author: verner@uchicago.edu
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Phys. Rev. D 113, 123503 – Published 1 June, 2026

DOI: https://doi.org/10.1103/3st2-jhx5

Abstract

We study the minimal gravitational portal for a massive spin-2 dark matter candidate Xμν produced during perturbative reheating. The dark sector couples to the visible sector only via gravity, and we analyze two unavoidable channels: (i) inflaton condensate annihilation, ϕ+ϕ→X+X, and (ii) thermal scatterings, SM+SM→X+X, both mediated by graviton exchange. Working in the Fierz-Pauli framework for a free massive spin-2 field of mass m2, we derive the graviton-mediated amplitudes and perform a full helicity decomposition of the final state. The relic abundance is obtained analytically in terms of m2 and the reheating temperature TRH. In the light mass regime m2≪mϕ (with mϕ the inflaton mass during oscillations), production is overwhelmingly dominated by the longitudinal (helicity-0) mode: the 2→2 cross section is parametrically enhanced, scaling as ∼(mϕ/m2)4, and yields efficient dark matter production despite purely gravitational couplings. Compared to lower-spin cases (spin-0, 1/2, 1, and 3/2), massive spin-2 production is substantially more efficient for the same reheating history. Over most of the parameter space the inflaton condensate channel dominates the yield, while the thermal contribution is negligible. Avoiding overproduction typically requires either a relatively low TRH or a spin-2 mass near threshold, m2≲mϕ. This places the spin-2 portal on similar footing to other higher spins in reheating scenarios, while emphasizing the central role of the helicity-0 mode and the reheating history in setting the dark matter density.

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References (90)

  1. F. Zwicky, Helv. Phys. Acta 6, 110 (1933).
  2. Y. Mambrini, Particles in the Dark Universe. A Student’s Guide to Particle Physics and Cosmology (Springer, New York, 2021).
  3. J. Aalbers et al. (LZ Collaboration), Phys. Rev. Lett. 131, 041002 (2023).
  4. E. Aprile et al. (XENON Collaboration), Phys. Rev. Lett. 135, 221003 (2025).
  5. Z. Bo et al. (PandaX Collaboration), Phys. Rev. Lett. 134, 011805 (2025).
  6. D. S. Akerib et al. (LUX Collaboration), Phys. Rev. Lett. 118, 021303 (2017).
  7. G. Arcadi, D. Cabo-Almeida, M. Dutra, P. Ghosh, M. Lindner, Y. Mambrini, J. P. Neto, M. Pierre, S. Profumo, and F. S. Queiroz, Eur. Phys. J. C 85, 152 (2025).
  8. A. D. Dolgov and A. D. Linde, Phys. Lett. 116B, 329 (1982).
  9. L. F. Abbott, E. Farhi, and M. B. Wise, Phys. Lett. 117B, 29 (1982).
  10. D. V. Nanopoulos, K. A. Olive, and M. Srednicki, Phys. Lett. 127B, 30 (1983).
  11. G. F. Giudice, E. W. Kolb, and A. Riotto, Phys. Rev. D 64, 023508 (2001).
  12. D. J. H. Chung, E. W. Kolb, and A. Riotto, Phys. Rev. D 60, 063504 (1999).
  13. M. A. G. Garcia, Y. Mambrini, K. A. Olive, and M. Peloso, Phys. Rev. D 96, 103510 (2017).
  14. M. A. G. Garcia, K. Kaneta, Y. Mambrini, and K. A. Olive, Phys. Rev. D 101, 123507 (2020).
  15. M. A. G. Garcia, K. Kaneta, Y. Mambrini, and K. A. Olive, J. Cosmol. Astropart. Phys. 04 (2021) 012.
  16. N. Bernal, J. Cosmol. Astropart. Phys. 10 (2020) 006.
  17. Y. Mambrini and K. A. Olive, Phys. Rev. D 103, 115009 (2021).
  18. B. Barman and N. Bernal, J. Cosmol. Astropart. Phys. 06 (2021) 011.
  19. S. Clery, Y. Mambrini, K. A. Olive, and S. Verner, Phys. Rev. D 105, 075005 (2022).
  20. S. Clery, Y. Mambrini, K. A. Olive, A. Shkerin, and S. Verner, Phys. Rev. D 105, 095042 (2022).
  21. R. T. Co, Y. Mambrini, and K. A. Olive, Phys. Rev. D 106, 075006 (2022).
  22. B. Barman, S. Cléry, R. T. Co, Y. Mambrini, and K. A. Olive, J. High Energy Phys. 12 (2022) 072.
  23. N. Bernal, J. Harz, M. A. Mojahed, and Y. Xu, Phys. Rev. D 111, 043517 (2025).
  24. S. E. Henrich, Y. Mambrini, and K. A. Olive, Phys. Rev. D 111, 083501 (2025).
  25. N. Bernal and C. S. Fong, J. Cosmol. Astropart. Phys. 06 (2021) 028.
  26. M. R. Haque and D. Maity, Phys. Rev. D 107, 043531 (2023).
  27. K. Kaneta, W. Ke, Y. Mambrini, K. A. Olive, and S. Verner, Phys. Rev. D 108, 115027 (2023).
  28. Y. Ema, R. Jinno, K. Mukaida, and K. Nakayama, J. Cosmol. Astropart. Phys. 05 (2015) 038.
  29. Y. Ema, R. Jinno, K. Mukaida, and K. Nakayama, Phys. Rev. D 94, 063517 (2016).
  30. Y. Ema, K. Nakayama, and Y. Tang, J. High Energy Phys. 09 (2018) 135.
  31. P. W. Graham, J. Mardon, and S. Rajendran, Phys. Rev. D 93, 103520 (2016).
  32. M. Garny, M. Sandora, and M. S. Sloth, Phys. Rev. Lett. 116, 101302 (2016).
  33. M. Garny, A. Palessandro, M. Sandora, and M. S. Sloth, J. Cosmol. Astropart. Phys. 02 (2018) 027.
  34. Y. Tang and Y.-L. Wu, Phys. Lett. B 774, 676 (2017).
  35. Y. Ema, K. Nakayama, and Y. Tang, J. High Energy Phys. 07 (2019) 060.
  36. M. Chianese, B. Fu, and S. F. King, J. Cosmol. Astropart. Phys. 06 (2020) 019.
  37. M. Chianese, B. Fu, and S. F. King, J. Cosmol. Astropart. Phys. 01 (2021) 034.
  38. A. Ahmed, B. Grzadkowski, and A. Socha, J. High Energy Phys. 08 (2020) 059.
  39. E. W. Kolb and A. J. Long, J. High Energy Phys. 03 (2021) 283.
  40. M. Redi, A. Tesi, and H. Tillim, J. High Energy Phys. 05 (2021) 010.
  41. S. Ling and A. J. Long, Phys. Rev. D 103, 103532 (2021).
  42. A. Ahmed, B. Grzadkowski, and A. Socha, Phys. Lett. B 831, 137201 (2022).
  43. M. R. Haque and D. Maity, Phys. Rev. D 106, 023506 (2022).
  44. S. Aoki, H. M. Lee, A. G. Menkara, and K. Yamashita, J. High Energy Phys. 05 (2022) 121.
  45. A. Ahmed, B. Grzadkowski, and A. Socha, J. High Energy Phys. 02 (2023) 196.
  46. M. R. Haque, D. Maity, and R. Mondal, J. High Energy Phys. 09 (2023) 012.
  47. K. Kaneta, S. M. Lee, and K.-y. Oda, J. Cosmol. Astropart. Phys. 09 (2022) 018.
  48. E. W. Kolb, S. Ling, A. J. Long, and R. A. Rosen, J. High Energy Phys. 05 (2023) 181.
  49. E. W. Kolb and A. J. Long, Rev. Mod. Phys. 96, 045005 (2024).
  50. M. A. G. Garcia, K. Kaneta, W. Ke, Y. Mambrini, K. A. Olive, and S. Verner, J. Cosmol. Astropart. Phys. 06 (2024) 014.
  51. K. Kaneta and K.-y. Oda, J. Cosmol. Astropart. Phys. 10 (2023) 048.
  52. M. A. G. Garcia, M. Pierre, and S. Verner, Phys. Rev. D 107, 043530 (2023).
  53. M. A. G. Garcia, M. Pierre, and S. Verner, Phys. Rev. D 107, 123508 (2023).
  54. M. A. G. Garcia, M. Pierre, and S. Verner, Phys. Rev. D 108, 115024 (2023).
  55. R. Zhang, Z. Xu, and S. Zheng, J. Cosmol. Astropart. Phys. 07 (2023) 048.
  56. O. Özsoy and G. Tasinato, J. Cosmol. Astropart. Phys. 06 (2024) 003.
  57. J. A. R. Cembranos, L. J. Garay, Á. Parra-López, and J. M. Sánchez Velázquez, J. Cosmol. Astropart. Phys. 02 (2024) 013.
  58. B. Barman, A. Das, P. Sarmah, and R. K. SivaKumar, arXiv:2512.09997.
  59. K. Aoki and S. Mukohyama, Phys. Rev. D 94, 024001 (2016).
  60. E. Babichev, L. Marzola, M. Raidal, A. Schmidt-May, F. Urban, H. Veermäe, and M. von Strauss, Phys. Rev. D 94, 084055 (2016).
  61. M. A. Gorji, J. Cosmol. Astropart. Phys. 11 (2023) 081.
  62. Y. Manita, K. Aoki, T. Fujita, and S. Mukohyama, Phys. Rev. D 107, 104007 (2023).
  63. R. S. Chivukula, J. A. Gill, K. A. Mohan, D. Sengupta, E. H. Simmons, and X. Wang, Phys. Rev. D 109, 015033 (2024).
  64. R. S. Chivukula, J. A. Gill, K. A. Mohan, G. Sanamyan, D. Sengupta, E. H. Simmons, and X. Wang, Phys. Rev. D 111, 075030 (2025).
  65. D. Blas, J. Carlton, and C. McCabe, Phys. Rev. D 111, 115020 (2025).
  66. A. Higuchi, Nucl. Phys. B282, 397 (1987).
  67. M. Fierz, Helv. Phys. Acta 12, 3 (1939).
  68. M. Fierz and W. Pauli, Proc. R. Soc. A 173, 211 (1939).
  69. K. Hinterbichler, Rev. Mod. Phys. 84, 671 (2012).
  70. C. de Rham, Living Rev. Relativity 17, 7 (2014).
  71. S. Folkerts, C. Germani, and N. Wintergerst, Massive spin-2 theories, in Cosmology and Particle Physics Beyond Standard Models: Ten Years of the SEENET-MTP Network, edited by L. Álvarez-Gaumé, G. S. Djordjevic, and D. Stojkovic (, Geneva, 2014), pp. 87–97.
  72. A. Koenigstein, F. Giacosa, and D. H. Rischke, Ann. Phys. (Amsterdam) 368, 16 (2016).
  73. L. Farolfi and F. Fecit, Eur. Phys. J. C 85, 356 (2025).
  74. R. J. Rivers, Nuovo Cimento 34, 386 (1964).
  75. G. F. Giudice, R. Rattazzi, and J. D. Wells, Nucl. Phys. B544, 3 (1999).
  76. N. Aghanim et al. (Planck Collaboration), Astron. Astrophys. 641, A6 (2020); 652, C4(E) (2021).
  77. A. A. Starobinsky, Phys. Lett. 91B, 99 (1980).
  78. R. Kallosh and A. Linde, J. Cosmol. Astropart. Phys. 07 (2013) 002.
  79. J. Ellis, M. A. G. Garcia, K. A. Olive, and S. Verner, arXiv:2510.18656.
  80. E. W. Kolb, A. J. Long, and E. McDonough, Phys. Rev. D 104, 075015 (2021).
  81. N. Arkani-Hamed, H. Georgi, and M. D. Schwartz, Ann. Phys. (Amsterdam) 305, 96 (2003).
  82. A. Aubert, Phys. Rev. D 69, 087502 (2004).
  83. Y. Shtanov, J. H. Traschen, and R. H. Brandenberger, Phys. Rev. D 51, 5438 (1995).
  84. K. Ichikawa, T. Suyama, T. Takahashi, and M. Yamaguchi, Phys. Rev. D 78, 063545 (2008).
  85. K. Kainulainen, S. Nurmi, T. Tenkanen, K. Tuominen, and V. Vaskonen, J. Cosmol. Astropart. Phys. 06 (2016) 022.
  86. M. S. Turner, Phys. Rev. D 28, 1243 (1983).
  87. E. Dudas, Y. Mambrini, and K. Olive, Phys. Rev. Lett. 119, 051801 (2017).
  88. K. Benakli, Y. Chen, E. Dudas, and Y. Mambrini, Phys. Rev. D 95, 095002 (2017).
  89. T. Hasegawa, N. Hiroshima, K. Kohri, R. S. L. Hansen, T. Tram, and S. Hannestad, J. Cosmol. Astropart. Phys. 12 (2019) 012.
  90. S. Nurmi, T. Tenkanen, and K. Tuominen, J. Cosmol. Astropart. Phys. 11 (2015) 001.

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Phys. Rev. D 113, 123503– Published 1 June, 2026
  • Received 23 January 2026
  • Accepted 23 March 2026
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