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Probing quadratically coupled ultralight dark matter with pulsar timing arrays

Xucheng Gan, Hyungjin Kim, and Andrea Mitridate

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Phys. Rev. D 113, 063034 – Published 19 March, 2026

DOI: https://doi.org/10.1103/15g1-gklx

Abstract

Ultralight dark matter may couple quadratically to Standard Model particles. Such quadratic interactions give rise to both coherent and stochastic signals in pulsar timing array (PTA) observations. In this work, we characterize these signals, including the effects of dark matter propagation in a finite-density medium, and assess the sensitivity of current and upcoming PTA observations to their detection. For coherent signals, we find that the sensitivity of current PTA observations competes with and sometimes exceeds that of other probes, such as equivalence principle tests and atomic clocks. For stochastic signals, we find that PTA sensitivities underperform equivalence principle constraints for both existing and upcoming PTA datasets.

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References (114)

  1. R. D. Peccei and H. R. Quinn, CP conservation in the presence of instantons, Phys. Rev. Lett. 38, 1440 (1977).
  2. R. D. Peccei and H. R. Quinn, Constraints imposed by CP conservation in the presence of instantons, Phys. Rev. D 16, 1791 (1977).
  3. S. Weinberg, A new light boson?, Phys. Rev. Lett. 40, 223 (1978).
  4. F. Wilczek, Problem of strong P and T invariance in the presence of instantons, Phys. Rev. Lett. 40, 279 (1978).
  5. J. E. Kim, Weak interaction singlet and strong CP invariance, Phys. Rev. Lett. 43, 103 (1979).
  6. M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Can confinement ensure natural CP invariance of strong interactions?, Nucl. Phys. B166, 493 (1980).
  7. A. R. Zhitnitsky, On possible suppression of the axion hadron interactions (in Russian), Sov. J. Nucl. Phys. 31, 260 (1980).
  8. M. Dine, W. Fischler, and M. Srednicki, A simple solution to the strong CP problem with a harmless axion, Phys. Lett. 104B, 199 (1981).
  9. J. Preskill, M. B. Wise, and F. Wilczek, Cosmology of the invisible axion, Phys. Lett. 120B, 127 (1983).
  10. L. F. Abbott and P. Sikivie, A cosmological bound on the invisible axion, Phys. Lett. 120B, 133 (1983).
  11. M. Dine and W. Fischler, The not so harmless axion, Phys. Lett. 120B, 137 (1983).
  12. P. W. Graham, D. E. Kaplan, and S. Rajendran, Cosmological relaxation of the electroweak scale, Phys. Rev. Lett. 115, 221801 (2015).
  13. A. Arvanitaki, S. Dimopoulos, V. Gorbenko, J. Huang, and K. Van Tilburg, A small weak scale from a small cosmological constant, J. High Energy Phys. 05 (2017) 071.
  14. M. Geller, Y. Hochberg, and E. Kuflik, Inflating to the weak scale, Phys. Rev. Lett. 122, 191802 (2019).
  15. A. Banerjee, H. Kim, and G. Perez, Coherent relaxion dark matter, Phys. Rev. D 100, 115026 (2019).
  16. N. Arkani-Hamed, R. T. D’Agnolo, and H. D. Kim, Weak scale as a trigger, Phys. Rev. D 104, 095014 (2021).
  17. A. Banerjee, H. Kim, O. Matsedonskyi, G. Perez, and M. S. Safronova, Probing the relaxed relaxion at the luminosity and precision frontiers, J. High Energy Phys. 07 (2020) 153.
  18. R. Tito D’Agnolo and D. Teresi, Sliding naturalness: New solution to the strong-CP and electroweak-hierarchy problems, Phys. Rev. Lett. 128, 021803 (2022).
  19. R. Tito D’Agnolo and D. Teresi, Sliding naturalness: Cosmological selection of the weak scale, J. High Energy Phys. 02 (2022) 023.
  20. A. Chatrchyan and G. Servant, The stochastic relaxion, J. High Energy Phys. 06 (2023) 107.
  21. A. Chatrchyan and G. Servant, Relaxion dark matter from stochastic misalignment, J. Cosmol. Astropart. Phys. 06 (2023) 036.
  22. C. Csaki, R. Tito D’Agnolo, E. Kuflik, and P. Sesma, A cosmological solution to the doublet-triplet splitting problem, J. High Energy Phys. 02 (2025) 048.
  23. A. Khmelnitsky and V. Rubakov, Pulsar timing signal from ultralight scalar dark matter, J. Cosmol. Astropart. Phys. 020 (2014) 019.
  24. N. K. Porayko et al., Parkes Pulsar Timing Array constraints on ultralight scalar-field dark matter, Phys. Rev. D 98, 102002 (2018).
  25. A. Afzal, G. Agazie et al. (NANOGrav Collaboration), The NANOGrav 15-year data set: Search for signals from new physics, Astrophys. J. Lett. 951, L11 (2023).
  26. C. Smarra, B. Goncharov et al. (EPTA Collaboration), The second data release from the European Pulsar Timing Array: VI. Challenging the ultralight dark matter paradigm, Phys. Rev. Lett. 131, 171001 (2023).
  27. H. Kim, Gravitational interaction of ultralight dark matter with interferometers, J. Cosmol. Astropart. Phys. 12 (2023) 018.
  28. H. Kim, A. Lenoci, G. Perez, and W. Ratzinger, Probing an ultralight QCD axion with electromagnetic quadratic interaction, Phys. Rev. D 109, 015030 (2024).
  29. H. Kim and A. Mitridate, Stochastic ultralight dark matter fluctuations in pulsar timing arrays, Phys. Rev. D 109, 055017 (2024).
  30. H. Kim, Astrometric search for ultralight dark matter, Phys. Rev. D 110, 083031 (2024).
  31. H.-Y. Schive, T. Chiueh, and T. Broadhurst, Cosmic structure as the quantum interference of a coherent dark wave, Nat. Phys. 10, 496 (2014).
  32. L. Hui, Wave dark matter, Annu. Rev. Astron. Astrophys. 59, 247 (2021).
  33. D. Antypas et al., New horizons: Scalar and vector ultralight dark matter, arXiv:2203.14915.
  34. J.-P. Uzan, Fundamental constants: From measurement to the universe, a window on gravitation and cosmology, Living Rev. Relativity 28, 6 (2025).
  35. P. W. Graham, D. E. Kaplan, J. Mardon, S. Rajendran, and W. A. Terrano, Dark matter direct detection with accelerometers, Phys. Rev. D 93, 075029 (2016).
  36. D. E. Kaplan, A. Mitridate, and T. Trickle, Constraining fundamental constant variations from ultralight dark matter with pulsar timing arrays, Phys. Rev. D 106, 035032 (2022).
  37. H. Kim and G. Perez, Oscillations of atomic energy levels induced by QCD axion dark matter, Phys. Rev. D 109, 015005 (2024).
  38. C. Beadle, S. A. R. Ellis, J. Quevillon, and P. N. Hoa Vuong, Quadratic coupling of the axion to photons, Phys. Rev. D 110, 035019 (2024).
  39. X. Gan and D. Liu, Cosmologically varying kinetic mixing, J. High Energy Phys. 11 (2023) 031.
  40. X. Gan, D. Liu, D. Liu, X. Luo, and B. Yu, Detecting ultralight dark matter with matter effect, arXiv:2504.11522.
  41. C. Smarra et al., Constraints on conformal ultralight dark matter couplings from the European Pulsar Timing Array, Phys. Rev. D 110, 043033 (2024).
  42. T. Damour and J. F. Donoghue, Phenomenology of the equivalence principle with light scalars, Classical Quantum Gravity 27, 202001 (2010).
  43. T. Damour and J. F. Donoghue, Equivalence principle violations and couplings of a light dilaton, Phys. Rev. D 82, 084033 (2010).
  44. A. Hees, O. Minazzoli, E. Savalle, Y. V. Stadnik, and P. Wolf, Violation of the equivalence principle from light scalar dark matter, Phys. Rev. D 98, 064051 (2018).
  45. A. Banerjee, G. Perez, M. Safronova, I. Savoray, and A. Shalit, The phenomenology of quadratically coupled ultra light dark matter, J. High Energy Phys. 10 (2023) 042.
  46. M. Asplund, N. Grevesse, A. J. Sauval, and P. Scott, The chemical composition of the Sun, Annu. Rev. Astron. Astrophys. 47, 481 (2009).
  47. N. F. Bell, G. Busoni, and S. Robles, Capture of leptophilic dark matter in neutron stars, J. Cosmol. Astropart. Phys. 06 (2019) 054.
  48. K. K. Boddy, J. A. Dror, and A. Lam, Ultralight dark matter statistics for pulsar timing detection, Phys. Rev. Lett. 135, 101001 (2025).
  49. V. A. Dzuba, V. V. Flambaum, and J. K. Webb, Calculations of the relativistic effects in many electron atoms and space-time variation of fundamental constants, Phys. Rev. A 59, 230 (1999).
  50. V. V. Flambaum and A. F. Tedesco, Dependence of nuclear magnetic moments on quark masses and limits on temporal variation of fundamental constants from atomic clock experiments, Phys. Rev. C 73, 055501 (2006).
  51. Z. Arzoumanian et al. (NANOGRAV Collaboration), The NANOGrav 11-year data set: Pulsar-timing constraints on the stochastic gravitational-wave background, Astrophys. J. 859, 47 (2018).
  52. S. R. Taylor, The nanohertz gravitational wave astronomer, arXiv:2105.13270.
  53. R. w. Hellings and G. s. Downs, Upper limits on the isotropic gravitational radiation background from pulsar timing analysis, Astrophys. J. Lett. 265, L39 (1983).
  54. H. N. Luu, T. Liu, J. Ren, T. Broadhurst, R. Yang, J.-S. Wang, and Z. Xie, Stochastic wave dark matter with Fermi-LAT γ-Ray Pulsar Timing Array, Astrophys. J. Lett. 963, L46 (2024).
  55. X. Li et al., Probing ultralight axion-like dark matter − A PTA-PPA synergy, arXiv:2506.04871.
  56. A. D. Johnson et al. (NANOGrav Collaboration), NANOGrav 15-year gravitational-wave background methods, Phys. Rev. D 109, 103012 (2024).
  57. A. Mitridate, D. Wright, R. von Eckardstein, T. Schröder, J. Nay, K. Olum, K. Schmitz, and T. Trickle, ptarcade, arXiv:2306.16377.
  58. J. A. Ellis, M. Vallisneri, S. R. Taylor, and P. T. Baker, enterprise: Enhanced Numerical Toolbox Enabling a Robust PulsaR Inference SuitE, Zendo, 10.5281/zenodo.4059815 (2020).
  59. M. F. Alam et al. (NANOGrav Collaboration), The NANOGrav 12.5 yr data set: Observations and narrowband timing of 47 millisecond pulsars, Astrophys. J. Suppl. Ser. 252, 4 (2021).
  60. M. Vallisneri, libstempo: python wrapper for tempo2, Astrophysics Source Code Library, record ascl:2002.017 (2020).
  61. R. T. Edwards, G. B. Hobbs, and R. N. Manchester, tempo2, A new pulsar timing package. 2. The timing model and precision estimates, Mon. Not. R. Astron. Soc. 372, 1549 (2006).
  62. G. Hobbs, R. Edwards, and R. Manchester, tempo2, A new pulsar timing package. 1. Overview, Mon. Not. R. Astron. Soc. 369, 655 (2006).
  63. G. Agazie et al. (NANOGrav Collaboration), The NANOGrav 15 yr data set: Observations and timing of 68 millisecond pulsars, Astrophys. J. Lett. 951, L9 (2023).
  64. G. Agazie et al. (NANOGrav Collaboration), The NANOGrav 15 yr data set: Evidence for a gravitational-wave background, Astrophys. J. Lett. 951, L8 (2023).
  65. H. Kim and A. Mitridate, Stochastic ultralight dark matter fluctuations in pulsar timing arrays—Model files for ptarcade, 2024, 10.5281/zenodo.10534322.
  66. P. Touboul et al. (MICROSCOPE Collaboration), Space test of the equivalence principle: First results of the MICROSCOPE mission, Classical Quantum Gravity 36, 225006 (2019).
  67. P. Touboul et al. (MICROSCOPE Collaboration), MICROSCOPE mission: Final results of the test of the equivalence principle, Phys. Rev. Lett. 129, 121102 (2022).
  68. Y. V. Stadnik and V. V. Flambaum, Can dark matter induce cosmological evolution of the fundamental constants of nature?, Phys. Rev. Lett. 115, 201301 (2015).
  69. S. Sibiryakov, P. Sørensen, and T.-T. Yu, BBN constraints on universally-coupled ultralight scalar dark matter, J. High Energy Phys. 12 (2020) 075.
  70. T. Bouley, P. Sørensen, and T.-T. Yu, Constraints on ultralight scalar dark matter with quadratic couplings, J. High Energy Phys. 03 (2023) 104.
  71. K. Van Tilburg, N. Leefer, L. Bougas, and D. Budker, Search for ultralight scalar dark matter with atomic spectroscopy, Phys. Rev. Lett. 115, 011802 (2015).
  72. A. Hees, J. Guéna, M. Abgrall, S. Bize, and P. Wolf, Searching for an oscillating massive scalar field as a dark matter candidate using atomic hyperfine frequency comparisons, Phys. Rev. Lett. 117, 061301 (2016).
  73. C. J. Kennedy, E. Oelker, J. M. Robinson, T. Bothwell, D. Kedar, W. R. Milner, G. E. Marti, A. Derevianko, and J. Ye, Precision metrology meets cosmology: Improved constraints on ultralight dark matter from atom-cavity frequency comparisons, Phys. Rev. Lett. 125, 201302 (2020).
  74. K. Beloy et al. (BACON Collaboration), Frequency ratio measurements at 18-digit accuracy using an optical clock network, Nature (London) 591, 564 (2021).
  75. N. Sherrill et al., Analysis of atomic-clock data to constrain variations of fundamental constants, New J. Phys. 25, 093012 (2023).
  76. M. Filzinger, S. Dörscher, R. Lange, J. Klose, M. Steinel, E. Benkler, E. Peik, C. Lisdat, and N. Huntemann, Improved limits on the coupling of ultralight bosonic dark matter to photons from optical atomic clock comparisons, Phys. Rev. Lett. 130, 253001 (2023).
  77. J. Gué, P. Wolf, and A. Hees, Search for QCD coupled axion dark matter with the MICROSCOPE space experiment, Phys. Rev. Lett. 135, 171001 (2025).
  78. C. Delaunay, M. Geller, Z. Heller-Algazi, G. Perez, and K. Springmann, Natural ultralight dark matter: The quadratic twin, arXiv:2507.12514.
  79. K. Bartnick, K. Springmann, S. Stelzl, and A. Weiler, ϕ-dwarfs: White dwarfs probe quadratically coupled scalars, arXiv:2509.25305.
  80. A. Banerjee, I. M. Bloch, Q. Bonnefoy, S. A. R. Ellis, G. Perez, I. Savoray, K. Springmann, and Y. V. Stadnik, Momentum and matter matter for axion dark matter matters on earth, arXiv:2502.04455.
  81. Y. G. del Castillo, B. Hammett, and J. Jaeckel, Enhanced axion-wind near Earth’s surface, J. Cosmol. Astropart. Phys. 10 (2025) 022.
  82. Z. Bai, V. Cardoso, Y. Chen, T. Do, A. Hees, H. Xiao, and X. Xue, Probing axions via spectroscopic measurements of S-stars at the galactic center, arXiv:2507.07482.
  83. L. Ubaldi, Effects of theta on the deuteron binding energy and the triple-alpha process, Phys. Rev. D 81, 025011 (2010).
  84. A. Hook, Solving the hierarchy problem discretely, Phys. Rev. Lett. 120, 261802 (2018).
  85. L. Di Luzio, B. Gavela, P. Quilez, and A. Ringwald, An even lighter QCD axion, J. High Energy Phys. 05 (2021) 184.
  86. L. Di Luzio, B. Gavela, P. Quilez, and A. Ringwald, Dark matter from an even lighter QCD axion: Trapped misalignment, J. Cosmol. Astropart. Phys. 10 (2021) 001.
  87. C. Abel et al., Search for axionlike dark matter through nuclear spin precession in electric and magnetic fields, Phys. Rev. X 7, 041034 (2017).
  88. A. Arvanitaki, M. Baryakhtar, and X. Huang, Discovering the QCD axion with black holes and gravitational waves, Phys. Rev. D 91, 084011 (2015).
  89. M. Baryakhtar, M. Galanis, R. Lasenby, and O. Simon, Black hole superradiance of self-interacting scalar fields, Phys. Rev. D 103, 095019 (2021).
  90. S. Hoof, D. J. E. Marsh, J. Sisk-Reynés, J. H. Matthews, and C. Reynolds, Getting more out of black hole superradiance: A statistically rigorous approach to ultralight boson constraints, Mon. Not. R. Astron. Soc. 546, staf1564 (2026).
  91. J. Zhang, Z. Lyu, J. Huang, M. C. Johnson, L. Sagunski, M. Sakellariadou, and H. Yang, First constraints on nuclear coupling of axionlike particles from the binary neutron star gravitational wave event GW170817, Phys. Rev. Lett. 127, 161101 (2021).
  92. R. Balkin, J. Serra, K. Springmann, S. Stelzl, and A. Weiler, White dwarfs as a probe of exceptionally light QCD axions, Phys. Rev. D 109, 095032 (2024).
  93. K. Blum, R. T. D’Agnolo, M. Lisanti, and B. R. Safdi, Constraining axion dark matter with big bang nucleosynthesis, Phys. Lett. B 737, 30 (2014).
  94. R. Oswald et al., Search for dark-matter-induced oscillations of fundamental constants using molecular spectroscopy, Phys. Rev. Lett. 129, 031302 (2022).
  95. H. Day, D. Liu, M. A. Luty, and Y. Zhao, Blowing in the dark matter wind, J. High Energy Phys. 07 (2024) 136.
  96. C. Burrage, A. Macdonald, M. P. Ross, G. Rybka, and E. Todarello, Impact of cavities on the detection of quadratically coupled ultra-light dark matter, arXiv:2507.16526.
  97. A. Zic et al., The Parkes Pulsar Timing Array third data release, Publ. Astron. Soc. Aust. 40, e049 (2023).
  98. J. Antoniadis et al. (EPTA Collaboration), The second data release from the European Pulsar Timing Array—I. The dataset and timing analysis, Astron. Astrophys. 678, A48 (2023).
  99. G. Panfilo and F. Arias, The coordinated Universal Time (UTC), Metrologia 56, 042001 (2019).
  100. J. Azoubib, M. Granveaud, and B. Guinot, Estimation of the scale unit duration of time scales, Metrologia 13, 87 (1977).
  101. A. Derevianko, Detecting dark-matter waves with a network of precision-measurement tools, Phys. Rev. A 97, 042506 (2018).
  102. J. W. Foster, N. L. Rodd, and B. R. Safdi, Revealing the dark matter halo with axion direct detection, Phys. Rev. D 97, 123006 (2018).
  103. G. P. Centers et al., Stochastic fluctuations of bosonic dark matter, Nat. Commun. 12, 7321 (2021).
  104. H. Kim and A. Lenoci, Gravitational focusing of wave dark matter, Phys. Rev. D 105, 063032 (2022).
  105. D. Y. Cheong, N. L. Rodd, and L.-T. Wang, Quantum description of wave dark matter, Phys. Rev. D 111, 015028 (2025).
  106. A. Hook and J. Huang, Probing axions with neutron star inspirals and other stellar processes, J. High Energy Phys. 06 (2018) 036.
  107. Y. Aoki et al. (Flavour Lattice Averaging Group (FLAG) Collaboration), FLAG review 2024, Phys. Rev. D 113, 014508 (2026).
  108. M. Kumamoto, J. Huang, C. Drischler, M. Baryakhtar, and S. Reddy, Neutron stars with exceptionally light QCD axions, Phys. Rev. D 112, 043008 (2025).
  109. J. Gué, A. Hees, and P. Wolf, Violation of the equivalence principle induced by oscillating rest mass and transition frequency, and its detection in atom interferometers, Phys. Rev. D 110, 035005 (2024).
  110. M. Bauer, S. Chakraborti, and G. Rostagni, Axion bounds from quantum technology, J. High Energy Phys. 05 (2025) 023.
  111. K. Van Tilburg, Wake forces in a background of quadratically coupled mediators, Phys. Rev. D 109, 096036 (2024).
  112. S. Barbosa and S. Fichet, Background-induced forces from dark relics, J. High Energy Phys. 01 (2025) 021.
  113. Y. Cheng and S. Ge, Background-enhanced axion force by axion dark matter, arXiv:2504.02702.
  114. Y. Grossman, B. Yu, and S. Zhou, Axion forces in axion backgrounds, J. High Energy Phys. 01 (2026) 145.

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Phys. Rev. D 113, 063034– Published 19 March, 2026
  • Received 29 October 2025
  • Accepted 29 January 2026
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