Charting the Nanohertz Gravitational Wave Sky with Pulsar Timing Arrays

Author(s)

Bernardo, Reginald Christian, Ng, Kin-Wang

Abstract

In the summer of 2023, the pulsar timing arrays (PTAs) announced a compelling evidence for the existence of a nanohertz stochastic gravitational wave background (SGWB). Despite this breakthrough, however, several critical questions remain unanswered: What is the source of the signal? How can cosmic variance be accounted for? To what extent can we constrain nanohertz gravity? When will individual supermassive black hole binaries become observable? And how can we achieve a stronger detection? These open questions have spurred significant interests in PTA science, making this an opportune moment to revisit the astronomical and theoretical foundations of the field, as well as the data analysis techniques employed. In this review, we focus on the theoretical aspects of the SGWB as detected by PTAs. We provide a comprehensive derivation of the expected signal and its correlation, presented in a pedagogical manner, while also addressing current constraints. Looking ahead, we explore future milestones in the field, with detailed discussions on emerging theoretical considerations such as cosmic variance, the cumulants of the one- and two-point functions, subluminal gravitational waves, and the anisotropy and polarization of the SGWB.

Figures

Red noise parameters in the NANOGrav 15-year data set (red whiskers) and the corresponding SGWB power spectrum (blue horizontal bands) \cite{NANOGrav:2023gor}.

Red noise parameters in the NANOGrav 15-year data set (red whiskers) and the corresponding SGWB power spectrum (blue horizontal bands) \cite{NANOGrav:2023gor}.


Sky distribution of pulsars in the NANOGrav 15-year data set on a projection of a SGWB \cite{NANOGrav:2023hde}.

Sky distribution of pulsars in the NANOGrav 15-year data set on a projection of a SGWB \cite{NANOGrav:2023hde}.


An inferred common spectrum in the NANOGrav 15-year data set \cite{NANOGrav:2023gor} (blue violins) and the SMBHB spectrum (gray dashed line).

An inferred common spectrum in the NANOGrav 15-year data set \cite{NANOGrav:2023gor} (blue violins) and the SMBHB spectrum (gray dashed line).


The overlap reduction function in the NANOGrav 15-year data set \cite{NANOGrav:2023gor} (blue violins) and the Hellings and Downs curve (gray dashed line).

The overlap reduction function in the NANOGrav 15-year data set \cite{NANOGrav:2023gor} (blue violins) and the Hellings and Downs curve (gray dashed line).


The Hellings and Downs curve \cite{Hellings:1983fr} and its cosmic variance \cite{Allen:2022dzg, Bernardo:2022xzl}.

The Hellings and Downs curve \cite{Hellings:1983fr} and its cosmic variance \cite{Allen:2022dzg, Bernardo:2022xzl}.


Mean (top left), variance (top right), skewness (lower left), and kurtosis (lower right) of the two-point function, in the 1st $\alpha$-bin (Eq. \eqref{eq:residual_fourier_series}), in a noise-free PTA simulation (blue points and error bars/cosmic variance). This is constructed by repeatedly computing the sample statistics with 100 pulsars randomly distributed over the sphere that are correlated by a SGWB ($A_{\rm gw}\sim 10^{-15}$ and $\gamma_{\rm gw}=13/3$). The red curves are given by Eqs. (\ref{eq:rarb_sgwb_mean}-\ref{eq:rarb_sgwb_kurtosis}).

Mean (top left), variance (top right), skewness (lower left), and kurtosis (lower right) of the two-point function, in the 1st $\alpha$-bin (Eq. \eqref{eq:residual_fourier_series}), in a noise-free PTA simulation (blue points and error bars/cosmic variance). This is constructed by repeatedly computing the sample statistics with 100 pulsars randomly distributed over the sphere that are correlated by a SGWB ($A_{\rm gw}\sim 10^{-15}$ and $\gamma_{\rm gw}=13/3$). The red curves are given by Eqs. (\ref{eq:rarb_sgwb_mean}-\ref{eq:rarb_sgwb_kurtosis}).


The SGWB angular power spectrum for tensor (top), vector (middle), and scalar (bottom) GW degrees of freedom at luminal and subluminal group speeds. For tensor, $C_0=C_1=0$; for vector, $C_0=0$; for scalar, the monopole, $C_0 \neq 0$, is excluded in the plot, with $C_0/C_1\sim 9, 25, 225$ for $v_{\rm S}\sim 1.0, 0.6, 0.2$, respectively. Red circles and error bars (2-sigma) are the standard HD correlation and its cosmic variance.

The SGWB angular power spectrum for tensor (top), vector (middle), and scalar (bottom) GW degrees of freedom at luminal and subluminal group speeds. For tensor, $C_0=C_1=0$; for vector, $C_0=0$; for scalar, the monopole, $C_0 \neq 0$, is excluded in the plot, with $C_0/C_1\sim 9, 25, 225$ for $v_{\rm S}\sim 1.0, 0.6, 0.2$, respectively. Red circles and error bars (2-sigma) are the standard HD correlation and its cosmic variance.


References
  • 1. LIGO Scientific, Virgo Collaboration (B. P. Abbott et al.), Phys. Rev. Lett. 116 (2016) 061102, arXiv:1602.03837 [gr-qc].
  • 2. KAGRA, VIRGO, LIGO Scientific Collaboration (R. Abbott et al.), Phys. Rev. X 13 (2023) 041039, arXiv:2111.03606 [gr-qc].
  • 3. LIGO Scientific, VIRGO, KAGRA Collaboration (R. Abbott et al.) (12 2021) arXiv:2112.06861 [gr-qc].
  • 4. C. Caprini and D. G. Figueroa, Class. Quant. Grav. 35 (2018) 163001, arXiv:1801.04268 [astro-ph.CO].
  • 5. N. Christensen, Rept. Prog. Phys. 82 (2019) 016903, arXiv:1811.08797 [gr-qc].
  • 6. J. D. Romano, Searches for stochastic gravitational-wave backgrounds (8 2019). arXiv:1909.00269 [gr-qc].
  • 7. C. J. Moore and A. Vecchio, Nature Astron. 5 (2021) 1268, arXiv:2104.15130 [astro-ph.CO].
  • 8. NANOGrav Collaboration (N. S. Pol et al.), Astrophys. J. Lett. 911 (2021) L34, arXiv:2010.11950 [astro-ph.HE].
  • 9. N. Pol, S. R. Taylor and J. D. Romano, Astrophys. J. 940 (2022) 173, arXiv:2206.09936 [astro-ph.HE].
  • 10. S. Staelens and G. Nelemans, Astron. Astrophys. 683 (2024) A139, arXiv:2310.19448 [astro-ph.HE].
  • 11. LIGO Scientific, VIRGO Collaboration (B. P. Abbott et al.), Nature 460 (2009) 990, arXiv:0910.5772 [astro-ph.CO].
  • 12. LIGO Scientific Collaboration (D. Shoemaker) (4 2019) arXiv:1904.03187 [gr-qc].
  • 13. L. Lehoucq, I. Dvorkin, R. Srinivasan, C. Pellouin and A. Lamberts, Mon. Not. Roy. Astron. Soc. 526 (2023) 4378, arXiv:2306.09861 [astro-ph.HE].
  • 14. LISA Cosmology Working Group Collaboration (N. Bartolo et al.), JCAP 11 (2022) 009, arXiv:2201.08782 [astro-ph.CO].
  • 15. J. Cheng, E.-K. Li, Y.-M. Hu, Z.-C. Liang, J.-d. Zhang and J. Mei, Phys. Rev. D 106 (2022) 124027, arXiv:2208.11615 [gr-qc].
  • 16. Z.-C. Liang, Y.-M. Hu, Y. Jiang, J. Cheng, J.-d. Zhang and J. Mei, Phys. Rev. D 105 (2022) 022001, arXiv:2107.08643 [astro-ph.CO].
  • 17. M. Muratore, J. Gair and L. Speri, Phys. Rev. D 109 (2024) 042001, arXiv:2308.01056 [gr-qc].
  • 18. NANOGrav Collaboration (G. Agazie et al.), Astrophys. J. Lett. 951 (2023) L8, arXiv:2306.16213 [astro-ph.HE].
  • 19. D. J. Reardon et al., Astrophys. J. Lett. 951 (2023) L6, arXiv:2306.16215 [astro-ph.HE].
  • 20. EPTA, InPTA: Collaboration (J. Antoniadis et al.), Astron. Astrophys. 678 (2023) A50, arXiv:2306.16214 [astro-ph.HE].
  • 21. H. Xu et al., Res. Astron. Astrophys. 23 (2023) 075024, arXiv:2306.16216 [astro-ph.HE].
  • 22. R. W. Hellings and G. W. Downs, Astrophys. J. Lett. 265 (1983) L39.
  • 23. M. V. Sazhin, Sov. Astron. 22 (1978) 36.
  • 24. S. L. Detweiler, Astrophys. J. 234 (1979) 1100.
  • 25. E. S. Phinney (7 2001) arXiv:astro-ph/0108028.
  • 26. J. S. B. Wyithe and A. Loeb, Astrophys. J. 590 (2003) 691, arXiv:astro-ph/0211556.
  • 27. A. Sesana, F. Haardt, P. Madau and M. Volonteri, Astrophys. J. 611 (2004) 623, arXiv:astro-ph/0401543.
  • 28. A. Sesana, A. Vecchio and C. N. Colacino, Mon. Not. Roy. Astron. Soc. 390 (2008) 192, arXiv:0804.4476 [astro-ph].
  • 29. S. Burke-Spolaor et al., Astron. Astrophys. Rev. 27 (2019) 5, arXiv:1811.08826 [astro-ph.HE].
  • 30. G. Sato-Polito, M. Zaldarriaga and E. Quataert (12 2023) arXiv:2312.06756 [astro-ph.CO].
  • 31. G. Sato-Polito and M. Kamionkowski, Phys. Rev. D 109 (2024) 123544, arXiv:2305.05690 [astro-ph.CO].
  • 32. Y.-C. Bi, Y.-M. Wu, Z.-C. Chen and Q.-G. Huang, Sci. China Phys. Mech. Astron. 66 (2023) 120402, arXiv:2307.00722 [astro-ph.CO].
  • 33. G. Sato-Polito and M. Zaldarriaga (6 2024) arXiv:2406.17010 [astro-ph.CO].
  • 34. J. Raidal, J. Urrutia, V. Vaskonen and H. Veermäe (6 2024) arXiv:2406.05125 [astro-ph.CO].
  • 35. Z.-C. Chen, C. Yuan and Q.-G. Huang, Phys. Rev. Lett. 124 (2020) 251101, arXiv:1910.12239 [astro-ph.CO].
  • 36. J. Ellis and M. Lewicki, Phys. Rev. Lett. 126 (2021) 041304, arXiv:2009.06555 [astro-ph.CO].
  • 37. S. Vagnozzi, Mon. Not. Roy. Astron. Soc. 502 (2021) L11, arXiv:2009.13432 [astro-ph.CO].
  • 38. NANOGrav Collaboration (Z. Arzoumanian et al.), Phys. Rev. Lett. 127 (2021) 251302, arXiv:2104.13930 [astro-ph.CO].
  • 39. W. Buchmuller, V. Domcke and K. Schmitz, JCAP 12 (2021) 006, arXiv:2107.04578 [hep-ph].
  • 40. X. Xue et al., Phys. Rev. Lett. 127 (2021) 251303, arXiv:2110.03096 [astro-ph.CO].
  • 41. R. Sharma, Phys. Rev. D 105 (2022) L041302, arXiv:2102.09358 [astro-ph.CO].
  • 42. EPTA, InPTA Collaboration (J. Antoniadis et al.), Astron. Astrophys. 685 (2024) A94, arXiv:2306.16227 [astro-ph.CO].
  • 43. S. Vagnozzi, JHEAp 39 (2023) 81, arXiv:2306.16912 [astro-ph.CO].
  • 44. D. G. Figueroa, M. Pieroni, A. Ricciardone and P. Simakachorn, Phys. Rev. Lett. 132 (2024) 171002, arXiv:2307.02399 [astro-ph.CO].
  • 45. J. Ellis, M. Fairbairn, G. Hütsi, J. Raidal, J. Urrutia, V. Vaskonen and H. Veermäe, Phys. Rev. D 109 (2024) L021302, arXiv:2306.17021 [astro-ph.CO].
  • 46. V. Saeedzadeh, S. Mukherjee, A. Babul, M. Tremmel and T. R. Quinn, Mon. Not. Roy. Astron. Soc. 529 (2024) 4295, arXiv:2309.08683 [astro-ph.GA].
  • 47. L. Liu, Z.-C. Chen and Q.-G. Huang (7 2023) arXiv:2307.14911 [astro-ph.CO].
  • 48. L. Liu, Y. Wu and Z.-C. Chen, JCAP 04 (2024) 011, arXiv:2310.16500 [astro-ph.CO].
  • 49. Z.-C. Chen, S.-L. Li, P. Wu and H. Yu, Phys. Rev. D 109 (2024) 043022, arXiv:2312.01824 [astro-ph.CO].
  • 50. L. Liu, Z.-C. Chen and Q.-G. Huang, Phys. Rev. D 109 (2024) L061301, arXiv:2307.01102 [astro-ph.CO].
  • 51. J.-H. Jin, Z.-C. Chen, Z. Yi, Z.-Q. You, L. Liu and Y. Wu, JCAP 09 (2023) 016, arXiv:2307.08687 [astro-ph.CO].
  • 52. H.-L. Huang, Y. Cai, J.-Q. Jiang, J. Zhang and Y.-S. Piao (6 2023) arXiv:2306.17577 [gr-qc].
  • 53. G. Ye, M. Zhu and Y. Cai, JHEP 02 (2024) 008, arXiv:2312.10685 [gr-qc].
  • 54. S. Wang, Z.-C. Zhao and Q.-H. Zhu, Phys. Rev. Res. 6 (2024) 013207, arXiv:2307.03095 [astro-ph.CO].
  • 55. S. Wang, Z.-C. Zhao, J.-P. Li and Q.-H. Zhu, Phys. Rev. Res. 6 (2024) L012060, arXiv:2307.00572 [astro-ph.CO].
  • 56. M. Zhu, G. Ye and Y. Cai, Eur. Phys. J. C 83 (2023) 816, arXiv:2307.16211 [astro-ph.CO].
  • 57. J.-Q. Jiang, Y. Cai, G. Ye and Y.-S. Piao, JCAP 05 (2024) 004, arXiv:2307.15547 [astro-ph.CO].
  • 58. L. Bian, S. Ge, J. Shu, B. Wang, X.-Y. Yang and J. Zong (6 2023) arXiv:2307.02376 [astro-ph.HE].
  • 59. J.-Q. Jiang and Y.-S. Piao (1 2024) arXiv:2401.16950 [gr-qc].
  • 60. M. W. Winkler and K. Freese (1 2024) arXiv:2401.13729 [astro-ph.CO].
  • 61. G. Agazie et al. (8 2024) arXiv:2408.10166 [astro-ph.HE].
  • 62. S. J. Chamberlin and X. Siemens, Phys. Rev. D 85 (2012) 082001, arXiv:1111.5661 [astro-ph.HE].
  • 63. C. Powell and G. Tasinato, JCAP 01 (2020) 017, arXiv:1910.04758 [gr-qc].
  • 64. G. Tasinato, Phys. Rev. D 105 (2022) 083506, arXiv:2203.15440 [gr-qc].
  • 65. W. Qin, K. K. Boddy and M. Kamionkowski, Phys. Rev. D 103 (2021) 024045, arXiv:2007.11009 [gr-qc].
  • 66. A. Boı̂tier, T. Giroud, S. Tiwari and P. Jetzer, Phys. Rev. D 105 (2022) 084006, arXiv:2111.12563 [gr-qc].
  • 67. Z.-C. Chen, Y.-M. Wu and Q.-G. Huang, Commun. Theor. Phys. 74 (2022) 105402, arXiv:2109.00296 [astro-ph.CO].
  • 68. Z.-C. Chen, C. Yuan and Q.-G. Huang, Sci. China Phys. Mech. Astron. 64 (2021) 120412, arXiv:2101.06869 [astro-ph.CO].
  • 69. Y.-M. Wu, Z.-C. Chen and Q.-G. Huang, Astrophys. J. 925 (2022) 37, arXiv:2108.10518 [astro-ph.CO].
  • 70. Q. Liang and M. Trodden, Phys. Rev. D 104 (2021) 084052, arXiv:2108.05344 [astro-ph.CO].
  • 71. R. C. Bernardo and K.-W. Ng, Phys. Lett. B 841 (2023) 137939, arXiv:2206.01056 [astro-ph.CO].
  • 72. R. C. Bernardo and K.-W. Ng, Phys. Rev. D 107 (2023) 044007, arXiv:2208.12538 [gr-qc].
  • 73. Q.-H. Zhu, Phys. Rev. D 107 (2023) 103519, arXiv:2301.00311 [gr-qc].
  • 74. R. C. Bernardo and K.-W. Ng, Phys. Rev. D 107 (2023) L101502, arXiv:2302.11796 [gr-qc].
  • 75. Y.-M. Wu, Z.-C. Chen and Q.-G. Huang, Phys. Rev. D 107 (2023) 042003, arXiv:2302.00229 [gr-qc].
  • 76. Q. Liang, M.-X. Lin and M. Trodden, JCAP 11 (2023) 042, arXiv:2304.02640 [astro-ph.CO].
  • 77. Q. Liang, M.-X. Lin, M. Trodden and S. S. C. Wong, Phys. Rev. D 109 (2024) 083028, arXiv:2309.16666 [astro-ph.CO].
  • 78. N. Cordes, A. Mitridate, K. Schmitz, T. Schröder and K. Wassner (7 2024) arXiv:2407.04464 [gr-qc].
  • 79. Q. Liang, I. Obata and M. Sasaki (5 2024) arXiv:2405.11755 [astro-ph.CO].
  • 80. G. Domènech and A. Tsabodimos (7 2024) arXiv:2407.21567 [gr-qc].
  • 81. B. Atkins, A. Malhotra and G. Tasinato (8 2024) arXiv:2408.10122 [gr-qc].
  • 82. V. Di Marco, A. Zic, R. M. Shannon and E. Thrane, Mon. Not. Roy. Astron. Soc. 532 (2024) 4026, arXiv:2403.13175 [astro-ph.HE].
  • 83. G. Agazie et al. (7 2024) arXiv:2407.20510 [astro-ph.HE].
  • 84. B. Goncharov, S. Sardana, A. Sesana, J. Antoniadis, A. Chalumeau, D. Champion, S. Chen, E. F. Keane, G. Shaifullah and L. Speri (9 2024) arXiv:2409.03627 [astro-ph.HE].
  • 85. B. Goncharov and S. Sardana (9 2024) arXiv:2409.03661 [astro-ph.HE].
  • 86. R. C. Bernardo and K.-W. Ng (9 2024) arXiv:2409.01218 [astro-ph.CO].
  • 87. B. C. Joshi, Int. J. Mod. Phys. D 22 (2013) 1341008, arXiv:1301.5730 [astro-ph.IM].
  • 88. M. McLaughlin, Gen. Rel. Grav. 46 (2014) 1810, arXiv:1409.4579 [astro-ph.IM].
  • 89. R. N. Manchester, Int. J. Mod. Phys. D 24 (2015) 1530018, arXiv:1502.05474 [gr-qc].
  • 90. A. N. Lommen, Rept. Prog. Phys. 78 (2015) 124901.
  • 91. J. D. Romano and N. J. Cornish, Living Rev. Rel. 20 (2017) 2, arXiv:1608.06889 [gr-qc].
  • 92. W. Becker, M. Kramer and A. Sesana, Space Sci. Rev. 214 (2018) 30, arXiv:1705.11022 [astro-ph.IM].
  • 93. J. P. W. Verbiest, S. Oslowski and S. Burke-Spolaor, Pulsar Timing Array Experiments 2022.
  • 94. S. R. Taylor (5 2021) arXiv:2105.13270 [astro-ph.HE].
  • 95. J. P. W. Verbiest, S. J. Vigeland, N. K. Porayko, S. Chen and D. J. Reardon, Results Phys. 61 (2024) 107719, arXiv:2404.19529 [astro-ph.HE].
  • 96. N. Yunes, X. Siemens and K. Yagi (8 2024) arXiv:2408.05240 [gr-qc].
  • 97. C. M. F. Mingarelli, T. Sidery, I. Mandel and A. Vecchio, Phys. Rev. D 88 (2013) 062005, arXiv:1306.5394 [astro-ph.HE].
  • 98. J. Gair, J. D. Romano, S. Taylor and C. M. F. Mingarelli, Phys. Rev. D 90 (2014) 082001, arXiv:1406.4664 [gr-qc].
  • 99. E. Roebber and G. Holder, Astrophys. J. 835 (2017) 21, arXiv:1609.06758 [astro-ph.CO].
  • 100. W. Qin, K. K. Boddy, M. Kamionkowski and L. Dai, Phys. Rev. D 99 (2019) 063002, arXiv:1810.02369 [astro-ph.CO].
  • 101. S. C. Hotinli, M. Kamionkowski and A. H. Jaffe, Open J. Astrophys. 2 (2019) 8, arXiv:1904.05348 [astro-ph.CO].
  • 102. E. Belgacem and M. Kamionkowski, Phys. Rev. D 102 (2020) 023004, arXiv:2004.05480 [astro-ph.CO].
  • 103. K.-W. Ng, Phys. Rev. D 106 (2022) 043505, arXiv:2106.12843 [astro-ph.CO].
  • 104. G.-C. Liu and K.-W. Ng, Phys. Rev. D 106 (2022) 064004, arXiv:2201.06767 [gr-qc].
  • 105. B. Allen, Phys. Rev. D 107 (2023) 043018, arXiv:2205.05637 [gr-qc].
  • 106. B. Allen and J. D. Romano (8 2022) arXiv:2208.07230 [gr-qc].
  • 107. R. C. Bernardo and K.-W. Ng, JCAP 11 (2022) 046, arXiv:2209.14834 [gr-qc].
  • 108. N. Anil Kumar and M. Kamionkowski (11 2023) arXiv:2311.14159 [astro-ph.CO].
  • 109. R. C. Bernardo, G.-C. Liu and K.-W. Ng, JCAP 04 (2024) 034, arXiv:2312.03383 [gr-qc].
  • 110. N. Anil Kumar, M. Çalışkan, G. Sato-Polito, M. Kamionkowski and L. Ji, Phys. Rev. D 110 (2024) 043501, arXiv:2312.03056 [astro-ph.CO].
  • 111. B. Allen and S. Valtolina, Phys. Rev. D 109 (2024) 083038, arXiv:2401.14329 [gr-qc].
  • 112. W. G. Lamb and S. R. Taylor (7 2024) arXiv:2407.06270 [gr-qc].
  • 113. R. C. Bernardo, S. Appleby and K.-W. Ng (7 2024) arXiv:2407.17987 [astro-ph.CO].
  • 114. X. Siemens, V. Mandic and J. Creighton, Phys. Rev. Lett. 98 (2007) 111101, arXiv:astro-ph/0610920.
  • 115. A. Khmelnitsky and V. Rubakov, JCAP 02 (2014) 019, arXiv:1309.5888 [astro-ph.CO].
  • 116. N. K. Porayko and K. A. Postnov, Phys. Rev. D 90 (2014) 062008, arXiv:1408.4670 [astro-ph.CO].
  • 117. M. C. Guzzetti, N. Bartolo, M. Liguori and S. Matarrese, Riv. Nuovo Cim. 39 (2016) 399, arXiv:1605.01615 [astro-ph.CO].
  • 118. H. W. H. Tahara and T. Kobayashi, Phys. Rev. D 102 (2020) 123533, arXiv:2011.01605 [gr-qc].
  • 119. P. Adshead, K. D. Lozanov and Z. J. Weiner, JCAP 10 (2021) 080, arXiv:2105.01659 [astro-ph.CO].
  • 120. Y.-K. Chu, G.-C. Liu and K.-W. Ng, Phys. Rev. D 104 (2021) 124018, arXiv:2107.00536 [gr-qc].
  • 121. S. Garcia-Saenz, L. Pinol, S. Renaux-Petel and D. Werth, JCAP 03 (2023) 057, arXiv:2207.14267 [astro-ph.CO].
  • 122. J. T. Acuña and P.-Y. Tseng, JHEP 08 (2023) 117, arXiv:2304.10084 [hep-ph].
  • 123. J.-c. Hwang, D. Jeong, H. Noh and C. Smarra, JCAP 02 (2024) 014, arXiv:2311.00234 [astro-ph.CO].
  • 124. M. Çalışkan, Y. Chen, L. Dai, N. Anil Kumar, I. Stomberg and X. Xue, JCAP 05 (2024) 030, arXiv:2312.03069 [gr-qc].
  • 125. P. F. Depta, V. Domcke, G. Franciolini and M. Pieroni (7 2024) arXiv:2407.14460 [astro-ph.CO].
  • 126. K. Inomata, M. Kamionkowski, C. M. Toral and S. R. Taylor (5 2024) arXiv:2406.00096 [astro-ph.CO].
  • 127. W. Hu, Q. Liang, M.-X. Lin and M. Trodden (8 2024) arXiv:2408.11774 [astro-ph.CO].
  • 128. F. A. Jenet and J. D. Romano, Am. J. Phys. 83 (2015) 635, arXiv:1412.1142 [gr-qc].
  • 129. J. D. Romano and B. Allen, Class. Quant. Grav. 41 (2024) 175008, arXiv:2308.05847 [gr-qc].
  • 130. G. Hobbs, R. Edwards and R. Manchester, Mon. Not. Roy. Astron. Soc. 369 (2006) 655, arXiv:astro-ph/0603381.
  • 131. R. T. Edwards, G. B. Hobbs and R. N. Manchester, Mon. Not. Roy. Astron. Soc. 372 (2006) 1549, arXiv:astro-ph/0607664.
  • 132. G. Hobbs, F. Jenet, K. J. Lee, J. P. W. Verbiest, D. Yardley, R. Manchester, A. Lommen, W. Coles, R. Edwards and C. Shettigara, Mon. Not. Roy. Astron. Soc. 394 (2009) 1945, arXiv:0901.0592 [astro-ph.SR].
  • 133. J. A. Ellis, M. Vallisneri, S. R. Taylor and P. T. Baker, Enterprise: Enhanced numerical toolbox enabling a robust pulsar inference suite Zenodo (September, 2020).
  • 134. R. C. Bernardo and K.-W. Ng, PTAfast: PTA correlations from stochastic gravitational wave background Astrophysics Source Code Library, record ascl:2211.001 (November, 2022).
  • 135. A. Mitridate, D. Wright, R. von Eckardstein, T. Schröder, J. Nay, K. Olum, K. Schmitz and T. Trickle (6 2023) arXiv:2306.16377 [hep-ph].
  • 136. R. Blandford, R. Narayan and R. W. Romani, Journal of Astrophysics and Astronomy 5 (December 1984) 369.
  • 137. J. S. Hazboun, J. D. Romano and T. L. Smith, Phys. Rev. D 100 (2019) 104028, arXiv:1907.04341 [gr-qc].
  • 138. J. Antoniadis et al. (6 2023) arXiv:2306.16224 [astro-ph.HE].
  • 139. NANOGrav Collaboration (G. Agazie et al.), Astrophys. J. Lett. 951 (2023) L9, arXiv:2306.16217 [astro-ph.HE].
  • 140. W. DeRocco and J. A. Dror, Phys. Rev. Lett. 132 (2024) 101403, arXiv:2212.09751 [astro-ph.HE].
  • 141. W. DeRocco and J. A. Dror, Phys. Rev. D 108 (2023) 103011, arXiv:2304.13042 [astro-ph.HE].
  • 142. L. G. Book and E. E. Flanagan, Phys. Rev. D 83 (2011) 024024, arXiv:1009.4192 [astro-ph.CO].
  • 143. S. A. Klioner, Class. Quant. Grav. 35 (2018) 045005, arXiv:1710.11474 [astro-ph.HE].
  • 144. J. Darling, A. E. Truebenbach and J. Paine, Astrophys. J. 861 (2018) 113, arXiv:1804.06986 [astro-ph.IM].
  • 145. S. Jaraba, J. Garcı́a-Bellido, S. Kuroyanagi, S. Ferraiuolo and M. Braglia, Mon. Not. Roy. Astron. Soc. 524 (2023) 3609, arXiv:2304.06350 [astro-ph.CO].
  • 146. R. Reyes and C. C. Bernido, arXiv e-prints (March 2023) arXiv:2303.00931, arXiv:2303.00931 [astro-ph.IM].
  • 146. R. Reyes and C. C. Bernido, arXiv e-prints (March 2023) arXiv:2303.00931, arXiv:2303.00931 [astro-ph.IM].
  • 147. R. van Haasteren and Y. Levin, Mon. Not. Roy. Astron. Soc. 428 (2013) 1147, arXiv:1202.5932 [astro-ph.IM].
  • 148. R. van Haasteren and M. Vallisneri, Phys. Rev. D 90 (2014) 104012, arXiv:1407.1838 [gr-qc].
  • 149. C. Tiburzi, G. Hobbs, M. Kerr, W. A. Coles, S. Dai, R. N. Manchester, A. Possenti, R. M. Shannon and X. P. You, Mon. Not. Roy. Astron. Soc. 455 (February 2016) 4339, arXiv:1510.02363 [astro-ph.IM].
  • 150. E. Roebber, Astrophys. J. 876 (2019) 55, arXiv:1901.05468 [astro-ph.HE].
  • 151. NANOGrav Collaboration (M. Vallisneri et al.) (1 2020) arXiv:2001.00595 [astro-ph.HE].
  • 152. R. K. Sachs and A. M. Wolfe, Astrophys. J. 147 (1967) 73.
  • 153. N. M. J. Cruz, A. Malhotra, G. Tasinato and I. Zavala (6 2024) arXiv:2406.04957 [astro-ph.CO].
  • 154. C. M. F. Mingarelli, K. Grover, T. Sidery, R. J. E. Smith and A. Vecchio, Phys. Rev. Lett. 109 (2012) 081104, arXiv:1207.5645 [astro-ph.HE].
  • 155. A. Susobhanan, A. Gopakumar, G. Hobbs and S. R. Taylor, Phys. Rev. D 101 (2020) 043022, arXiv:2002.03285 [gr-qc].
  • 156. A. Susobhanan, Class. Quant. Grav. 40 (2023) 155014, arXiv:2210.11454 [gr-qc].
  • 157. V. De Falco and E. Battista, Phys. Rev. D 108 (2023) 064032, arXiv:2309.00319 [gr-qc].
  • 158. V. De Falco, E. Battista, D. Usseglio and S. Capozziello, Eur. Phys. J. C 84 (2024) 137, arXiv:2401.13374 [gr-qc].
  • 159. NANOGrav Collaboration (A. Afzal et al.), Astrophys. J. Lett. 951 (2023) L11, arXiv:2306.16219 [astro-ph.HE].
  • 160. Y.-M. Wu, Z.-C. Chen and Q.-G. Huang, Sci. China Phys. Mech. Astron. 67 (2024) 240412, arXiv:2307.03141 [astro-ph.CO].
  • 161. J. Ellis, M. Fairbairn, G. Franciolini, G. Hütsi, A. Iovino, M. Lewicki, M. Raidal, J. Urrutia, V. Vaskonen and H. Veermäe, Phys. Rev. D 109 (2024) 023522, arXiv:2308.08546 [astro-ph.CO].
  • 162. R. C. Bernardo and K.-W. Ng, JCAP 08 (2023) 028, arXiv:2304.07040 [gr-qc].
  • 163. B. Allen and J. D. Romano (7 2024) arXiv:2407.10968 [gr-qc].
  • 164. B. Allen (4 2024) arXiv:2404.05677 [gr-qc].
  • 165. Y.-M. Wu, Y.-C. Bi and Q.-G. Huang (7 2024) arXiv:2407.07319 [astro-ph.CO].
  • 166. K.-W. Ng, Int. J. Mod. Phys. D 7 (January 1998) 89.
  • 167. K.-W. Ng and G.-C. Liu, Int. J. Mod. Phys. D 8 (1999) 61, arXiv:astro-ph/9710012.
  • 168. R. C. Bernardo and K.-W. Ng (10 2023) arXiv:2310.07537 [gr-qc].
  • 169. R. C. Bernardo and K.-W. Ng, Phys. Rev. D 109 (2024) L101502, arXiv:2306.13593 [gr-qc].
  • 170. S. Weinberg, Cosmology (OUP Oxford, 2008).
  • 171. L. ISSERLIS, Biometrika 12 (11 1918) 134.
  • 172. M. Srednicki, Astrophys. J. Lett. 416 (1993) L1, arXiv:astro-ph/9306012.
  • 173. A. Gangui and L. Perivolaropoulos, Astrophys. J. 447 (1995) 1, arXiv:astro-ph/9408034.
  • 174. Planck Collaboration (Y. Akrami et al.), Astron. Astrophys. 641 (2020) A7, arXiv:1906.02552 [astro-ph.CO].
  • 175. T. Clifton, P. G. Ferreira, A. Padilla and C. Skordis, Phys. Rept. 513 (2012) 1, arXiv:1106.2476 [astro-ph.CO].
  • 176. A. Joyce, B. Jain, J. Khoury and M. Trodden, Phys. Rept. 568 (2015) 1, arXiv:1407.0059 [astro-ph.CO].
  • 177. S. Nojiri, S. D. Odintsov and V. K. Oikonomou, Phys. Rept. 692 (2017) 1, arXiv:1705.11098 [gr-qc].
  • 178. R. Kase and S. Tsujikawa, Int. J. Mod. Phys. D 28 (2019) 1942005, arXiv:1809.08735 [gr-qc].
  • 179. P. G. Ferreira, Ann. Rev. Astron. Astrophys. 57 (2019) 335, arXiv:1902.10503 [astro-ph.CO].
  • 180. J. Nay, K. K. Boddy, T. L. Smith and C. M. F. Mingarelli, Phys. Rev. D 110 (2024) 044062, arXiv:2306.06168 [gr-qc].
  • 181. S. Babak, M. Falxa, G. Franciolini and M. Pieroni (4 2024) arXiv:2404.02864 [astro-ph.CO].
  • 182. S. Wang and Z.-C. Zhao, Phys. Rev. D 109 (2024) L061502, arXiv:2307.04680 [astro-ph.HE].
  • 183. Y.-M. Wu, Z.-C. Chen, Y.-C. Bi and Q.-G. Huang, Class. Quant. Grav. 41 (2024) 075002, arXiv:2310.07469 [astro-ph.CO].
  • 184. Y.-C. Bi, Y.-M. Wu, Z.-C. Chen and Q.-G. Huang, Phys. Rev. D 109 (2024) L061101, arXiv:2310.08366 [astro-ph.CO].
  • 185. Z.-C. Chen, Y.-M. Wu, Y.-C. Bi and Q.-G. Huang, Phys. Rev. D 109 (2024) 084045, arXiv:2310.11238 [astro-ph.CO].
  • 186. C. Choi, J. Magallanes, M. Gurgenidze and T. Kahniashvili (12 2023) arXiv:2312.03932 [astro-ph.CO].
  • 187. Z.-C. Chen, J. Li, L. Liu and Z. Yi, Phys. Rev. D 109 (2024) L101302, arXiv:2401.09818 [gr-qc].
  • 188. S. R. Taylor and J. R. Gair, Phys. Rev. D 88 (2013) 084001, arXiv:1306.5395 [gr-qc].
  • 189. R. Kato and J. Soda, Phys. Rev. D 93 (2016) 062003, arXiv:1512.09139 [gr-qc].
  • 190. G. Sato-Polito and M. Kamionkowski, Phys. Rev. D 106 (2022) 023004, arXiv:2111.05867 [astro-ph.CO].
  • 191. NANOGrav Collaboration (G. Agazie et al.), Astrophys. J. Lett. 956 (2023) L3, arXiv:2306.16221 [astro-ph.HE].
  • 192. G. Tasinato, Phys. Rev. D 108 (2023) 103521, arXiv:2309.00403 [gr-qc].
  • 193. N. M. J. Cruz, A. Malhotra, G. Tasinato and I. Zavala (2 2024) arXiv:2402.17312 [gr-qc].
  • 194. X. Siemens, J. Ellis, F. Jenet and J. D. Romano, Class. Quant. Grav. 30 (2013) 224015, arXiv:1305.3196 [astro-ph.IM].
  • 195. C. J. Moore, S. R. Taylor and J. R. Gair, Class. Quant. Grav. 32 (2015) 055004, arXiv:1406.5199 [astro-ph.IM].
  • 196. S. J. Vigeland and X. Siemens, Phys. Rev. D 94 (2016) 123003, arXiv:1609.03656 [astro-ph.IM].
  • 197. B. Allen, D. Agarwal, J. D. Romano and S. Valtolina (6 2024) arXiv:2406.16031 [gr-qc].
  • 198. EPTA Collaboration (J. Antoniadis et al.) (6 2023) arXiv:2306.16226 [astro-ph.HE].
  • 199. NANOGrav Collaboration (G. Agazie et al.), Astrophys. J. 963 (2024) 144, arXiv:2309.17438 [astro-ph.HE].
  • 200. NANOGrav Collaboration (G. Agazie et al.), Astrophys. J. Lett. 951 (2023) L50, arXiv:2306.16222 [astro-ph.HE].
  • 201. L. O’Beirne, N. J. Cornish, S. J. Vigeland and S. R. Taylor, Phys. Rev. D 99 (2019) 124039, arXiv:1904.02744 [gr-qc].
  • 202. T. J. W. Lazio, Class. Quant. Grav. 30 (2013) 224011.
  • 203. A. Weltman et al., Publ. Astron. Soc. Austral. 37 (2020) e002, arXiv:1810.02680 [astro-ph.CO].
  • 204. B. Chandra Joshi et al., J. Astrophys. Astron. 43 (2022) 98, arXiv:2207.06461 [astro-ph.HE].