Impact of Correlated Test-Mass Noise on Stochastic Gravitational-Wave Background Searches with Space-Based Interferometers

Author(s)

Wu, Jing-yi, Tang, Yong

Abstract

The stochastic gravitational-wave background (SGWB) is a major science target of future space-based interferometers because it encodes information about unresolved astrophysical populations and physical processes in the early Universe. Its stochastic nature, however, means that its detection and parameter inference rely on distinguishing its statistical contribution to the data from that of instrumental noise. In particular, both the SGWB and instrumental noise contribute to the data covariance, so an incomplete noise model can be misinterpreted as part of the SGWB and thereby bias the inferred spectrum and its parameters. Here we investigate this effect in the presence of correlations between the acceleration (ACC) noises of adjacent test masses (TMs) in each spacecraft. We simulate the correlated noise at the single-link level, propagate it through the moving-orbit response into the time-delay interferometry (TDI) observables, and analyze the resulting TDI auto spectra with two Bayesian frequency-domain templates: the Corr. Template includes the correlated-noise covariance, whereas the Non-Corr. Template neglects it. We find that, when correlated ACC noise is present in the mock data, the Non-Corr. Template exhibits systematic parameter offsets that generally become more pronounced with increasing correlation amplitude. In contrast, the Corr. Template recovers the injected SGWB and instrumental-noise parameters substantially more accurately. These results demonstrate that correlations in test-mass ACC noise can constitute a relevant systematic for space-based SGWB searches and motivate their explicit inclusion in mission-realistic noise models and inference pipelines.

Figures

Taiji constellation and single-link notation, where the three spacecrafts are labeled as $\vb{x}_1$, $\vb{x}_2$ and $\vb{x}_3$. The arrows along the arms indicate the six directed optical paths $\boldsymbol{L}_{ij}$ from emitting spacecraft $\vb{x}_j$ to receiving spacecraft $\vb{x}_i$. The jitter motions of the TMs and OBs are denoted by $\boldsymbol{\delta}_{ij}$ and $\boldsymbol{\Delta}_{ij}$, with indices directed from the local to the distant spacecraft.
Caption Taiji constellation and single-link notation, where the three spacecrafts are labeled as $\vb{x}_1$, $\vb{x}_2$ and $\vb{x}_3$. The arrows along the arms indicate the six directed optical paths $\boldsymbol{L}_{ij}$ from emitting spacecraft $\vb{x}_j$ to receiving spacecraft $\vb{x}_i$. The jitter motions of the TMs and OBs are denoted by $\boldsymbol{\delta}_{ij}$ and $\boldsymbol{\Delta}_{ij}$, with indices directed from the local to the distant spacecraft.
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Marginalized $AE$-channel posteriors for the WN model in Eq.~\eqref{eq:cacc_wn}. The panels show $\log_{10}\Omega_0$ (top left), $\gamma$ (top right), $s_{\rm oms}$ (bottom left), and $s_{\rm acc}$ (bottom right). Blue and green half-violins denote the Corr. and Non-Corr. Templates, respectively. The internal black bars mark posterior quartiles, and the horizontal black dashed line marks the injected value. The lower axis gives the injected correlation $\rho$, and the upper axis gives the corresponding network SNR.
Caption Marginalized $AE$-channel posteriors for the WN model in Eq.~\eqref{eq:cacc_wn}. The panels show $\log_{10}\Omega_0$ (top left), $\gamma$ (top right), $s_{\rm oms}$ (bottom left), and $s_{\rm acc}$ (bottom right). Blue and green half-violins denote the Corr. and Non-Corr. Templates, respectively. The internal black bars mark posterior quartiles, and the horizontal black dashed line marks the injected value. The lower axis gives the injected correlation $\rho$, and the upper axis gives the corresponding network SNR.
Marginalized $AE$-channel posteriors for the WN model in Eq.~\eqref{eq:cacc_wn}. The panels show $\log_{10}\Omega_0$ (top left), $\gamma$ (top right), $s_{\rm oms}$ (bottom left), and $s_{\rm acc}$ (bottom right). Blue and green half-violins denote the Corr. and Non-Corr. Templates, respectively. The internal black bars mark posterior quartiles, and the horizontal black dashed line marks the injected value. The lower axis gives the injected correlation $\rho$, and the upper axis gives the corresponding network SNR.
Caption Marginalized $AE$-channel posteriors for the WN model in Eq.~\eqref{eq:cacc_wn}. The panels show $\log_{10}\Omega_0$ (top left), $\gamma$ (top right), $s_{\rm oms}$ (bottom left), and $s_{\rm acc}$ (bottom right). Blue and green half-violins denote the Corr. and Non-Corr. Templates, respectively. The internal black bars mark posterior quartiles, and the horizontal black dashed line marks the injected value. The lower axis gives the injected correlation $\rho$, and the upper axis gives the corresponding network SNR.
Marginalized $AE$-channel posteriors for the WN model in Eq.~\eqref{eq:cacc_wn}. The panels show $\log_{10}\Omega_0$ (top left), $\gamma$ (top right), $s_{\rm oms}$ (bottom left), and $s_{\rm acc}$ (bottom right). Blue and green half-violins denote the Corr. and Non-Corr. Templates, respectively. The internal black bars mark posterior quartiles, and the horizontal black dashed line marks the injected value. The lower axis gives the injected correlation $\rho$, and the upper axis gives the corresponding network SNR.
Caption Marginalized $AE$-channel posteriors for the WN model in Eq.~\eqref{eq:cacc_wn}. The panels show $\log_{10}\Omega_0$ (top left), $\gamma$ (top right), $s_{\rm oms}$ (bottom left), and $s_{\rm acc}$ (bottom right). Blue and green half-violins denote the Corr. and Non-Corr. Templates, respectively. The internal black bars mark posterior quartiles, and the horizontal black dashed line marks the injected value. The lower axis gives the injected correlation $\rho$, and the upper axis gives the corresponding network SNR.
Marginalized $AE$-channel posteriors for the WN model in Eq.~\eqref{eq:cacc_wn}. The panels show $\log_{10}\Omega_0$ (top left), $\gamma$ (top right), $s_{\rm oms}$ (bottom left), and $s_{\rm acc}$ (bottom right). Blue and green half-violins denote the Corr. and Non-Corr. Templates, respectively. The internal black bars mark posterior quartiles, and the horizontal black dashed line marks the injected value. The lower axis gives the injected correlation $\rho$, and the upper axis gives the corresponding network SNR.
Caption Marginalized $AE$-channel posteriors for the WN model in Eq.~\eqref{eq:cacc_wn}. The panels show $\log_{10}\Omega_0$ (top left), $\gamma$ (top right), $s_{\rm oms}$ (bottom left), and $s_{\rm acc}$ (bottom right). Blue and green half-violins denote the Corr. and Non-Corr. Templates, respectively. The internal black bars mark posterior quartiles, and the horizontal black dashed line marks the injected value. The lower axis gives the injected correlation $\rho$, and the upper axis gives the corresponding network SNR.
Marginalized $AE$-channel posteriors for the PS model in Eq.~\eqref{eq:cacc_ps}. The panels show $\log_{10}\Omega_0$ (top left), $\gamma$ (top right), $s_{\rm oms}$ (bottom left), and $s_{\rm acc}$ (bottom right). Blue and green half-violins denote the Corr. and Non-Corr. Templates, respectively. The internal black bars mark posterior quartiles, and the horizontal black dashed line marks the injected value. The lower axis gives the injected correlation $\rho$, and the upper axis gives the corresponding network SNR.
Caption Marginalized $AE$-channel posteriors for the PS model in Eq.~\eqref{eq:cacc_ps}. The panels show $\log_{10}\Omega_0$ (top left), $\gamma$ (top right), $s_{\rm oms}$ (bottom left), and $s_{\rm acc}$ (bottom right). Blue and green half-violins denote the Corr. and Non-Corr. Templates, respectively. The internal black bars mark posterior quartiles, and the horizontal black dashed line marks the injected value. The lower axis gives the injected correlation $\rho$, and the upper axis gives the corresponding network SNR.
Marginalized $AE$-channel posteriors for the PS model in Eq.~\eqref{eq:cacc_ps}. The panels show $\log_{10}\Omega_0$ (top left), $\gamma$ (top right), $s_{\rm oms}$ (bottom left), and $s_{\rm acc}$ (bottom right). Blue and green half-violins denote the Corr. and Non-Corr. Templates, respectively. The internal black bars mark posterior quartiles, and the horizontal black dashed line marks the injected value. The lower axis gives the injected correlation $\rho$, and the upper axis gives the corresponding network SNR.
Caption Marginalized $AE$-channel posteriors for the PS model in Eq.~\eqref{eq:cacc_ps}. The panels show $\log_{10}\Omega_0$ (top left), $\gamma$ (top right), $s_{\rm oms}$ (bottom left), and $s_{\rm acc}$ (bottom right). Blue and green half-violins denote the Corr. and Non-Corr. Templates, respectively. The internal black bars mark posterior quartiles, and the horizontal black dashed line marks the injected value. The lower axis gives the injected correlation $\rho$, and the upper axis gives the corresponding network SNR.
Marginalized $AE$-channel posteriors for the PS model in Eq.~\eqref{eq:cacc_ps}. The panels show $\log_{10}\Omega_0$ (top left), $\gamma$ (top right), $s_{\rm oms}$ (bottom left), and $s_{\rm acc}$ (bottom right). Blue and green half-violins denote the Corr. and Non-Corr. Templates, respectively. The internal black bars mark posterior quartiles, and the horizontal black dashed line marks the injected value. The lower axis gives the injected correlation $\rho$, and the upper axis gives the corresponding network SNR.
Caption Marginalized $AE$-channel posteriors for the PS model in Eq.~\eqref{eq:cacc_ps}. The panels show $\log_{10}\Omega_0$ (top left), $\gamma$ (top right), $s_{\rm oms}$ (bottom left), and $s_{\rm acc}$ (bottom right). Blue and green half-violins denote the Corr. and Non-Corr. Templates, respectively. The internal black bars mark posterior quartiles, and the horizontal black dashed line marks the injected value. The lower axis gives the injected correlation $\rho$, and the upper axis gives the corresponding network SNR.
Marginalized $AE$-channel posteriors for the PS model in Eq.~\eqref{eq:cacc_ps}. The panels show $\log_{10}\Omega_0$ (top left), $\gamma$ (top right), $s_{\rm oms}$ (bottom left), and $s_{\rm acc}$ (bottom right). Blue and green half-violins denote the Corr. and Non-Corr. Templates, respectively. The internal black bars mark posterior quartiles, and the horizontal black dashed line marks the injected value. The lower axis gives the injected correlation $\rho$, and the upper axis gives the corresponding network SNR.
Caption Marginalized $AE$-channel posteriors for the PS model in Eq.~\eqref{eq:cacc_ps}. The panels show $\log_{10}\Omega_0$ (top left), $\gamma$ (top right), $s_{\rm oms}$ (bottom left), and $s_{\rm acc}$ (bottom right). Blue and green half-violins denote the Corr. and Non-Corr. Templates, respectively. The internal black bars mark posterior quartiles, and the horizontal black dashed line marks the injected value. The lower axis gives the injected correlation $\rho$, and the upper axis gives the corresponding network SNR.
Pearson correlation coefficients between $\rho$ and the other parameters for the WN model, evaluated with the diagonal $AE$ likelihood. The strongest local degeneracy is between $\rho$ and $\gamma$, consistent with the leading posterior shift in Fig.~\ref{fig:violin_wn}.Fisher PCCs between $\rho$ and the remaining parameters for the WN model (left) and PS model (right), evaluated with the diagonal $AE$ likelihood. Blue, green, red, and purple solid lines denote $\log_{10}\Omega_0$, $\gamma$, $s_{\rm oms}$, and $s_{\rm acc}$, respectively. The horizontal black dashed line marks zero correlation.
Caption Pearson correlation coefficients between $\rho$ and the other parameters for the WN model, evaluated with the diagonal $AE$ likelihood. The strongest local degeneracy is between $\rho$ and $\gamma$, consistent with the leading posterior shift in Fig.~\ref{fig:violin_wn}.Fisher PCCs between $\rho$ and the remaining parameters for the WN model (left) and PS model (right), evaluated with the diagonal $AE$ likelihood. Blue, green, red, and purple solid lines denote $\log_{10}\Omega_0$, $\gamma$, $s_{\rm oms}$, and $s_{\rm acc}$, respectively. The horizontal black dashed line marks zero correlation.
Pearson correlation coefficients between $\rho$ and the other parameters for the PS model, evaluated with the diagonal $AE$ likelihood. The strongest local degeneracy is between $\rho$ and $s_{\rm acc}$, consistent with the dominant posterior shift in Fig.~\ref{fig:violin_ps}.Fisher PCCs between $\rho$ and the remaining parameters for the WN model (left) and PS model (right), evaluated with the diagonal $AE$ likelihood. Blue, green, red, and purple solid lines denote $\log_{10}\Omega_0$, $\gamma$, $s_{\rm oms}$, and $s_{\rm acc}$, respectively. The horizontal black dashed line marks zero correlation.
Caption Pearson correlation coefficients between $\rho$ and the other parameters for the PS model, evaluated with the diagonal $AE$ likelihood. The strongest local degeneracy is between $\rho$ and $s_{\rm acc}$, consistent with the dominant posterior shift in Fig.~\ref{fig:violin_ps}.Fisher PCCs between $\rho$ and the remaining parameters for the WN model (left) and PS model (right), evaluated with the diagonal $AE$ likelihood. Blue, green, red, and purple solid lines denote $\log_{10}\Omega_0$, $\gamma$, $s_{\rm oms}$, and $s_{\rm acc}$, respectively. The horizontal black dashed line marks zero correlation.
Representative TDI cross-power diagnostic for the PS model with $\rho=0.5$ in STFT segment 100. In the top panel, blue, orange, and green solid lines denote the $A$, $E$, and $T$ auto spectra. In the middle and bottom panels, the same colors denote the $AE$, $AT$, and $ET$ channel pairs, respectively. The middle panel shows $\abs{S_{UV}}$, and the bottom panel shows $\abs{S_{UV}}/\sqrt{S_U S_V}$.
Caption Representative TDI cross-power diagnostic for the PS model with $\rho=0.5$ in STFT segment 100. In the top panel, blue, orange, and green solid lines denote the $A$, $E$, and $T$ auto spectra. In the middle and bottom panels, the same colors denote the $AE$, $AT$, and $ET$ channel pairs, respectively. The middle panel shows $\abs{S_{UV}}$, and the bottom panel shows $\abs{S_{UV}}/\sqrt{S_U S_V}$.
Ensemble validation of the single-link ACC CSD for the WN model with $\rho=0.5$. In the top panel, the black solid line is the injected $C_{\rm acc}(f)$ and the blue dashed line is the ensemble mean of $\Re[\hat{C}]$. In the bottom panel, the black dotted line marks the theoretical value zero and the green dashed line is the ensemble mean of $\Im[\hat{C}]$.
Caption Ensemble validation of the single-link ACC CSD for the WN model with $\rho=0.5$. In the top panel, the black solid line is the injected $C_{\rm acc}(f)$ and the blue dashed line is the ensemble mean of $\Re[\hat{C}]$. In the bottom panel, the black dotted line marks the theoretical value zero and the green dashed line is the ensemble mean of $\Im[\hat{C}]$.
References
  • [1] P. Amaro-Seoane, H. Audley, S. Babak, J. Baker, E. Barausse, P. Bender, E. Berti, P. Binetruy, M. Born, D. Bortoluzzi, et al., arXiv e-prints arXiv:1702.00786 (2017), 1702.00786.
  • [1] P. Amaro-Seoane, H. Audley, S. Babak, J. Baker, E. Barausse, P. Bender, E. Berti, P. Binetruy, M. Born, D. Bortoluzzi, et al., arXiv e-prints arXiv:1702.00786 (2017), 1702.00786.
  • [2] W.-R. Hu and Y.-L. Wu, National Science Review 4, 685 (2017), ISSN 2095-5138, https://academic.oup.com/nsr/article-pdf/4/5/685/31566708/nwx116.pdf, URL https:// doi.org/10.1093/nsr/nwx116.
  • [3] J. Luo, L.-S. Chen, H.-Z. Duan, Y.-G. Gong, S. Hu, J. Ji, Q. Liu, J. Mei, V. Milyukov, M. Sazhin, et al., Classical and Quantum Gravity 33, 035010 (2016), ISSN 1361-6382, URL http://dx.doi.org/10.1088/0264-9381/33/3/035010.
  • [4] E. S. Phinney (2001), astro-ph/0108028, URL https://arxiv.org/abs/astro-ph/0108028.
  • [5] T. Regimbau and J. A. de Freitas Pacheco, The Astrophysical Journal 642, 455–461 (2006), ISSN 1538-4357, URL http://dx.doi.org/10.1086/500190.
  • [6] T. Regimbau, Physical Review D 75 (2007), ISSN 1550-2368, URL http://dx.doi.org/10. 1103/PhysRevD.75.043002.
  • [7] S. Marassi, R. Schneider, G. Corvino, V. Ferrari, and S. P. Zwart, Physical Review D 84 (2011), ISSN 1550-2368, URL http://dx.doi.org/10.1103/PhysRevD.84.124037.
  • [8] T. Regimbau, Research in Astronomy and Astrophysics 11, 369 (2011), URL https://doi. org/10.1088/1674-4527/11/4/001.
  • [9] X.-J. Zhu, E. J. Howell, D. G. Blair, and Z.-H. Zhu, Monthly Notices of the Royal Astronomical Society 431, 882–899 (2013), ISSN 1365-2966, URL http://dx.doi.org/10.1093/mnras/ stt207.
  • [10] X.-L. Fan and Y. Chen, Physical Review D 98, 044020 (2018), ISSN 2470-0029, URL http: //dx.doi.org/10.1103/PhysRevD.98.044020.
  • [11] S. Babak, C. Caprini, D. G. Figueroa, N. Karnesis, P. Marcoccia, G. Nardini, M. Pieroni, A. Ricciardone, A. Sesana, and J. Torrado, Journal of Cosmology and Astroparticle Physics 2023, 034 (2023), URL https://doi.org/10.1088/1475-7516/2023/08/034.
  • [12] M. Bonetti and A. Sesana, Physical Review D 102 (2020), ISSN 2470-0029, URL http: //dx.doi.org/10.1103/PhysRevD.102.103023.
  • [13] F. Pozzoli, S. Babak, A. Sesana, M. Bonetti, and N. Karnesis, Phys. Rev. D 108, 103039 (2023), URL https://link.aps.org/doi/10.1103/PhysRevD.108.103039.
  • [14] A. A. Starobinsky, JETP Lett. 30, 682 (1979).
  • [15] K. N. Ananda, C. Clarkson, and D. Wands, Physical Review D 75, 123518 (2007), ISSN 1550-2368, URL http://dx.doi.org/10.1103/PhysRevD.75.123518.
  • [16] R. Easther and E. A. Lim, Journal of Cosmology and Astroparticle Physics 2006, 010–010 (2006), ISSN 1475-7516, URL http://dx.doi.org/10.1088/1475-7516/2006/04/010.
  • [17] R. Easther, J. T. Giblin, and E. A. Lim, Physical Review Letters 99 (2007), ISSN 1079-7114, URL http://dx.doi.org/10.1103/PhysRevLett.99.221301.
  • [18] J. Garcı́a-Bellido and D. G. Figueroa, Physical Review Letters 98, 061302 (2007), ISSN 10797114, URL http://dx.doi.org/10.1103/PhysRevLett.98.061302.
  • [19] J.-F. Dufaux, A. Bergman, G. Felder, L. Kofman, and J.-P. Uzan, Physical Review D 76, 123517 (2007), ISSN 1550-2368, URL http://dx.doi.org/10.1103/PhysRevD.76.123517.
  • [20] J. Garcı́a-Bellido, D. G. Figueroa, and A. Sastre, Physical Review D 77 (2008), ISSN 15502368, URL http://dx.doi.org/10.1103/PhysRevD.77.043517.
  • [21] L. Bethke, D. G. Figueroa, and A. Rajantie, Physical Review Letters 111 (2013), ISSN 10797114, URL http://dx.doi.org/10.1103/PhysRevLett.111.011301.
  • [22] L. Bethke, D. G. Figueroa, and A. Rajantie, Journal of Cosmology and Astroparticle Physics 2014, 047–047 (2014), ISSN 1475-7516, URL http://dx.doi.org/10.1088/1475-7516/ 2014/06/047.
  • [23] M. Kamionkowski, A. Kosowsky, and M. S. Turner, Physical Review D 49, 2837–2851 (1994), ISSN 0556-2821, URL http://dx.doi.org/10.1103/PhysRevD.49.2837.
  • [24] R. Apreda, M. Maggiore, A. Nicolis, and A. Riotto, Nuclear Physics B 631, 342–368 (2002), ISSN 0550-3213, URL http://dx.doi.org/10.1016/S0550-3213(02)00264-X.
  • [25] C. Grojean and G. Servant, Physical Review D 75 (2007), ISSN 1550-2368, URL http: //dx.doi.org/10.1103/PhysRevD.75.043507.
  • [26] C. Caprini, R. Durrer, and G. Servant, Physical Review D 77 (2008), ISSN 1550-2368, URL http://dx.doi.org/10.1103/PhysRevD.77.124015.
  • [27] S. J. Huber and T. Konstandin, Journal of Cosmology and Astroparticle Physics 2008, 022 (2008), ISSN 1475-7516, URL http://dx.doi.org/10.1088/1475-7516/2008/09/022.
  • [28] C. Caprini, R. Durrer, T. Konstandin, and G. Servant, Physical Review D 79 (2009), ISSN 1550-2368, URL http://dx.doi.org/10.1103/PhysRevD.79.083519.
  • [29] L. Leitao, A. Mégevand, and A. D. Sánchez, Journal of Cosmology and Astroparticle Physics 2012, 024–024 (2012), ISSN 1475-7516, URL http://dx.doi.org/10.1088/1475-7516/ 2012/10/024.
  • [30] M. Hindmarsh, S. J. Huber, K. Rummukainen, and D. J. Weir, Physical Review Letters 112, 041301 (2014), ISSN 1079-7114, URL http://dx.doi.org/10.1103/PhysRevLett.112. 041301.
  • [31] J. T. Giblin and J. B. Mertens, Physical Review D 90 (2014), ISSN 1550-2368, URL http: //dx.doi.org/10.1103/PhysRevD.90.023532.
  • [32] R. Jinno and M. Takimoto, Physical Review D 95 (2017), ISSN 2470-0029, URL http://dx. doi.org/10.1103/PhysRevD.95.024009.
  • [33] C. Caprini, M. Hindmarsh, S. Huber, T. Konstandin, J. Kozaczuk, G. Nardini, J. M. No, A. Petiteau, P. Schwaller, G. Servant, et al., Journal of Cosmology and Astroparticle Physics 2016, 001–001 (2016), ISSN 1475-7516, URL http://dx.doi.org/10.1088/1475-7516/2016/04/ 001.
  • [34] W. Chao, H.-K. Guo, and J. Shu, Journal of Cosmology and Astroparticle Physics 2017, 009–009 (2017), ISSN 1475-7516, URL http://dx.doi.org/10.1088/1475-7516/2017/09/ 009.
  • [35] D. J. Weir, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, 20170126 (2018), ISSN 1471-2962, URL http://dx.doi.org/10. 1098/rsta.2017.0126.
  • [36] M. Hindmarsh and M. Hijazi, Journal of Cosmology and Astroparticle Physics 2019, 062–062 (2019), ISSN 1475-7516, URL http://dx.doi.org/10.1088/1475-7516/2019/12/062.
  • [37] T. Vachaspati and A. Vilenkin, Phys. Rev. D 31, 3052 (1985), URL https://link.aps.org/ doi/10.1103/PhysRevD.31.3052.
  • [38] M. Sakellariadou, Phys. Rev. D 42, 354 (1990), URL https://link.aps.org/doi/10.1103/ PhysRevD.42.354.
  • [39] R. R. Caldwell and B. Allen, Phys. Rev. D 45, 3447 (1992), URL https://link.aps.org/ doi/10.1103/PhysRevD.45.3447.
  • [40] R. R. Caldwell, R. A. Battye, and E. P. S. Shellard, Physical Review D 54, 7146–7152 (1996), ISSN 1089-4918, URL http://dx.doi.org/10.1103/PhysRevD.54.7146.
  • [41] D. G. Figueroa, M. Hindmarsh, and J. Urrestilla, Physical Review Letters 110 (2013), ISSN 1079-7114, URL http://dx.doi.org/10.1103/PhysRevLett.110.101302.
  • [42] J. J. Blanco-Pillado and K. D. Olum, Physical Review D 96 (2017), ISSN 2470-0029, URL http://dx.doi.org/10.1103/PhysRevD.96.104046.
  • [43] J. Liu, R.-G. Cai, and Z.-K. Guo, Physical Review Letters 126, 141303 (2021), ISSN 10797114, URL http://dx.doi.org/10.1103/PhysRevLett.126.141303.
  • [44] C. Caprini and D. G. Figueroa, Classical and Quantum Gravity 35, 163001 (2018), ISSN 1361-6382, URL http://dx.doi.org/10.1088/1361-6382/aac608.
  • [45] P. Auclair, D. Bacon, T. Baker, T. Barreiro, N. Bartolo, E. Belgacem, N. Bellomo, I. BenDayan, D. Bertacca, M. Besancon, et al., Living Reviews in Relativity 26 (2023), ISSN 14338351, URL http://dx.doi.org/10.1007/s41114-023-00045-2.
  • [46] M. R. Adams and N. J. Cornish, Physical Review D 82, 022002 (2010), ISSN 1550-2368, URL http://dx.doi.org/10.1103/PhysRevD.82.022002.
  • [47] M. R. Adams and N. J. Cornish, Physical Review D 89 (2014), ISSN 1550-2368, URL http: //dx.doi.org/10.1103/PhysRevD.89.022001.
  • [48] C. Caprini, D. G. Figueroa, R. Flauger, G. Nardini, M. Peloso, M. Pieroni, A. Ricciardone, and G. Tasinato, Journal of Cosmology and Astroparticle Physics 2019, 017–017 (2019), ISSN 1475-7516, URL http://dx.doi.org/10.1088/1475-7516/2019/11/017.
  • [49] R. Flauger, N. Karnesis, G. Nardini, M. Pieroni, A. Ricciardone, and J. Torrado, Journal of Cosmology and Astroparticle Physics 2021, 059–059 (2021), ISSN 1475-7516, URL http: //dx.doi.org/10.1088/1475-7516/2021/01/059.
  • [50] Q. Baghi, N. Karnesis, J.-B. Bayle, M. Besançon, and H. Inchauspé, Journal of Cosmology and Astroparticle Physics 2023, 066 (2023), ISSN 1475-7516, URL http://dx.doi.org/10. 1088/1475-7516/2023/04/066.
  • [51] M. Muratore, J. Gair, and L. Speri, Phys. Rev. D 109, 042001 (2024), URL https://link. aps.org/doi/10.1103/PhysRevD.109.042001.
  • [52] F. Pozzoli, R. Buscicchio, C. J. Moore, F. Haardt, and A. Sesana, Physical Review D 109, 083029 (2024), ISSN 2470-0029, URL http://dx.doi.org/10.1103/PhysRevD.109.083029.
  • [53] A. Santini, M. Muratore, J. Gair, and O. Hartwig, Phys. Rev. D 112, 084050 (2025), URL https://link.aps.org/doi/10.1103/csx9-9trp.
  • [54] O. Hartwig, M. Lilley, M. Muratore, and M. Pieroni, Phys. Rev. D 107, 123531 (2023), URL https://link.aps.org/doi/10.1103/PhysRevD.107.123531.
  • [55] G. Wang, B. Li, P. Xu, and X. Fan, Physical Review D 106, 044054 (2022), ISSN 2470-0029, URL http://dx.doi.org/10.1103/PhysRevD.106.044054.
  • [56] M. Braglia, G. Calcagni, G. Franciolini, J. Fumagalli, G. Nardini, M. Peloso, M. Pieroni, S. Renaux-Petel, A. Ricciardone, G. Tasinato, et al., Journal of Cosmology and Astroparticle Physics 2024, 032 (2024), URL https://doi.org/10.1088/1475-7516/2024/11/032.
  • [57] C. Caprini, R. Jinno, M. Lewicki, E. Madge, M. Merchand, G. Nardini, M. Pieroni, A. R. Pol, and V. Vaskonen, Journal of Cosmology and Astroparticle Physics 2024, 020 (2024), ISSN 1475-7516, URL http://dx.doi.org/10.1088/1475-7516/2024/10/020.
  • [58] J. J. Blanco-Pillado, Y. Cui, S. Kuroyanagi, M. Lewicki, G. Nardini, M. Pieroni, I. Y. Rybak, L. Sousa, J. M. Wachter, and for the LISA Cosmology Working Group, Journal of Cosmology and Astroparticle Physics 2025, 006 (2025), URL https://doi.org/10.1088/1475-7516/ 2025/05/006.
  • [59] J. E. Gammal, A. Ghaleb, G. Franciolini, T. Papanikolaou, M. Peloso, G. Perna, M. Pieroni, A. Ricciardone, R. Rosati, G. Tasinato, et al., Journal of Cosmology and Astroparticle Physics 2025, 062 (2025), URL https://doi.org/10.1088/1475-7516/2025/05/062.
  • [60] O. Hartwig and M. Muratore, Phys. Rev. D 105, 062006 (2022), URL https://link.aps. org/doi/10.1103/PhysRevD.105.062006.
  • [61] M. Muratore, O. Hartwig, D. Vetrugno, S. Vitale, and W. J. Weber, Phys. Rev. D 107, 082004 (2023), URL https://link.aps.org/doi/10.1103/PhysRevD.107.082004.
  • [62] G. Wang, Physical Review D 110, 064085 (2024), ISSN 2470-0029, URL http://dx.doi. org/10.1103/PhysRevD.110.064085.
  • [63] J. Kume, M. Peloso, M. Pieroni, and A. Ricciardone, Journal of Cosmology and Astroparticle Physics 2025, 030 (2025), ISSN 1475-7516, URL http://dx.doi.org/10.1088/1475-7516/ 2025/06/030.
  • [64] N. Aimen, P. Maturana-Russel, A. Vajpeyi, N. Christensen, and R. Meyer, Phys. Rev. D 113, 024022 (2026), URL https://link.aps.org/doi/10.1103/dcb6-1jsl.
  • [65] M. C. Digman and N. J. Cornish, The Astrophysical Journal 940, 10 (2022), ISSN 1538-4357, URL http://dx.doi.org/10.3847/1538-4357/ac9139.
  • [66] Z. Ren, T. Zhao, Z. Cao, Z.-K. Guo, W.-B. Han, H.-B. Jin, and Y.-L. Wu, Frontiers of Physics 18, 64302 (2023), ISSN 2095-0470, URL http://dx.doi.org/10.1007/s11467-023-1318-y.
  • [67] M. Du et al., Sci. China Phys. Mech. Astron. 69, 249501 (2026), 2505.16500, URL https: //doi.org/10.1007/s11433-025-2870-8.
  • [68] E.-K. Li, H. Wang, H.-Y. Chen, H. Fan, Y.-N. Li, Z.-Y. Li, Z.-C. Liang, X.-Y. Lyu, T.-X. Wang, Z. Wu, et al., Classical and Quantum Gravity 42, 165005 (2025), ISSN 1361-6382, URL http://dx.doi.org/10.1088/1361-6382/adf409.
  • [69] M. Du, Z. Luo, and P. Xu, Phys. Rev. D 112, 083036 (2025), URL https://link.aps.org/ doi/10.1103/gpmh-1hqx.
  • [70] S. Banagiri, A. Criswell, T. Kuan, V. Mandic, J. D. Romano, and S. R. Taylor, Monthly Notices of the Royal Astronomical Society 507, 5451 (2021), 2103.00826, URL https://doi. org/10.1093/mnras/stab2479.
  • [71] S. Rieck, A. W. Criswell, V. Korol, M. A. Keim, M. Bloom, and V. Mandic, Monthly Notices of the Royal Astronomical Society 531, 2642 (2024), 2308.12437, URL https://doi.org/10. 1093/mnras/stae1283.
  • [72] C. Tian, R. Ding, and X.-X. Kou, Journal of Cosmology and Astroparticle Physics 2025, 037 (2025), 2412.01219, URL https://doi.org/10.1088/1475-7516/2025/08/037.
  • [73] M. Bloom, A. W. Criswell, and V. Mandic, Physical Review D 112, 083021 (2025), 2412.16372, URL https://link.aps.org/doi/10.1103/jp2c-jb67.
  • [74] M. C. Digman and N. J. Cornish, Physical Review D 108, 023022 (2023), ISSN 2470-0029, URL http://dx.doi.org/10.1103/PhysRevD.108.023022.
  • [75] A. W. Criswell, S. Banagiri, J. Lawrence, L. Schult, S. Rieck, S. R. Taylor, and V. Mandic (2025), 2508.20308, URL https://arxiv.org/abs/2508.20308.
  • [76] G. Boileau, N. Christensen, and R. Meyer, Phys. Rev. D 106, 063025 (2022), 2204.03867, URL https://link.aps.org/doi/10.1103/PhysRevD.106.063025.
  • [77] M. Armano, H. Audley, J. Baird, P. Binetruy, M. Born, D. Bortoluzzi, E. Castelli, A. Cavalleri, A. Cesarini, A. M. Cruise, et al., Phys. Rev. Lett. 120, 061101 (2018), URL https://link. aps.org/doi/10.1103/PhysRevLett.120.061101.
  • [78] D. Quang Nam, J. Martino, Y. Lemière, A. Petiteau, J.-B. Bayle, O. Hartwig, and M. Staab, Phys. Rev. D 108, 082004 (2023), 2211.02539, URL https://link.aps.org/doi/10.1103/ PhysRevD.106.044054.
  • [79] M. Otto, Ph.D. thesis, Leibniz U., Hannover (2015).
  • [80] M. Tinto and S. V. Dhurandhar, Living Reviews in Relativity 24, 1 (2021), gr-qc/0409034, URL https://doi.org/10.1007/s41114-020-00029-6.
  • [81] M. Colpi, K. Danzmann, M. Hewitson, K. Holley-Bockelmann, P. Jetzer, G. Nelemans, A. Petiteau, D. Shoemaker, C. Sopuerta, R. Stebbins, et al., arXiv e-prints arXiv:2402.07571 (2024), 2402.07571.
  • [81] M. Colpi, K. Danzmann, M. Hewitson, K. Holley-Bockelmann, P. Jetzer, G. Nelemans, A. Petiteau, D. Shoemaker, C. Sopuerta, R. Stebbins, et al., arXiv e-prints arXiv:2402.07571 (2024), 2402.07571.
  • [82] J.-C. Yu, Y.-H. Yao, Y. Tang, and Y.-L. Wu, Phys. Rev. D 108, 083007 (2023), URL https: //link.aps.org/doi/10.1103/PhysRevD.108.083007.
  • [83] M. Armano, H. Audley, G. Auger, J. T. Baird, M. Bassan, P. Binetruy, M. Born, D. Bortoluzzi, N. Brandt, M. Caleno, et al., Phys. Rev. Lett. 116, 231101 (2016), URL https://link.aps. org/doi/10.1103/PhysRevLett.116.231101.
  • [84] B. Allen and J. D. Romano, Physical Review D 59, 102001 (1999), ISSN 1089-4918, URL http://dx.doi.org/10.1103/PhysRevD.59.102001.
  • [85] T. A. Prince, M. Tinto, S. L. Larson, and J. W. Armstrong, Physical Review D 66, 122002 (2002), ISSN 1089-4918, URL http://dx.doi.org/10.1103/PhysRevD.66.122002.
  • [86] M. Tinto, D. A. Shaddock, J. Sylvestre, and J. W. Armstrong, Physical Review D 67, 122003 (2003), ISSN 1089-4918, URL http://dx.doi.org/10.1103/PhysRevD.67.122003.
  • [87] S. Babak, M. Hewitson, and A. Petiteau (2021), 2108.01167, URL https://arxiv.org/abs/ 2108.01167.
  • [87] S. Babak, M. Hewitson, and A. Petiteau (2021), 2108.01167, URL https://arxiv.org/abs/ 2108.01167.
  • [88] N. Karnesis, M. Lilley, and A. Petiteau, Classical and Quantum Gravity 37, 215017 (2020), ISSN 1361-6382, URL http://dx.doi.org/10.1088/1361-6382/abb637.
  • [89] J. D. Romano and N. J. Cornish, Living Rev. Rel. 20, 2 (2017), 1608.06889, URL https: //doi.org/10.1007/s41114-017-0004-1.