A superconducting tensor detector for mid-frequency gravitational waves: its multi-channel nature and main astrophysical targets

Author(s)

Bae, Yeong-Bok, Park, Chan, Son, Edwin J., Ahn, Sang-Hyeon, Jeong, Minjoong, Kang, Gungwon, Kim, Chunglee, Kim, Dong Lak, Kim, Jaewan, Kim, Whansun, Lee, Hyung Mok, Lee, Yong-Ho, Norton, Ronald S., Oh, John J., Oh, Sang Hoon, Paik, Ho Jung

Abstract

Mid-frequency band gravitational-wave detectors will be complementary for the existing Earth-based detectors (sensitive above 10 Hz or so) and the future space-based detectors such as LISA, which will be sensitive below around 10 mHz. A ground-based superconducting omnidirectional gravitational radiation observatory (SOGRO) has recently been proposed along with several design variations for the frequency band of 0.1 to 10 Hz. For three conceptual designs of SOGRO (e.g., pSOGRO, SOGRO and aSOGRO), we examine their multi-channel natures, sensitivities and science cases. One of the key characteristics of the SOGRO concept is its six detection channels. The response functions of each channel are calculated for all possible gravitational wave polarizations including scalar and vector modes. Combining these response functions, we also confirm the omnidirectional nature of SOGRO. Hence, even a single SOGRO detector will be able to determine the position of a source and polarizations of gravitational waves, if detected. Taking into account SOGRO's sensitivity and technical requirements, two main targets are most plausible: gravitational waves from compact binaries and stochastic backgrounds. Based on assumptions we consider in this work, detection rates for intermediate-mass binary black holes (in the mass range of hundreds up to $10^{4}$$M_\odot$) are expected to be $0.0014-2.5 \,\, {\rm yr}^{-1}$. In order to detect stochastic gravitational wave background, multiple detectors are required. Two aSOGRO detector networks may be able to put limits on the stochastic background beyond the indirect limit from cosmological observations.

Figures

Responses of each channel in the SOGRO detector for plus ($+$), cross ($\times$), $x$, $y$, breathing ($b$), and longitudinal ($\ell$) polarization modes. The $+$ and $\times$ polarizations correspond to tensor modes, the $x$ and $y$ are vector modes, and the $b$ and $\ell$ polarizations are relevant to scalar modes. The responses of the (22) and the (31) channels are the same with the (11) and the (23) channels, respectively, up to $\pi/2$ rotation along the $z$-axis. The (33) channel is not shown as this is not used in sensitivity calculation. Combined (total) responses of all 5 channels are also depicted. The total responses show the omnidirectional nature of SOGRO; however, the total response of a terrestrial SOGRO to the $\ell$ mode vanishes in the direction of $z$-axis. In the last column, the general responses of a laser interferometer such as LIGO, Virgo or KAGRA are presented for comparison.

Responses of each channel in the SOGRO detector for plus ($+$), cross ($\times$), $x$, $y$, breathing ($b$), and longitudinal ($\ell$) polarization modes. The $+$ and $\times$ polarizations correspond to tensor modes, the $x$ and $y$ are vector modes, and the $b$ and $\ell$ polarizations are relevant to scalar modes. The responses of the (22) and the (31) channels are the same with the (11) and the (23) channels, respectively, up to $\pi/2$ rotation along the $z$-axis. The (33) channel is not shown as this is not used in sensitivity calculation. Combined (total) responses of all 5 channels are also depicted. The total responses show the omnidirectional nature of SOGRO; however, the total response of a terrestrial SOGRO to the $\ell$ mode vanishes in the direction of $z$-axis. In the last column, the general responses of a laser interferometer such as LIGO, Virgo or KAGRA are presented for comparison.


Responses of each channel in the SOGRO detector for plus ($+$), cross ($\times$), $x$, $y$, breathing ($b$), and longitudinal ($\ell$) polarization modes. The $+$ and $\times$ polarizations correspond to tensor modes, the $x$ and $y$ are vector modes, and the $b$ and $\ell$ polarizations are relevant to scalar modes. The responses of the (22) and the (31) channels are the same with the (11) and the (23) channels, respectively, up to $\pi/2$ rotation along the $z$-axis. The (33) channel is not shown as this is not used in sensitivity calculation. Combined (total) responses of all 5 channels are also depicted. The total responses show the omnidirectional nature of SOGRO; however, the total response of a terrestrial SOGRO to the $\ell$ mode vanishes in the direction of $z$-axis. In the last column, the general responses of a laser interferometer such as LIGO, Virgo or KAGRA are presented for comparison.

Responses of each channel in the SOGRO detector for plus ($+$), cross ($\times$), $x$, $y$, breathing ($b$), and longitudinal ($\ell$) polarization modes. The $+$ and $\times$ polarizations correspond to tensor modes, the $x$ and $y$ are vector modes, and the $b$ and $\ell$ polarizations are relevant to scalar modes. The responses of the (22) and the (31) channels are the same with the (11) and the (23) channels, respectively, up to $\pi/2$ rotation along the $z$-axis. The (33) channel is not shown as this is not used in sensitivity calculation. Combined (total) responses of all 5 channels are also depicted. The total responses show the omnidirectional nature of SOGRO; however, the total response of a terrestrial SOGRO to the $\ell$ mode vanishes in the direction of $z$-axis. In the last column, the general responses of a laser interferometer such as LIGO, Virgo or KAGRA are presented for comparison.


Responses of each channel in the SOGRO detector for plus ($+$), cross ($\times$), $x$, $y$, breathing ($b$), and longitudinal ($\ell$) polarization modes. The $+$ and $\times$ polarizations correspond to tensor modes, the $x$ and $y$ are vector modes, and the $b$ and $\ell$ polarizations are relevant to scalar modes. The responses of the (22) and the (31) channels are the same with the (11) and the (23) channels, respectively, up to $\pi/2$ rotation along the $z$-axis. The (33) channel is not shown as this is not used in sensitivity calculation. Combined (total) responses of all 5 channels are also depicted. The total responses show the omnidirectional nature of SOGRO; however, the total response of a terrestrial SOGRO to the $\ell$ mode vanishes in the direction of $z$-axis. In the last column, the general responses of a laser interferometer such as LIGO, Virgo or KAGRA are presented for comparison.

Responses of each channel in the SOGRO detector for plus ($+$), cross ($\times$), $x$, $y$, breathing ($b$), and longitudinal ($\ell$) polarization modes. The $+$ and $\times$ polarizations correspond to tensor modes, the $x$ and $y$ are vector modes, and the $b$ and $\ell$ polarizations are relevant to scalar modes. The responses of the (22) and the (31) channels are the same with the (11) and the (23) channels, respectively, up to $\pi/2$ rotation along the $z$-axis. The (33) channel is not shown as this is not used in sensitivity calculation. Combined (total) responses of all 5 channels are also depicted. The total responses show the omnidirectional nature of SOGRO; however, the total response of a terrestrial SOGRO to the $\ell$ mode vanishes in the direction of $z$-axis. In the last column, the general responses of a laser interferometer such as LIGO, Virgo or KAGRA are presented for comparison.


Responses of each channel in the SOGRO detector for plus ($+$), cross ($\times$), $x$, $y$, breathing ($b$), and longitudinal ($\ell$) polarization modes. The $+$ and $\times$ polarizations correspond to tensor modes, the $x$ and $y$ are vector modes, and the $b$ and $\ell$ polarizations are relevant to scalar modes. The responses of the (22) and the (31) channels are the same with the (11) and the (23) channels, respectively, up to $\pi/2$ rotation along the $z$-axis. The (33) channel is not shown as this is not used in sensitivity calculation. Combined (total) responses of all 5 channels are also depicted. The total responses show the omnidirectional nature of SOGRO; however, the total response of a terrestrial SOGRO to the $\ell$ mode vanishes in the direction of $z$-axis. In the last column, the general responses of a laser interferometer such as LIGO, Virgo or KAGRA are presented for comparison.

Responses of each channel in the SOGRO detector for plus ($+$), cross ($\times$), $x$, $y$, breathing ($b$), and longitudinal ($\ell$) polarization modes. The $+$ and $\times$ polarizations correspond to tensor modes, the $x$ and $y$ are vector modes, and the $b$ and $\ell$ polarizations are relevant to scalar modes. The responses of the (22) and the (31) channels are the same with the (11) and the (23) channels, respectively, up to $\pi/2$ rotation along the $z$-axis. The (33) channel is not shown as this is not used in sensitivity calculation. Combined (total) responses of all 5 channels are also depicted. The total responses show the omnidirectional nature of SOGRO; however, the total response of a terrestrial SOGRO to the $\ell$ mode vanishes in the direction of $z$-axis. In the last column, the general responses of a laser interferometer such as LIGO, Virgo or KAGRA are presented for comparison.


Responses of each channel in the SOGRO detector for plus ($+$), cross ($\times$), $x$, $y$, breathing ($b$), and longitudinal ($\ell$) polarization modes. The $+$ and $\times$ polarizations correspond to tensor modes, the $x$ and $y$ are vector modes, and the $b$ and $\ell$ polarizations are relevant to scalar modes. The responses of the (22) and the (31) channels are the same with the (11) and the (23) channels, respectively, up to $\pi/2$ rotation along the $z$-axis. The (33) channel is not shown as this is not used in sensitivity calculation. Combined (total) responses of all 5 channels are also depicted. The total responses show the omnidirectional nature of SOGRO; however, the total response of a terrestrial SOGRO to the $\ell$ mode vanishes in the direction of $z$-axis. In the last column, the general responses of a laser interferometer such as LIGO, Virgo or KAGRA are presented for comparison.

Responses of each channel in the SOGRO detector for plus ($+$), cross ($\times$), $x$, $y$, breathing ($b$), and longitudinal ($\ell$) polarization modes. The $+$ and $\times$ polarizations correspond to tensor modes, the $x$ and $y$ are vector modes, and the $b$ and $\ell$ polarizations are relevant to scalar modes. The responses of the (22) and the (31) channels are the same with the (11) and the (23) channels, respectively, up to $\pi/2$ rotation along the $z$-axis. The (33) channel is not shown as this is not used in sensitivity calculation. Combined (total) responses of all 5 channels are also depicted. The total responses show the omnidirectional nature of SOGRO; however, the total response of a terrestrial SOGRO to the $\ell$ mode vanishes in the direction of $z$-axis. In the last column, the general responses of a laser interferometer such as LIGO, Virgo or KAGRA are presented for comparison.


Responses of each channel in the SOGRO detector for plus ($+$), cross ($\times$), $x$, $y$, breathing ($b$), and longitudinal ($\ell$) polarization modes. The $+$ and $\times$ polarizations correspond to tensor modes, the $x$ and $y$ are vector modes, and the $b$ and $\ell$ polarizations are relevant to scalar modes. The responses of the (22) and the (31) channels are the same with the (11) and the (23) channels, respectively, up to $\pi/2$ rotation along the $z$-axis. The (33) channel is not shown as this is not used in sensitivity calculation. Combined (total) responses of all 5 channels are also depicted. The total responses show the omnidirectional nature of SOGRO; however, the total response of a terrestrial SOGRO to the $\ell$ mode vanishes in the direction of $z$-axis. In the last column, the general responses of a laser interferometer such as LIGO, Virgo or KAGRA are presented for comparison.

Responses of each channel in the SOGRO detector for plus ($+$), cross ($\times$), $x$, $y$, breathing ($b$), and longitudinal ($\ell$) polarization modes. The $+$ and $\times$ polarizations correspond to tensor modes, the $x$ and $y$ are vector modes, and the $b$ and $\ell$ polarizations are relevant to scalar modes. The responses of the (22) and the (31) channels are the same with the (11) and the (23) channels, respectively, up to $\pi/2$ rotation along the $z$-axis. The (33) channel is not shown as this is not used in sensitivity calculation. Combined (total) responses of all 5 channels are also depicted. The total responses show the omnidirectional nature of SOGRO; however, the total response of a terrestrial SOGRO to the $\ell$ mode vanishes in the direction of $z$-axis. In the last column, the general responses of a laser interferometer such as LIGO, Virgo or KAGRA are presented for comparison.


Responses of each channel in the SOGRO detector for plus ($+$), cross ($\times$), $x$, $y$, breathing ($b$), and longitudinal ($\ell$) polarization modes. The $+$ and $\times$ polarizations correspond to tensor modes, the $x$ and $y$ are vector modes, and the $b$ and $\ell$ polarizations are relevant to scalar modes. The responses of the (22) and the (31) channels are the same with the (11) and the (23) channels, respectively, up to $\pi/2$ rotation along the $z$-axis. The (33) channel is not shown as this is not used in sensitivity calculation. Combined (total) responses of all 5 channels are also depicted. The total responses show the omnidirectional nature of SOGRO; however, the total response of a terrestrial SOGRO to the $\ell$ mode vanishes in the direction of $z$-axis. In the last column, the general responses of a laser interferometer such as LIGO, Virgo or KAGRA are presented for comparison.

Responses of each channel in the SOGRO detector for plus ($+$), cross ($\times$), $x$, $y$, breathing ($b$), and longitudinal ($\ell$) polarization modes. The $+$ and $\times$ polarizations correspond to tensor modes, the $x$ and $y$ are vector modes, and the $b$ and $\ell$ polarizations are relevant to scalar modes. The responses of the (22) and the (31) channels are the same with the (11) and the (23) channels, respectively, up to $\pi/2$ rotation along the $z$-axis. The (33) channel is not shown as this is not used in sensitivity calculation. Combined (total) responses of all 5 channels are also depicted. The total responses show the omnidirectional nature of SOGRO; however, the total response of a terrestrial SOGRO to the $\ell$ mode vanishes in the direction of $z$-axis. In the last column, the general responses of a laser interferometer such as LIGO, Virgo or KAGRA are presented for comparison.


Responses of each channel in the SOGRO detector for plus ($+$), cross ($\times$), $x$, $y$, breathing ($b$), and longitudinal ($\ell$) polarization modes. The $+$ and $\times$ polarizations correspond to tensor modes, the $x$ and $y$ are vector modes, and the $b$ and $\ell$ polarizations are relevant to scalar modes. The responses of the (22) and the (31) channels are the same with the (11) and the (23) channels, respectively, up to $\pi/2$ rotation along the $z$-axis. The (33) channel is not shown as this is not used in sensitivity calculation. Combined (total) responses of all 5 channels are also depicted. The total responses show the omnidirectional nature of SOGRO; however, the total response of a terrestrial SOGRO to the $\ell$ mode vanishes in the direction of $z$-axis. In the last column, the general responses of a laser interferometer such as LIGO, Virgo or KAGRA are presented for comparison.

Responses of each channel in the SOGRO detector for plus ($+$), cross ($\times$), $x$, $y$, breathing ($b$), and longitudinal ($\ell$) polarization modes. The $+$ and $\times$ polarizations correspond to tensor modes, the $x$ and $y$ are vector modes, and the $b$ and $\ell$ polarizations are relevant to scalar modes. The responses of the (22) and the (31) channels are the same with the (11) and the (23) channels, respectively, up to $\pi/2$ rotation along the $z$-axis. The (33) channel is not shown as this is not used in sensitivity calculation. Combined (total) responses of all 5 channels are also depicted. The total responses show the omnidirectional nature of SOGRO; however, the total response of a terrestrial SOGRO to the $\ell$ mode vanishes in the direction of $z$-axis. In the last column, the general responses of a laser interferometer such as LIGO, Virgo or KAGRA are presented for comparison.


: $h_{ij}$ only

: $h_{ij}$ only


: SNR: 167.9

: SNR: 167.9


: SNR: 83.94

: SNR: 83.94


: SNR: 55.96

: SNR: 55.96


: SNR: 41.97

: SNR: 41.97


: SNR: 33.57

: SNR: 33.57


Amplitude spectral densities (ASDs) of pSOGRO, SOGRO, and aSOGRO are overlaid with the expected signals from BBHs with different masses. The inclinations of BBHs are assumed to be zero (face-on). GW150914 signal is based on its observed masses and distance given in the literature\cite{GWTC1}. GW190426\_190642-like source is a BBH whose masses are adopted from GW190426\_190642 (105.5 and 76.0 M$_{\odot}$) but located at 500 Mpc which is closer than its actual distance (4.58 Gpc)\cite{GWTC2-1}. For the BBHs with IMBH, the masses and distances are assumed to be as indicated in the figure.

Amplitude spectral densities (ASDs) of pSOGRO, SOGRO, and aSOGRO are overlaid with the expected signals from BBHs with different masses. The inclinations of BBHs are assumed to be zero (face-on). GW150914 signal is based on its observed masses and distance given in the literature\cite{GWTC1}. GW190426\_190642-like source is a BBH whose masses are adopted from GW190426\_190642 (105.5 and 76.0 M$_{\odot}$) but located at 500 Mpc which is closer than its actual distance (4.58 Gpc)\cite{GWTC2-1}. For the BBHs with IMBH, the masses and distances are assumed to be as indicated in the figure.


Horizon distances versus the total mass of BBHs based on the sensitivities of pSOGRO (green), SOGRO (orange), and aSOGRO (blue), respectively. We consider the cases where the mass ratios are $m_{1}/m_{2} = 1$ and 10, and additionally when the secondary mass is fixed to be 30M$_{\odot}$. All BBHs are assumed to be face-on and $\rho=8$.

Horizon distances versus the total mass of BBHs based on the sensitivities of pSOGRO (green), SOGRO (orange), and aSOGRO (blue), respectively. We consider the cases where the mass ratios are $m_{1}/m_{2} = 1$ and 10, and additionally when the secondary mass is fixed to be 30M$_{\odot}$. All BBHs are assumed to be face-on and $\rho=8$.


Left: Expected GW signal from an equal-mass (1 M$_{\odot}$) BBH coalescence and the sensitivity curves of aLIGO, eLISA, and aSOGRO. Right: Horizon distance with respect to the total mass of BBHs when the mass ratio is one. The curve is based on the fixed SNR $\rho=8$.

Left: Expected GW signal from an equal-mass (1 M$_{\odot}$) BBH coalescence and the sensitivity curves of aLIGO, eLISA, and aSOGRO. Right: Horizon distance with respect to the total mass of BBHs when the mass ratio is one. The curve is based on the fixed SNR $\rho=8$.


Power-law integrated (PI) sensitivity curves for different detectors for SGWB. In addition, theoretical predictions and constraints are shown. The two-detector configurations with SOGRO (blue dotted line) and aSOGRO (orange dotted line) are presented in a frequency range of $0.1\,\mathrm{Hz}\leq f\leq10\,\mathrm{Hz}$.

Power-law integrated (PI) sensitivity curves for different detectors for SGWB. In addition, theoretical predictions and constraints are shown. The two-detector configurations with SOGRO (blue dotted line) and aSOGRO (orange dotted line) are presented in a frequency range of $0.1\,\mathrm{Hz}\leq f\leq10\,\mathrm{Hz}$.


We compare expected sensitivities for SGWB observation considering network of N detectors. We present results for networks consisting of N SOGRO's or N aSOGRO's. We assume that SGWB follows the power-law model with $\alpha=0$.

We compare expected sensitivities for SGWB observation considering network of N detectors. We present results for networks consisting of N SOGRO's or N aSOGRO's. We assume that SGWB follows the power-law model with $\alpha=0$.


Platform thermal strain noises of pSOGRO for the first six lowest XX-modes (solid) and first two lowest XY-modes (dashed). $T_{\rm pl}=0.1$ K and $Q_{\rm pl}=10^6$. The upper-most curve is the sum.

Platform thermal strain noises of pSOGRO for the first six lowest XX-modes (solid) and first two lowest XY-modes (dashed). $T_{\rm pl}=0.1$ K and $Q_{\rm pl}=10^6$. The upper-most curve is the sum.


Sketch of the pSOGRO's cooling system. All system is in a vacuum isolated in the cryostat. Radiation heat load from room temperature (R.T., 300 K) is shielded by 80 K liquid nitrogen shield plate. The platform is in the 4 K chamber and cooled by 4 K surface through helium heat exchange gas.

Sketch of the pSOGRO's cooling system. All system is in a vacuum isolated in the cryostat. Radiation heat load from room temperature (R.T., 300 K) is shielded by 80 K liquid nitrogen shield plate. The platform is in the 4 K chamber and cooled by 4 K surface through helium heat exchange gas.


References
  • [1] B. P. Abbott et al., Phys. Rev. Lett., 116(6), 061102 (2016), arXiv:1602.03837.
  • [2] LIGO Scientific Collaboration et al., Classical and Quantum Gravity, 32(7), 074001 (April 2015), arXiv:1411.4547.
  • [3] F. Acernese et al., Classical and Quantum Gravity, 32(2), 024001 (January 2015), arXiv:1408.3978.
  • [4] Kagra Collaboration, T. Akutsu, et al., Nature Astronomy, 3, 35–40 (January 2019), arXiv:1811.08079.
  • [5] J. Antoniadis et al., arXiv e-prints, page arXiv:2306.16214 (June 2023), arXiv:2306.16214.
  • [5] J. Antoniadis et al., arXiv e-prints, page arXiv:2306.16214 (June 2023), arXiv:2306.16214.
  • [6] J. Antoniadis et al., arXiv e-prints, page arXiv:2306.16226 (June 2023), arXiv:2306.16226.
  • [6] J. Antoniadis et al., arXiv e-prints, page arXiv:2306.16226 (June 2023), arXiv:2306.16226.
  • [7] Daniel J. Reardon et al., Astrophysical Journal Letters, 951(1), L6 (July 2023), arXiv:2306.16215.
  • [8] Gabriella Agazie et al., Astrophysical Journal Letters, 951(1), L8 (July 2023), arXiv:2306.16213.
  • [9] Heng Xu et al., Research in Astronomy and Astrophysics, 23(7), 075024 (July 2023), arXiv:2306.16216.
  • [10] Pau Amaro-Seoane et al., arXiv e-prints, page arXiv:1702.00786 (February 2017), arXiv:1702.00786.
  • [10] Pau Amaro-Seoane et al., arXiv e-prints, page arXiv:1702.00786 (February 2017), arXiv:1702.00786.
  • [11] M. Punturo et al., Classical and Quantum Gravity, 27(19), 194002 (October 2010).
  • [12] Jan Harms, Bram J. J. Slagmolen, Rana X. Adhikari, M. Coleman Miller, Matthew Evans, Yanbei Chen, Holger Müller, and Masaki Ando, Phys. Rev. D, 88, 122003 (Dec 2013).
  • [13] Ho Jung Paik, Cornelius E. Griggs, M. Vol Moody, Krishna Venkateswara, Hyung Mok Lee, Alex B. Nielsen, Ettore Majorana, and Jan Harms, Class. Quant. Grav., 33(7), 075003 (2016).
  • [14] Ho Jung Paik, M. Vol Moody, and Ronald S. Norton, International Journal of Modern Physics D, 29(4), 1940001–265 (January 2020).
  • [15] Seiji Kawamura et al., Progress of Theoretical and Experimental Physics, 2021(5), 05A105 (May 2021), arXiv:2006.13545.
  • [16] Masaki Ando, Koji Ishidoshiro, Kazuhiro Yamamoto, Kent Yagi, Wataru Kokuyama, Kimio Tsubono, and Akiteru Takamori, Physical Review Letters, 105(16), 161101 (October 2010).
  • [17] B. Canuel et al., Scientific Reports, 8, 14064 (September 2018), arXiv:1703.02490.
  • [18] Jeff Crowder and Neil J. Cornish, Physical Review D, 72(8), 083005 (October 2005), arXiv:gr-qc/0506015.
  • [19] Jan Harms and Ho Jung Paik, Phys. Rev. D, 92, 022001 (Jul 2015).
  • [20] R. V. Wagoner, C. M. Will, and H. J. Paik, Phys. Rev. D, 19, 2325 (1979).
  • [21] Ho Jung Paik, Sensitivity and bandwidth of resonant-mass gravitational wave detectors, In S-W Kim, editor, Int. Workshop on Gravitation and Fifth Force, pages 1–20 (1993).
  • [22] H. J. Paik, Nuovo Cim. B, 55, 15–36 (1980).
  • [23] H. J. Paik, Phys. Rev. D, 33, 309–318 (1986).
  • [24] H. A. Chan and H. J. Paik, Phys. Rev. D, 35, 3551–3571 (1987).
  • [25] H. A. Chan, M. V. Moody, and H. J. Paik, Phys. Rev. D, 35, 3572–3597 (1987).
  • [26] M. Vol Moody, Ho Jung Paik, and Edgar R. Canavan, Review of Scientific Instruments, 73(11), 3957–3974 (November 2002).
  • [27] Ho Jung Paik, EPJ Web Conf., 168, 01005 (January 2018).
  • [28] Atsushi Nishizawa, Atsushi Taruya, Kazuhiro Hayama, Seiji Kawamura, and Masa-aki Sakagami, Phys. Rev., D79, 082002 (2009), arXiv:0903.0528.
  • [29] C. Cinquegrana, P. Rapagnani, F. Ricci, and E. Majorana, Phys. Rev. D, 48, 448–465 (1993).
  • [30] N. Christensen, Phys. Rev. D, 46, 5250–5266 (1992).
  • [31] B. S. Sathyaprakash and B. F. Schutz, Living Rev. Rel., 12, 2 (2009), arXiv:0903.0338.
  • [32] Vijay Varma, Parameswaran Ajith, Sascha Husa, Juan Calderon Bustillo, Mark Hannam, and Michael Pürrer, Phys. Rev. D, 90(12), 124004 (2014), arXiv:1409.2349.
  • [33] W. W. Johnson and S. M. Merkowitz, Phys. Rev. Lett., 70, 2367–2370 (1993).
  • [34] Mini-GRAIL, http://www.minigrail.nl/ (), Accessed: 2018-05-29.
  • [35] Young-Hwan Hyun, Yoonbai Kim, and Seokcheon Lee, Phys. Rev. D, 99(12), 124002 (2019), arXiv:1810.09316.
  • [36] Lee Samuel Finn, Phys. Rev. D, 63, 102001 (2001), gr-qc/0010033.
  • [37] B. P. Abbott et al., Living Rev. Rel., 23(1), 3 (2020).
  • [38] Carl Z. Zhou and Peter F. Michelson, Phys. Rev. D, 51, 2517–2545 (Mar 1995).
  • [39] Bruce Allen and Joseph D. Romano, Phys. Rev. D, 59, 102001 (1999), gr-qc/9710117.
  • [40] B. P. Abbott, R. Abbott, T. D. Abbott, and et al., Physical Review X, 9(3), 031040 (July 2019), arXiv:1811.12907.
  • [41] R. Abbott, T. D. Abbott, S. Abraham, et al., Physical Review X, 11(2), 021053 (April 2021), arXiv:2010.14527.
  • [42] The LIGO Scientific Collaboration and the Virgo Collaboration, arXiv e-prints, page arXiv:2108.01045 (August 2021), arXiv:2108.01045.
  • [42] The LIGO Scientific Collaboration and the Virgo Collaboration, arXiv e-prints, page arXiv:2108.01045 (August 2021), arXiv:2108.01045.
  • [43] The LIGO Scientific Collaboration, the Virgo Collaboration, and the KAGRA Collaboration, arXiv e-prints, page arXiv:2111.03606 (November 2021), arXiv:2111.03606.
  • [43] The LIGO Scientific Collaboration, the Virgo Collaboration, and the KAGRA Collaboration, arXiv e-prints, page arXiv:2111.03606 (November 2021), arXiv:2111.03606.
  • [44] J. Abadie et al., Class. Quant. Grav., 27, 173001 (2010), arXiv:1003.2480.
  • [45] David Merritt, Black Holes and Galaxy Evolution, In Francoise Combes, Gary A. Mamon, and Vassilis Charmandaris, editors, Dynamics of Galaxies: from the Early Universe to the Present, volume 197 of Astronomical Society of the Pacific Conference Series, page 221 (January 2000), arXiv:astro-ph/9910546.
  • [46] Laura Ferrarese and David Merritt, Astrophysical Journal Letters, 539(1), L9–L12 (August 2000), arXiv:astroph/0006053.
  • [47] Karl Gebhardt, Ralf Bender, Gary Bower, Alan Dressler, S. M. Faber, Alexei V. Filippenko, Richard Green, Carl Grillmair, Luis C. Ho, John Kormendy, Tod R. Lauer, John Magorrian, Jason Pinkney, Douglas Richstone, and Scott Tremaine, Astrophysical Journal Letters, 539(1), L13–L16 (August 2000), arXiv:astro-ph/0006289.
  • [48] Nicholas J. McConnell, Chung-Pei Ma, Karl Gebhardt, Shelley A. Wright, Jeremy D. Murphy, Tod R. Lauer, James R. Graham, and Douglas O. Richstone, Nature, 480(7376), 215–218 (December 2011), arXiv:1112.1078.
  • [49] John Kormendy and Luis C. Ho, Annu. Rev. Astron. Astrophys., 51(1), 511–653 (August 2013), arXiv:1304.7762.
  • [50] A. Patruno, S. Portegies Zwart, J. Dewi, and C. Hopman, Monthly Notices of the RAS, 370(1), L6–L9 (July 2006), arXiv:astro-ph/0602230.
  • [51] Thomas J. Maccarone, Arunav Kundu, Stephen E. Zepf, and Katherine L. Rhode, Nature, 445(7124), 183–185 (January 2007), arXiv:astro-ph/0701310.
  • [52] Igor V. Chilingarian, Ivan Yu. Katkov, Ivan Yu. Zolotukhin, Kirill A. Grishin, Yuri Beletsky, Konstantina Boutsia, and David J. Osip, Astrophysical Journal, 863(1), 1 (August 2018), arXiv:1805.01467.
  • [53] Shunya Takekawa, Tomoharu Oka, Yuhei Iwata, Shiho Tsujimoto, and Mariko Nomura, Astrophysical Journal Letters, 871(1), L1 (January 2019), arXiv:1812.10733.
  • [54] Jong-Hak Woo, Hojin Cho, Elena Gallo, Edmund Hodges-Kluck, Huynh Anh N. Le, Jaejin Shin, Donghoon Son, and John C. Horst, Nature Astronomy, 3, 755–759 (June 2019), arXiv:1905.00145.
  • [55] Jenny E. Greene, Jay Strader, and Luis C. Ho, Annu. Rev. Astron. Astrophys., 58, 257–312 (August 2020), arXiv:1911.09678.
  • [56] P. Ajith et al., Phys. Rev. Lett., 106, 241101 (2011), arXiv:0909.2867.
  • [57] P. A. R. Ade et al., Astron. Astrophys., 594, A13 (2016), arXiv:1502.01589.
  • [58] M. Giersz, N. Leigh, A. Hypki, A. Askar, and N. Lützgendorf, Mem. Soc. Astron. Italiana, 87, 555 (January 2016), arXiv:1607.08384.
  • [59] John M. Fregeau, Shane L. Larson, M. Coleman Miller, Richard O’Shaughnessy, and Frederic A. Rasio, Astrophysical Journal Letters, 646(2), L135–L138 (August 2006), arXiv:astro-ph/0605732.
  • [60] Pau Amaro-Seoane and Lucı́a Santamarı́a, Astrophysical Journal, 722(2), 1197–1206 (October 2010), arXiv:0910.0254.
  • [61] S. F. Portegies Zwart, J. Makino, S. L. W. McMillan, and P. Hut, Astronomy and Astrophysics, 348, 117–126 (August 1999), arXiv:astro-ph/9812006.
  • [62] Toshikazu Ebisuzaki, Junichiro Makino, Takeshi Go Tsuru, Yoko Funato, Simon Portegies Zwart, Piet Hut, Steve McMillan, Satoki Matsushita, Hironori Matsumoto, and Ryohei Kawabe, Astrophysical Journal Letters, 562(1), L19–L22 (November 2001), arXiv:astro-ph/0106252.
  • [63] Simon F. Portegies Zwart and Stephen L. W. McMillan, Astrophysical Journal, 576(2), 899–907 (September 2002), arXiv:astro-ph/0201055.
  • [64] Simon F. Portegies Zwart, Holger Baumgardt, Piet Hut, Junichiro Makino, and Stephen L. W. McMillan, Nature, 428(6984), 724–726 (April 2004), arXiv:astro-ph/0402622.
  • [65] M. Atakan Gürkan, Marc Freitag, and Frederic A. Rasio, Astrophysical Journal, 604(2), 632–652 (April 2004), arXiv:astro-ph/0308449.
  • [66] Piero Madau and Lucia Pozzetti, Monthly Notices of the RAS, 312(2), L9–L15 (February 2000), arXiv:astroph/9907315.
  • [67] Charles C. Steidel, Kurt L. Adelberger, Mauro Giavalisco, Mark Dickinson, and Max Pettini, Astrophysical Journal, 519(1), 1–17 (July 1999), arXiv:astro-ph/9811399.
  • [68] A. W. Blain, J. P. Kneib, R. J. Ivison, and Ian Smail, Astrophysical Journal Letters, 512(2), L87–L90 (February 1999), arXiv:astro-ph/9812412.
  • [69] Cristiano Porciani and Piero Madau, Astrophysical Journal, 548(2), 522–531 (February 2001), arXiv:astroph/0008294.
  • [70] Dean E. McLaughlin, Astron. J., 117(5), 2398–2427 (May 1999), arXiv:astro-ph/9901283.
  • [71] Marc Freitag, Frederic A. Rasio, and Holger Baumgardt, Monthly Notices of the RAS, 368(1), 121–140 (May 2006), arXiv:astro-ph/0503129.
  • [72] Marc Freitag, M. Atakan Gürkan, and Frederic A. Rasio, Monthly Notices of the RAS, 368(1), 141–161 (May 2006), arXiv:astro-ph/0503130.
  • [73] Qing Zhang and S. Michael Fall, Astrophysical Journal Letters, 527(2), L81–L84 (December 1999), arXiv:astroph/9911229.
  • [74] Pau Amaro-Seoane and Marc Freitag, Astrophysical Journal Letters, 653(1), L53–L56 (December 2006), arXiv:astro-ph/0610478.
  • [75] Manuel Arca Sedda et al., Classical and Quantum Gravity, 37(21), 215011 (November 2020), arXiv:1908.11375.
  • [76] Davide Gerosa and Emanuele Berti, Physical Review D, 95(12), 124046 (June 2017), arXiv:1703.06223.
  • [77] Hisa-Aki Shinkai, Nobuyuki Kanda, and Toshikazu Ebisuzaki, Astrophys. J., 835(2), 276 (2017), arXiv:1610.09505.
  • [78] Takashi Nakamura et al., Progress of Theoretical and Experimental Physics, 2016(9), 093E01 (September 2016), arXiv:1607.00897.
  • [79] Bernard Carr and Florian Kühnel, Annual Review of Nuclear and Particle Science, 70, 355–394 (October 2020), arXiv:2006.02838.
  • [80] Sarah Shandera, Donghui Jeong, and Henry S. Grasshorn Gebhardt, Physical Review Letters, 120(24), 241102 (June 2018), arXiv:1802.08206.
  • [81] Alexander H. Nitz and Yi-Fan Wang, Physical Review Letters, 127(15), 151101 (October 2021), arXiv:2106.08979.
  • [82] Eric Thrane and Joseph D. Romano, Physical Review D, 88(12), 124032 (December 2013).
  • [83] Vuk Mandic and Erik Floden, An interactive plotter for energy spectrum of stochastic gravitational wave backgrounds from various theoretical models, https://homepages.spa.umn.edu/gwplotter/ ().
  • [84] R. Abbott et al., Phys. Rev. D, 104, 022004 (Jul 2021).
  • [85] Paul D. Lasky et al., Phys. Rev. X, 6, 011035 (Mar 2016).
  • [86] Neil Cornish and Travis Robson, Journal of Physics: Conference Series, 840, 012024 (may 2017).
  • [87] Y. Akrami et al., A&A, 641, A10 (2020).
  • [88] C. E. Griggs, M. V. Moody, R. S. Norton, H. J. Paik, and K. Venkateswara, Phys. Rev. Applied, 8, 064024 (Dec 2017).
  • [89] P. Falferi, M. Bonaldi, Massimo Cerdonio, R. Mezzena, G. Prodi, A. Vinante, and S. Vitale, Applied Physics Letters, 93, 172506 – 172506 (11 2008). A Response functions for extra polarizations of gravitational waves Alternative theories of gravity could allow extra polarization degrees of freedom for gravitational waves in addition to the plus and cross ones in general relativity. Here, the response functions for such extra modes are summarized for the laser interferometer and SOGRO detectors. First of all, the three unit vectors mentioned in Sect. 2.1 are explicitly given by û = (sin ϕ cos ψ + cos θ cos ϕ sin ψ, − cos ϕ cos ψ + cos θ sin ϕ sin ψ, − sin θ sin ψ), v̂ = (cos θ cos ϕ cos ψ − sin ϕ sin ψ, cos θ sin ϕ cos ψ + cos ϕ sin ψ, − sin θ cos ψ), n̂ = (sin θ cos ϕ, sin θ sin ϕ, cos θ), (A1) where θ and ϕ are the polar and azimuthal angles of the propagating direction n̂, respectively, and ψ are the polarization angle. For the laser interferometer, the response functions for GWs having vector, breathing and longitudinal polarizations are given by Fx(θ, ϕ, ψ) = sin θ sin 2ϕ cos ψ + 1 2 sin 2θ cos 2ϕ sin ψ, Fy(θ, ϕ, ψ) = 1 2 sin 2θ cos 2ϕ cos ψ − sin θ sin 2ϕ sin ψ, Fb(θ, ϕ, ψ) = − 1 2