# Stochastic Gravitational-Wave Backgrounds: Current Detection Efforts and Future Prospects

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## Abstract

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## 1. Introduction

## 2. Theory of Stochastic Backgrounds

#### 2.1. Gravitational-Wave Strain and Stokes Parameters

#### 2.2. The Energy Density of Gravitational Waves

## 3. Sources of Stochastic Backgrounds

#### 3.1. Astrophysical Backgrounds

#### 3.2. Primordial Backgrounds

#### 3.3. Anisotropies in Stochastic Backgrounds

#### 3.4. Observational Properties of Stochastic Backgrounds

## 4. Detection Approaches and Methodologies

#### 4.1. Interdetector and Spatial Correlations

#### 4.2. Isotropic Background Search Methods

#### 4.2.1. Ground-Based Detectors

#### 4.2.2. Pulsar Timing Arrays

#### 4.3. Anisotropic Background Detection Methods

#### 4.4. The Approach towards Non-Gaussian Backgrounds

## 5. Current Detection Efforts of SGWBs

`O3`) of the 2G detectors: the Advanced LIGO and the Advanced Virgo. These results build on the previous data runs

`O1`and

`O2`of the 2G detectors. The overall sensitivity of the 2G detector network may be observed in Figure 7. We then discuss future detection prospects for the 2G network, which should reach design sensitivity with the next few upgrades of the instruments [218]. When representing the sensitivity to a GWB, we use “power-law integrated” sensitivity curves, which show the sensitivity to a GWB at each frequency (at a chosen SNR level) with a power law spectral shape that is tangent to the curve at that frequency [219].

#### 5.1. Searches with Ground-Based Laser Interferometers

#### 5.1.1. Search Results for an Isotropic Background by LVK

`O3`. In addition, frequencies where narrow spectral artifacts (“lines”) occur and are caused by known instrumental disturbances or show coherence between detectors are removed [243] and typically cut ∼15–20% of the frequency band. More details on the methodology, exact values of data excised for each baseline in

`O3`, and implementation of the quality checks can be found in [56,243]. Note that this is the first advanced-detector era stochastic analysis to include multiple baselines13; we highlight in this section the important implications of adding two long baselines to the network. Specifically,

`O3`analysis spans Hanford–Livingston (H–L), Hanford–Virgo (H–V), and Livingston–Virgo (L–V).

`O3`data [244,245]. These were not as frequent nor problematic in previous data runs, and investigations are still under way to determine the sources of these noise artifacts. Without their removal, loud glitches would bias the PSD of a 192-second segment, which would then not pass the stationarity cut, reducing the effective live time of the LIGO detectors to less than 50%, whereas gated datasets are reduced by about 1% of the original total length.

`O3`,2-year projections for the Advanced LIGO-Virgo network operating at design sensitivity [218], and a 2-year, 50% duty cycle A+ network at design sensitivity [248]. Figure 2 shown in Section 3 suggests that by the time design sensitivity is reached for Advanced LIGO-Virgo detectors, these will be marginally sensitive to the background. By the A+ phase, the vast majority of the background signal should be within reach of detection.

#### 5.1.2. Search Results for an Anisotropic Background by LVK

#### 5.2. Stochastic Searches with Pulsar Timing Arrays

#### 5.2.1. Search Results for an Isotropic Nanohertz Background

#### 5.2.2. Challenges in GWB Searches with PTAs

#### 5.2.3. Search Results for an Anisotropic Nanohertz Background

#### 5.2.4. Search Results for a Nanohertz Background Not Related to Supermassive Black Hole Binaries

#### 5.3. Other Stochastic Background Searches

## 6. Stochastic Background Detection Prospects with Future Gravitational-Wave Detectors

#### 6.1. Stochastic Searches with Third Generation Interferometers

#### 6.2. Stochastic Searches with the Laser Interferometer Space Antenna

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notes

1 | There are several theories that postulate modifications to this statement, but these modifications are strongly constrained by multimessenger observations of the binary neutron star merger GW170817 [36]. |

2 | Strictly speaking, GWs with wavelengths larger than the size of the cosmological horizon are frozen out by Hubble friction and, thus, do not contribute to the effective energy density. As a result, the density parameter appearing on the left-hand side of Equation (14) should be interpreted as an integral over ${\Omega}_{\mathrm{GW}}\left(f\right)$ from a minimum frequency of $f\sim {10}^{-15}\phantom{\rule{0.166667em}{0ex}}\mathrm{Hz}$, which corresponds to the size of the horizon at the epoch when the CMB photons were emitted. |

3 | Often in the literature, the letter $\alpha $ is used to denote a different spectral index, hereby denoted ${\alpha}^{\prime}$, of the characteristic strain spectrum of the stochastic gravitational wave background $h\left(f\right)\equiv \sqrt{fS\left(f\right)}\propto {f}^{{\alpha}^{\prime}}$, where $S\left(f\right)$ is the power spectral density of the background. This leads to the power-law index of $2{\alpha}^{\prime}+2$ for ${\Omega}_{\mathrm{GW}}$ in Equation (25) instead of $\alpha $. |

4 | |

5 | As mentioned in the footnote 3, the energy density spectral index $\alpha =2/3$ is related to the strain amplitude spectral index ${\alpha}^{\prime}$ such that $\alpha =2{\alpha}^{\prime}+2$. Therefore ${\alpha}^{\prime}=-2/3$, which is the quantity often referred to as $\alpha $ in publications. Due to the parametric choices made in the different search pipelines, $\alpha $ is most frequently used in LVK literature, while ${\alpha}^{\prime}$ is most frequently used in the PTA literature. |

6 | There are, in fact, sources of spatially correlated noise. They can be distinguished from SGWBs by either a deterministic component or by an overlap reduction function different to those of SGWBs. We provide more details in Section 5. |

7 | Time segments with non-stationary noise are removed from analyses of real data [56], the stationarity is usually determined empirically. Note that when applying a non-trivial windowing function to time-domain data, e.g., for computing a discrete Fourier transform, we introduce correlations between frequency bins. This effect and the methods to mitigate it are described in [130]. |

8 | |

9 | Note that directly interfering laser beams in the case of LISA is impossible due to energy dispersion along the arm [29]. |

10 | Note that this is no longer the case in the presence of temporal shot noise, e.g., for the astrophysical GWB from compact binary coalescences [122]. In this case, each time segment will have random GWB intensity fluctuations due to the finite number of sources. The statistical independence of these fluctuations at different times can be leveraged to mitigate the impact of shot noise on measurements of the angular power spectrum [123]. |

11 | Note that in these two cases ${I}_{p}$ and ${I}_{\ell m}$ have different units because in (114) the basis function carry units ${\mathrm{sr}}^{-1}$. |

12 | In fact, in [215] the authors discuss the extension of their optimal search method in the presence of non-Gaussian noise as well. The implementation of the method becomes substantially more involved, however practical considerations about the nature and behaviour of the non-Gaussian noise can simplify it considerably. |

13 | |

14 | Often used in the literature to quantify noise in a given pulsar. For rms ${\sigma}^{2}$ of Gaussian white noise, the power spectral density of residuals $P=2\pi {\sigma}^{2}\Delta t$, where $\Delta t$ is the time between observations, which is typically a couple of weeks. Note, rms residuals may increase with longer data spans due to more sources of noise becoming prominent at longer time scales. Sometimes rms residuals are provided for a particular source of noise. |

## References

- Davis, D.; Walker, M. Detector Characterization and Mitigation of Noise in Ground-Based Gravitational-Wave Interferometers. Galaxies
**2022**, 10, 12. [Google Scholar] [CrossRef] - Sun, L.; Goetz, E.; Kissel, J.S.; Betzwieser, J.; Karki, S.; Viets, A.; Wade, M.; Bhattacharjee, D.; Bossilkov, V.; Covas, P.B.; et al. Characterization of systematic error in Advanced LIGO calibration. Class. Quant. Grav.
**2020**, 37, 225008. [Google Scholar] [CrossRef] - Sun, L.; Goetz, E.; Kissel, J.S.; Betzwieser, J.; Karki, S.; Bhattacharjee, D.; Covas, P.B.; Datrier, L.E.; Kandhasamy, S.; Lecoeuche, Y.K.; et al. Characterization of systematic error in Advanced LIGO calibration in the second half of O3. arXiv
**2021**, arXiv:2107.00129. [Google Scholar] - Acernese, F.; Agathos, M.; Ain, A.; Albanesi, S.; Allocca, A.; Amato, A.; Andrade, T.; Andres, N.; Andrić, T.; Ansoldi, S.; et al. Calibration of Advanced Virgo and reconstruction of detector strain h(t) during the Observing Run O3. Class. Quant. Grav.
**2022**, 39, 045006. [Google Scholar] [CrossRef] - Akutsu, T.; Ando, M.; Arai, K.; Arai, Y.; Araki, S.; Araya, A.; Aritomi, N.; Asada, H.; Aso, Y.; Bae, S.; et al. Overview of KAGRA: Calibration, detector characterization, physical environmental monitors, and the geophysics interferometer. PTEP
**2021**, 2021, 05A102. [Google Scholar] [CrossRef] - Verbiest, J.P.W.; Osłowski, S.; Burke-Spolaor, S. Pulsar Timing Array Experiments. In Handbook of Gravitational Wave Astronomy; Bambi, C., Katsanevas, S., Kokkotas, K.D., Eds.; Springer: Singapore, 2020; pp. 1–42. [Google Scholar] [CrossRef]
- Tiburzi, C. Pulsars Probe the Low-Frequency Gravitational Sky: Pulsar Timing Arrays Basics and Recent Results. PASA
**2018**, 35, e013. [Google Scholar] [CrossRef] [Green Version] - Hobbs, G.; Dai, S. Gravitational wave research using pulsar timing arrays. Natl. Sci. Rev.
**2017**, 4, 707–717. [Google Scholar] [CrossRef] [Green Version] - Abbott, B.P.; Abbott, R.; Abbott, T.D.; Abernathy, M.R.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.X.; et al. GW150914: The Advanced LIGO Detectors in the Era of First Discoveries. Phys. Rev. Lett.
**2016**, 116, 131103. [Google Scholar] [CrossRef] [Green Version] - Accadia, T.; Acernese, F.; Antonucci, F.; Astone, P.; Ballardin, G.; Barone, F.; Barsuglia, M.; Basti, A.; Bauer, T.S.; Bebronne, M.; et al. Status of the Virgo project. Class. Quantum Gravity
**2011**, 28, 114002. [Google Scholar] [CrossRef] - Affeldt, C.; Danzmann, K.; Dooley, K.L.; Grote, H.; Hewitson, M.; Hild, S.; Hough, J.; Leong, J.; Lück, H.; Prijatelj, M.; et al. Advanced techniques in GEO 600. Class. Quantum Gravity
**2014**, 31, 224002. [Google Scholar] [CrossRef] - Castelvecchi, D. Japan’s pioneering detector set to join hunt for gravitational waves. Nature
**2019**, 562, 9–10. [Google Scholar] [CrossRef] [PubMed] - LIGO-India, Proposal of the Consortium for Indian Initiative in Gravitational-Wave Observations (IndIGO). Available online: https://dcc.ligo.org/LIGO-M1100296/public (accessed on 1 November 2021).
- Punturo, M.; Abernathy, M.; Acernese, F.; Allen, B.; Andersson, N.; Arun, K.; Barone, F.; Barr, B.; Barsuglia, M.; Beker, M.; et al. The Einstein Telescope: A third-generation gravitational wave observatory. Class. Quant. Grav.
**2010**, 27, 194002. [Google Scholar] [CrossRef] - Sathyaprakash, B.; Abernathy, M.; Acernese, F.; Amaro-Seoane, P.; Andersson, N.; Arun, K.; Barone, F.; Barr, B.; Barsuglia, M.; Beker, M.; et al. Scientific Potential of Einstein Telescope. In Proceedings of the 46th Rencontres de Moriond on Gravitational Waves and Experimental Gravity, La Thuile, Italy, 20–27 March 2011; pp. 127–136. [Google Scholar]
- Abadie, J.; Abbott, B.P.; Abbott, R.; Abbott, T.D.; Abernathy, M.; Accadia, T.; Acernese, F.; Adams, C.; Adhikari, R.; Affeldt, C.; et al. Upper limits on a stochastic gravitational-wave background using LIGO and Virgo interferometers at 600–1000 Hz. Phys. Rev. D
**2012**, 85, 122001. [Google Scholar] [CrossRef] [Green Version] - Reitze, D.; Adhikari, R.X.; Ballmer, S.; Barish, B.; Barsotti, L.; Billingsley, G.; Brown, D.A.; Chen, Y.; Coyne, D.; Eisenstein, R.; et al. Cosmic Explorer: The U.S. Contribution to Gravitational-Wave Astronomy beyond LIGO. Bull. Am. Astron. Soc.
**2019**, 51, 035. [Google Scholar] - Hall, E.D.; Kuns, K.; Smith, J.R.; Bai, Y.; Wipf, C.; Biscans, S.; Adhikari, R.X.; Arai, K.; Ballmer, S.; Barsotti, L.; et al. Gravitational-wave physics with Cosmic Explorer: Limits to low-frequency sensitivity. Phys. Rev. D
**2021**, 103, 122004. [Google Scholar] [CrossRef] - McLaughlin, M.A. The North American Nanohertz Observatory for Gravitational Waves. Class. Quantum Gravity
**2013**, 30, 224008. [Google Scholar] [CrossRef] [Green Version] - Manchester, R.N.; Hobbs, G.; Bailes, M.; Coles, W.A.; van Straten, W.; Keith, M.J.; Shannon, R.M.; Bhat, N.D.R.; Brown, A.; Burke-Spolaor, S.G.; et al. The Parkes Pulsar Timing Array Project. PASA
**2013**, 30, e017. [Google Scholar] [CrossRef] [Green Version] - Kramer, M.; Champion, D.J. The European Pulsar Timing Array and the Large European Array for Pulsars. Class. Quantum Gravity
**2013**, 30, 224009. [Google Scholar] [CrossRef] - Desvignes, G.; Caballero, R.N.; Lentati, L.; Verbiest, J.P.W.; Champion, D.J.; Stappers, B.W.; Janssen, G.H.; Lazarus, P.; Osłowski, S.; Babak, S.; et al. High-precision timing of 42 millisecond pulsars with the European Pulsar Timing Array. Mon. Not. R. Astron. Soc.
**2016**, 458, 3341–3380. [Google Scholar] [CrossRef] - Hobbs, G.; Archibald, A.; Arzoumanian, Z.; Backer, D.; Bailes, M.; Bhat, N.D.R.; Burgay, M.; Burke-Spolaor, S.; Champion, D.; Cognard, I.; et al. The International Pulsar Timing Array project: Using pulsars as a gravitational wave detector. Class. Quantum Gravity
**2010**, 27, 084013. [Google Scholar] [CrossRef] - Verbiest, J.P.W.; Lentati, L.; Hobbs, G.; van Haasteren, R.; Demorest, P.B.; Janssen, G.H.; Wang, J.B.; Desvignes, G.; Caballero, R.N.; Keith, M.J.; et al. The International Pulsar Timing Array: First data release. MNRAS
**2016**, 458, 1267–1288. [Google Scholar] [CrossRef] - Joshi, B.C.; Arumugasamy, P.; Bagchi, M.; Bandyopadhyay, D.; Basu, A.; Dhanda Batra, N.; Bethapudi, S.; Choudhary, A.; De, K.; Dey, L.; et al. Precision pulsar timing with the ORT and the GMRT and its applications in pulsar astrophysics. J. Astrophys. Astron.
**2018**, 39, 51. [Google Scholar] [CrossRef] - Goncharov, B.; Shannon, R.M.; Reardon, D.J.; Hobbs, G.; Zic, A.; Bailes, M.; Curyło, M.; Dai, S.; Kerr, M.; Lower, M.E.; et al. On the Evidence for a Common-spectrum Process in the Search for the Nanohertz Gravitational-wave Background with the Parkes Pulsar Timing Array. ApJ
**2021**, 917, L19. [Google Scholar] [CrossRef] - Dewdney, P.E.; Hall, P.J.; Schilizzi, R.T.; Lazio, T.J.L.W. The Square Kilometre Array. IEEE Proc.
**2009**, 97, 1482–1496. [Google Scholar] [CrossRef] - Nan, R.; Li, D.; Jin, C.; Wang, Q.; Zhu, L.; Zhu, W.; Zhang, H.; Yue, Y.; Qian, L. The Five-Hundred Aperture Spherical Radio Telescope (fast) Project. Int. J. Mod. Phys. D
**2011**, 20, 989–1024. [Google Scholar] [CrossRef] [Green Version] - Amaro-Seoane, P.; Audley, H.; Babak, S.; Baker, J.; Barausse, E.; Bender, P.; Berti, E.; Binetruy, P.; Born, M.; Bortoluzzi, D.; et al. Laser Interferometer Space Antenna. arXiv
**2017**, arXiv:1702.00786. [Google Scholar] - Wald, R.M. General Relativity; University of Chicago Press: Chicago, IL, USA, 1984. [Google Scholar]
- Cornish, N.J. Mapping the gravitational-wave background. Class. Quantum Gravity
**2001**, 18, 4277–4291. [Google Scholar] [CrossRef] [Green Version] - Allen, B.; Ottewill, A.C. Detection of Anisotropies in the Gravitational-Wave Stochastic Background. Phys. Rev. D
**1996**, 56, 545–563. [Google Scholar] [CrossRef] [Green Version] - Romano, J.D.; Cornish, N.J. Detection methods for stochastic gravitational-wave backgrounds: A unified treatment. Living Rev. Relativ.
**2017**, 20, 1–223. [Google Scholar] [CrossRef] - Jackson, J.D. Classical Electrodynamics, 3rd ed.; Wiley: New York, NY, USA, 1999. [Google Scholar]
- Renzini, A.I. Mapping the Gravitational-Wave Background. Ph.D. Thesis, Imperial College London, London, UK, 2020. [Google Scholar] [CrossRef]
- Abbott, B.P.; Abbott, R.; Abbott, T.D.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.X.; Adya, V.B.; et al. Gravitational Waves and Gamma-rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A. Astrophys. J. Lett.
**2017**, 848, L13. [Google Scholar] [CrossRef] - Kolb, E.W.; Turner, M.S. The Early Universe; Addison-Wesley: Boston, MA, USA, 1990; Volume 69. [Google Scholar]
- Pagano, L.; Salvati, L.; Melchiorri, A. New constraints on primordial gravitational waves from Planck 2015. Phys. Lett. B
**2016**, 760, 823–825. [Google Scholar] [CrossRef] [Green Version] - Christensen, N. Stochastic Gravitational Wave Backgrounds. Rep. Prog. Phys.
**2019**, 82, 016903. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Phinney, E.S. A Practical Theorem on Gravitational Wave Backgrounds. arXiv
**2001**, arXiv:0108028. [Google Scholar] - Isaacson, R.A. Gravitational Radiation in the Limit of High Frequency. I. The Linear Approximation and Geometrical Optics. Phys. Rev.
**1968**, 166, 1263–1271. [Google Scholar] [CrossRef] - Isaacson, R.A. Gravitational Radiation in the Limit of High Frequency. II. Nonlinear Terms and the Ef fective Stress Tensor. Phys. Rev.
**1968**, 166, 1272–1279. [Google Scholar] [CrossRef] - Renzini, A.I.; Romano, J.D.; Contaldi, C.R.; Cornish, N.J. A comparison of maximum likelihood mapping methods for gravitational-wave backgrounds. Phys. Rev. D
**2022**, 105, 023519. [Google Scholar] [CrossRef] - Grishchuk, L. Amplification of gravitational waves in an isotropic universe. Sov. J. Exp. Theor. Phys.
**1975**, 40, 409. [Google Scholar] - Maggiore, M. Gravitational wave experiments and early universe cosmology. Phys. Rep.
**2000**, 331, 283–367. [Google Scholar] [CrossRef] [Green Version] - Hogan, C.J. Gravitational radiation from cosmological phase transitions. Mon. Not. R. Astron. Soc.
**1986**, 218, 629–636. [Google Scholar] [CrossRef] - Battye, R.A.; Caldwell, R.R.; Shellard, E.P.S. Gravitational waves from cosmic strings. In Topological Defects in Cosmology; World Scientific Pub. Co. Inc.: Singapore, 1997; pp. 11–31. [Google Scholar]
- Vilenkin, A.; Shellard, E.S. Cosmic Strings and Other Topological Defects; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Bartolo, N.; De Luca, V.; Franciolini, G.; Lewis, A.; Peloso, M.; Riotto, A. Primordial Black Hole Dark Matter: LISA Serendipity. Phys. Rev. Lett.
**2019**, 122, 211301. [Google Scholar] [CrossRef] - Bartolo, N.; Bertacca, D.; De Luca, V.; Franciolini, G.; Matarrese, S.; Peloso, M.; Ricciardone, A.; Riotto, A.; Tasinato, G. Gravitational wave anisotropies from primordial black holes. J. Cosmol. Astropart. Phys.
**2020**, 02, 028. [Google Scholar] [CrossRef] [Green Version] - Margalit, A.; Contaldi, C.R.; Pieroni, M. Phase Decoherence of Gravitational Wave Backgrounds. Phys. Rev. D
**2020**, 102, 083506. [Google Scholar] [CrossRef] - Maggiore, M. Gravitational Waves. Vol. 2: Astrophysics and Cosmology; Oxford University Press: Oxford, UK, 2018. [Google Scholar]
- Auclair, P.; Blanco-Pillado, J.J.; Figueroa, D.G.; Jenkins, A.C.; Lewicki, M.; Sakellariadou, M.; Sanidas, S.; Sousa, L.; Steer, D.A.; Wachter, J.M.; et al. Probing the gravitational wave background from cosmic strings with LISA. J. Cosmol. Astropart. Phys.
**2020**, 04, 034. [Google Scholar] [CrossRef] - Arzoumanian, Z.; Baker, P.T.; Blumer, H.; Bécsy, B.; Brazier, A.; Brook, P.R.; Burke-Spolaor, S.; Chatterjee, S.; Chen, S.; Cordes, J.M.; et al. The NANOGrav 12.5 yr Data Set: Search for an Isotropic Stochastic Gravitational-wave Background. ApJ
**2020**, 905, L34. [Google Scholar] [CrossRef] - Caprini, C.; Hindmarsh, M.; Huber, S.; Konstandin, T.; Kozaczuk, J.; Nardini, G.; No, J.M.; Petiteau, A.; Schwaller, P.; Servant, G.; et al. Science with the space-based interferometer eLISA. II: Gravitational waves from cosmological phase transitions. JCAP
**2016**, 04, 001. [Google Scholar] [CrossRef] - Abbott, R.; Abbott, T.D.; Abraham, S.; Acernese, F.; Ackley, K.; Adams, A.; Adams, C.; Adhikari, R.X.; Adya, V.B.; Affeldt, C.; et al. Upper limits on the isotropic gravitational-wave background from Advanced LIGO and Advanced Virgo’s third observing run. Phys. Rev. D
**2021**, 104, 022004. [Google Scholar] [CrossRef] - Kalogera, V.; Baym, G. The maximum mass of a neutron star. Astrophys. J. Lett.
**1996**, 470, L61–L64. [Google Scholar] [CrossRef] - Bailyn, C.D.; Jain, R.K.; Coppi, P.; Orosz, J.A. The Mass Distribution of Stellar Black Holes. ApJ
**1998**, 499, 367–374. [Google Scholar] [CrossRef] [Green Version] - Özel, F.; Psaltis, D.; Narayan, R.; McClintock, J.E. The Black Hole Mass Distribution in the Galaxy. ApJ
**2010**, 725, 1918–1927. [Google Scholar] [CrossRef] [Green Version] - Gupta, A.; Gerosa, D.; Arun, K.G.; Berti, E.; Farr, W.M.; Sathyaprakash, B.S. Black holes in the low mass gap: Implications for gravitational wave observations. Phys. Rev. D
**2020**, 101, 103036. [Google Scholar] [CrossRef] - Celotti, A.; Miller, J.C.; Sciama, D.W. Astrophysical evidence for the existence of black holes. Class. Quantum Gravity
**2000**, 16, A3. [Google Scholar] [CrossRef] [Green Version] - Cooray, A.R. Gravitational wave background of neutron star-white dwarf binaries. Mon. Not. R. Astron. Soc.
**2004**, 354, 25–30. [Google Scholar] [CrossRef] [Green Version] - Korol, V.; Belokurov, V.; Moore, C.J.; Toonen, S. Weighing Milky Way Satellites with LISA. Mon. Not. R. Astron. Soc.
**2021**, 502, L55–L60. [Google Scholar] [CrossRef] - Regimbau, T. The astrophysical gravitational wave stochastic background. Res. Astron. Astrophys.
**2011**, 11, 369. [Google Scholar] [CrossRef] - Callister, T.; Fishbach, M.; Holz, D.; Farr, W. Shouts and Murmurs: Combining Individual Gravitational-Wave Sources with the Stochastic Background to Measure the History of Binary Black Hole Mergers. Astrophys. J. Lett.
**2020**, 896, L32. [Google Scholar] [CrossRef] - Fishbach, M.; Holz, D.E.; Farr, W.M. Does the Black Hole Merger Rate Evolve with Redshift? Astrophys. J. Lett.
**2018**, 863, L41. [Google Scholar] [CrossRef] [Green Version] - Madau, P.; Dickinson, M. Cosmic Star Formation History. Annu. Rev. Astron. Astrophys.
**2014**, 52, 415–486. [Google Scholar] [CrossRef] [Green Version] - Boco, L.; Lapi, A.; Goswami, S.; Perrotta, F.; Baccigalupi, C.; Danese, L. Merging Rates of Compact Binaries in Galaxies: Perspectives for Gravitational Wave Detections. Astrophys. J.
**2019**, 881, 157. [Google Scholar] [CrossRef] [Green Version] - Boco, L.; Lapi, A.; Chruslinska, M.; Donevski, D.; Sicilia, A.; Danese, L. Evolution of Galaxy Star Formation and Metallicity: Impact on Double Compact Objects Mergers. Astrophys. J.
**2021**, 907, 110. [Google Scholar] [CrossRef] - Callister, T.; Sammut, L.; Qiu, S.; Mandel, I.; Thrane, E. The limits of astrophysics with gravitational-wave backgrounds. Phys. Rev. X
**2016**, 6, 031018. [Google Scholar] [CrossRef] [Green Version] - Sesana, A.; Vecchio, A.; Colacino, C.N. The stochastic gravitational-wave background from massive black hole binary systems: Implications for observations with Pulsar Timing Arrays. Mon. Not. R. Astron. Soc.
**2008**, 390, 192–209. [Google Scholar] [CrossRef] [Green Version] - Thorne, K.; Hawking, S.; Israel, W. Three Hundred Years of Gravitation; Cambridge University Press: Cambridge, UK, 1987; p. 330. [Google Scholar]
- Abbott, B.P.; Abbott, R.; Abbott, T.D.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.X.; Adya, V.B.; et al. GW170817: Implications for the Stochastic Gravitational-Wave Background from Compact Binary Coalescences. Phys. Rev. Lett.
**2018**, 120, 091101. [Google Scholar] [CrossRef] [PubMed] [Green Version] - LIGO Scientific Collaboration; Virgo Collaboration; KAGRA Scientific Collaboration. The population of merging compact binaries inferred using gravitational waves through GWTC-3. arXiv
**2021**, arXiv:2111.03634. [Google Scholar] - LIGO Scientific Collaboration; Virgo Collaboration; KAGRA Collaboration. The Population of Merging Compact Binaries Inferred Using Gravitational Waves through GWTC-3—Data Release. 2021. Available online: https://zenodo.org/record/5655785 (accessed on 1 November 2021).
- Caprini, C.; Figueroa, D.G. Cosmological backgrounds of gravitational waves. Class. Quantum Gravity
**2018**, 35, 163001. [Google Scholar] [CrossRef] [Green Version] - Guzzetti, M.C.; Bartolo, N.; Liguori, M.; Matarrese, S. Gravitational waves from inflation. Riv. Nuovo Cim.
**2016**, 39, 399–495. [Google Scholar] [CrossRef] - Contaldi, C.R.; Magueijo, J. Unsqueezing of standing waves due to inflationary domain structure. Phys. Rev. D
**2018**, 98, 043523. [Google Scholar] [CrossRef] [Green Version] - Kamionkowski, M.; Kosowsky, A.; Stebbins, A. A Probe of primordial gravity waves and vorticity. Phys. Rev. Lett.
**1997**, 78, 2058–2061. [Google Scholar] [CrossRef] [Green Version] - COrE Collaboration; Bouchet, F.R. COrE (Cosmic Origins Explorer) A White Paper. arXiv
**2011**, arXiv:1102.2181. [Google Scholar] - Matsumura, T.; Akiba, Y.; Borrill, J.; Chinone, Y.; Dobbs, M.; Fuke, H.; Ghribi, A.; Hasegawa, M.; Hattori, K.; Hattori, M.; et al. Mission design of LiteBIRD. J. Low Temp. Phys.
**2014**, 176, 733. [Google Scholar] [CrossRef] [Green Version] - Abazajian, K.N.; Adshead, P.; Ahmed, Z.; Allen, S.W.; Alonso, D.; Arnold, K.S.; Baccigalupi, C.; Bartlett, J.G.; Battaglia, N.; Benson, B.A.; et al. CMB-S4 Science Book, First Edition. arXiv
**2016**, arXiv:1610.02743. [Google Scholar] - Ade, P.; Aguirre, J.; Ahmed, Z.; Aiola, S.; Ali, A.; Alonso, D.; Alvarez, M.A.; Arnold, K.; Ashton, P.; Austermann, J.; et al. The Simons Observatory: Science goals and forecasts. JCAP
**2019**, 02, 056. [Google Scholar] [CrossRef] - Kuroyanagi, S.; Chiba, T.; Sugiyama, N. Precision calculations of the gravitational wave background spectrum from inflation. Phys. Rev. D
**2009**, 79, 103501. [Google Scholar] [CrossRef] [Green Version] - Chung, D.J.H.; Kolb, E.W.; Riotto, A.; Tkachev, I.I. Probing Planckian physics: Resonant production of particles during inflation and features in the primordial power spectrum. Phys. Rev. D
**2000**, 62, 043508. [Google Scholar] [CrossRef] [Green Version] - Ananda, K.N.; Clarkson, C.; Wands, D. The Cosmological gravitational wave background from primordial density perturbations. Phys. Rev. D
**2007**, 75, 123518. [Google Scholar] [CrossRef] [Green Version] - Clesse, S.; García-Bellido, J.; Orani, S. Detecting the Stochastic Gravitational Wave Background from Primordial Black Hole Formation. arXiv
**2018**, arXiv:1812.11011. [Google Scholar] - Garcia-Bellido, J.; Peloso, M.; Unal, C. Gravitational Wave signatures of inflationary models from Primordial Black Hole Dark Matter. JCAP
**2017**, 09, 013. [Google Scholar] [CrossRef] [Green Version] - Khlebnikov, S.Y.; Tkachev, I.I. Relic gravitational waves produced after preheating. Phys. Rev. D
**1997**, 56, 653–660. [Google Scholar] [CrossRef] [Green Version] - Easther, R.; Lim, E.A. Stochastic gravitational wave production after inflation. JCAP
**2006**, 04, 010. [Google Scholar] [CrossRef] [Green Version] - Garcia-Bellido, J.; Figueroa, D.G. A stochastic background of gravitational waves from hybrid preheating. Phys. Rev. Lett.
**2007**, 98, 061302. [Google Scholar] [CrossRef] [Green Version] - Easther, R.; Giblin, J.T.; Lim, E.A. Gravitational Wave Production at the End of Inflation. Phys. Rev. Lett.
**2007**, 99, 221301. [Google Scholar] [CrossRef] [Green Version] - Bartolo, N.; Caprini, C.; Domcke, V.; Figueroa, D.G.; Garcia-Bellido, J.; Guzzetti, M.C.; Liguori, M.; Matarrese, S.; Peloso, M.; Petiteau, A.; et al. Science with the space-based interferometer LISA. IV: Probing inflation with gravitational waves. J. Cosmol. Astropart. Phys.
**2016**, 12, 026. [Google Scholar] [CrossRef] - Jeannerot, R.; Rocher, J.; Sakellariadou, M. How generic is cosmic string formation in SUSY GUTs. Phys. Rev. D
**2003**, 68, 103514. [Google Scholar] [CrossRef] [Green Version] - García-Bellido, J.; Jaraba, S.; Kuroyanagi, S. The stochastic gravitational wave background from close hyperbolic encounters of primordial black holes in dense clusters. arXiv
**2021**, arXiv:2109.11376. [Google Scholar] - Alexander, S.; Marciano, A.; Smolin, L. Gravitational origin of the weak interaction’s chirality. Phys. Rev. D
**2014**, 89, 065017. [Google Scholar] [CrossRef] [Green Version] - Freidel, L.; Minic, D.; Takeuchi, T. Quantum gravity, torsion, parity violation and all that. Phys. Rev. D
**2005**, 72, 104002. [Google Scholar] [CrossRef] [Green Version] - Alexander, S.H. Isogravity: Toward an Electroweak and Gravitational Unification. arXiv
**2007**, arXiv:0706.4481. [Google Scholar] - Smolin, L. The Plebanski action extended to a unification of gravity and Yang-Mills theory. Phys. Rev. D
**2009**, 80, 124017. [Google Scholar] [CrossRef] [Green Version] - Contaldi, C.R. Anisotropies of Gravitational Wave Backgrounds: A Line Of Sight Approach. Phys. Rev. Lett.
**2017**, B771, 9–12. [Google Scholar] [CrossRef] - Jenkins, A.C.; Sakellariadou, M. Anisotropies in the stochastic gravitational-wave background: Formalism and the cosmic string case. Phys. Rev. D
**2018**, 98, 063509. [Google Scholar] [CrossRef] [Green Version] - Jenkins, A.C.; O’Shaughnessy, R.; Sakellariadou, M.; Wysocki, D. Anisotropies in the astrophysical gravitational-wave background: The impact of black hole distributions. Phys. Rev. Lett.
**2019**, 122, 111101. [Google Scholar] [CrossRef] [Green Version] - Durrer, R. The Cosmic Microwave Background; Cambridge University Press: Cambridge, UK, 2008. [Google Scholar] [CrossRef]
- Peebles, P.J.E. The Large-Scale Structure of the Universe; Princeton University Press: Princeton, NJ, USA, 2020. [Google Scholar]
- Sachs, R.; Wolfe, A. Perturbations of a cosmological model and angular variations of the microwave background. Astrophys. J.
**1967**, 147, 73–90. [Google Scholar] [CrossRef] - Bartolo, N.; Bertacca, D.; Matarrese, S.; Peloso, M.; Ricciardone, A.; Riotto, A.; Tasinato, G. Anisotropies and non-Gaussianity of the Cosmological Gravitational Wave Background. Phys. Rev. D
**2019**, 100, 121501. [Google Scholar] [CrossRef] [Green Version] - Cusin, G.; Dvorkin, I.; Pitrou, C.; Uzan, J.P. First predictions of the angular power spectrum of the astrophysical gravitational wave background. Phys. Rev. Lett.
**2018**, 120, 231101. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Jenkins, A.C.; Sakellariadou, M.; Regimbau, T.; Slezak, E. Anisotropies in the astrophysical gravitational-wave background: Predictions for the detection of compact binaries by LIGO and Virgo. Phys. Rev. D
**2018**, 98, 063501. [Google Scholar] [CrossRef] [Green Version] - Capurri, G.; Lapi, A.; Baccigalupi, C.; Boco, L.; Scelfo, G.; Ronconi, T. Intensity and anisotropies of the stochastic gravitational wave background from merging compact binaries in galaxies. JCAP
**2021**, 11, 032. [Google Scholar] [CrossRef] - Geller, M.; Hook, A.; Sundrum, R.; Tsai, Y. Primordial Anisotropies in the Gravitational Wave Background from Cosmological Phase Transitions. Phys. Rev. Lett.
**2018**, 121, 201303. [Google Scholar] [CrossRef] [Green Version] - Bertacca, D.; Ricciardone, A.; Bellomo, N.; Jenkins, A.C.; Matarrese, S.; Raccanelli, A.; Regimbau, T.; Sakellariadou, M. Projection effects on the observed angular spectrum of the astrophysical stochastic gravitational wave background. Phys. Rev. D
**2020**, 101, 103513. [Google Scholar] [CrossRef] - Bellomo, N.; Bertacca, D.; Jenkins, A.C.; Matarrese, S.; Raccanelli, A.; Regimbau, T.; Ricciardone, A.; Sakellariadou, M. CLASS_GWB: Robust modeling of the astrophysical gravitational wave background anisotropies. arXiv
**2021**, arXiv:2110.15059. [Google Scholar] - Cusin, G.; Dvorkin, I.; Pitrou, C.; Uzan, J.P. Comment on the article “Anisotropies in the astrophysical gravitational-wave background: The impact of black hole distributions” by A.C. Jenkins et al. [arXiv:1810.13435]. arXiv
**2018**, arXiv:1811.03582. [Google Scholar] - Jenkins, A.C.; Sakellariadou, M.; Regimbau, T.; Slezak, E.; O’Shaughnessy, R.; Wysocki, D. Response to Cusin et al’s comment on arXiv:1810.13435. arXiv
**2019**, arXiv:1901.01078. [Google Scholar] - Pitrou, C.; Cusin, G.; Uzan, J.P. Unified view of anisotropies in the astrophysical gravitational-wave background. Phys. Rev. D
**2020**, 101, 081301. [Google Scholar] [CrossRef] [Green Version] - Alonso, D.; Contaldi, C.R.; Cusin, G.; Ferreira, P.G.; Renzini, A.I. The N
_{ℓ}of gravitational wave background experiments. arXiv**2020**, arXiv:2005.03001. [Google Scholar] - Regimbau, T.; Dent, T.; Del Pozzo, W.; Giampanis, S.; Li, T.G.F.; Robinson, C.; Van Den Broeck, C.; Meacher, D.; Rodriguez, C.; Sathyaprakash, B.S.; et al. Mock data challenge for the Einstein Gravitational-Wave Telescope. Phys. Rev. D
**2012**, 86, 122001. [Google Scholar] [CrossRef] [Green Version] - Regimbau, T.; Meacher, D.; Coughlin, M. Second Einstein Telescope mock science challenge: Detection of the gravitational-wave stochastic background from compact binary coalescences. Phys. Rev. D
**2014**, 89, 084046. [Google Scholar] [CrossRef] [Green Version] - Cornish, N.J.; Romano, J.D. When is a gravitational-wave signal stochastic? Phys. Rev. D
**2015**, 92, 042001. [Google Scholar] [CrossRef] [Green Version] - Rosado, P.A.; Sesana, A.; Gair, J. Expected properties of the first gravitational wave signal detected with pulsar timing arrays. Mon. Not. R. Astron. Soc.
**2015**, 451, 2417–2433. [Google Scholar] [CrossRef] - Meacher, D.; Thrane, E.; Regimbau, T. Statistical properties of astrophysical gravitational-wave backgrounds. Phys. Rev. D
**2014**, 89, 084063. [Google Scholar] [CrossRef] [Green Version] - Jenkins, A.C.; Sakellariadou, M. Shot noise in the astrophysical gravitational-wave background. Phys. Rev. D
**2019**, 100, 063508. [Google Scholar] [CrossRef] [Green Version] - Jenkins, A.C.; Romano, J.D.; Sakellariadou, M. Estimating the angular power spectrum of the gravitational-wave background in the presence of shot noise. Phys. Rev. D
**2019**, 100, 083501. [Google Scholar] [CrossRef] [Green Version] - Alonso, D.; Cusin, G.; Ferreira, P.G.; Pitrou, C. Detecting the anisotropic astrophysical gravitational wave background in the presence of shot noise through cross-correlations. Phys. Rev. D
**2020**, 102, 023002. [Google Scholar] [CrossRef] - Canas-Herrera, G.; Contigiani, O.; Vardanyan, V. Cross-correlation of the astrophysical gravitational-wave background with galaxy clustering. Phys. Rev. D
**2020**, 102, 043513. [Google Scholar] [CrossRef] - Dvorkin, I.; Uzan, J.P.; Vangioni, E.; Silk, J. Exploring stellar evolution with gravitational-wave observations. Mon. Not. R. Astron. Soc.
**2018**, 479, 121–129. [Google Scholar] [CrossRef] [Green Version] - Mukherjee, S.; Silk, J. Time-dependence of the astrophysical stochastic gravitational wave background. Mon. Not. R. Astron. Soc.
**2020**, 491, 4690–4701. [Google Scholar] [CrossRef] - Abbott, R.; Abbott, T.D.; Acernese, F.; Ackley, K.; Adams, C.; Adhikari, N.; Adhikari, R.X.; Adya, V.B.; Affeldt, C.; Agarwal, D.; et al. Constraints on dark photon dark matter using data from LIGO’s and Virgo’s third observing run. arXiv
**2021**, arXiv:2105.13085. [Google Scholar] - Abbott, R.; Abbott, T.D.; Abraham, S.; Acernese, F.; Ackley, K.; Adams, A.; Adams, C.; Adhikari, R.X.; Adya, V.B.; Affeldt, C.; et al. Constraints on Cosmic Strings Using Data from the Third Advanced LIGO–Virgo Observing Run. Phys. Rev. Lett.
**2021**, 126, 241102. [Google Scholar] [CrossRef] [PubMed] - Talbot, C.; Thrane, E.; Biscoveanu, S.; Smith, R. Inference with finite time series: Observing the gravitational Universe through windows. Phys. Rev. Res.
**2021**, 3, 043049. [Google Scholar] [CrossRef] - Matas, A.; Romano, J.D. Frequentist versus Bayesian analyses: Cross-correlation as an approximate sufficient statistic for LIGO-Virgo stochastic background searches. Phys. Rev. D
**2021**, 103, 062003. [Google Scholar] [CrossRef] - Littenberg, T.; Cornish, N.; Lackeos, K.; Robson, T. Global Analysis of the Gravitational Wave Signal from Galactic Binaries. Phys. Rev. D
**2020**, 101, 123021. [Google Scholar] [CrossRef] - Aasi, J.; Abbott, B.P.; Abbott, R.; Abbott, T.; Abernathy, M.R.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.X.; et al. Advanced LIGO. Class. Quantum Gravity
**2015**, 32, 074001. [Google Scholar] [CrossRef] - Armstrong, J.W.; Estabrook, F.B.; Tinto, M. Time-Delay Interferometry for Space-based Gravitational Wave Searches. Astrophys. J.
**1999**, 527, 814–826. [Google Scholar] [CrossRef] [Green Version] - Tinto, M.; Armstrong, J.W. Cancellation of laser noise in an unequal-arm interferometer detector of gravitational radiation. Phys. Rev. D
**1999**, 59, 102003. [Google Scholar] [CrossRef] [Green Version] - Seto, N.; Taruya, A. Measuring a parity-violation signature in the early universe via ground-based laser interferometers. Phys. Rev. Lett.
**2007**, 99, 121101. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Seto, N.; Taruya, A. Polarization analysis of gravitational-wave backgrounds from the correlation signals of ground-based interferometers: Measuring a circular-polarization mode. Phys. Rev. D Part. Fields, Gravit. Cosmol.
**2008**, 77, 1–27. [Google Scholar] [CrossRef] [Green Version] - Bayle, J.B.; Hartwig, O.; Staab, M. Adapting time-delay interferometry for LISA data in frequency. Phys. Rev. D
**2021**, 104, 023006. [Google Scholar] [CrossRef] - Bayle, J.B.; Lilley, M.; Petiteau, A.; Halloin, H. Effect of filters on the time-delay interferometry residual laser noise for LISA. PRD
**2019**, 99, 084023. [Google Scholar] [CrossRef] [Green Version] - Vallisneri, M.; Bayle, J.B.; Babak, S.; Petiteau, A. Time-delay interferometry without delays. Phys. Rev. D
**2021**, 103, 082001. [Google Scholar] [CrossRef] - Bayle, J.B.; Vallisneri, M.; Babak, S.; Petiteau, A. On the matrix formulation of time-delay interferometry. arXiv
**2021**, arXiv:2106.03976. [Google Scholar] - Tinto, M.; Dhurandhar, S.; Joshi, P. Matrix representation of time-delay interferometry. PRD
**2021**, 104, 044033. [Google Scholar] [CrossRef] - Page, J.; Littenberg, T.B. Bayesian time delay interferometry. PRD
**2021**, 104, 084037. [Google Scholar] [CrossRef] - Baghi, Q.; Thorpe, J.I.; Slutsky, J.; Baker, J. Statistical inference approach to time-delay interferometry for gravitational-wave detection. PRD
**2021**, 103, 042006. [Google Scholar] [CrossRef] - LISA Data Challenge Manual. 2018. Available online: https://lisa-ldc.lal.in2p3.fr/static/data/pdf/LDC-manual-001.pdf (accessed on 1 November 2021).
- Tinto, M.; Dhurandhar, S.V. TIME DELAY. Living Rev. Relativ.
**2005**, 8, 4. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Adams, M.R.; Cornish, N.J. Discriminating between a stochastic gravitational wave background and instrument noise. Phys. Rev. D
**2010**, 82, 022002. [Google Scholar] [CrossRef] [Green Version] - Adams, M.R.; Cornish, N.J. Detecting a stochastic gravitational wave background in the presence of a galactic foreground and instrument noise. Phys. Rev. D
**2014**, 89, 022001. [Google Scholar] [CrossRef] [Green Version] - Anholm, M.; Ballmer, S.; Creighton, J.D.E.; Price, L.R.; Siemens, X. Optimal strategies for gravitational wave stochastic background searches in pulsar timing data. PRD
**2009**, 79, 084030. [Google Scholar] [CrossRef] [Green Version] - Hellings, R.W.; Downs, G.S. Upper limits on the isotropic gravitational radiation background from pulsar timing analysis. ApJ
**1983**, 265, L39–L42. [Google Scholar] [CrossRef] - Allen, B.; Romano, J.D. Detecting a stochastic background of gravitational radiation: Signal processing strategies and sensitivities. Phys. Rev. D
**1999**, 59, 102001. [Google Scholar] [CrossRef] [Green Version] - Renzini, A.I.; Contaldi, C.R. Mapping Incoherent Gravitational Wave Backgrounds. Mon. Not. R. Astron. Soc.
**2018**, 481, 4650–4661. [Google Scholar] [CrossRef] [Green Version] - Renzini, A.I.; Contaldi, C.R. Gravitational Wave Background Sky Maps from Advanced LIGO O1 Data. Phys. Rev. Lett.
**2019**, 122, 081102. [Google Scholar] [CrossRef] [Green Version] - Renzini, A.; Contaldi, C. Improved limits on a stochastic gravitational-wave background and its anisotropies from Advanced LIGO O1 and O2 runs. Phys. Rev.
**2019**, D100, 063527. [Google Scholar] [CrossRef] [Green Version] - Chatziioannou, K.; Haster, C.J.; Littenberg, T.B.; Farr, W.M.; Ghonge, S.; Millhouse, M.; Clark, J.A.; Cornish, N. Noise spectral estimation methods and their impact on gravitational wave measurement of compact binary mergers. Phys. Rev. D
**2019**, 100, 104004. [Google Scholar] [CrossRef] [Green Version] - Cornish, N.J.; Romano, J.D. Towards a unified treatment of gravitational-wave data analysis. Phys. Rev. D
**2013**, 87, 122003. [Google Scholar] [CrossRef] [Green Version] - Cornish, N.J. Detecting a stochastic gravitational wave background with the Laser Interferometer Space Antenna. Phys. Rev. D
**2002**, 65, 022004. [Google Scholar] [CrossRef] [Green Version] - Coughlin, M.; Christensen, N.; Gair, J.; Kandhasamy, S.; Thrane, E. Method for estimation of gravitational-wave transient model parameters in frequency–time maps. Class. Quantum Gravity
**2014**, 31, 165012. [Google Scholar] [CrossRef] [Green Version] - Mandic, V.; Thrane, E.; Giampanis, S.; Regimbau, T. Parameter Estimation in Searches for the Stochastic Gravitational-Wave Background. Phys. Rev. Lett.
**2012**, 109, 171102. [Google Scholar] [CrossRef] [Green Version] - Callister, T.; Biscoveanu, A.S.; Christensen, N.; Isi, M.; Matas, A.; Minazzoli, O.; Regimbau, T.; Sakellariadou, M.; Tasson, J.; Thrane, E. Polarization-based Tests of Gravity with the Stochastic Gravitational-Wave Background. Phys. Rev. X
**2017**, 7, 041058. [Google Scholar] [CrossRef] [Green Version] - Abbott, B.P.; Abbott, R.; Abbott, T.D.; Abraham, S.; Acernese, F.; Ackley, K.; Adams, C.; Adya, V.B.; Affeldt, C.; Agathos, M.; et al. Search for the isotropic stochastic background using data from Advanced LIGO’s second observing run. Phys. Rev. D
**2019**, 100, 061101. [Google Scholar] [CrossRef] [Green Version] - Tsukada, L.; Callister, T.; Matas, A.; Meyers, P. First search for a stochastic gravitational-wave background from ultralight bosons. Phys. Rev. D
**2019**, 99, 103015. [Google Scholar] [CrossRef] [Green Version] - Tsukada, L.; Brito, R.; East, W.E.; Siemonsen, N. Modeling and searching for a stochastic gravitational-wave background from ultralight vector bosons. Phys. Rev. D
**2021**, 103, 083005. [Google Scholar] [CrossRef] - Romero, A.; Martinovic, K.; Callister, T.A.; Guo, H.K.; Martínez, M.; Sakellariadou, M.; Yang, F.W.; Zhao, Y. Implications for First-Order Cosmological Phase Transitions from the Third LIGO-Virgo Observing Run. Phys. Rev. Lett.
**2021**, 126, 151301. [Google Scholar] [CrossRef] - Romero-Rodriguez, A.; Martinez, M.; Pujolàs, O.; Sakellariadou, M.; Vaskonen, V. Search for a scalar induced stochastic gravitational wave background in the third LIGO-Virgo observing run. Phys. Rev. Lett.
**2022**, 128, 051301. [Google Scholar] [CrossRef] - Martinovic, K.; Badger, C.; Sakellariadou, M.; Mandic, V. Searching for parity violation with the LIGO-Virgo-KAGRA network. Phys. Rev. D
**2021**, 104, L081101. [Google Scholar] [CrossRef] - Meyers, P.M.; Martinovic, K.; Christensen, N.; Sakellariadou, M. Detecting a stochastic gravitational-wave background in the presence of correlated magnetic noise. PRD
**2020**, 102, 102005. [Google Scholar] [CrossRef] - Martinovic, K.; Meyers, P.M.; Sakellariadou, M.; Christensen, N. Simultaneous estimation of astrophysical and cosmological stochastic gravitational-wave backgrounds with terrestrial detectors. Phys. Rev. D
**2021**, 103, 043023. [Google Scholar] [CrossRef] - Parida, A.; Suresh, J.; Mitra, S.; Jhingan, S. Component separation map-making for stochastic gravitational wave background. arXiv
**2019**, arXiv:1904.05056. [Google Scholar] - Meacher, D.; Coughlin, M.; Morris, S.; Regimbau, T.; Christensen, N.; Kandhasamy, S.; Mandic, V.; Romano, J.D.; Thrane, E. Mock data and science challenge for detecting an astrophysical stochastic gravitational-wave background with Advanced LIGO and Advanced Virgo. Phys. Rev. D
**2015**, 92, 063002. [Google Scholar] [CrossRef] [Green Version] - Biwer, C.; Barker, D.; Batch, J.C.; Betzwieser, J.; Fisher, R.P.; Goetz, E.; Kandhasamy, S.; Karki, S.; Kissel, J.S.; Lundgren, A.P.; et al. Validating gravitational-wave detections: The Advanced LIGO hardware injection system. Phys. Rev. D
**2017**, 95, 062002. [Google Scholar] [CrossRef] [Green Version] - van Haasteren, R.; Levin, Y.; McDonald, P.; Lu, T. On measuring the gravitational-wave background using Pulsar Timing Arrays. MNRAS
**2009**, 395, 1005–1014. [Google Scholar] [CrossRef] [Green Version] - van Haasteren, R.; Levin, Y.; Janssen, G.H.; Lazaridis, K.; Kramer, M.; Stappers, B.W.; Desvignes, G.; Purver, M.B.; Lyne, A.G.; Ferdman, R.D.; et al. Placing limits on the stochastic gravitational-wave background using European Pulsar Timing Array data. MNRAS
**2011**, 414, 3117–3128. [Google Scholar] [CrossRef] [Green Version] - Lentati, L.; Alexander, P.; Hobson, M.P.; Taylor, S.; Gair, J.; Balan, S.T.; van Haasteren, R. Hyper-efficient model-independent Bayesian method for the analysis of pulsar timing data. PRD
**2013**, 87, 104021. [Google Scholar] [CrossRef] [Green Version] - van Haasteren, R.; Vallisneri, M. New advances in the Gaussian-process approach to pulsar-timing data analysis. PRD
**2014**, 90, 104012. [Google Scholar] [CrossRef] [Green Version] - Lentati, L.; Alexander, P.; Hobson, M.P.; Feroz, F.; van Haasteren, R.; Lee, K.J.; Shannon, R.M. TEMPONEST: A Bayesian approach to pulsar timing analysis. MNRAS
**2014**, 437, 3004–3023. [Google Scholar] [CrossRef] [Green Version] - Hobbs, G.; Jenet, F.; Lee, K.J.; Verbiest, J.P.W.; Yardley, D.; Manchester, R.; Lommen, A.; Coles, W.; Edwards, R.; Shettigara, C. TEMPO2: A new pulsar timing package—III. Gravitational wave simulation. MNRAS
**2009**, 394, 1945–1955. [Google Scholar] [CrossRef] [Green Version] - Arzoumanian, Z.; Brazier, A.; Burke-Spolaor, S.; Chamberlin, S.J.; Chatterjee, S.; Christy, B.; Cordes, J.M.; Cornish, N.J.; Crowter, K.; Demorest, P.B.; et al. The NANOGrav Nine-year Data Set: Limits on the Isotropic Stochastic Gravitational Wave Background. ApJ
**2016**, 821, 13. [Google Scholar] [CrossRef] - Taylor, S.R.; Lentati, L.; Babak, S.; Brem, P.; Gair, J.R.; Sesana, A.; Vecchio, A. All correlations must die: Assessing the significance of a stochastic gravitational-wave background in pulsar timing arrays. PRD
**2017**, 95, 042002. [Google Scholar] [CrossRef] [Green Version] - Arzoumanian, Z.; Baker, P.T.; Brazier, A.; Burke-Spolaor, S.; Chamberlin, S.J.; Chatterjee, S.; Christy, B.; Cordes, J.M.; Cornish, N.J.; Crawford, F.; et al. The NANOGrav 11 Year Data Set: Pulsar-timing Constraints on the Stochastic Gravitational-wave Background. ApJ
**2018**, 859, 47. [Google Scholar] [CrossRef] - Chamberlin, S.J.; Creighton, J.D.E.; Siemens, X.; Demorest, P.; Ellis, J.; Price, L.R.; Romano, J.D. Time-domain implementation of the optimal cross-correlation statistic for stochastic gravitational-wave background searches in pulsar timing data. PRD
**2015**, 91, 044048. [Google Scholar] [CrossRef] [Green Version] - Vigeland, S.J.; Islo, K.; Taylor, S.R.; Ellis, J.A. Noise-marginalise optimal statistic: A robust hybrid frequentist-Bayesian statistic for the stochastic gravitational-wave background in pulsar timing arrays. PRD
**2018**, 98, 044003. [Google Scholar] [CrossRef] [Green Version] - Cornish, N.J.; Sampson, L. Towards robust gravitational wave detection with pulsar timing arrays. PRD
**2016**, 93, 104047. [Google Scholar] [CrossRef] [Green Version] - Bécsy, B.; Cornish, N.J. Joint search for isolated sources and an unresolved confusion background in pulsar timing array data. Class. Quantum Gravity
**2020**, 37, 135011. [Google Scholar] [CrossRef] - Prix, R.; Itoh, Y. Global parameter-space correlations of coherent searches for continuous gravitational waves. Class. Quantum Gravity
**2005**, 22, S1003–S1012. [Google Scholar] [CrossRef] [Green Version] - Ain, A.; Dalvi, P.; Mitra, S. Fast gravitational wave radiometry using data folding. Phys. Rev. D Part. Fields, Gravit. Cosmol.
**2015**, 92, 022003. [Google Scholar] [CrossRef] [Green Version] - Ain, A.; Suresh, J.; Mitra, S. Very fast stochastic gravitational wave background map making using folded data. PRD
**2018**, 98, 024001. [Google Scholar] [CrossRef] [Green Version] - Goncharov, B.; Thrane, E. All-sky radiometer for narrowband gravitational waves using folded data. PRD
**2018**, 98, 064018. [Google Scholar] [CrossRef] [Green Version] - Mitra, S.; Dhurandhar, S.; Souradeep, T.; Lazzarini, A.; Mandic, V.; Bose, S.; Ballmer, S. Gravitational wave radiometry: Mapping a stochastic gravitational wave background. Phys. Rev. D
**2008**, 77, 042002. [Google Scholar] [CrossRef] [Green Version] - Thrane, E.; Ballmer, S.; Romano, J.D.; Mitra, S.; Talukder, D.; Bose, S.; Mandic, V. Probing the anisotropies of a stochastic gravitational-wave background using a network of ground-based laser interferometers. Phys. Rev. D Part. Fields, Gravit. Cosmol.
**2009**, 80, 122002. [Google Scholar] [CrossRef] [Green Version] - Mingarelli, C.M.F.; Sidery, T.; Mandel, I.; Vecchio, A. Characterizing gravitational wave stochastic background anisotropy with pulsar timing arrays. PRD
**2013**, 88, 062005. [Google Scholar] [CrossRef] [Green Version] - Taylor, S.R.; Gair, J.R. Searching for anisotropic gravitational-wave backgrounds using pulsar timing arrays. PRD
**2013**, 88, 084001. [Google Scholar] [CrossRef] [Green Version] - Contaldi, C.R.; Pieroni, M.; Renzini, A.I.; Cusin, G.; Karnesis, N.; Peloso, M.; Ricciardone, A.; Tasinato, G. Maximum likelihood map-making with the Laser Interferometer Space Antenna. Phys. Rev. D
**2020**, 102, 043502. [Google Scholar] [CrossRef] - Banagiri, S.; Criswell, A.; Kuan, T.; Mandic, V.; Romano, J.D.; Taylor, S.R. Mapping the gravitational-wave sky with LISA: A Bayesian spherical harmonic approach. MNRAS
**2021**, 507, 5451–5462. [Google Scholar] [CrossRef] - Bond, J.R.; Jaffe, A.H.; Knox, L. Estimating the power spectrum of the cosmic microwave background. Phys. Rev.
**1998**, D57, 2117–2137. [Google Scholar] [CrossRef] [Green Version] - Rocha, G.; Contaldi, C.; Bond, J.; Gorski, K. Application of XFaster power spectrum and likelihood estimator to Planck. Mon. Not. R. Astron. Soc.
**2011**, 414, 823–846. [Google Scholar] [CrossRef] [Green Version] - Cornish, N.J.; van Haasteren, R. Mapping the nano-Hertz gravitational wave sky. arXiv
**2014**, arXiv:1406.4511. [Google Scholar] - Ballmer, S.W. A radiometer for stochastic gravitational waves. Class. Quantum Gravity
**2006**, 23, S179–S185. [Google Scholar] [CrossRef] [Green Version] - Gair, J.; Romano, J.D.; Taylor, S.; Mingarelli, C.M.F. Mapping gravitational-wave backgrounds using methods from CMB analysis: Application to pulsar timing arrays. PRD
**2014**, 90, 082001. [Google Scholar] [CrossRef] [Green Version] - Gair, J.R.; Romano, J.D.; Taylor, S.R. Mapping gravitational-wave backgrounds of arbitrary polarisation using pulsar timing arrays. PRD
**2015**, 92, 102003. [Google Scholar] [CrossRef] [Green Version] - Romano, J.D.; Taylor, S.R.; Cornish, N.J.; Gair, J.; Mingarelli, C.M.F.; van Haasteren, R. Phase-coherent mapping of gravitational-wave backgrounds using ground-based laser interferometers. PRD
**2015**, 92, 042003. [Google Scholar] [CrossRef] [Green Version] - Taylor, S.R.; Mingarelli, C.M.F.; Gair, J.R.; Sesana, A.; Theureau, G.; Babak, S.; Bassa, C.G.; Brem, P.; Burgay, M.; Caballero, R.N.; et al. Limits on Anisotropy in the Nanohertz Stochastic Gravitational Wave Background. PRL
**2015**, 115, 041101. [Google Scholar] [CrossRef] - Ali-Haïmoud, Y.; Smith, T.L.; Mingarelli, C.M.F. Fisher formalism for anisotropic gravitational-wave background searches with pulsar timing arrays. PRD
**2020**, 102, 122005. [Google Scholar] [CrossRef] - Ali-Haïmoud, Y.; Smith, T.L.; Mingarelli, C.M.F. Insights into searches for anisotropies in the nanohertz gravitational-wave background. PRD
**2021**, 103, 042009. [Google Scholar] [CrossRef] - Suresh, J.; Agarwal, D.; Mitra, S. Jointly setting upper limits on multiple components of an anisotropic stochastic gravitational-wave background. PRD
**2021**, 104, 102003. [Google Scholar] [CrossRef] - Gorski, K.M.; Hivon, E.; Banday, A.J.; Wandelt, B.D.; Hansen, F.K.; Reinecke, M.; Bartelman, M. HEALPix—A Framework for high resolution discretization, and fast analysis of data distributed on the sphere. Astrophys. J.
**2005**, 622, 759–771. [Google Scholar] [CrossRef] - Hotinli, S.C.; Kamionkowski, M.; Jaffe, A.H. The search for statistical anisotropy in the gravitational-wave background with pulsar timing arrays. Open J. Astrophys.
**2019**, 2, 8. [Google Scholar] [CrossRef] - Ivezić, Ž.; Connolly, A.; VanderPlas, J.; Gray, A. Statistics, Data Mining, and Machine Learning in Astronomy: A Practical Python Guide for the Analysis of Survey Data; Princeton Series in Modern Observational Astronomy; Princeton University Press: Princeton, NJ, USA, 2014. [Google Scholar]
- Suresh, J.; Ain, A.; Mitra, S. Unified mapmaking for an anisotropic stochastic gravitational wave background. PRD
**2021**, 103, 083024. [Google Scholar] [CrossRef] - Myers, S.T.; Contaldi, C.R.; Bond, J.R.; Pen, U.L.; Pogosyan, D.; Prunet, S.; Sievers, J.L.; Mason, B.S.; Pearson, T.J.; Readhead, A.C.S.; et al. A fast gridded method for the estimation of the power spectrum of the CMB from interferometer data with application to the cosmic background imager. Astrophys. J.
**2003**, 591, 575–598. [Google Scholar] [CrossRef] [Green Version] - Abbott, R.; Abbott, T.D.; Abraham, S.; Acernese, F.; Ackley, K.; Adams, A.; Adams, C.; Adhikari, R.X.; Adya, V.B.; Affeldt, C.; et al. Search for anisotropic gravitational-wave backgrounds using data from Advanced LIGO and Advanced Virgo’s first three observing runs. Phys. Rev. D
**2021**, 104, 022005. [Google Scholar] [CrossRef] - Drasco, S.; Flanagan, E.E. Detection methods for non-Gaussian gravitational wave stochastic backgrounds. Phys. Rev.
**2003**, D67, 082003. [Google Scholar] [CrossRef] [Green Version] - Abbott, R.; Abbott, T.D.; Acernese, F.; Ackley, K.; Adams, C.; Adhikari, N.; Adhikari, R.X.; Adya, V.B.; Affeldt, C.; Agarwal, D.; et al. GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo During the Second Part of the Third Observing Run. arXiv
**2021**, arXiv:2111.03606. [Google Scholar] - Thrane, E. Measuring the non-Gaussian stochastic gravitational-wave background: A method for realistic interferometer data. Phys. Rev. D
**2013**, 87, 043009. [Google Scholar] [CrossRef] [Green Version] - Smith, R.; Thrane, E. Optimal Search for an Astrophysical Gravitational-Wave Background. Phys. Rev. X
**2018**, 8, 021019. [Google Scholar] [CrossRef] [Green Version] - Lawrence, J.; Turbang, K.; Matas, A.; Renzini, A.; Van Remortel, N.; Romano, J. A stochastic-signal-based search for intermittent gravitational-wave backgrounds. 2022; in preparation. [Google Scholar]
- Banagiri, S.; Mandic, V.; Scarlata, C.; Yang, K.Z. Measuring angular N-point correlations of binary black hole merger gravitational-wave events with hierarchical Bayesian inference. Phys. Rev. D
**2020**, 102, 063007. [Google Scholar] [CrossRef] - Barsotti, L.; Fritschel, P.; Evans, M.; Gras, S. Advanced LIGO Design Sensitivity Curve. 2018. Available online: https://dcc.ligo.org/LIGO-T1800044/public (accessed on 1 November 2021).
- Thrane, E.; Romano, J.D. Sensitivity curves for searches for gravitational-wave backgrounds. Phys. Rev.
**2013**, D88, 124032. [Google Scholar] [CrossRef] [Green Version] - Coughlin, M.; Harms, J. Constraining the gravitational wave energy density of the Universe using Earth’s ring. Phys. Rev. D
**2014**, 90, 042005. [Google Scholar] [CrossRef] [Green Version] - Armstrong, J.W.; Iess, L.; Tortora, P.; Bertotti, B. Stochastic gravitational wave background: Upper limits in the 10**-6-Hz 10**-3-Hz band. Astrophys. J.
**2003**, 599, 806–813. [Google Scholar] [CrossRef] [Green Version] - Lasky, P.D.; Mingarelli, C.M.; Smith, T.L.; Giblin, J.T., Jr.; Thrane, E.; Reardon, D.J.; Caldwell, R.; Bailes, M.; Bhat, N.R.; Burke-Spolaor, S.; et al. Gravitational-wave cosmology across 29 decades in frequency. Phys. Rev. X
**2016**, 6, 011035. [Google Scholar] [CrossRef] [Green Version] - Akrami, Y.; Arroja, F.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Ballardini, M.; Banday, A.J.; Barreiro, R.B.; Bartolo, N.; Basak, S.; et al. Planck 2018 results. X. Constraints on inflation. Astron. Astrophys.
**2020**, 641, A10. [Google Scholar] [CrossRef] [Green Version] - Badurina, L.; Bentine, E.; Blas, D.; Bongs, K.; Bortoletto, D.; Bowcock, T.; Bridges, K.; Bowden, W.; Buchmueller, O.; Burrage, C.; et al. AION: An Atom Interferometer Observatory and Network. JCAP
**2020**, 05, 011. [Google Scholar] [CrossRef] - Blas, D.; Jenkins, A.C. Detecting stochastic gravitational waves with binary resonance. arXiv
**2021**, arXiv:2107.04063. [Google Scholar] - Blas, D.; Jenkins, A.C. Bridging the μHz gap in the gravitational-wave landscape with binary resonance. arXiv
**2021**, arXiv:2107.04601. [Google Scholar] - Janssen, G.H.; Hobbs, G.; McLaughlin, M.; Bassa, C.G.; Deller, A.T.; Kramer, M.; Lee, K.J.; Mingarelli, C.M.F.; Rosado, P.A.; Sanidas, S.; et al. Gravitational wave astronomy with the SKA. PoS
**2015**, AASKA14, 037. [Google Scholar] [CrossRef] [Green Version] - Abbott, B.; Abbott, R.; Adhikari, R.; Ageev, A.; Allen, B.; Amin, R.; Anderson, S.B.; Anderson, W.G.; Araya, M.; Armandula, H.; et al. Analysis of first LIGO science data for stochastic gravitational waves. Phys. Rev. D
**2004**, 69, 122004. [Google Scholar] [CrossRef] [Green Version] - Abbott, B.; Abbott, R.; Adhikari, R.; Agresti, J.; Ajith, P.; Allen, B.; Allen, J.; Amin, R.; Anderson, S.B.; Anderson, W.G.; et al. Upper limits on a stochastic background of gravitational waves. Phys. Rev. Lett.
**2005**, 95, 221101. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Abbott, B.; Abbott, R.; Adhikari, R.; Agresti, J.; Ajith, P.; Allen, B.; Amin, R.; Anderson, S.B.; Anderson, W.G.; Araya, M.; et al. Searching for a Stochastic Background of Gravitational Waves with LIGO. Astrophys. J.
**2007**, 659, 918–930. [Google Scholar] [CrossRef] [Green Version] - Aasi, J.; Abbott, B.P.; Abbott, R.; Abbott, T.; Abernathy, M.R.; Accadia, T.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; et al. Improved Upper Limits on the Stochastic Gravitational-Wave Background from 2009–2010 LIGO and Virgo Data. Phys. Rev. Lett.
**2014**, 113, 231101. [Google Scholar] [CrossRef] [Green Version] - Abbott, B.P.; Abbott, R.; Abbott, T.D.; Abernathy, M.R.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.X.; et al. Upper Limits on the Stochastic Gravitational-Wave Background from Advanced LIGO’s First Observing Run. Phys. Rev. Lett.
**2017**, 118, 121101, Erratum on**2017**, 119, 029901. [Google Scholar] [CrossRef] - Abbott, B.; Abbott, R.; Adhikari, R.; Agresti, J.; Ajith, P.; Allen, B.; Amin, R.; Anderson, S.B.; Anderson, W.G.; Arain, M.; et al. Upper limit map of a background of gravitational waves. Phys. Rev. D
**2007**, 76, 082003. [Google Scholar] [CrossRef] [Green Version] - Abadie, J.; Abbott, B.P.; Abbott, R.; Abernathy, M.; Accadia, T.; Acernese, F.; Adams, C.; Adhikari, R.; Ajith, P.; Allen, B.; et al. Directional Limits on Persistent Gravitational Waves Using LIGO S5 Science Data. Phys. Rev. Lett.
**2011**, 107, 271102. [Google Scholar] [CrossRef] [Green Version] - Abbott, B.P.; Abbott, R.; Abbott, T.D.; Abernathy, M.R.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.X.; et al. Directional Limits on Persistent Gravitational Waves from Advanced LIGO’s First Observing Run. Phys. Rev. Lett.
**2017**, 118, 121102. [Google Scholar] [CrossRef] [Green Version] - Abbott, B.P.; Abbott, R.; Abbott, T.D.; Abraham, S.; Acernese, F.; Ackley, K.; Adams, C.; Adhikari, R.X.; Adya, V.B.; Affeldt, C.; et al. Directional limits on persistent gravitational waves using data from Advanced LIGO’s first two observing runs. Phys. Rev. D
**2019**, 100, 062001. [Google Scholar] [CrossRef] [Green Version] - Aasi, J.; Abadie, J.; Abbott, B.P.; Abbott, R.; Abbott, T.; Abernathy, M.R.; Accadia, T.; Acernese, F.; Adams, C.; Adams, T.; et al. Constraints on Cosmic Strings from the LIGO-Virgo Gravitational-Wave Detectors. Phys. Rev. Lett.
**2014**, 112, 131101. [Google Scholar] [CrossRef] [Green Version] - Abbott, B.P.; Abbott, R.; Abbott, T.D.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.X.; Adya, V.B.; et al. Constraints on cosmic strings using data from the first Advanced LIGO observing run. Phys. Rev. D
**2018**, 97, 102002. [Google Scholar] [CrossRef] [Green Version] - Abbott, B.P.; Abbott, R.; Abbott, T.D.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.X.; Adya, V.B.; et al. Search for Tensor, Vector, and Scalar Polarizations in the Stochastic Gravitational-Wave Background. Phys. Rev. Lett.
**2018**, 120, 201102. [Google Scholar] [CrossRef] [Green Version] - Agarwal, D.; Suresh, J.; Mitra, S.; Ain, A. Upper limits on persistent gravitational waves using folded data and the full covariance matrix from Advanced LIGO’s first two observing runs. PRD
**2021**, 104, 123018. [Google Scholar] [CrossRef] - Yang, K.Z.; Mandic, V.; Scarlata, C.; Banagiri, S. Searching for Cross-Correlation Between Stochastic Gravitational Wave Background and Galaxy Number Counts. Mon. Not. R. Astron. Soc.
**2020**, 500, 1666–1672. [Google Scholar] [CrossRef] - Kapadia, S.J.; Lal Pandey, K.; Suyama, T.; Kandhasamy, S.; Ajith, P. Search for the Stochastic Gravitational-wave Background Induced by Primordial Curvature Perturbations in LIGO’s Second Observing Run. Astrophys. J. Lett.
**2021**, 910, L4. [Google Scholar] [CrossRef] - Covas, P.B.; Effler, A.; Goetz, E.; Meyers, P.M.; Neunzert, A.; Oliver, M.; Pearlstone, B.L.; Roma, V.J.; Schofield, R.M.; Adya, V.B.; et al. Identification and mitigation of narrow spectral artifacts that degrade searches for persistent gravitational waves in the first two observing runs of Advanced LIGO. Phys. Rev. D
**2018**, 97. [Google Scholar] [CrossRef] [Green Version] - LIGO Document T2000384-v4. Available online: https://dcc.ligo.org/T2000384/public (accessed on 1 November 2021).
- Matas, A.; Dvorkin, I.; Regimbau, T.; Romero, A. Applying Gating to Stochastic Searches in O3. 2021. Available online: https://dcc.ligo.org/P2000546/public (accessed on 1 November 2021).
- LIGO Scientific Collaboration; Virgo Collaboration; KAGRA Scientific Collaboration. Data for Upper Limits on the Isotropic Gravitational-Wave Background from Advanced LIGO’s and Advanced Virgo’s Third Observing Run. 2021. Available online: https://dcc.ligo.org/LIGO-G2001287 (accessed on 1 November 2021).
- Will, C.M. The Confrontation between General Relativity and Experiment. Living Rev. Relativ.
**2014**, 17, 4. [Google Scholar] [CrossRef] [Green Version] - Abbott, B.P.; Abbott, R.; Abbott, T.D.; Abraham, S.; Acernese, F.; Ackley, K.; Adams, C.; Adya, V.B.; Affeldt, C.; Agathos, M.; et al. Prospects for observing and localizing gravitational-wave transients with Advanced LIGO, Advanced Virgo and KAGRA. Living Rev. Relativ.
**2020**, 23, 3. [Google Scholar] [CrossRef] - Mukherjee, S.; Broadhurst, T.; Diego, J.M.; Silk, J.; Smoot, G.F. Inferring the lensing rate of LIGO-Virgo sources from the stochastic gravitational wave background. Mon. Not. R. Astron. Soc.
**2021**, 501, 2451–2466. [Google Scholar] [CrossRef] - Buscicchio, R.; Moore, C.J.; Pratten, G.; Schmidt, P.; Bianconi, M.; Vecchio, A. Constraining the lensing of binary black holes from their stochastic background. Phys. Rev. Lett.
**2020**, 125, 141102. [Google Scholar] [CrossRef] - Buscicchio, R.; Moore, C.J.; Pratten, G.; Schmidt, P.; Vecchio, A. Constraining the lensing of binary neutron stars from their stochastic background. Phys. Rev. D
**2020**, 102, 081501. [Google Scholar] [CrossRef] - Abbott, R.; Abbott, T.D.; Abraham, S.; Acernese, F.; Ackley, K.; Adams, A.; Adams, C.; Adhikari, R.X.; Adya, V.B.; Affeldt, C.; et al. GWTC-2: Compact Binary Coalescences Observed by LIGO and Virgo During the First Half of the Third Observing Run. Phys. Rev. X
**2021**, 11, 021053. [Google Scholar] [CrossRef] - Ade, P.A.; Aghanim, N.; Armitage-Caplan, C.; Arnaud, M.; Ashdown, M.; Atrio-Barandela, F.; Aumont, J.; Baccigalupi, C.; Banday, A.J.; Barreiro, R.B.; et al. Planck 2013 results. XXV. Searches for cosmic strings and other topological defects. Astron. Astrophys.
**2014**, 571, A25. [Google Scholar] [CrossRef] [Green Version] - Vaskonen, V.; Veermäe, H. Lower bound on the primordial black hole merger rate. Phys. Rev. D
**2020**, 101, 043015. [Google Scholar] [CrossRef] [Green Version] - Carr, B.; Kohri, K.; Sendouda, Y.; Yokoyama, J. Constraints on Primordial Black Holes. Rep. Prog. Phys.
**2021**, 84, 116902. [Google Scholar] [CrossRef] - LIGO Scientific Collaboration; Virgo Collaboration; KAGRA Scientific Collaboration. Data Products and Supplemental Information for O3 Stochastic Directional Paper. 2021. Available online: https://dcc.ligo.org/LIGO-G2002165 (accessed on 1 November 2021).
- Abbott, B.; Abbott, R.; Abbott, T.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.; Adya, V.; et al. Search for gravitational waves from Scorpius X-1 in the first Advanced LIGO observing run with a hidden Markov model. Phys. Rev. D
**2017**, 95, 122003. [Google Scholar] [CrossRef] [Green Version] - Sun, L.; Melatos, A.; Lasky, P.D.; Chung, C.T.Y.; Darman, N.S. Cross-correlation search for continuous gravitational waves from a compact object in SNR 1987A in LIGO Science run 5. Phys. Rev. D
**2016**, 94, 082004. [Google Scholar] [CrossRef] [Green Version] - Aasi, J.; Abadie, J.; Abbott, B.P.; Abbott, R.; Abbott, T.; Abernathy, M.R.; Accadia, T.; Acernese, F.; Adams, C.; Adams, T.; et al. Directed search for continuous gravitational waves from the Galactic center. Phys. Rev. D
**2013**, 88, 102002. [Google Scholar] [CrossRef] - Abbott, R.; Abbott, T.D.; Acernese, F.; Ackley, K.; Adams, C.; Adhikari, N.; Adhikari, R.X.; Adya, V.B.; Affeldt, C.; Agarwal, D.; et al. All-sky, all-frequency directional search for persistent gravitational-waves from Advanced LIGO’s and Advanced Virgo’s first three observing runs. arXiv
**2021**, arXiv:2110.09834. [Google Scholar] - Tenorio, R.; Keitel, D.; Sintes, A.M. Application of a hierarchical MCMC follow-up to Advanced LIGO continuous gravitational-wave candidates. Phys. Rev. D
**2021**, 104, 084012. [Google Scholar] [CrossRef] - Burke-Spolaor, S.; Taylor, S.R.; Charisi, M.; Dolch, T.; Hazboun, J.S.; Holgado, A.M.; Kelley, L.Z.; Lazio, T.J.W.; Madison, D.R.; McMann, N.; et al. The astrophysics of nanohertz gravitational waves. A&A Rev.
**2019**, 27, 5. [Google Scholar] [CrossRef] [Green Version] - Taylor, S.R. The Nanohertz Gravitational Wave Astronomer. arXiv
**2021**, arXiv:2105.13270. [Google Scholar] - Sazhin, M.V. Opportunities for detecting ultralong gravitational waves. Sov. Astron.
**1978**, 22, 36–38. [Google Scholar] - Detweiler, S. Pulsar timing measurements and the search for gravitational waves. ApJ
**1979**, 234, 1100–1104. [Google Scholar] [CrossRef] - Siemens, X.; Ellis, J.; Jenet, F.; Romano, J.D. The stochastic background: Scaling laws and time to detection for pulsar timing arrays. Class. Quantum Gravity
**2013**, 30, 224015. [Google Scholar] [CrossRef] [Green Version] - Bailes, M.; Jameson, A.; Abbate, F.; Barr, E.D.; Bhat, N.D.R.; Bondonneau, L.; Burgay, M.; Buchner, S.J.; Camilo, F.; Champion, D.J.; et al. The MeerKAT telescope as a pulsar facility: System verification and early science results from MeerTime. PASA
**2020**, 37, e028. [Google Scholar] [CrossRef] - Jonas, J.L. MeerKAT - The South African Array With Composite Dishes and Wide-Band Single Pixel Feeds. IEEE Proc.
**2009**, 97, 1522–1530. [Google Scholar] [CrossRef] - Jenet, F.A.; Hobbs, G.B.; van Straten, W.; Manchester, R.N.; Bailes, M.; Verbiest, J.P.W.; Edwards, R.T.; Hotan, A.W.; Sarkissian, J.M.; Ord, S.M. Upper Bounds on the Low-Frequency Stochastic Gravitational Wave Background from Pulsar Timing Observations: Current Limits and Future Prospects. ApJ
**2006**, 653, 1571–1576. [Google Scholar] [CrossRef] - Demorest, P.B.; Ferdman, R.D.; Gonzalez, M.E.; Nice, D.; Ransom, S.; Stairs, I.H.; Arzoumanian, Z.; Brazier, A.; Burke-Spolaor, S.; Chamberlin, S.J.; et al. Limits on the Stochastic Gravitational Wave Background from the North American Nanohertz Observatory for Gravitational Waves. ApJ
**2013**, 762, 94. [Google Scholar] [CrossRef] [Green Version] - Shannon, R.M.; Ravi, V.; Coles, W.A.; Hobbs, G.; Keith, M.J.; Manchester, R.N.; Wyithe, J.S.B.; Bailes, M.; Bhat, N.D.R.; Burke-Spolaor, S.; et al. Gravitational-wave limits from pulsar timing constrain supermassive black hole evolution. Science
**2013**, 342, 334–337. [Google Scholar] [CrossRef] [Green Version] - Lentati, L.; Taylor, S.R.; Mingarelli, C.M.F.; Sesana, A.; Sanidas, S.A.; Vecchio, A.; Caballero, R.N.; Lee, K.J.; van Haasteren, R.; Babak, S.; et al. European Pulsar Timing Array limits on an isotropic stochastic gravitational-wave background. MNRAS
**2015**, 453, 2576–2598. [Google Scholar] [CrossRef] - NANOGrav Collaboration; Arzoumanian, Z.; Brazier, A.; Burke-Spolaor, S.; Chamberlin, S.; Chatterjee, S.; Christy, B.; Cordes, J.M.; Cornish, N.; Crowter, K.; et al. The NANOGrav Nine-year Data Set: Observations, Arrival Time Measurements, and Analysis of 37 Millisecond Pulsars. ApJ
**2015**, 813, 65. [Google Scholar] [CrossRef] - Shannon, R.M.; Ravi, V.; Lentati, L.T.; Lasky, P.D.; Hobbs, G.; Kerr, M.; Manchester, R.N.; Coles, W.A.; Levin, Y.; Bailes, M.; et al. Gravitational waves from binary supermassive black holes missing in pulsar observations. Science
**2015**, 349, 1522–1525. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chen, S.; Caballero, R.N.; Guo, Y.J.; Chalumeau, A.; Liu, K.; Shaifullah, G.; Lee, K.J.; Babak, S.; Desvignes, G.; Parthasarathy, A.; et al. Common-red-signal analysis with 24-yr high-precision timing of the European Pulsar Timing Array: Inferences in the stochastic gravitational-wave background search. MNRAS
**2021**, 508, 4970–4993. [Google Scholar] [CrossRef] - Antoniadis, J.; Arzoumanian, Z.; Babak, S.; Bailes, M.; Bak Nielsen, A.S.; Baker, P.T.; Bassa, C.G.; Bécsy, B.; Berthereau, A.; Bonetti, M.; et al. The International Pulsar Timing Array second data release: Search for an isotropic gravitational wave background. MNRAS
**2022**, 510, 4873–4887. [Google Scholar] [CrossRef] - Shannon, R.M.; Osłowski, S.; Dai, S.; Bailes, M.; Hobbs, G.; Manchester, R.N.; van Straten, W.; Raithel, C.A.; Ravi, V.; Toomey, L.; et al. Limitations in timing precision due to single-pulse shape variability in millisecond pulsars. MNRAS
**2014**, 443, 1463–1481. [Google Scholar] [CrossRef] [Green Version] - Lam, M.T.; McLaughlin, M.A.; Arzoumanian, Z.; Blumer, H.; Brook, P.R.; Cromartie, H.T.; Demorest, P.B.; DeCesar, M.E.; Dolch, T.; Ellis, J.A.; et al. The NANOGrav 12.5 yr Data Set: The Frequency Dependence of Pulse Jitter in Precision Millisecond Pulsars. ApJ
**2019**, 872, 193. [Google Scholar] [CrossRef] [Green Version] - Kerr, M.; Reardon, D.J.; Hobbs, G.; Shannon, R.M.; Manchester, R.N.; Dai, S.; Russell, C.J.; Zhang, S.; van Straten, W.; Osłowski, S.; et al. The Parkes Pulsar Timing Array project: Second data release. PASA
**2020**, 37, e020. [Google Scholar] [CrossRef] - Lentati, L.; Shannon, R.M.; Coles, W.A.; Verbiest, J.P.W.; van Haasteren, R.; Ellis, J.A.; Caballero, R.N.; Manchester, R.N.; Arzoumanian, Z.; Babak, S.; et al. From spin noise to systematics: Stochastic processes in the first International Pulsar Timing Array data release. MNRAS
**2016**, 458, 2161–2187. [Google Scholar] [CrossRef] - Goncharov, B.; Zhu, X.J.; Thrane, E. Is there a spectral turnover in the spin noise of millisecond pulsars? MNRAS
**2020**, 497, 3264–3272. [Google Scholar] [CrossRef] - Chalumeau, A.; Babak, S.; Petiteau, A.; Chen, S.; Samajdar, A.; Caballero, R.N.; Theureau, G.; Guillemot, L.; Desvignes, G.; Parthasarathy, A.; et al. Noise analysis in the European Pulsar Timing Array data release 2 and its implications on the gravitational-wave background search. MNRAS
**2022**, 509, 5538–5558. [Google Scholar] [CrossRef] - Haskell, B.; Melatos, A. Models of pulsar glitches. Int. J. Mod. Phys. D
**2015**, 24, 1530008. [Google Scholar] [CrossRef] - Hazboun, J.S.; Simon, J.; Siemens, X.; Romano, J.D. Model Dependence of Bayesian Gravitational-wave Background Statistics for Pulsar Timing Arrays. ApJ
**2020**, 905, L6. [Google Scholar] [CrossRef] - Goncharov, B.; Reardon, D.J.; Shannon, R.M.; Zhu, X.J.; Thrane, E.; Bailes, M.; Bhat, N.D.R.; Dai, S.; Hobbs, G.; Kerr, M.; et al. Identifying and mitigating noise sources in precision pulsar timing data sets. MNRAS
**2021**, 502, 478–493. [Google Scholar] [CrossRef] - Meyers, P.M.; O’Neill, N.J.; Melatos, A.; Evans, R.J. Rapid parameter estimation of a two-component neutron star model with spin wandering using a Kalman filter. MNRAS
**2021**, 506, 3349–3363. [Google Scholar] [CrossRef] - Tiburzi, C.; Hobbs, G.; Kerr, M.; Coles, W.A.; Dai, S.; Manchester, R.N.; Possenti, A.; Shannon, R.M.; You, X.P. A study of spatial correlations in pulsar timing array data. MNRAS
**2016**, 455, 4339–4350. [Google Scholar] [CrossRef] [Green Version] - Guo, Y.J.; Li, G.Y.; Lee, K.J.; Caballero, R.N. Studying the Solar system dynamics using pulsar timing arrays and the LINIMOSS dynamical model. MNRAS
**2019**, 489, 5573–5581. [Google Scholar] [CrossRef] - Vallisneri, M.; Taylor, S.R.; Simon, J.; Folkner, W.M.; Park, R.S.; Cutler, C.; Ellis, J.A.; Lazio, T.J.W.; Vigeland, S.J.; Aggarwal, K.; et al. Modeling the Uncertainties of Solar System Ephemerides for Robust Gravitational-wave Searches with Pulsar-timing Arrays. ApJ
**2020**, 893, 112. [Google Scholar] [CrossRef] [Green Version] - Hazboun, J.S.; Romano, J.D.; Smith, T.L. Realistic sensitivity curves for pulsar timing arrays. PRD
**2019**, 100, 104028. [Google Scholar] [CrossRef] [Green Version] - Hazboun, J.; Romano, J.; Smith, T. Hasasia: A Python package for Pulsar Timing Array Sensitivity Curves. J. Open Source Softw.
**2019**, 4, 1775. [Google Scholar] [CrossRef] [Green Version] - Verbiest, J.P.W.; Shaifullah, G.M. Measurement uncertainty in pulsar timing array experiments. Class. Quantum Gravity
**2018**, 35, 133001. [Google Scholar] [CrossRef] - Lommen, A.N. Pulsar timing arrays: The promise of gravitational wave detection. Rep. Prog. Phys.
**2015**, 78, 124901. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Sesana, A.; Vecchio, A. Measuring the parameters of massive black hole binary systems with pulsar timing array observations of gravitational waves. PRD
**2010**, 81, 104008. [Google Scholar] [CrossRef] [Green Version] - Mingarelli, C.M.F.; Lazio, T.J.W.; Sesana, A.; Greene, J.E.; Ellis, J.A.; Ma, C.P.; Croft, S.; Burke-Spolaor, S.; Taylor, S.R. The local nanohertz gravitational-wave landscape from supermassive black hole binaries. Nat. Astron.
**2017**, 1, 886–892. [Google Scholar] [CrossRef] - Arzoumanian, Z.; Baker, P.T.; Blumer, H.; Bécsy, B.; Brazier, A.; Brook, P.R.; Burke-Spolaor, S.; Charisi, M.; Chatterjee, S.; Chen, S.; et al. Searching for Gravitational Waves from Cosmological Phase Transitions with the NANOGrav 12.5-Year Dataset. PRL
**2021**, 127, 251302. [Google Scholar] [CrossRef] [PubMed] - Xue, X.; Bian, L.; Shu, J.; Yuan, Q.; Zhu, X.; Bhat, N.D.R.; Dai, S.; Feng, Y.; Goncharov, B.; Hobbs, G.; et al. Constraining Cosmological Phase Transitions with the Parkes Pulsar Timing Array. PRL
**2021**, 127, 251303. [Google Scholar] [CrossRef] [PubMed] - Moore, C.J.; Vecchio, A. Ultra-low-frequency gravitational waves from cosmological and astrophysical processes. Nat. Astron.
**2021**, 5, 1268–1274. [Google Scholar] [CrossRef] - Chen, Z.C.; Yuan, C.; Huang, Q.G. Non-tensorial gravitational wave background in NANOGrav 12.5-year data set. Sci. China Physics, Mech. Astron.
**2021**, 64, 120412. [Google Scholar] [CrossRef] - Chen, Z.C.; Wu, Y.M.; Huang, Q.G. Searching for Isotropic Stochastic Gravitational-Wave Background in the International Pulsar Timing Array Second Data Release. arXiv
**2021**, arXiv:2109.00296. [Google Scholar] - Wu, Y.M.; Chen, Z.C.; Huang, Q.G. Constraining the Polarization of Gravitational Waves with the Parkes Pulsar Timing Array Second Data Release. arXiv
**2021**, arXiv:2108.10518. [Google Scholar] [CrossRef] - Arzoumanian, Z.; Baker, P.T.; Blumer, H.; Bécsy, B.; Brazier, A.; Brook, P.R.; Burke-Spolaor, S.; Charisi, M.; Chatterjee, S.; Chen, S.; et al. The NANOGrav 12.5-year Data Set: Search for Non-Einsteinian Polarization Modes in the Gravitational-wave Background. ApJ
**2021**, 923, L22. [Google Scholar] [CrossRef] - Aggarwal, N.; Aguiar, O.D.; Bauswein, A.; Cella, G.; Clesse, S.; Cruise, A.M.; Domcke, V.; Figueroa, D.G.; Geraci, A.; Goryachev, M.; et al. Challenges and opportunities of gravitational-wave searches at MHz to GHz frequencies. Living Rev. Relativ.
**2021**, 24, 4. [Google Scholar] [CrossRef] - Weber, J. Detection and Generation of Gravitational Waves. Phys. Rev.
**1960**, 117, 306–313. [Google Scholar] [CrossRef] - Weber, J. Evidence for discovery of gravitational radiation. Phys. Rev. Lett.
**1969**, 22, 1320–1324. [Google Scholar] [CrossRef] [Green Version] - Aguiar, O.D. The Past, Present and Future of the Resonant-Mass Gravitational Wave Detectors. Res. Astron. Astrophys.
**2011**, 11, 1–42. [Google Scholar] [CrossRef] - Harms, J.; Ambrosino, F.; Angelini, L.; Braito, V.; Branchesi, M.; Brocato, E.; Cappellaro, E.; Coccia, E.; Coughlin, M.; Della Ceca, R.; et al. Lunar Gravitational-wave Antenna. Astrophys. J.
**2021**, 910, 1. [Google Scholar] [CrossRef] - Jani, K.; Loeb, A. Gravitational-Wave Lunar Observatory for Cosmology. J. Cosmol. Astropart. Phys.
**2021**, 2021, 044. [Google Scholar] [CrossRef] - Sesana, A.; Korsakova, N.; Sedda, M.A.; Baibhav, V.; Barausse, E.; Barke, S.; Berti, E.; Bonetti, M.; Capelo, P.R.; Caprini, C.; et al. Unveiling the gravitational universe at μ-Hz frequencies. Exper. Astron.
**2021**, 51, 1333–1383. [Google Scholar] [CrossRef] - Book, L.G.; Flanagan, E.E. Astrometric Effects of a Stochastic Gravitational Wave Background. Phys. Rev. D
**2011**, 83, 024024. [Google Scholar] [CrossRef] [Green Version] - Moore, C.J.; Mihaylov, D.P.; Lasenby, A.; Gilmore, G. Astrometric Search Method for Individually Resolvable Gravitational Wave Sources with Gaia. Phys. Rev. Lett.
**2017**, 119, 261102. [Google Scholar] [CrossRef] [Green Version] - Garcia-Bellido, J.; Murayama, H.; White, G. Exploring the Early Universe with Gaia and THEIA. J. Cosmol. Astropart. Phys.
**2021**, 2021, 023. [Google Scholar] [CrossRef] - Graham, P.W.; Hogan, J.M.; Kasevich, M.A.; Rajendran, S.; Romani, R.W. Mid-band gravitational wave detection with precision atomic sensors. arXiv
**2017**, arXiv:1711.02225. [Google Scholar] - Abe, M.; Adamson, P.; Borcean, M.; Bortoletto, D.; Bridges, K.; Carman, S.P.; Chattopadhyay, S.; Coleman, J.; Curfman, N.M.; DeRose, K.; et al. Matter-wave Atomic Gradiometer Interferometric Sensor (MAGIS-100). Quantum Sci. Technol.
**2021**, 6, 044003. [Google Scholar] [CrossRef] - Vitale, S.; Farr, W.M.; Ng, K.K.Y.; Rodriguez, C.L. Measuring the Star Formation Rate with Gravitational Waves from Binary Black Holes. ApJ
**2019**, 886, L1. [Google Scholar] [CrossRef] [Green Version] - Regimbau, T.; Evans, M.; Christensen, N.; Katsavounidis, E.; Sathyaprakash, B.; Vitale, S. Digging deeper: Observing primordial gravitational waves below the binary black hole produced stochastic background. Phys. Rev. Lett.
**2017**, 118, 151105. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ackley, K.; Adya, V.B.; Agrawal, P.; Altin, P.; Ashton, G.; Bailes, M.; Baltinas, E.; Barbuio, A.; Beniwal, D.; Blair, C.; et al. Neutron Star Extreme Matter Observatory: A kilohertz-band gravitational-wave detector in the global network. Publ. Astron. Soc. Austral.
**2020**, 37, e047. [Google Scholar] [CrossRef] - Maggiore, M.; Van Den Broeck, C.; Bartolo, N.; Belgacem, E.; Bertacca, D.; Bizouard, M.A.; Branchesi, M.; Clesse, S.; Foffa, S.; García-Bellido, J.; et al. Science Case for the Einstein Telescope. JCAP
**2020**, 03, 050. [Google Scholar] [CrossRef] [Green Version] - Cutler, C.; Harms, J. BBO and the neutron-star-binary subtraction problem. Phys. Rev. D
**2006**, 73, 042001. [Google Scholar] [CrossRef] [Green Version] - Sachdev, S.; Regimbau, T.; Sathyaprakash, B.S. Subtracting compact binary foreground sources to reveal primordial gravitational-wave backgrounds. PRD
**2020**, 102, 024051. [Google Scholar] [CrossRef] - Sharma, A.; Harms, J. Searching for cosmological gravitational-wave backgrounds with third-generation detectors in the presence of an astrophysical foreground. PRD
**2020**, 102, 063009. [Google Scholar] [CrossRef] - Raidal, M.; Spethmann, C.; Vaskonen, V.; Veermäe, H. Formation and evolution of primordial black hole binaries in the early universe. J. Cosmol. Astropart. Phys.
**2019**, 2019, 018. [Google Scholar] [CrossRef] [Green Version] - Mandic, V.; Bird, S.; Cholis, I. Stochastic Gravitational-Wave Background due to Primordial Binary Black Hole Mergers. Phys. Rev. Lett.
**2016**, 117, 201102. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ali-Haïmoud, Y.; Kovetz, E.D.; Kamionkowski, M. Merger rate of primordial black-hole binaries. Phys. Rev. D
**2017**, 96, 123523. [Google Scholar] [CrossRef] [Green Version] - De Luca, V.; Franciolini, G.; Pani, P.; Riotto, A. The Minimum Testable Abundance of Primordial Black Holes at Future Gravitational-Wave Detectors. J. Cosmol. Astropart. Phys.
**2021**, 2021, 039. [Google Scholar] [CrossRef] - Ng, K.K.Y.; Chen, S.; Goncharov, B.; Dupletsa, U.; Borhanian, S.; Branchesi, M.; Harms, J.; Maggiore, M.; Sathyaprakash, B.S.; Vitale, S. On the single-event-based identification of primordial black hole mergers at cosmological distances. arXiv
**2021**, arXiv:2108.07276. [Google Scholar] - Biscoveanu, S.; Talbot, C.; Thrane, E.; Smith, R. Measuring the primordial gravitational-wave background in the presence of astrophysical foregrounds. Phys. Rev. Lett.
**2020**, 125, 241101. [Google Scholar] [CrossRef] [PubMed] - Mukherjee, S.; Silk, J. Can we distinguish astrophysical from primordial black holes via the stochastic gravitational wave background? Mon. Not. R. Astron. Soc.
**2021**, 506, 3977–3985. [Google Scholar] [CrossRef] - Mukherjee, S.; Meinema, M.S.P.; Silk, J. Prospects of discovering sub-solar primordial black holes using the stochastic gravitational wave background from third-generation detectors. arXiv
**2021**, arXiv:2107.02181. [Google Scholar] - Mukherjee, S.; Silk, J. Fundamental physics using the temporal gravitational wave background. Phys. Rev. D
**2021**, 104, 063518. [Google Scholar] [CrossRef] - Schumann, W.O. Über die strahlungslosen Eigenschwingungen einer leitenden Kugel, die von einer Luftschicht und einer Ionosphärenhülle umgeben ist. Z. Naturforschung Teil A
**1952**, 7, 149–154. [Google Scholar] [CrossRef] - Schumann, W.O.; König, H. Über die Beobachtung von “atmospherics” bei geringsten Frequenzen. Naturwissenschaften
**1954**, 41, 183–184. [Google Scholar] [CrossRef] - Christensen, N. Measuring the stochastic gravitational radiation background with laser interferometric antennas. Phys. Rev. D
**1992**, 46, 5250–5266. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Thrane, E.; Christensen, N.; Schofield, R.M.S. Correlated magnetic noise in global networks of gravitational-wave detectors: Observations and implications. PRD
**2013**, 87, 123009. [Google Scholar] [CrossRef] [Green Version] - Thrane, E.; Christensen, N.; Schofield, R.M.S.; Effler, A. Correlated noise in networks of gravitational-wave detectors: Subtraction and mitigation. PRD
**2014**, 90, 023013. [Google Scholar] [CrossRef] [Green Version] - Coughlin, M.W.; Christensen, N.L.; De Rosa, R.; Fiori, I.; Gołkowski, M.; Guidry, M.; Harms, J.; Kubisz, J.; Kulak, A.; Mlynarczyk, J.; et al. Subtraction of correlated noise in global networks of gravitational-wave interferometers. Class. Quantum Gravity
**2016**, 33, 224003. [Google Scholar] [CrossRef] [Green Version] - Himemoto, Y.; Taruya, A. Impact of correlated magnetic noise on the detection of stochastic gravitational waves: Estimation based on a simple analytical model. PRD
**2017**, 96, 022004. [Google Scholar] [CrossRef] [Green Version] - Coughlin, M.W.; Cirone, A.; Meyers, P.; Atsuta, S.; Boschi, V.; Chincarini, A.; Christensen, N.L.; De Rosa, R.; Effler, A.; Fiori, I.; et al. Measurement and subtraction of Schumann resonances at gravitational-wave interferometers. PRD
**2018**, 97, 102007. [Google Scholar] [CrossRef] [Green Version] - Himemoto, Y.; Taruya, A. Correlated magnetic noise from anisotropic lightning sources and the detection of stochastic gravitational waves. PRD
**2019**, 100, 082001. [Google Scholar] [CrossRef] [Green Version] - Himemoto, Y.; Nishizawa, A.; Taruya, A. Impacts of overlapping gravitational-wave signals on the parameter estimation: Toward the search for cosmological backgrounds. PRD
**2021**, 104, 044010. [Google Scholar] [CrossRef] - Janssens, K.; Martinovic, K.; Christensen, N.; Meyers, P.M.; Sakellariadou, M. Impact of Schumann resonances on the Einstein Telescope and projections for the magnetic coupling function. PRD
**2021**, 104, 122006. [Google Scholar] [CrossRef] - Saulson, P.R. Terrestrial gravitational noise on a gravitational wave antenna. PRD
**1984**, 30, 732–736. [Google Scholar] [CrossRef] - Hughes, S.A.; Thorne, K.S. Seismic gravity-gradient noise in interferometric gravitational-wave detectors. PRD
**1998**, 58, 122002. [Google Scholar] [CrossRef] [Green Version] - Harms, J. Terrestrial gravity fluctuations. Living Rev. Relativ.
**2019**, 22, 6. [Google Scholar] [CrossRef] [Green Version] - Amann, F.; Bonsignorio, F.; Bulik, T.; Bulten, H.J.; Cuccuru, S.; Dassargues, A.; DeSalvo, R.; Fenyvesi, E.; Fidecaro, F.; Fiori, I.; et al. Site-selection criteria for the Einstein Telescope. Rev. Sci. Instruments
**2020**, 91, 094504. [Google Scholar] [CrossRef] [PubMed] - Driggers, J.C.; Harms, J.; Adhikari, R.X. Subtraction of Newtonian noise using optimized sensor arrays. PRD
**2012**, 86, 102001. [Google Scholar] [CrossRef] [Green Version] - Coughlin, M.; Harms, J.; Christensen, N.; Dergachev, V.; DeSalvo, R.; Kandhasamy, S.; Mandic, V. Wiener filtering with a seismic underground array at the Sanford Underground Research Facility. Class. Quantum Gravity
**2014**, 31, 215003. [Google Scholar] [CrossRef] - Coughlin, M.; Mukund, N.; Harms, J.; Driggers, J.; Adhikari, R.; Mitra, S. Towards a first design of a Newtonian-noise cancellation system for Advanced LIGO. Class. Quantum Gravity
**2016**, 33, 244001. [Google Scholar] [CrossRef] - Coughlin, M.W.; Harms, J.; Driggers, J.; McManus, D.J.; Mukund, N.; Ross, M.P.; Slagmolen, B.J.J.; Venkateswara, K. Implications of Dedicated Seismometer Measurements on Newtonian-Noise Cancellation for Advanced LIGO. PRL
**2018**, 121, 221104. [Google Scholar] [CrossRef] [Green Version] - Badaracco, F.; Harms, J.; Bertolini, A.; Bulik, T.; Fiori, I.; Idzkowski, B.; Kutynia, A.; Nikliborc, K.; Paoletti, F.; Paoli, A.; et al. Machine learning for gravitational-wave detection: Surrogate Wiener filtering for the prediction and optimized cancellation of Newtonian noise at Virgo. Class. Quantum Gravity
**2020**, 37, 195016. [Google Scholar] [CrossRef] - Harms, J.; Hild, S. Passive Newtonian noise suppression for gravitational-wave observatories based on shaping of the local topography. Class. Quantum Gravity
**2014**, 31, 185011. [Google Scholar] [CrossRef] - Badaracco, F.; Harms, J. Optimization of seismometer arrays for the cancellation of Newtonian noise from seismic body waves. Class. Quantum Gravity
**2019**, 36, 145006. [Google Scholar] [CrossRef] [Green Version] - Harms, J.; Venkateswara, K. Newtonian-noise cancellation in large-scale interferometric GW detectors using seismic tiltmeters. Class. Quantum Gravity
**2016**, 33, 234001. [Google Scholar] [CrossRef] - Creighton, T. Tumbleweeds and airborne gravitational noise sources for LIGO. Class. Quantum Gravity
**2008**, 25, 125011. [Google Scholar] [CrossRef] [Green Version] - Fiorucci, D.; Harms, J.; Barsuglia, M.; Fiori, I.; Paoletti, F. Impact of infrasound atmospheric noise on gravity detectors used for astrophysical and geophysical applications. PRD
**2018**, 97, 062003. [Google Scholar] [CrossRef] [Green Version] - Muratore, M.; Vetrugno, D.; Vitale, S. Revisitation of time delay interferometry combinations that suppress laser noise in LISA. Class. Quantum Gravity
**2020**, 37, 185019. [Google Scholar] [CrossRef] - Estabrook, F.B.; Tinto, M.; Armstrong, J.W. Time-delay analysis of LISA gravitational wave data: Elimination of spacecraft motion effects. Phys. Rev. D
**2000**, 62, 042002. [Google Scholar] [CrossRef] - Prince, T.A.; Tinto, M.; Larson, S.L.; Armstrong, J. The LISA optimal sensitivity. Phys. Rev. D
**2002**, 66, 122002. [Google Scholar] [CrossRef] [Green Version] - Sylvestre, J.; Tinto, M. Noise characterization for LISA. Phys. Rev. D
**2003**, 68, 102002. [Google Scholar] [CrossRef] [Green Version] - Crowder, J.; Cornish, N.J. LISA source confusion. Phys. Rev. D
**2004**, 70, 082004. [Google Scholar] [CrossRef] [Green Version] - Chen, Z.C.; Huang, F.; Huang, Q.G. Stochastic Gravitational-wave Background from Binary Black Holes and Binary Neutron Stars and Implications for LISA. Astrophys. J.
**2019**, 871, 97. [Google Scholar] [CrossRef] [Green Version] - Caprini, C.; Chala, M.; Dorsch, G.C.; Hindmarsh, M.; Huber, S.J.; Konstandin, T.; Kozaczuk, J.; Nardini, G.; No, J.M.; Rummukainen, K.; et al. Detecting gravitational waves from cosmological phase transitions with LISA: An update. JCAP
**2020**, 03, 024. [Google Scholar] [CrossRef] [Green Version] - Boileau, G.; Jenkins, A.C.; Sakellariadou, M.; Meyer, R.; Christensen, N. Ability of LISA to detect a gravitational-wave background of cosmological origin: The cosmic string case. Phys. Rev. D
**2022**, 105, 023510. [Google Scholar] [CrossRef] - Crowder, J.; Cornish, N. A Solution to the Galactic Foreground Problem for LISA. Phys. Rev. D
**2007**, 75, 043008. [Google Scholar] [CrossRef] [Green Version] - LISA Data Challenge Manual. Available online: https://lisa-ldc.lal.in2p3.fr/static/data/pdf/LDC-manual-Sangria.pdf (accessed on 1 November 2021).

**Figure 1.**An overview of potential GWB signals across the frequency spectrum. The light blue curve shows the prediction for single-field slow-roll inflation with a canonical kinetic term, with tensor-to-scalar ratio ${r}_{0.002}=0.1$ [52]. The pink curve shows a GWB from Nambu–Goto cosmic strings, using “model 2” of the loop network, with a dimensionless string tension of $G\mu ={10}^{-11}$ [53]. The brown curve shows a GWB from inspiralling supermassive BBHs, with the amplitude and shaded region shown here corresponding to the common noise process in the NANOGrav 12.5-year data set [54]. The two grey curves show GWBs generated by first-order phase transitions at the electroweak scale (∼200 GeV) and the QCD scale (∼200 MeV), respectively [55]. The yellow curve shows a GWB generated by stellar-mass compact binaries, based on the mass distributions and local merger rates inferred by LVK detections [56]. The dashed curves show various observational constraints, as described further in Section 5 (this includes the PPTA constraint, which intersects the possible NANOGrav SMBBH signal); the dotted curve shows the integrated constraint from measurements of ${N}_{\mathrm{eff}}$, which cannot be directly compared with the frequency-dependent constraint curves but is shown here for indicative purposes.

**Figure 2.**Projections for the CBC stochastic background using the third GW transient catalogue (GWTC-3), first presented in [74]. On the right, the total expected background is compared to the integrated sensitivity of current and future configurations of the 2G detector network. This plot was obtained using open data published in [75].

**Figure 3.**Examples of interferometer timestreams populated by GWBs with different time-domain properties, inspired by [64]: Poisson noise (top panel), popcorn (middle panel), and Gaussian (bottom panel). The strain on the y-axis is normalised to show the fractional contribution of each burst/wavelet to the signal, while the $\Delta $ parameter is the duty cycle, i.e., the mean ratio between event duration and the time interval between consecutive events.

**Figure 4.**Overlap reduction functions for different GW interferometer detectors: ground-based (

**left**) and space-based (

**right**). Note that, in panel (

**b**), the positive and negative parts of the LISA TDI overlap function ${\Gamma}_{XY}$ are plotted separately, as the y-axis is log-scaled.

**Figure 5.**Geometric components of the overlap functions for different ground-based detector combinations mapped on the sky in geocentric celestial coordinates at a common time.

**Figure 6.**The dashed line is the Hellings–Downs curve described by Equation (63). $\zeta $ is the angle between Earth–pulsars baselines, and $\Gamma $ measures the spatial correlation. At $\zeta =0$, the function assumes the pulsars located at different distances. When distances are the same, $\Gamma \left(0\right)=1$. Shaded regions correspond to constraints on spatial correlations obtained with the second data release of the Parkes Pulsar Timing Array except PSR J0437–4715 [26].

**Figure 7.**A survey of constraints (all at 95% confidence) on the GWB across the frequency spectrum. Solid curves indicate existing results from LIGO/Virgo’s first three observing runs [56], monitoring of the Earth’s normal modes [220], Doppler tracking of the Cassini satellite [221], pulsar timing observations by the PPTA [222], and CMB temperature and polarisation spectra measured by Planck [222,223]. Dashed curves are forecast constraints for LIGO at A+ sensitivity, Einstein Telescope [14], AION-km [224], LISA [29], binary resonance searches [225,226], and pulsar timing with the Square Kilometre Array [227]. The dotted curve indicates the level of the integrated constraint from measurements of ${N}_{\mathrm{eff}}$ [38]; note that this is a constraint on the total GW energy density over a broad frequency range and cannot be directly compared to the other constraints. Note also that both the Planck and ${N}_{\mathrm{eff}}$ constraints apply only to primordial GWs emitted before the epoch of BBN. See Figure 1 for various GWB signal predictions in relation to these constraints.

**Figure 8.**Combined cross-correlation spectra for the three O3 baselines. The grey lines here mark $1\sigma $ contours. The data, by eye, appears to follow the Gaussian distribution. Note that adding the Virgo baselines has filled in the gap in the H–L data present around the 60 Hz power line (this broad line is present in both USA-based detectors, but not in Italy). The original version of this plot is presented in [56]; the one shown here was obtained using open data published in [246].

**Figure 9.**Posterior distributions for the GWB energy density ${\Omega}_{\mathrm{GW}}$ at reference frequency $f=25$ Hz and spectral index $\alpha $. The dashed grey lines show the priors used in parameter estimation. Note that a log-uniform prior was chosen here on ${\Omega}_{\mathrm{GW}}$, and a mean–zero Gaussian was chosen for $\alpha $ as the detector noise PSD is approximately flat. The original version of this plot is presented in [56]; the one shown here was obtained using the open data published in [246].

**Figure 10.**Collection of all posterior draws for the merger rate as a function of redshift ${\dot{N}}_{\mathrm{BBH}}\left(z\right)$ inferred from the combination of the resolved BBH detections in the GWTC-2 catalog [252] and the upper limits on the stochastic background obtained with LIGO/Virgo O3 data. The original version of this plot is presented in [56]; the one shown here was obtained using open data published in [246].

**Figure 11.**SNR maps obtained with the “broad–band radiometer” search method applied to

`O1`,

`O2`, and

`O3`data, for different fixed values of the spectral index $\alpha $, as explained in the text. It is clear there is no significant high SNR signal in the maps. The original version of this plot is presented in [211]; the one shown here was obtained using open data published in [256].

**Figure 12.**Angular power spectrum upper limits obtained with the “spherical harmonic decomposition” search method applied to

`O1`,

`O2`, and

`O3`data, for different fixed values of the spectral index $\alpha $, as explained in the text. The original version of this plot is presented in [211]; the one shown here was obtained by using open data published in [256].

**Figure 14.**Power-law-integrated sensitivity [290,291] at the level of SNR = 3 of simulated PTAs to the SGWB strain $h\left(t\right)$. We demonstrate the effects of timing precision, red noise, and a number of array pulsars. Pulsars are distributed isotropically across the sky and observed biweekly. Orange lines represent variations of the 15-year-long PPTA DR2, with real pulsar ${T}_{\mathrm{obs}}$, white noise levels based on the rms residuals ranging from 240 ns to 24,050 ns [279] and best-fit red noise parameters from [285]. The dashed line represents PPTA DR2 extended by 5 more years of observation, whereas the dotted line represents PPTA DR2 with a doubled number of pulsars with the same noise properties. Red line is an example of the future IPTA with 20 years of observations of 100 pulsars with white noise rms drawn uniformly between 20 and 500 ns. Red noise ${log}_{10}A$ is drawn uniformly from $[-14,-20]$ and $\gamma $ drawn uniformly from $[1,7]$, a fiducial choice based on [285]. Green lines represent SKA observing 50 pulsars with white noise rms of 30 ns for 15 years, with (solid line) and without (dashed line) red noise assumed for the IPTA.

**Table 1.**Results of searches for the nanohertz-frequency isotropic GWB with the amplitude A and the energy-density spectral index $\alpha =2/3$ in chronological order. The first three columns list details of corresponding publications. Publications that adopted alphabetical author list are denoted by asterisks. The fourth column contains 95% upper limits on A for the GWB and measurements of A of the common-spectrum process (CP) with credible levels of $1\sigma $ in [26] and 5–95% in [54,275,276]. The values in the last two columns are the characteristics of the data set: the total observation time ${T}_{\mathrm{obs}}$ and the number of pulsars in the array ${N}_{\mathrm{psr}}$.

Publication | Collaboration | Year | $\mathit{A}\phantom{\rule{3.33333pt}{0ex}}(\times {10}^{-15})$ | ${\mathit{T}}_{\mathbf{obs}}\left(\mathbf{yr}\right)$ | ${\mathit{N}}_{\mathbf{psr}}$ |
---|---|---|---|---|---|

Jenet et al. [269] | PPTA | 2006 | <11 | 20 | 7 |

van Haasteren et al. [173] | EPTA | 2011 | <6 | 11 | 5 |

Demorest et al. [270] | NANOGrav | 2013 | <7 | 5 | 17 |

Shannon et al. [271] | PPTA | 2013 | <2.4 | 25 | 6 |

Lentati et al. [272] | EPTA | 2015 | <3.0 | 18 | 6 |

* Arzoumanian et al. [178] | NANOGrav | 2015 | <1.5 | 9 | 37 |

Shannon et al. [274] | PPTA | 2015 | <1.0 | 11 | 4 |

Verbiest et al. [24] | IPTA | 2016 | <1.7 | 21 | 4 |

* Arzoumanian et al. [180] | NANOGrav | 2018 | <1.45 | 11.4 | 45 |

* Arzoumanian et al. [54] | NANOGrav | 2020 | CP: $1.{9}_{-0.6}^{+0.8}$ | 12.5 | 45 |

Goncharov et al. [26] | PPTA | 2021 | CP: $2.{2}_{-0.3}^{+0.4}$ | 15.0 | 26 |

Chen et al. [275] | EPTA | 2021 | CP: $3.{0}_{-0.7}^{+0.9}$ | 24.0 | 6 |

* Antoniadis et al. [276] | IPTA | 2022 | CP: $2.{8}_{-0.8}^{+1.2}$ | 30.2 | 53 |

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Renzini, A.I.; Goncharov, B.; Jenkins, A.C.; Meyers, P.M.
Stochastic Gravitational-Wave Backgrounds: Current Detection Efforts and Future Prospects. *Galaxies* **2022**, *10*, 34.
https://doi.org/10.3390/galaxies10010034

**AMA Style**

Renzini AI, Goncharov B, Jenkins AC, Meyers PM.
Stochastic Gravitational-Wave Backgrounds: Current Detection Efforts and Future Prospects. *Galaxies*. 2022; 10(1):34.
https://doi.org/10.3390/galaxies10010034

**Chicago/Turabian Style**

Renzini, Arianna I., Boris Goncharov, Alexander C. Jenkins, and Patrick M. Meyers.
2022. "Stochastic Gravitational-Wave Backgrounds: Current Detection Efforts and Future Prospects" *Galaxies* 10, no. 1: 34.
https://doi.org/10.3390/galaxies10010034