Gravitational-wave lensing in Einstein-aether theory

Author(s)

Streibert, Julius, Silva, Hector O., Zumalacárregui, Miguel

Abstract

Einstein-aether theory provides a model to test the validity of local Lorentz invariance in gravitational interactions. The speed of gravitational waves as measured from the binary neutron star event GW170817 sets stringent limits on Einstein-aether theory, but only on a combination of the theory's free parameters. For this reason, a significant part of the theory's parameter space remains unconstrained by observations. Motivated by this, we explore the propagation of gravitational waves in Einstein-aether theory over an inhomogeneous background (i.e., gravitational wave lensing) as a potential mechanism to break the degeneracies between the theory's free parameters, and hence enable new constraints on the theory to be obtained. By bringing the field equations into the form of the so-called kinetic matrix and applying a formalism known as the propagation eigenstate framework, we find that the speed of gravitational waves is modified by inhomogeneities in the aether field. However, the modification is common to both gravitational polarizations and vanishes in the limit in which gravitational waves propagate with luminal speed. This lens-dependent gravitational wave speed contrasts with the lens-induced birefringence observed in other theories beyond general relativity, like Horndeski's theory. While the potential to improve tests based on gravitational-wave speed is limited, our formalism sets the basis to fully describe signal propagation over inhomogeneous spacetimes in Einstein-aether theory and other extensions of general relativity.

Figures

Qualitative effect of GWs beyond GR on circles and spheres composed of test particles. We assume the GW wavelengths to be much larger than the diameter of any such circle or sphere. Tensor and vector waves deform circles into ellipses in different planes and along different axes. These modes preserve the circle areas. The scalar modes instead expand and contract circles and spheres, changing their area and volume, respectively : in the $xy$ plane for the breathing mode and in the $z$ direction for the longitudinal mode. Figure modified from Ref.~\cite{Takeda_2023}.

Qualitative effect of GWs beyond GR on circles and spheres composed of test particles. We assume the GW wavelengths to be much larger than the diameter of any such circle or sphere. Tensor and vector waves deform circles into ellipses in different planes and along different axes. These modes preserve the circle areas. The scalar modes instead expand and contract circles and spheres, changing their area and volume, respectively : in the $xy$ plane for the breathing mode and in the $z$ direction for the longitudinal mode. Figure modified from Ref.~\cite{Takeda_2023}.


Effect of scalar waves with amplitude 0.3 and different values of $\alpha$ on an a sphere of test particles with diameter $1$ and a scalar wavelength much larger than $1$. Time is measured in fractions of the period $T$, and different curves correspond to $\alpha = 0$ and $\pm 1$. The sphere expands and contracts both in the $xy$ plane and in the $z$ direction. The effect differs based on the value of $\alpha$. For example, at $t=T/4$ the sphere expands in the $z$ direction for $\alpha=1$, while it contracts for $\alpha=-1$.

Effect of scalar waves with amplitude 0.3 and different values of $\alpha$ on an a sphere of test particles with diameter $1$ and a scalar wavelength much larger than $1$. Time is measured in fractions of the period $T$, and different curves correspond to $\alpha = 0$ and $\pm 1$. The sphere expands and contracts both in the $xy$ plane and in the $z$ direction. The effect differs based on the value of $\alpha$. For example, at $t=T/4$ the sphere expands in the $z$ direction for $\alpha=1$, while it contracts for $\alpha=-1$.


Differences between lens-induced effects in different theories. The propagation of a wave packet from left to right is shown. Time is measured in arbitrary units. Top and middle: lens-dependent GW speed as observed in Einstein-aether theory. There is no difference in speed between the two tensor modes. There is no birefringence, any one signal does not get scrambled. However, signals lensed by different lenses propagate with different speeds. Bottom: Lens-induced birefringence as observed in Horndeski's theory. The modified cross and plus mode propagate with different speeds. Birefringence scrambles the signal.

Differences between lens-induced effects in different theories. The propagation of a wave packet from left to right is shown. Time is measured in arbitrary units. Top and middle: lens-dependent GW speed as observed in Einstein-aether theory. There is no difference in speed between the two tensor modes. There is no birefringence, any one signal does not get scrambled. However, signals lensed by different lenses propagate with different speeds. Bottom: Lens-induced birefringence as observed in Horndeski's theory. The modified cross and plus mode propagate with different speeds. Birefringence scrambles the signal.


Deviation of the fifth order modified tensor mode speed $c_T'$ from the background value $c_T$ as a function of the longitudinal aether component $\mu_L$ and transverse aether component $\mu_T$. Top: Upper limit of $c_+$ permitted by observations. Bottom: Lower limit of $c_+$. A faster speed compared to the background value is indicated by reds, a slower speed is indicated by blues. Higher values of $\mu_L$ slow tensor waves down for the upper limit of $c_+$ and speed them up for the lower limit. Higher magnitudes of $\mu_T$ increase speed for the upper limit and decrease it for the lower limit.

Deviation of the fifth order modified tensor mode speed $c_T'$ from the background value $c_T$ as a function of the longitudinal aether component $\mu_L$ and transverse aether component $\mu_T$. Top: Upper limit of $c_+$ permitted by observations. Bottom: Lower limit of $c_+$. A faster speed compared to the background value is indicated by reds, a slower speed is indicated by blues. Higher values of $\mu_L$ slow tensor waves down for the upper limit of $c_+$ and speed them up for the lower limit. Higher magnitudes of $\mu_T$ increase speed for the upper limit and decrease it for the lower limit.


References
  • [1] E. Berti et al., Testing General Relativity with Present and Future Astrophysical Observations, Class. Quant. Grav. 32, 243001 (2015), arXiv:1501.07274 [gr-qc].
  • [2] T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis, Modified Gravity and Cosmology, Phys. Rept. 513, 1 (2012), arXiv:1106.2476 [astro-ph.CO].
  • [3] S. Shankaranarayanan and J. P. Johnson, Modified theories of gravity: Why, how and what?, Gen. Rel. Grav. 54, 44 (2022), arXiv:2204.06533 [gr-qc].
  • [4] T. Jacobson and D. Mattingly, Gravity with a dynamical preferred frame, Phys. Rev. D 64, 024028 (2001), arXiv:grqc/0007031.
  • [5] D. Mattingly, Modern tests of Lorentz invariance, Living Rev. Rel. 8, 5 (2005), arXiv:gr-qc/0502097.
  • [6] J. Oost, S. Mukohyama, and A. Wang, Constraints on Einstein-aether theory after GW170817, Phys. Rev. D 97, 124023 (2018), arXiv:1802.04303 [gr-qc].
  • [7] O. Sarbach, E. Barausse, and J. A. Preciado-López, Well-posed Cauchy formulation for Einstein-æther theory, Class. Quant. Grav. 36, 165007 (2019), arXiv:1902.05130 [gr-qc].
  • [8] C. Eling and T. Jacobson, Static postNewtonian equivalence of GR and gravity with a dynamical preferred frame, Phys. Rev. D 69, 064005 (2004), arXiv:gr-qc/0310044.
  • [9] C. Eling and T. Jacobson, Spherical solutions in Einsteinaether theory: Static aether and stars, Class. Quant. Grav. 23, 5625 (2006), [Erratum: Class.Quant.Grav. 27, 049801 (2010)], arXiv:gr-qc/0603058.
  • [9] C. Eling and T. Jacobson, Spherical solutions in Einsteinaether theory: Static aether and stars, Class. Quant. Grav. 23, 5625 (2006), [Erratum: Class.Quant.Grav. 27, 049801 (2010)], arXiv:gr-qc/0603058.
  • [10] C. Eling and T. Jacobson, Black Holes in Einstein-Aether Theory, Class. Quant. Grav. 23, 5643 (2006), [Erratum: Class.Quant.Grav. 27, 049802 (2010)], arXiv:grqc/0604088.
  • [10] C. Eling and T. Jacobson, Black Holes in Einstein-Aether Theory, Class. Quant. Grav. 23, 5643 (2006), [Erratum: Class.Quant.Grav. 27, 049802 (2010)], arXiv:grqc/0604088.
  • [11] T. Jacobson, Extended Horava gravity and Einsteinaether theory, Phys. Rev. D 81, 101502 (2010), [Erratum: Phys.Rev.D 82, 129901 (2010)], arXiv:1001.4823 [hep-th].
  • [11] T. Jacobson, Extended Horava gravity and Einsteinaether theory, Phys. Rev. D 81, 101502 (2010), [Erratum: Phys.Rev.D 82, 129901 (2010)], arXiv:1001.4823 [hep-th].
  • [12] S. M. Carroll and E. A. Lim, Lorentz-violating vector fields slow the universe down, Phys. Rev. D 70, 123525 (2004), arXiv:hep-th/0407149.
  • [13] M. L. Graesser, A. Jenkins, and M. B. Wise, Spontaneous Lorentz violation and the long-range gravitational preferred-frame effect, Phys. Lett. B 613, 5 (2005), arXiv:hep-th/0501223.
  • [14] B. Z. Foster and T. Jacobson, Post-Newtonian parameters and constraints on Einstein-aether theory, Phys. Rev. D 73, 064015 (2006), arXiv:gr-qc/0509083.
  • [15] K. Yagi, D. Blas, E. Barausse, and N. Yunes, Constraints on Einstein-Æther theory and Hořava gravity from binary pulsar observations, Phys. Rev. D 89, 084067 (2014), [Erratum: Phys.Rev.D 90, 069902 (2014), Erratum: Phys.Rev.D 90, 069901 (2014)], arXiv:1311.7144 [gr-qc].
  • [15] K. Yagi, D. Blas, E. Barausse, and N. Yunes, Constraints on Einstein-Æther theory and Hořava gravity from binary pulsar observations, Phys. Rev. D 89, 084067 (2014), [Erratum: Phys.Rev.D 90, 069902 (2014), Erratum: Phys.Rev.D 90, 069901 (2014)], arXiv:1311.7144 [gr-qc].
  • [15] K. Yagi, D. Blas, E. Barausse, and N. Yunes, Constraints on Einstein-Æther theory and Hořava gravity from binary pulsar observations, Phys. Rev. D 89, 084067 (2014), [Erratum: Phys.Rev.D 90, 069902 (2014), Erratum: Phys.Rev.D 90, 069901 (2014)], arXiv:1311.7144 [gr-qc].
  • [16] T. Jacobson and D. Mattingly, Einstein-Aether waves, Phys. Rev. D 70, 024003 (2004), arXiv:gr-qc/0402005.
  • [17] R. Abbott et al. (LIGO Scientific, Virgo), Tests of general relativity with binary black holes from the second LIGOVirgo gravitational-wave transient catalog, Phys. Rev. D 103, 122002 (2021), arXiv:2010.14529 [gr-qc].
  • [18] R. Abbott et al. (LIGO Scientific, VIRGO, KAGRA), Tests of General Relativity with GWTC-3 (2021), arXiv:2112.06861 [gr-qc].
  • [19] H. Takeda, S. Morisaki, and A. Nishizawa, Search for scalar-tensor mixed polarization modes of gravitational waves, Phys. Rev. D 105, 084019 (2022), arXiv:2105.00253 [gr-qc].
  • [20] H. Takeda, Y. Manita, H. Omiya, and T. Tanaka, Scalar polarization window in gravitational-wave signals, PTEP 2023, 073E01 (2023), arXiv:2304.14430 [gr-qc].
  • [21] C. Eling, Energy in the Einstein-aether theory, Phys. Rev. D 73, 084026 (2006), [Erratum: Phys.Rev.D 80, 129905 (2009)], arXiv:gr-qc/0507059.
  • [21] C. Eling, Energy in the Einstein-aether theory, Phys. Rev. D 73, 084026 (2006), [Erratum: Phys.Rev.D 80, 129905 (2009)], arXiv:gr-qc/0507059.
  • [22] J. W. Elliott, G. D. Moore, and H. Stoica, Constraining the new Aether: Gravitational Cerenkov radiation, JHEP 2005 (08), 066, arXiv:hep-ph/0505211.
  • [23] B. Z. Foster, Radiation damping in Einstein-aether theory, Phys. Rev. D 73, 104012 (2006), [Erratum: Phys.Rev.D 75, 129904 (2007)], arXiv:gr-qc/0602004.
  • [23] B. Z. Foster, Radiation damping in Einstein-aether theory, Phys. Rev. D 73, 104012 (2006), [Erratum: Phys.Rev.D 75, 129904 (2007)], arXiv:gr-qc/0602004.
  • [24] B. Z. Foster, Strong field effects on binary systems in Einstein-aether theory, Phys. Rev. D 76, 084033 (2007), arXiv:0706.0704 [gr-qc].
  • [25] J. M. Ezquiaga and M. Zumalacárregui, Dark Energy After GW170817: Dead Ends and the Road Ahead, Phys. Rev. Lett. 119, 251304 (2017), arXiv:1710.05901 [astroph.CO].
  • [26] P. Creminelli and F. Vernizzi, Dark Energy after GW170817 and GRB170817A, Phys. Rev. Lett. 119, 251302 (2017), arXiv:1710.05877 [astro-ph.CO].
  • [27] T. Baker, E. Bellini, P. G. Ferreira, M. Lagos, J. Noller, and I. Sawicki, Strong constraints on cosmological gravity from GW170817 and GRB 170817A, Phys. Rev. Lett. 119, 251301 (2017), arXiv:1710.06394 [astro-ph.CO].
  • [28] J. Sakstein and B. Jain, Implications of the Neutron Star Merger GW170817 for Cosmological ScalarTensor Theories, Phys. Rev. Lett. 119, 251303 (2017), arXiv:1710.05893 [astro-ph.CO].
  • [29] L. Lombriser and A. Taylor, Breaking a Dark Degeneracy with Gravitational Waves, JCAP 2016 (03), 031, arXiv:1509.08458 [astro-ph.CO].
  • [30] D. Bettoni, J. M. Ezquiaga, K. Hinterbichler, and M. Zumalacárregui, Speed of Gravitational Waves and the Fate of Scalar-Tensor Gravity, Phys. Rev. D 95, 084029 (2017), arXiv:1608.01982 [gr-qc].
  • [31] B. P. Abbott et al. (LIGO Scientific, Virgo), GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral, Phys. Rev. Lett. 119, 161101 (2017), arXiv:1710.05832 [gr-qc].
  • [32] B. P. Abbott et al. (LIGO Scientific, Virgo, FermiGBM, INTEGRAL), Gravitational Waves and Gammarays from a Binary Neutron Star Merger: GW170817 and GRB 170817A, Astrophys. J. Lett. 848, L13 (2017), arXiv:1710.05834 [astro-ph.HE].
  • [33] B. P. Abbott et al. (LIGO Scientific, Virgo, Fermi GBM, INTEGRAL, IceCube, AstroSat Cadmium Zinc Telluride Imager Team, IPN, Insight-Hxmt, ANTARES, Swift, AGILE Team, 1M2H Team, Dark Energy Camera GW-EM, DES, DLT40, GRAWITA, Fermi-LAT, ATCA, ASKAP, Las Cumbres Observatory Group, OzGrav, DWF (Deeper Wider Faster Program), AST3, CAASTRO, VINROUGE, MASTER, J-GEM, GROWTH, JAGWAR, CaltechNRAO, TTU-NRAO, NuSTAR, Pan-STARRS, MAXI Team, TZAC Consortium, KU, Nordic Optical Telescope, ePESSTO, GROND, Texas Tech University, SALT Group, TOROS, BOOTES, MWA, CALET, IKI-GW Follow-up, H.E.S.S., LOFAR, LWA, HAWC, Pierre Auger, ALMA, Euro VLBI Team, Pi of Sky, Chandra Team at McGill University, DFN, ATLAS Telescopes, High Time Resolution Universe Survey, RIMAS, RATIR, SKA South Africa/MeerKAT), Multi-messenger Observations of a Binary Neutron Star Merger, Astrophys. J. Lett. 848, L12 (2017), arXiv:1710.05833 [astro-ph.HE].
  • [34] B. P. Abbott et al. (LIGO Scientific, Virgo), Tests of General Relativity with GW170817, Phys. Rev. Lett. 123, 011102 (2019), arXiv:1811.00364 [gr-qc].
  • [35] Y. Gong, S. Hou, D. Liang, and E. Papantonopoulos, Gravitational waves in Einstein-æther and generalized TeVeS theory after GW170817, Phys. Rev. D 97, 084040 (2018), arXiv:1801.03382 [gr-qc].
  • [36] T. Gupta, M. Herrero-Valea, D. Blas, E. Barausse, N. Cornish, K. Yagi, and N. Yunes, New binary pulsar constraints on Einstein-æther theory after GW170817, Class. Quant. Grav. 38, 195003 (2021), arXiv:2104.04596 [gr-qc].
  • [37] K. Schumacher, S. E. Perkins, A. Shaw, K. Yagi, and N. Yunes, Gravitational wave constraints on Einsteinæther theory with LIGO/Virgo data, Phys. Rev. D 108, 104053 (2023), arXiv:2304.06801 [gr-qc].
  • [38] J. M. Ezquiaga and M. Zumalacárregui, Dark Energy in light of Multi-Messenger Gravitational-Wave astronomy, Front. Astron. Space Sci. 5, 44 (2018), arXiv:1807.09241 [astro-ph.CO].
  • [39] R. Reyes, R. Mandelbaum, U. Seljak, T. Baldauf, J. E. Gunn, L. Lombriser, and R. E. Smith, Confirmation of general relativity on large scales from weak lensing and galaxy velocities, Nature 464, 256 (2010), arXiv:1003.2185 [astro-ph.CO].
  • [40] T. E. Collett, L. J. Oldham, R. J. Smith, M. W. Auger, K. B. Westfall, D. Bacon, R. C. Nichol, K. L. Masters, K. Koyama, and R. van den Bosch, A precise extragalactic test of General Relativity, Science 360, 1342 (2018), arXiv:1806.08300 [astro-ph.CO].
  • [41] S. Goyal, K. Haris, A. K. Mehta, and P. Ajith, Testing the nature of gravitational-wave polarizations using strongly lensed signals, Phys. Rev. D 103, 024038 (2021), arXiv:2008.07060 [gr-qc].
  • [42] H. Narola, J. Janquart, L. Haegel, K. Haris, O. A. Hannuksela, and C. Van Den Broeck, How well can modified gravitational wave propagation be constrained with strong lensing? (2023), arXiv:2308.01709 [gr-qc].
  • [43] J. M. Ezquiaga and M. Zumalacárregui, Gravitational wave lensing beyond general relativity: birefringence, echoes and shadows, Phys. Rev. D 102, 124048 (2020), arXiv:2009.12187 [gr-qc].
  • [44] S. Goyal, A. Vijaykumar, J. M. Ezquiaga, and M. Zumalacárregui, Probing lens-induced gravitational-wave birefringence as a test of general relativity, Phys. Rev. D 108, 024052 (2023), arXiv:2301.04826 [gr-qc].
  • [45] W. Zhao, T. Zhu, J. Qiao, and A. Wang, Waveform of gravitational waves in the general parity-violating gravities, Phys. Rev. D 101, 024002 (2020), arXiv:1909.10887 [gr-qc].
  • [46] Y.-F. Wang, S. M. Brown, L. Shao, and W. Zhao, Tests of gravitational-wave birefringence with the open gravitational-wave catalog, Phys. Rev. D 106, 084005 (2022), arXiv:2109.09718 [astro-ph.HE].
  • [47] L. Jenks, L. Choi, M. Lagos, and N. Yunes, Parametrized parity violation in gravitational wave propagation, Phys. Rev. D 108, 044023 (2023), arXiv:2305.10478 [gr-qc].
  • [48] M. Lagos, L. Jenks, M. Isi, K. Hotokezaka, B. D. Metzger, E. Burns, W. M. Farr, S. Perkins, K. W. K. Wong, and N. Yunes, Birefringence tests of gravity with multimessenger binaries (2024), arXiv:2402.05316 [gr-qc].
  • [49] C. M. Will, Theory and Experiment in Gravitational Physics (Cambridge University Press, 1993).
  • [50] T. Jacobson, Einstein-aether gravity: A Status report, PoS QG-PH, 020 (2007), arXiv:0801.1547 [gr-qc].
  • [51] H. Takeda, Testing theories of gravity in extreme environments with gravitational-wave polarization from compact binary systems (2023), unpublished slides of a presentation for the Cosmology meeting of the Max Planck Institute for Gravitational Physics.
  • [52] K. Nordtvedt, Equivalence Principle for Massive Bodies. 2. Theory, Phys. Rev. 169, 1017 (1968).
  • [53] K. Nordtvedt, Equivalence Principle for Massive Bodies Including Rotational Energy and Radiation Pressure, Phys. Rev. 180, 1293 (1969).
  • [54] C. M. Will, Theoretical Frameworks for Testing Relativistic Gravity. 2. Parametrized Post-Newtonian Hydrodynamics, and the Nordtvedt Effect, Astrophys. J. 163, 611 (1971).
  • [55] C. M. Will and K. Nordtvedt, Jr., Conservation Laws and Preferred Frames in Relativistic Gravity. I. PreferredFrame Theories and an Extended PPN Formalism, Astrophys. J. 177, 757 (1972).
  • [56] J. Muller, K. Nordtvedt, and D. Vokrouhlicky, Improved constraint on the alpha-1 PPN parameter from lunar motion, Phys. Rev. D 54, R5927 (1996).
  • [57] L. Shao and N. Wex, New tests of local Lorentz invariance of gravity with small-eccentricity binary pulsars, Class. Quant. Grav. 29, 215018 (2012), arXiv:1209.4503 [gr-qc].
  • [58] L. Shao, R. N. Caballero, M. Kramer, N. Wex, D. J. Champion, and A. Jessner, A new limit on local Lorentz invariance violation of gravity from solitary pulsars, Class. Quant. Grav. 30, 165019 (2013), arXiv:1307.2552 [gr-qc].
  • [59] U. Muller, C. Schubert, and A. M. E. van de Ven, A Closed formula for the Riemann normal coordinate expansion, Gen. Rel. Grav. 31, 1759 (1999), arXiv:gr-qc/9712092.
  • [60] L. Laboratory, LIGO R&D (2015), accessed: 2023-04-14.
  • [61] J. Bland-Hawthorn and O. Gerhard, The Galaxy in Context: Structural, Kinematic, and Integrated Properties, Annu. Rev. Astron. Astrophys. 54, 529 (2016), arXiv:1602.07702 [astro-ph.GA].
  • [62] G. Tambalo, M. Zumalacárregui, L. Dai, and M. H.-Y. Cheung, Gravitational wave lensing as a probe of halo properties and dark matter, Phys. Rev. D 108, 103529 (2023), arXiv:2212.11960 [astro-ph.CO].
  • [63] P. Schneider, J. Ehlers, and E. E. Falco, Gravitational Lenses (Springer, 1992).
  • [64] R. Takahashi and T. Nakamura, Wave effects in gravitational lensing of gravitational waves from chirping binaries, Astrophys. J. 595, 1039 (2003), arXiv:astro-ph/0305055.
  • [65] G. Tambalo, M. Zumalacárregui, L. Dai, and M. H.-Y. Cheung, Lensing of gravitational waves: Efficient waveoptics methods and validation with symmetric lenses, Phys. Rev. D 108, 043527 (2023), arXiv:2210.05658 [grqc].
  • [66] C. Dalang, P. Fleury, and L. Lombriser, Scalar and tensor gravitational waves, Phys. Rev. D 103, 064075 (2021), arXiv:2009.11827 [gr-qc].
  • [67] S. Boran, S. Desai, E. O. Kahya, and R. P. Woodard, GW170817 Falsifies Dark Matter Emulators, Phys. Rev. D 97, 041501 (2018), arXiv:1710.06168 [astro-ph.HE].
  • [68] C. Guépin, K. Kotera, and F. Oikonomou, Highenergy neutrino transients and the future of multimessenger astronomy, Nature Rev. Phys. 4, 697 (2022), arXiv:2207.12205 [astro-ph.HE].
  • [69] K. Max, M. Platscher, and J. Smirnov, Gravitational Wave Oscillations in Bigravity, Phys. Rev. Lett. 119, 111101 (2017), arXiv:1703.07785 [gr-qc].
  • [70] C. M. Will, Bounding the mass of the graviton using gravitational wave observations of inspiralling compact binaries, Phys. Rev. D 57, 2061 (1998), arXiv:gr-qc/9709011.
  • [71] C. de Rham, J. T. Deskins, A. J. Tolley, and S.-Y. Zhou, Graviton Mass Bounds, Rev. Mod. Phys. 89, 025004 (2017), arXiv:1606.08462 [astro-ph.CO].
  • [72] L. Andersson, J. Joudioux, M. A. Oancea, and A. Raj, Propagation of polarized gravitational waves, Phys. Rev. D 103, 044053 (2021), arXiv:2012.08363 [gr-qc].
  • [73] M. A. Oancea, R. Stiskalek, and M. Zumalacárregui, From the gates of the abyss: Frequency- and polarizationdependent lensing of gravitational waves in strong gravitational fields (2022), arXiv:2209.06459 [gr-qc].
  • [74] M. A. Oancea, R. Stiskalek, and M. Zumalacárregui, Probing general relativistic spin-orbit coupling with gravitational waves from hierarchical triple systems (2023), arXiv:2307.01903 [gr-qc].
  • [75] P. Horava, Quantum Gravity at a Lifshitz Point, Phys. Rev. D 79, 084008 (2009), arXiv:0901.3775 [hep-th].
  • [76] D. Blas, O. Pujolas, and S. Sibiryakov, Consistent Extension of Horava Gravity, Phys. Rev. Lett. 104, 181302 (2010), arXiv:0909.3525 [hep-th].
  • [77] M. Maggiore et al., Science Case for the Einstein Telescope, JCAP 2020 (03), 050, arXiv:1912.02622 [astroph.CO].
  • [78] M. Evans et al., A Horizon Study for Cosmic Explorer: Science, Observatories, and Community (2021), arXiv:2109.09882 [astro-ph.IM].
  • [79] V. Kalogera et al., The Next Generation Global Gravitational Wave Observatory: The Science Book (2021), arXiv:2111.06990 [gr-qc].
  • [80] P. Amaro-Seoane et al. (LISA), Laser Interferometer Space Antenna (2017), arXiv:1702.00786 [astro-ph.IM].
  • [81] P. Auclair et al. (LISA Cosmology Working Group), Cosmology with the Laser Interferometer Space Antenna (2022), arXiv:2204.05434 [astro-ph.CO].
  • [82] Y. Gong, J. Luo, and B. Wang, Concepts and status of Chinese space gravitational wave detection projects, Nature Astron. 5, 881 (2021), arXiv:2109.07442 [astroph.IM].
  • [83] J. Antoniadis et al. (EPTA, InPTA:), The second data release from the European Pulsar Timing Array - III. Search for gravitational wave signals, Astron. Astrophys. 678, A50 (2023), arXiv:2306.16214 [astro-ph.HE].
  • [84] G. Agazie et al. (NANOGrav), The NANOGrav 15 yr Data Set: Evidence for a Gravitational-wave Background, Astrophys. J. Lett. 951, L8 (2023), arXiv:2306.16213 [astro-ph.HE].
  • [85] D. J. Reardon et al., Search for an Isotropic Gravitationalwave Background with the Parkes Pulsar Timing Array, Astrophys. J. Lett. 951, L6 (2023), arXiv:2306.16215 [astro-ph.HE].
  • [86] H. Xu et al., Searching for the Nano-Hertz Stochastic Gravitational Wave Background with the Chinese Pulsar Timing Array Data Release I, Res. Astron. Astrophys. 23, 075024 (2023), arXiv:2306.16216 [astro-ph.HE].
  • [87] G. B. Folland, Introduction to Partial Differential Equations: Second Edition (Princeton University Press, 1995).
  • [88] B. Shoshany, OGRe: An Object-Oriented General Relativity Package for Mathematica, J. Open Source Softw. 6, 3416 (2021), arXiv:2109.04193 [cs.MS].