Tighter bounds on a hypothetical graviton screening mass from the gravitational wave observation GW150914 at LIGO
While quantum gravity is not solved yet, a screening mass for the graviton remains theoretically possible. If such a mass would screen gravity at distances of the order of the cluster galaxy radius, it could account for the universe expansion. The modified Newtonian dynamics model also could be related to a screening graviton mass at inter-galactic scales. Moreover, massive spin-2 theories constitute a very active theoretical topic. We briefly show how the very recent LIGO gravitational wave observation GW150914, emitted by a binary black hole merger distant ly from the Earth, tightens the phenomenological bound on a hypothetical graviton screening mass, or on the effective screening of gravity.
Quantum field theory has different ways to create a mass for the interaction bosons. Thus, while the problem of quantifying gravitation remains to be solved, a screening mass for the graviton remains theoretically possible. The very recent observation the 14th of September 2015 of a gravitational wave at both the LIGO detectors at Hanford, Washington and Livingston, Louisiana, from a binary black hole merger, GW150914 Abbott:2016blz () tightly constrains any screening of gravity. More detail on the LIGO observation is found in Refs. Abbott:2016jsd (); Abbott:2016nhf (); TheLIGOScientific:2016src (); TheLIGOScientific:2016wyq (); TheLIGOScientific:2016htt (); TheLIGOScientific:2016uux (); TheLIGOScientific:2016wfe (); TheLIGOScientific:2016qqj (); TheLIGOScientific:2016agk (). Here we briefly review how the screening of gravity was phenomenologically possible before the GW150914 observation, and how the LIGO observation rules out most phenomenological interest in gravity screening or in modified Newtonian dynamics.
The discovery of the , the lightest boson mediating the nuclear strong interaction, predicted by Yukawa Yukawa:1935xg () in 1935, is the first experimental example of a boson screening mass, of the order of MeV Agashe:2014kda (). Nambu and Jona-Lasinio Nambu:1961tp (); Nambu:1961fr () modelled chiral symmetry breaking, in 1961, leading to the mass generation of hadrons. The mass origin, in the Gell-Mann Oakes and Renner relation GellMann:1968rz () , is actually due to the interplay of spontaneous breaking of chiral symmetry, and to the quarks mass generation in the weak sector. More modernly, chiral symmetry breaking, leading to the mass relation and to other partially conserved axial current theorems, are directly related to Quantum Chromo Dynamics (QCD) with non-perturbative techniques such as the Dyson-Schwinger equations and the Bethe-Salpeter Equation Bicudo:2003fp ().
In the electroweak sector of the Standard Model, the Higgs mechanism Higgs:1964ia (); Higgs:1964pj (), not only produces masses for the fermion quarks and leptons, but also for the gauge bosons first observed at LEP in CERN and with screening masses starting at GeV Agashe:2014kda (). The recent discovery of the Higgs boson Chatrchyan:2012xdj (); Aad:2012tfa () absolutely confirmed the Standard Model of particle physics and the Higgs mechanism.
|strong confinement||0.5 to 0.8 GeV||linear|
|weak||W, Z||80 GeV||Yukawa|
|strong nuclear||0.14 GeV||Yukawa|
Back to the strong interactions, at smaller scales than the hadronic interactions mediated by the boson, QCD also has another interesting screening mass, of a second type. This second, more microscopic, screening mass is present in the flux tubes leading to the confinement of quarks and gluons in colour singlets. In Table 1, we separate the two different QCD scales of screening, the effective one of nuclear strong interactions related to the mass, and the more fundamental one of confinement. Unlike the mass generation in chiral symmetry breaking and in the Higgs mechanism, the mechanism of confinement is not yet theoretically understood in full detail, and remains a difficult open problem in theoretical particle physics. Notice the quantization of QCD, either in pure gauge QCD or in QCD with massless quarks which have no dimensional scale in the Lagrangian, breaks conformal invariance and this was computed since the onset of lattice QCD Creutz:1979dw (); Creutz:1980zw (). Nevertheless the phenomenology of confinement is well known in lattice QCD computations of flux tubes and of the gluon propagator. The QCD flux tubes are gauge invariant and screening is related to the width of the flux tubes Cardoso:2013lla () (after accounting for the flux tube quantum vibrations) with a penetration length to 0.24 fm. The inverse of the penetration length
indicates an effective screening mass for the gluon (or another related quantum degree of freedom) of to 0.9 GeV. Besides, the transverse gluon propagator computed in Landau gauge saturates in the infrared, qualitatively consistent with a gluon screening mass of the order of GeV in the Landau gauge Oliveira:2010xc (). Although the gluon propagator is gauge dependent, this saturation in the Landau gauge is of the same order of the saturation recently computed in different gauges Bicudo:2015rma (). The screening in QCD is most interesting because it preserves Gauss law, and nevertheless it enhances the interaction between charges by squeezing the colour fields in flux tubes; similar to the screening of the electromagnetic field in type II superconductors Cardoso:2006mf (). This results in a linear potential between static charges, much stronger than the Coulomb potential. Moreover confinement screening is absolute, in the sense the gluons are unable to propagate in the vacuum. Thus, in contradistinction with the weak and nuclear types of screenings, in confinement screening waves do not propagate.
Nevertheless the quantization of a theory does not, of course, imply screening. The main case without screening is the electromagnetic sector of the Standard Model: it remains totally unscreened upon quantization. But even the photon may have a mass, See Ref. Goldhaber:2008xy () for a review of direct and indirect bounds on the photon mass. Assuming a propagation with no dispersion or damping in a distance with the radius of all the visible universe, with a diameter of ly, of the Cosmic Microwave Background, since the universe became transparent, would correspond to a nearly vanishing mass eV. The different screening masses of the quantum interactions of particle physics are listed in Table 1.
Possibly inspired in particle physics, or in condensed matter physics, the idea of a graviton mass has been considered both theoretically and in phenomenology, since the early work of Fierz and Pauli in 1939 Fierz:1939ix ().
Theoretically, the graviton mass is a very active topic, because quantized gravity is a spin 2 theory. Inasmuch as the Proca Proca:1988ii () and Stueckelberg Stueckelberg:1900zz () theories are unavoidable to fully understand spin 1 theories, studying a graviton mass is unavoidable in the mapping of all possible sub-classes of spin 2 theories. Moreover, a tantalizing motivation to screen gravity is the gravitational constant problem Weinberg:1988cp (); Jackiw:2005yc (); Polyakov:2006bz (); Porto:2009xj (). The massive Fierz-Pauli theory suffers from a discontinuity vanDam:1970vg (); Zakharov:1970cc (); Iwasaki:1971uz (): in the limit of a vanishing graviton mass, the Fierz–Pauli theory is not equivalent to the linearized general relativity. The discontinuity cure led to several developments of spin 2 theories, starting with the non-linear Vainshtein mechanism Vainshtein:1972sx ().
Clearly, spin 2 theories are more complex than Abelian spin 1 theories, and the massive has only been understood recently. Numerically Damour:2002gp (), the Vainshtein mechanism was worked out in Refs Babichev:2009jt (); Babichev:2010jd (). see Ref. Babichev:2013usa () for a recent review of the Vainshtein mechanism. Theoretically, inasmuch as a possible realization of massive spin 1 theories include a longitudinally polarized photon, in spin 2 massive theories, helicities 0 and 1 are included together with the spin 2 of general gravity. For recent theoretical reviews see Refs. Hinterbichler:2011tt (); deRham:2014zqa (). A ghost was believed to be present in massive gravity, until the model of Dvali, Gabadadze and Porrat (DGP) was proposed, as the first theory without a ghost Dvali:2000rv (); Dvali:2000hr (), and where one could understand explicitly how the Vainshtein mechanism is implemented Deffayet:2001uk (); Dvali:2002vf (). In the decoupling limit of DGP the interactions of the helicity 0 mode become important for a finite mass, whereas the corrections to GR tend to 0 in the limit where the graviton mass goes to zero Luty:2003vm (); Nicolis:2004qq (). The helicity 0 mode was also considered as a scalar field theory in own right, the Galileon Nicolis:2008in (); Brax:2011sv (), reminiscent of the Abelian Higgs model used in the Stueckelberg photons. Finally, the DGP model has been successfully extended to the massive gravity theory deRham:2010ik (); deRham:2010kj (); deRham:2010tw () that implements the Vainshtein mechanism, fully solves the ghost problem and where the graviton is a real massive particle with a pole in the propagator. Moreover massive gravity has several interesting properties, leading to an intense research deRham:2014zqa () and also inspiring solutions to problems of other approaches, such as the di-metric theories Hossenfelder:2008bg (); Hassan:2011hr ().
We now discuss the phenomenological motivation to a hypothetical graviton screening mass Goldhaber:1974wg (), which is appealing at two different scales. Notice at distances of the solar systems and smaller, Newtonian gravity is correct in first order, and general gravity is 100% compatible with experiment in all observations, and thus is assumed to be correct. Newtonian dynamics is a first order approximation to general gravity, and is a direct result of the Gauss law. Thus no screening exists at this scale, and screening is only phenomenologically interesting at much larger scales.
At inter-galactic distances, considering the visible stellar masses distributions and velocities only, the naive Newtonian gravity fails. Two different main solutions have been proposed: dark matter (DM) and modified Newtonian gravity (MOND). While DM has been accumulating observational evidences from lensing and cluster collisions, but has not yet been produced in the laboratory, MOND has been trying to comply with the observations Milgrom:1983ca (); Milgrom:1983pn (); Milgrom:1983zz (); Bugg:2014bka (). Notice MOND corresponds to a potential stronger than the Coulomb potential, apparently in excess of the Gauss law Moody:1993ir () of gravity field flux conservation. In order not to violate the Gauss law, MOND could possibly be due to an effective graviton screening. For distances of the inter-galactic order, a screening similar to the one of the confinement of QCD in strong interactions, where the colour fields are squeezed in flux tubes, could enhance the gravitation force, thus leading to a stronger gravitation force, compatible with MOND. This is sketched in Fig. 1. Notice screening here would not be exactly identical to the one in QCD confinement, where the potential is linear, and thus even stronger than the one in MOND.
Moreover at larger distances than the galaxy cluster scale, the universe expansion has two different solutions as well. One is the dark energy (DE), counter-acting the gravitational pull, and the other is screening. This hypothetical screening is different from the one leading to flux tubes inside the galaxy, since it totally saturates the gravitic potential. It is reminiscent of the screening in effective strong nuclear interactions, mediated by mesons, see Table 1, where the strong interaction is short range because the interaction range is of the order of the inverse meson mass. Such a screening should break the Gauss law conservation of the gravitational field flux, eventually suppressing it. In particular, due to the recent theoretical advances in massive gravity, this screening is already studied in Ref. deRham:2010tw (). The negative pressure causing the acceleration is due to a condensate of the helicity-0 component of the massive graviton, and the background evolution, in the approximation used in Ref. deRham:2010tw (), is indistinguishable from the CDM standard model of cosmology.
Using the relation between the screening mass and the characteristic distance or penetration length of Eq. (1), in units of a penetration length of 1 m corresponds to a screening mass of 0.197 eV and a penetration length of 1 light year (ly) corresponds to a screening mass of . We then arrive at the two different relations for the two hypothetical graviton screening masses. From the solar system characteristic scale Gott:2003pf () of the order of 1 ly,
This mass should squeeze the graviton field in flux tubes, larger than the solar system size, and approximately preserve the Gauss law. On the other hand from the galaxy typical radius Gott:2003pf (); Juric:2005zr ()of ly or super cluster typical radius of the order of ly Gott:2003pf (); Zehavi:2004ii (); Cole:2005sx (); Springel:2006vs (), we arrive at the absolute screening mass of the order of,
Thus, already inside the galaxy (if we assume a screening model of MOND) or beyond the super clusters (if we assume a screening model of the universe expansion), see Fig. 1 the gravitational field would be screened, and the gravitational wave would be unable to propagate.
However, due to the observed gravitational wave GW150914 at LIGO, assuming the general gravity equations and computer simulations at short distances are correct, then the signal is clearly produced by an extragalactic source. The source of the gravitational wave GW150914 Abbott:2016blz (); TheLIGOScientific:2016wfe () has a luminosity distance of 410 Mpc = ly from Earth, corresponding to a redshift of 0.09 . Using the linear approximation of the redshift to distance relation, we get a distance of ly from Earth, very close to the luminosity distance. This means the source is not only extragalactic, it is well beyond our local supercluster with radius of ly only. But it is close enough that the redshift is still approximately linear with distance, which does not depend strongly on the cosmological model.
Assuming the signal is able to propagate with no attenuation, then general gravity, which preserves Gauss law, is correct at least up to this distance. This imposes an upper bound for a screening graviton mass of,
where the screening mass should clearly be smaller that this bound, otherwise some attenuation effect would already be visible at the distance to the GW150914 source. It is important to notice that this bound is for the absolute type of screening only, similar to the confinement screening in QCD, where the wave propagation is damped in the vacuum.
The bound for a Yukawa-like screening, as in nuclear and weak interactions, is much looser, since the corresponding graviton waves propagate through the vacuum, and only the dispersion relation is changed Will:1997bb (). This bound was already determined by the LIGO Scientific and the Virgo Collaborations TheLIGOScientific:2016src (), who determined,
at 90 % confidence.
It seems clear a screening-driven MOND, with screening mass in Eq. (2), is absolutely excluded by the result in Eq.(4). Moreover, a universe expansion due to the screening of the gravitational interaction between superclusters, with screening mass in Eq. (3) remains possible if the screning is Yukawa-like with the bound in Eq. (5).
Notice the visible universe radius is only the distance to the GW150914 source. In the same sense, our bound is getting close to the Hubble constant in eV units, eV Xu:2013ega (). As noted in the review deRham:2014zqa () a graviton mass smaller to this bound would be unobservable in the visible Universe and such a mass would thus loose its phenomenological interest.
Let us now compare our bound with other bounds for the graviton mass or for the mass scale of gravity screening. From the observation of gravitational waves as we did here, two different sorts of bounds have been predicted. Comparing the speed of light with the speed of gravitational waves from a same source, bounds of eV would be expected form supernovae observations Will:2014kxa (). With similar ideas as in this work, from the observation of spiralling massive objects, it was expected that bounds of eV could be extracted in advanced LIGOWill:2014kxa () . Notice our bound turns out to be tighter by a factor of 1000 than the anticipated bound in Ref. Will:2014kxa (). Our bound in Eq. (4) is close to the one combining the Lunar Laser Ranging (LLR) experiment Williams:2004qba () with the present massive gravity theories deRham:2014zqa (). It is interesting that a numerical work with black holes already placed a bound on the graviton mass. With the numerical study of both Schwarzschild and slowly rotating Kerr black holes Brito:2013wya (), it was shown they are unstable for graviton masses eV.
To conclude, the event GW150914 observed at LIGO with 5.1 significance, deeply extends our knowledge of gravitation Calabrese:2016bnu (); Evans:2016mta (); Berti:2016iki (). GW150914 just tested general gravity from the size of the black hole merger Pretorius:2005gq (); Campanelli:2005dd (); Baker:2005vv (); Berti:2007fi (); Sperhake:2008ga () to the distance from the black hole merger to the Earth, of ly. Unlike microscopic QCD, nuclear forces and the weak sector of the Standard Model, gravitational wave observations may provide evidence that gravitation has no screening mechanism up to larger than the typical scale of superclusters. In what concerns damping as in QCD-like screening the bound is already very tight, as in Eq. (4). In what concerns the modification of the dispersion relation due to Yukawa-like screening, the present bound in Eq. (5) could further be tightened with more gravitational wave observations. This provides a tighter bound on screening, and further constrains alternative models to dark matter and dark energy. We are even more confident in Einstein’s general gravity Einstein:1915 (), although it still remains to be quantized.
Nevertheless, it would be interesting to extend massive gravity simulations of black holes Brito:2013wya () to compute numerically black hole mergers, and the resulting gravitational wave. Notice the GW150914 signal, including the luminosity distance, was fitted assuming general relativity, with no screening. A hypothetical screening could damp the luminosity, and then the effective distance to the GW150914 signal could be shorter than the one reported. This would change our conclusions and relax our bound.
Acknowledgements - P. B. is very grateful to Claudia de Rham for discussions on massive gravity and to David Bugg for discussions on MOND; is thankful to Ana Mourão, George Rupp, José Lemos and Vítor Cardoso for discussions on black holes, gravitation and cosmology; and acknowledges the support of CFTP (grant FCT UID/FIS/00777/2013).
- (1) B. P. Abbott et al. [LIGO Scientific and Virgo Collaborations], ”Observation of Gravitational wave from a Binary Black Hole Merger,” Phys. Rev. Lett. 116, no. 6, 061102 (2016) [Phys. Rev. Lett. 116, 061102 (2016)] doi:10.1103/PhysRevLett.116.061102 [arXiv:1602.03837 [gr-qc]].
- (2) B. P. Abbott et al. [The LIGO Scientific Collaboration], ”Calibration of the Advanced LIGO detectors for the discovery of the binary black-hole merger GW150914,” arXiv:1602.03845 [gr-qc].
- (3) B. P. Abbott et al. [The LIGO Scientific and the Virgo Collaboration], ”The Rate of Binary Black Hole Mergers Inferred from Advanced LIGO Observations Surrounding GW150914,” arXiv:1602.03842 [astro-ph.HE].
- (4) [The LIGO Scientific and the Virgo Collaborations], ”Tests of general relativity with GW150914,” arXiv:1602.03841 [gr-qc].
- (5) [The LIGO Scientific and the Virgo Collaborations], ”GW150914: Implications for the stochastic gravitational wave background from binary black holes,” arXiv:1602.03847 [gr-qc].
- (6) [The LIGO Scientific and the Virgo Collaborations], ”Astrophysical Implications of the Binary Black-Hole Merger GW150914,” Astrophys. J. 818, L22 (2016) [arXiv:1602.03846 [astro-ph.HE]].
- (7) [The LIGO Scientific and the Virgo Collaborations], ”Observing gravitational-wave transient GW150914 with minimal assumptions,” arXiv:1602.03843 [gr-qc].
- (8) [The LIGO Scientific and the Virgo Collaborations], ”Properties of the binary black hole merger GW150914,” arXiv:1602.03840 [gr-qc].
- (9) [The LIGO Scientific and the Virgo Collaborations], ”GW150914: First results from the search for binary black hole coalescence with Advanced LIGO,” arXiv:1602.03839 [gr-qc].
- (10) [The LIGO Scientific and The Virgo Collaborations], ”GW150914: The Advanced LIGO Detectors in the Era of First Discoveries,” arXiv:1602.03838 [gr-qc].
- (11) H. Yukawa, ”On the Interaction of Elementary Particles I,” Proc. Phys. Math. Soc. Jap. 17, 48 (1935) [Prog. Theor. Phys. Suppl. 1, 1]. doi:10.1143/PTPS.1.1
- (12) K. A. Olive et al. [Particle Data Group Collaboration], ”Review of Particle Physics,” Chin. Phys. C 38, 090001 (2014). doi:10.1088/1674-1137/38/9/090001
- (13) Y. Nambu and G. Jona-Lasinio, ”Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. 1.,” Phys. Rev. 122, 345 (1961). doi:10.1103/PhysRev.122.345
- (14) Y. Nambu and G. Jona-Lasinio, ”Dynamical Model Of Elementary Particles Based On An Analogy With Superconductivity. Ii,” Phys. Rev. 124, 246 (1961). doi:10.1103/PhysRev.124.246
- (15) M. Gell-Mann, R. J. Oakes and B. Renner, ”Behavior of current divergences under SU(3) x SU(3),” Phys. Rev. 175, 2195 (1968). doi:10.1103/PhysRev.175.2195
- (16) P. Bicudo, ”Analytic proof that the quark model complies with partially conserved axial current theorems,” Phys. Rev. C 67, 035201 (2003) doi:10.1103/PhysRevC.67.035201 [hep-ph/0311277].
- (17) P. W. Higgs, ”Broken symmetries, massless particles and gauge fields,” Phys. Lett. 12, 132 (1964). doi:10.1016/0031-9163(64)91136-9
- (18) P. W. Higgs, ”Broken Symmetries and the Masses of Gauge Bosons,” Phys. Rev. Lett. 13, 508 (1964). doi:10.1103/PhysRevLett.13.508
- (19) S. Chatrchyan et al. [CMS Collaboration], ”Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC,” Phys. Lett. B 716, 30 (2012) doi:10.1016/j.physletb.2012.08.021 [arXiv:1207.7235 [hep-ex]].
- (20) G. Aad et al. [ATLAS Collaboration], ”Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC,” Phys. Lett. B 716, 1 (2012) doi:10.1016/j.physletb.2012.08.020 [arXiv:1207.7214 [hep-ex]].
- (21) N. Cardoso, M. Cardoso and P. Bicudo, ”Inside the SU(3) quark-antiquark QCD flux tube: screening versus quantum widening,” Phys. Rev. D 88, 054504 (2013) doi:10.1103/PhysRevD.88.054504 [arXiv:1302.3633 [hep-lat]].
- (22) O. Oliveira and P. Bicudo, ”Running Gluon Mass from Landau Gauge Lattice QCD Propagator,” J. Phys. G 38, 045003 (2011) doi:10.1088/0954-3899/38/4/045003 [arXiv:1002.4151 [hep-lat]].
- (23) P. Bicudo, D. Binosi, N. Cardoso, O. Oliveira and P. J. Silva, ”Lattice gluon propagator in renormalizable gauges,” Phys. Rev. D 92, no. 11, 114514 (2015) doi:10.1103/PhysRevD.92.114514 [arXiv:1505.05897 [hep-lat]].
- (24) M. Cardoso, P. Bicudo and P. D. Sacramento, ”Confinement of monopole field lines in a superconductor at T does not equal 0,” Annals Phys. 323, 337 (2008) doi:10.1016/j.aop.2007.02.007 [hep-ph/0607218].
- (25) M. Creutz, ”Confinement and the Critical Dimensionality of Space-Time,” Phys. Rev. Lett. 43, 553 (1979) [Phys. Rev. Lett. 43, 890 (1979)]. doi:10.1103/PhysRevLett.43.553
- (26) M. Creutz, ”Monte Carlo Study of Quantized SU(2) Gauge Theory,” Phys. Rev. D 21, 2308 (1980). doi:10.1103/PhysRevD.21.2308
- (27) A. S. Goldhaber and M. M. Nieto, ”Photon and Graviton Mass Limits,” Rev. Mod. Phys. 82, 939 (2010) doi:10.1103/RevModPhys.82.939 [arXiv:0809.1003 [hep-ph]].
- (28) M. Fierz and W. Pauli, ”On relativistic wave equations for particles of arbitrary spin in an electromagnetic field,” Proc. Roy. Soc. Lond. A 173, 211 (1939). doi:10.1098/rspa.1939.0140
- (29) G. A. Proca, ”Alexandre Proca (1897 - 1955): Scientific publications. (Mostly in French),” Paris, France: Proca (1988) nonconsec. pag
- (30) E. C. G. Stueckelberg, ”Interaction energy in electrodynamics and in the field theory of nuclear forces,” Helv. Phys. Acta 11, 225 (1938). doi:10.5169/seals-110852
- (31) S. Weinberg, ”The Cosmological Constant Problem,” Rev. Mod. Phys. 61, 1 (1989). doi:10.1103/RevModPhys.61.1
- (32) R. Jackiw, C. Nunez and S.-Y. Pi, ”Quantum relaxation of the cosmological constant,” Phys. Lett. A 347, 47 (2005) doi:10.1016/j.physleta.2005.04.020 [hep-th/0502215].
- (33) A. M. Polyakov, ”Beyond space-time,” hep-th/0602011.
- (34) R. A. Porto and A. Zee, ”Relaxing the cosmological constant in the extreme ultra-infrared,” Class. Quant. Grav. 27, 065006 (2010) doi:10.1088/0264-9381/27/6/065006 [arXiv:0910.3716 [hep-th]].
- (35) H. van Dam and M. J. G. Veltman, ”Massive and massless Yang-Mills and gravitational fields,” Nucl. Phys. B 22, 397 (1970). doi:10.1016/0550-3213(70)90416-5
- (36) V. I. Zakharov, ”Linearized gravitation theory and the graviton mass,” JETP Lett. 12, 312 (1970) [Pisma Zh. Eksp. Teor. Fiz. 12, 447 (1970)].
- (37) Y. Iwasaki, ”Consistency condition for propagators,” Phys. Rev. D 2, 2255 (1970). doi:10.1103/PhysRevD.2.2255
- (38) A. I. Vainshtein, ”To the problem of nonvanishing gravitation mass,” Phys. Lett. B 39, 393 (1972). doi:10.1016/0370-2693(72)90147-5
- (39) T. Damour, I. I. Kogan and A. Papazoglou, ”Spherically symmetric space-times in massive gravity,” Phys. Rev. D 67, 064009 (2003) doi:10.1103/PhysRevD.67.064009 [hep-th/0212155].
- (40) E. Babichev, C. Deffayet and R. Ziour, ”Recovering General Relativity from massive gravity,” Phys. Rev. Lett. 103, 201102 (2009) doi:10.1103/PhysRevLett.103.201102 [arXiv:0907.4103 [gr-qc]].
- (41) E. Babichev, C. Deffayet and R. Ziour, ”The Recovery of General Relativity in massive gravity via the Vainshtein mechanism,” Phys. Rev. D 82, 104008 (2010) doi:10.1103/PhysRevD.82.104008 [arXiv:1007.4506 [gr-qc]].
- (42) E. Babichev and C. Deffayet, ”An introduction to the Vainshtein mechanism,” Class. Quant. Grav. 30, 184001 (2013) doi:10.1088/0264-9381/30/18/184001 [arXiv:1304.7240 [gr-qc]].
- (43) K. Hinterbichler, ”Theoretical Aspects of Massive Gravity,” Rev. Mod. Phys. 84, 671 (2012) doi:10.1103/RevModPhys.84.671 [arXiv:1105.3735 [hep-th]].
- (44) C. de Rham, ”Massive Gravity,” Living Rev. Rel. 17, 7 (2014) doi:10.12942/lrr-2014-7 [arXiv:1401.4173 [hep-th]].
- (45) G. R. Dvali, G. Gabadadze and M. Porrati, ”Metastable gravitons and infinite volume extra dimensions,” Phys. Lett. B 484, 112 (2000) doi:10.1016/S0370-2693(00)00631-6 [hep-th/0002190].
- (46) G. R. Dvali, G. Gabadadze and M. Porrati, ”4-D gravity on a brane in 5-D Minkowski space,” Phys. Lett. B 485, 208 (2000) doi:10.1016/S0370-2693(00)00669-9 [hep-th/0005016].
- (47) C. Deffayet, G. R. Dvali, G. Gabadadze and A. I. Vainshtein, ”Nonperturbative continuity in graviton mass versus perturbative discontinuity,” Phys. Rev. D 65, 044026 (2002) doi:10.1103/PhysRevD.65.044026 [hep-th/0106001].
- (48) G. Dvali, A. Gruzinov and M. Zaldarriaga, ”The Accelerated universe and the moon,” Phys. Rev. D 68, 024012 (2003) doi:10.1103/PhysRevD.68.024012 [hep-ph/0212069].
- (49) M. A. Luty, M. Porrati and R. Rattazzi, ”Strong interactions and stability in the DGP model,” JHEP 0309, 029 (2003) doi:10.1088/1126-6708/2003/09/029 [hep-th/0303116].
- (50) A. Nicolis and R. Rattazzi, ”Classical and quantum consistency of the DGP model,” JHEP 0406, 059 (2004) doi:10.1088/1126-6708/2004/06/059 [hep-th/0404159].
- (51) A. Nicolis, R. Rattazzi and E. Trincherini, ”The Galileon as a local modification of gravity,” Phys. Rev. D 79, 064036 (2009) doi:10.1103/PhysRevD.79.064036 [arXiv:0811.2197 [hep-th]].
- (52) P. Brax, C. Burrage and A. C. Davis, ”Laboratory Tests of the Galileon,” JCAP 1109, 020 (2011) doi:10.1088/1475-7516/2011/09/020 [arXiv:1106.1573 [hep-ph]].
- (53) C. de Rham and G. Gabadadze, ”Generalization of the Fierz-Pauli Action,” Phys. Rev. D 82, 044020 (2010) doi:10.1103/PhysRevD.82.044020 [arXiv:1007.0443 [hep-th]].
- (54) C. de Rham, G. Gabadadze and A. J. Tolley, ”Resummation of Massive Gravity,” Phys. Rev. Lett. 106, 231101 (2011) doi:10.1103/PhysRevLett.106.231101 [arXiv:1011.1232 [hep-th]].
- (55) C. de Rham, G. Gabadadze, L. Heisenberg and D. Pirtskhalava, ”Cosmic Acceleration and the Helicity-0 Graviton,” Phys. Rev. D 83, 103516 (2011) doi:10.1103/PhysRevD.83.103516 [arXiv:1010.1780 [hep-th]].
- (56) S. Hossenfelder, ”A Bi-Metric Theory with Exchange Symmetry,” Phys. Rev. D 78, 044015 (2008) doi:10.1103/PhysRevD.78.044015 [arXiv:0807.2838 [gr-qc]].
- (57) S. F. Hassan and R. A. Rosen, ”Resolving the Ghost Problem in non-Linear Massive Gravity,” Phys. Rev. Lett. 108, 041101 (2012) doi:10.1103/PhysRevLett.108.041101 [arXiv:1106.3344 [hep-th]].
- (58) A. S. Goldhaber and M. M. Nieto, ”Mass of the graviton,” Phys. Rev. D 9, 1119 (1974). doi:10.1103/PhysRevD.9.1119
- (59) M. Milgrom, ”A Modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis,” Astrophys. J. 270, 365 (1983). doi:10.1086/161130
- (60) M. Milgrom, ”A Modification of the Newtonian dynamics: Implications for galaxies,” Astrophys. J. 270, 371 (1983). doi:10.1086/161131
- (61) M. Milgrom, ”A modification of the Newtonian dynamics: implications for galaxy systems,” Astrophys. J. 270, 384 (1983). doi:10.1086/161132
- (62) D. V. Bugg, ”MOND — A review,” Can. J. Phys. 93, no. 2, 119 (2015) doi:10.1139/cjp-2014-0057 [arXiv:1405.1695 [physics.gen-ph]].
- (63) M. V. Moody and H. J. Paik, ”Gauss’s law test of gravity at short range,” Phys. Rev. Lett. 70, 1195 (1993). doi:10.1103/PhysRevLett.70.1195
- (64) J. R. Gott, III, M. Juric, D. Schlegel, F. Hoyle, M. Vogeley, M. Tegmark, N. A. Bahcall and J. Brinkmann, ”A map of the universe,” Astrophys. J. 624, 463 (2005) doi:10.1086/428890 [astro-ph/0310571].
- (65) M. Juric et al. [SDSS Collaboration], ”The Milky Way Tomography with SDSS. 1. Stellar Number Density Distribution,” Astrophys. J. 673, 864 (2008) doi:10.1086/523619 [astro-ph/0510520].
- (66) I. Zehavi et al. [SDSS Collaboration], ”The Luminosity and color dependence of the galaxy correlation function,” Astrophys. J. 630, 1 (2005) doi:10.1086/431891 [astro-ph/0408569].
- (67) S. Cole et al. [2dFGRS Collaboration], ”The 2dF Galaxy Redshift Survey: Power-spectrum analysis of the final dataset and cosmological implications,” Mon. Not. Roy. Astron. Soc. 362, 505 (2005) doi:10.1111/j.1365-2966.2005.09318.x [astro-ph/0501174].
- (68) V. Springel, C. S. Frenk and S. D. M. White, ”The large-scale structure of the Universe,” Nature 440, 1137 (2006) doi:10.1038/nature04805 [astro-ph/0604561].
- (69) C. M. Will, Phys. Rev. D 57, 2061 (1998) doi:10.1103/PhysRevD.57.2061 [gr-qc/9709011].
- (70) L. Xu, ”Confronting DGP braneworld gravity with cosmico observations after Planck data,” JCAP 1402, 048 (2014) doi:10.1088/1475-7516/2014/02/048 [arXiv:1312.4679 [astro-ph.CO]].
- (71) C. M. Will, ”The Confrontation between General Relativity and Experiment,” Living Rev. Rel. 17, 4 (2014) doi:10.12942/lrr-2014-4 [arXiv:1403.7377 [gr-qc]].
- (72) J. G. Williams, S. G. Turyshev and D. H. Boggs, ”Progress in lunar laser ranging tests of relativistic gravity,” Phys. Rev. Lett. 93, 261101 (2004) doi:10.1103/PhysRevLett.93.261101 [gr-qc/0411113].
- (73) R. Brito, V. Cardoso and P. Pani, ”Massive spin-2 fields on black hole spacetimes: Instability of the Schwarzschild and Kerr solutions and bounds on the graviton mass,” Phys. Rev. D 88, no. 2, 023514 (2013) doi:10.1103/PhysRevD.88.023514 [arXiv:1304.6725 [gr-qc]].
- (74) E. Calabrese, N. Battaglia and D. N. Spergel, ”Testing Gravity with Gravitational Wave Source Counts,” arXiv:1602.03883 [gr-qc].
- (75) P. A. Evans et al., ”Swift follow-up of the Gravitational Wave source GW150914,” arXiv:1602.03868 [astro-ph.HE].
- (76) E. Berti, ”Viewpoint: The First Sounds of Merging Black Holes,” APS Physics 9, 17 (2016).
- (77) F. Pretorius, ”Evolution of binary black hole spacetimes,” Phys. Rev. Lett. 95, 121101 (2005) doi:10.1103/PhysRevLett.95.121101 [gr-qc/0507014].
- (78) M. Campanelli, C. O. Lousto, P. Marronetti and Y. Zlochower, ”Accurate evolutions of orbiting black-hole binaries without excision,” Phys. Rev. Lett. 96, 111101 (2006) doi:10.1103/PhysRevLett.96.111101 [gr-qc/0511048].
- (79) J. G. Baker, J. Centrella, D. I. Choi, M. Koppitz and J. van Meter, ”Gravitational wave extraction from an inspiraling configuration of merging black holes,” Phys. Rev. Lett. 96, 111102 (2006) doi:10.1103/PhysRevLett.96.111102 [gr-qc/0511103].
- (80) E. Berti, V. Cardoso, J. A. Gonzalez, U. Sperhake, M. Hannam, S. Husa and B. Bruegmann, ”Inspiral, merger and ringdown of unequal mass black hole binaries: A Multipolar analysis,” Phys. Rev. D 76, 064034 (2007) doi:10.1103/PhysRevD.76.064034 [gr-qc/0703053 [GR-QC]].
- (81) U. Sperhake, V. Cardoso, F. Pretorius, E. Berti and J. A. Gonzalez, ”The High-energy collision of two black holes,” Phys. Rev. Lett. 101, 161101 (2008) doi:10.1103/PhysRevLett.101.161101 [arXiv:0806.1738 [gr-qc]].
- (82) A. Einstein, ”Die Feldgleichungen der Gravitation”, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ) 1915, 844-847 (1915).