If the photon had a mass it would no longer propagate at ‘the speed of light’, but at a lower speed. For the photon its speed of propagation is known with such an accuracy in so many different media that it can be used to put the most stringent constraints on the photon mass to  . In the rest of this review we will adopt the viewpoint that the photon is massless and that light does indeed propagate at the ‘speed of light’.
The earliest bounds on the graviton mass were based on the same idea. As described in [487*], (see also ), if the graviton had a mass, gravitational waves would propagate at a speed different than that of light, (assuming a speed of light ). This different velocity between the light and gravitational waves would manifest itself in observations of supernovae. Assuming the emission of a gravitational wave with frequency larger than the graviton mass, this could lead to a bound on the graviton mass of considering a frequency of 100 Hz and a supernovae located 200 Mpc away [487*] (assuming that the photon propagates at the speed of light).
Alternatively, another way to test the speed of gravitational waves and bound the graviton mass without relying on any assumptions on the photon is through the observation of inspiralling compact objects which allows to derive the frequency-dependence of GWs. The detection of GWs in Advanced LIGO could then bound the graviton mass potentially all the way down to [487*, 486, 71].
The graviton mass is also relevant for the production of primordial gravitational waves during inflation. Following the analysis of  it was shown that the graviton mass opens up the production of gravitational waves during inflation with a sharp peak with a height and position which depend on the graviton mass. See also  for the study of exact plane wave solutions in massive gravity.
Nevertheless, these bounds on the graviton mass are relatively weak compared to the typical value of considered till now in this review. The reason for this is because these bounds do not take into account the effects arising from the additional polarization in the gravitational waves which would be present if the graviton had a mass in a Lorentz-invariant theory. For the photon, if it had a mass, the additional polarization would decouple and would therefore be irrelevant (this is related to the absence of vDVZ discontinuity at the classical level for a Proca theory.) In massive gravity, however, the helicity-0 mode of the graviton couples to matter. As we shall see below, the bounds on the graviton mass inferred from the absence of fifth forces are typically much more stringent.
One of the predictions of GR is the existence of gravitational waves (GW) with two transverse independent polarizations.
While GWs have not been directly detected via interferometer yet, they have been detected through the spin-down of binary pulsar systems [322, 457, 485]. This detection via binary pulsars does not count as a direct detection, but it matches expectations from GWs with such an accuracy, and for now so many different systems of different relative masses that it seems unlikely that the spin-down could be due to something different than the emission of GWs.
As emphasized in the first part of this review, and particularly in Section 2.5, the sixth excitation, namely the longitudinal one, represents a ghost degree of freedom. Thus, if that mode is observed, it cannot be arising from a Lorentz-invariant massive graviton. Its presence could be linked for instance to new scalar degrees of freedom which are independent from the graviton itself. In massive gravity, only five polarizations are expected. Notice however that the helicity-1 mode does not couple directly to matter or external sources, so it is unlikely that GWs with polarizations which mix the transverse and longitudinal directions would be produced in a natural process.
Furthermore, any physical process which is expected to produce GWs would include very dense sources where the Vainshtein mechanism will thus be expected to be active and screen the effect of the helicity-0 mode. As a result the excitation of the breathing mode is expected to be suppressed in any theory of massive gravity which includes an active Vainshtein mechanism.
So, while one could in principle expect up to six polarizations for GWs in a modified theory of gravity, in massive gravity only the two helicity-2 polarizations are expected to be produced in a potentially observable amount by interferometers like advanced-LIGO . To summarize, in ghost-free massive gravity or DGP we expect the following:
- The helicity-2 modes are produced in the same way as in GR and would be indistinguishable if they travel distances smaller than the graviton Compton wavelength
- The helicity-1 modes are not produced
- The breathing or conformal mode is produced but suppressed by the Vainshtein mechanism and so the magnitude of this mode is suppressed compared to the helicity-2 polarization by many orders of magnitudes.
- The longitudinal mode does not exist in a ghost-free theory of massive gravity. If such a mode is observed it must be arise from another field independent from the graviton.
We will also discuss the implications for indirect detection of GWs via binary pulsar spin-down in Section 11.4. We will see that already in these setups the radiation in the breathing mode is suppressed by 8 orders of magnitude compared to that in the helicity-2 mode. In more relativistic systems such as black-hole mergers, this suppression will be even bigger as the Vainshtein mechanism is stronger in these cases, and so we do not expect to see the helicity-0 mode component of a GW emitted by such systems.
To summarize, while additional polarizations are present in massive gravity, we do not expect to be able to observe them in current interferometers. However, these additional polarizations, and in particular the breathing mode can have larger effects on solar-system tests of gravity (see Section 11.2) as well as for weak lensing (see Section 11.3), as we review in what follows. They also have important implications for black holes as we discuss in Section 11.5 and in cosmology in Section 12.
A lot of the phenomenology of massive gravity can be derived from its decoupling limit where it resembles a Galileon theory. Since the Galileon was first encountered in DGP most of the phenomenology was first derived for that model. The extension to massive gravity is usually relatively straightforward with a few subtleties which we mention at the end. We start by reviewing the phenomenology assuming a cubic Galileon decoupling limit, which is directly applicable for DGP and then extend to the quartic Galileon and ghost-free massive gravity.
Within the context of DGP, a lot of its phenomenology within the solar system was derived in [388*, 386*] using the full higher-dimensional picture as well as in [215*]. In these work the effect from the helicity-0 mode in the advanced of the perihelion were computed explicitly. In particular in [215*] it was shown how an infrared modification of gravity could have an effect on small solar system scales and in particular on the Moon. In what follows we review their approach.
Consider a point source of mass localized at . In GR (or rather Newtonian gravity as it is a sufficient approximation), the gravitational potential mediated by the point source is10.1, when the Vainshtein mechanism is active the contribution from the helicity-0 mode is very much suppressed. but measurements in the Solar system are reaching such a level of accuracy than even a small deviation could in principle be observable [488*].
In the decoupling limit of DGP, matter fields couple to the following perturbed metric10.1.3 and (10.21*)),
[388*, 386*] that in DGP the sign of this anomalous angle depends on whether on the branch studied (self-accelerating branch – or normal branch).  (for instance the accuracy quoted for the effective variation of the Gravitational constant is /year /orbit).
As already mentioned in Section 10.1.2, the Vainshtein mechanism is typically much stronger28 in the spherically symmetric configuration of the quartic Galileon and thus in massive gravity (see for instance the suppression of the force given in (10.17*)). Using the same values as before for a quartic Galileon we obtain10.1.3 for more precision) 29
As mentioned previously, one peculiarity of massive gravity not found in DGP nor in a typical Galileon theory (unless we derive the Galileons from a higher-dimensional brane picture ) is the new disformal coupling to matter of the form , which means that the helicity-0 mode also couples to conformal matter.
In the vacuum, for a static and spherically symmetric configuration the coupling plays no role. So to the level at which we are working when deriving the Vainshtein mechanism about a point-like mass this additional coupling to matter does not affect the background configuration of the field (see  for a discussion outside the vacuum, taking into account for the instance the effect of the Earth atmosphere). However, it does affect this disformal coupling does affect the effect metric seen by perturbed sources on top of this configuration. This could have some implications for structure formation is to the best of our knowledge have not been fully explored yet, and does affect the bending of light. This effect was pointed out in [490*] and the effects to gravitational lensing were explored. We review the key results in what follows and refer to [490*] for further discussions (see also ).
In GR, the relevant potential for lensing is , where we use the same notation as before, and . A conformal coupling of the form does not affect this lensing potential but the disformal coupling leads to a new contribution given by[490*], here .] This new contribution to the lensing potential leads to an anomalous fractional lensing of
At the level of galaxies or clusters of galaxy, the effect might be more tangible. The reason for that is that for the mass of a galaxy, the associated strong coupling radius is not much larger than the galaxy itself and thus at the edge of a galaxy these effects could be stronger. These effects were investigated in  where it was shown a few percent effect on the tangential shear caused by the helicity-0 mode of the graviton or of a disformal Galileon considering a Navarro–Frenk–White halo profile, for some parameters of the theory. Interestingly, the effect peaks at some specific radius which is the same for any halo when measured in units of the viral radius. Even though the effect is small, this peak could provide a smoking gun for such modifications of gravity.
Recently, another analysis was performed in Ref. , where the possibility to testing theories of modified gravity exhibiting the Vainshtein mechanism against observations of cluster lensing was explored. In such theories, like in massive gravity, the second derivative of the field can be large at the transition between the screened and unscreened region, leading to observational signatures in cluster lensing.
One of the main predictions of massive gravity is the presence of new polarizations for GWs. While these new polarization might not be detectable in GW interferometers as explained in Section 11.1.2, we could still expect them to lead to detectable effects in the binary pulsar systems whose spin-down is in extremely good agreement with GR. In this section, we thus consider the power emitted in the helicity-0 mode of the graviton in a binary-pulsar system. We use the effective action approach derived by Goldberger and Rothstein in  and start with the decoupling limit of DGP before exploring that of ghost-free massive gravity and discussing the subtleties that arise in that case. We mainly focus on the monopole and quadrupole radiation although the whole formalism can be derived for any multipoles. We follow the derivation of Refs. [158*, 151*], see also Refs. [100, 18] for related studies.
In order to account for the Vainshtein mechanism into account we perform a similar background-perturbation split as was performed in Section 10.1. The source is thus split as where is a static and spherically source representing the total mass localized at the center of mass and captures the motion of the companions with respect to the center of mass.
This matter profile leads to a profile for the helicity-0 mode (here mimicked as a cubic Galileon which is the case for DGP) as in (10.3*) as , where the background has the same static and spherical symmetry as and so has the same profile as in Section 10.1.2.
The background configuration of the field was derived in (10.13*) where accounts in this case for the total mass of both companions and is the distance to the center of mass. Following the same procedure, the fluctuation then follows a modified Klein–Gordon equation10.14*).)
Expanding the field in spherical harmonics the mode functions satisfy
The total power emitted via the field is given by the sum over these mode functions,
Without the Vainshtein mechanism, the mode functions would be the same as for a standard free-field in flat space-time, and the power emitted in the monopole would be larger than that emitted in GR, which would be clearly ruled out by observations. The Vainshtein mechanism is thus crucial here as well for the viability of DGP or ghost-free massive gravity.
Taking the prefactor into account, the zero mode for the monopole is given instead by
This is to be compared with the Peters–Mathews formula for the power emitted in GR (in the helicity-2 modes) in the quadrupole ,
We see that the radiation in the monopole is suppressed by a factor of compared with the GR result. For the Hulse–Taylor pulsar this is a suppression of 10 orders of magnitudes which is completely unobservable (at best the precision of the GR result is of 3 orders of magnitude).
Notice, however, that the suppression is far less than what was naively anticipated from the static approximation in Section 10.1.2.
The same analysis can be performed for the dipole emission with an even larger suppression of about 19 orders of magnitude compared the Peters–Mathews formula.
The quadrupole emission in the field is slightly larger than the monopole. The reason is that energy conservation makes the non-relativistic limit of the monopole radiation irrelevant and one needs to take the first relativistic correction into account to emit in that channel. This is not so for the quadrupole as it does not correspond to the charge associated with any Noether current even in the non-relativistic limit.
In the non-relativistic limit, the mode function for the quadrupole is simply
yielding a quadrupole emission
When extending the analysis to more general Galileons or to massive gravity which includes a quartic Galileon, we expect a priori by following the analysis of Section 10.1.2, to find a stronger Vainshtein suppression. This result is indeed correct when considering the power radiated in only one multipole. For instance in a quartic Galileon, the power emitted in the field via the quadrupole channel is suppressed by 12 orders of magnitude compared the GR emission.
However, this estimation does not account for the fact that there could be many multipoles contributing with the same strength in a quartic Galileon theory .
In a quartic Galileon theory, the effective metric in the strong coupling radius for a static and spherically symmetric background is
In situations where there is a large hierarchy between the mass of the two objects (which is the case for instance within the solar system), perturbation theory can be seen to remain under control and the power emitted in the quartic Galileon is completely negligible.
As in any gravitational theory, the existence and properties of black holes are crucially important for probing the non-perturbative aspects of gravity. The celebrated black-hole theorems of GR play a significant role in guiding understanding of non-perturbative aspects of quantum gravity. Furthermore, the phenomenology of black holes is becoming increasingly important as understanding of astrophysical black holes increases.
Massive gravity and its extensions certainly exhibit black-hole solutions and if the Vainshtein mechanism is successful then we would expect solutions which look arbitrary close to the Schwarzschild and Kerr solutions of GR. However, as in the case of cosmological solutions, the situation is more complicated due to the absence of a unique static spherically symmetric solution that arises from the existence of additional degrees of freedom, and also the existence of other branches of solutions which may or may not be physical. There are a handful of known exact solutions in massive gravity [413*, 363*, 365*, 277*, 105*, 56*, 477*, 90, 478*, 455*, 30*, 357*], but the most interesting and physically relevant solutions probably correspond to the generic case where exact analytic solutions cannot be obtained. A recent review of black-hole solutions in bi-gravity and massive gravity is given in [478*].
An interesting effect was recently found in the context of bi-gravity in Ref. . In that case, the Schwarzschild solutions were shown to be unstable (with a Gregory–Laflamme type of instability [268, 269]) at a scale dictated by the graviton mass, i.e., the instability rate is of the order of the age of the Universe. See also Ref.  where the analysis was generalized to the non-bidiagonal. In this more general situation, spherically symmetric perturbations were also found but generically no instabilities. Black-hole disappearance in massive gravity was explored in Ref. .
Since all black-hole solutions of massive gravity arise as decoupling limits of solutions in bi-gravity,30 we can consider from the outset the bi-gravity solutions and consider the massive gravity limit after the fact. Let us consider then the bi-gravity action expressed as
One immediate consequence of working with bi-gravity is that since the metric is sourced by polynomials of whereas the metric is sourced by polynomials of . We, thus, require that is invertible away from curvature singularities. This is equivalent to saying that the eigenvalues of and should not pass through zero away from a curvature singularity. This in turn means that if one metric is diagonal and admits a horizon, the second metric if it is diagonal must admit a horizon at the same place, i.e., two diagonal metrics have common horizons. This is a generic observation that is valid for any theory with more than one metric [167*] regardless of the field equations. Equivalently, this implies that if is a diagonal metric without horizons, e.g., Minkowski spacetime, then the metric for a black hole must be non-diagonal when working in unitary gauge. This is consistent with the known exact solutions. For certain solutions it may be possible by means of introducing Stückelberg fields to put both metrics in diagonal form, due to the Stückelberg fields absorbing the off-diagonal terms. However, for the generic solution we would expect that at least one metric to be non-diagonal even with Stückelberg fields present.
The -metric coordinates are related to those of the metric by (in other words the profiles of the Stückelberg fields)[413, 363*, 365, 277, 105, 56, 477*, 478*, 455*, 30, 357]. Note in particular that for every set of ’s there are two branches of solutions determined by the two possible values of .
These solutions describe black holes sourced by matter minimally coupled to metric with mass . An obvious generalization is to assume that the matter couples to both metrics, with effective masses and so that[477*]). We note only that in  a distinct class of solutions is obtained numerically in bi-gravity for which the two metrics take the diagonal form [167*] these solutions do not correspond to black holes in the massive gravity on Minkowski limit , however the limit can be taken and they correspond to black-hole solutions in a theory of massive gravity in which the reference metric is Schwarzschild (–de Sitter or anti-de Sitter). The arguments of  are then evaded since the reference metric itself admits a horizon.