Another way in which the speed of gravitational waves could differ from c is if gravitation were propagated by a massive field (a massive graviton), in which case would be given by, in a local inertial frame,
The simplest attempt to incorporate a massive graviton into general relativity in a ghost-free manner suffers from the so-called van Dam–Veltman–Zakharov (vDVZ) discontinuity [263, 299]. Because of the 3 additional helicity states available to the massive spin-2 graviton, the limit of small graviton mass does not coincide with pure GR, and the predicted perihelion advance, for example, violates experiment. A model theory by Visser  attempts to circumvent the vDVZ problem by introducing a non-dynamical flat-background metric. This theory is truly continuous with GR in the limit of vanishing graviton mass; on the other hand, its observational implications have been only partially explored. Braneworld scenarios predict a tower or a continuum of massive gravitons, and may avoid the vDVZ discontinuity, although the full details are still a work in progress [91, 66].
The most obvious way to test this is to compare the arrival times of a gravitational wave and an electromagnetic wave from the same event, e.g., a supernova. For a source at a distance , the resulting value of the difference is
For a massive graviton, if the frequency of the gravitational waves is such that , where is Planck’s constant, then , where is the graviton Compton wavelength, and the bound on can be converted to a bound on , given by
The foregoing discussion assumes that the source emits both gravitational and electromagnetic radiation in detectable amounts, and that the relative time of emission can be established to sufficient accuracy, or can be shown to be sufficiently small.
However, there is a situation in which a bound on the graviton mass can be set using gravitational radiation alone . That is the case of the inspiralling compact binary. Because the frequency of the gravitational radiation sweeps from low frequency at the initial moment of observation to higher frequency at the final moment, the speed of the gravitons emitted will vary, from lower speeds initially to higher speeds (closer to c) at the end. This will cause a distortion of the observed phasing of the waves and result in a shorter than expected overall time of passage of a given number of cycles. Furthermore, through the technique of matched filtering, the parameters of the compact binary can be measured accurately (assuming that GR is a good approximation to the orbital evolution, even in the presence of a massive graviton), and thereby the emission time can be determined accurately. Roughly speaking, the “phase interval” in Equation (101) can be measured to an accuracy , where is the signal-to-noise ratio.
Thus one can estimate the bounds on achievable for various compact inspiral systems, and for various detectors. For stellar-mass inspiral (neutron stars or black holes) observed by the LIGO/VIRGO class of ground-based interferometers, , , and . The result is . For supermassive binary black holes ( to ) observed by the proposed laser interferometer space antenna (LISA), , , and . The result is .
A full noise analysis using proposed noise curves for the advanced LIGO and for LISA weakens these crude bounds by factors between two and 10 [285, 292, 27, 28]. For example, for the inspiral of two black holes with aligned spins at a distance of 3 Gpc observed by LISA, a bound of 2 1016 km could be placed . Other possibilities include using binary pulsar data to bound modifications of gravitational radiation damping by a massive graviton , and using LISA observations of the phasing of waves from compact white-dwarf binaries, eccentric galactic binaries, and eccentric inspiral binaries [69, 142].
Bounds obtainable from gravitational radiation effects should be compared with the solid bound  derived from solar system dynamics, which limit the presence of a Yukawa modification of Newtonian gravity of the form.
© Max Planck Society and the author(s)