## 6.4 Speed of gravitational waves

According to GR, in the limit in which the wavelength of gravitational waves is small compared to the radius of curvature of the background spacetime, the waves propagate along null geodesics of the background spacetime,i.e. they have the same speed c as light (in this section, we do not set c =1). In other theories, the speed could differ from c because of coupling of gravitation to ``background'' gravitational fields. For example, in the Rosen bimetric theory with a flat background metric , gravitational waves follow null geodesics of , while light follows null geodesics of (TEGP 10.1 [147]).

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,

where and E are the graviton rest mass and energy, respectively. For a recent review of the idea of a massive graviton along with a model theory, see [132].

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 D, the resulting value of the difference is

where is the ``time difference'', where and are the differences in arrival time and emission time, respectively, of the two signals, and Z is the redshift of the source. In many cases, is unknown, so that the best one can do is employ an upper bound on based on observation or modelling. The result will then be a bound on .

For a massive graviton, if the frequency of the gravitational waves is such that , where h 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 [151]. 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 Eq. (76) 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, 200 Mpc, 100 Hz, and . The result is  km. For supermassive binary black holes ( to ) observed by the proposed laser interferometer space antenna (LISA), 3 Gpc,  Hz, and . The result is  km.

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. These potential bounds can be compared with the solid bound , [122] derived from solar system dynamics, which limit the presence of a Yukawa modification of Newtonian gravity of the form

and with the model-dependent bound from consideration of galactic and cluster dynamics [132].

 The Confrontation between General Relativity and Experiment Clifford M. Will http://www.livingreviews.org/lrr-2001-4 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de