Another way in which the speed of gravitational waves could differ from 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 [265] 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 [285]. 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
) 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
observed by LISA, a bound of
could be placed [27]. Other possibilities include using binary pulsar data to bound
modifications of gravitational radiation damping by a massive graviton [106], 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
[250] derived from solar system dynamics, which limit the presence of a Yukawa
modification of Newtonian gravity of the form
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