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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 η, gravitational waves follow null geodesics of η, while light follows null geodesics of g (TEGP 10.1 [281Jump To The Next Citation Point]).

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 vg would be given by, in a local inertial frame,

v2g m2gc4 -2-= 1 − ---2-, (99 ) c E
where mg and E are the graviton rest mass and energy, respectively.

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 [263299]. 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 [265Jump To The Next Citation Point] 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 [9166].

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 1 − vg∕c is

( ) ( ) 1 − vg-= 5 × 10−17 200-Mpc-- Δt- , (100 ) c D 1 s
where Δt ≡ Δt − (1 + Z)Δt a e is the “time difference”, where Δt a and Δt e are the differences in arrival time and emission time of the two signals, respectively, and Z is the redshift of the source. In many cases, Δte is unknown, so that the best one can do is employ an upper bound on Δte based on observation or modelling. The result will then be a bound on 1 − vg∕c.

For a massive graviton, if the frequency of the gravitational waves is such that hf ≫ mgc2, where h is Planck’s constant, then 1 2 vg∕c ≈ 1 − 2(c∕λgf ), where λg = h∕mgc is the graviton Compton wavelength, and the bound on 1 − vg∕c can be converted to a bound on λg, given by

( D 100 Hz )1∕2 ( 1 )1∕2 λg > 3 × 1012 km ---------------- ----- . (101 ) 200 Mpc f f Δt

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 [285Jump To The Next Citation Point]. 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 Δta 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 Δte can be determined accurately. Roughly speaking, the “phase interval” fΔt in Equation (101View Equation) can be measured to an accuracy 1∕ρ, where ρ is the signal-to-noise ratio.

Thus one can estimate the bounds on λg 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, D ≈ 200 Mpc, f ≈ 100 Hz, and −1 fΔt ∼ ρ ≈ 1∕10. The result is λg > 1013 km. For supermassive binary black holes (104 to 107 M ⊙) observed by the proposed laser interferometer space antenna (LISA), D ≈ 3 Gpc, f ≈ 10− 3 Hz, and fΔt ∼ ρ−1 ≈ 1∕1000. The result is λ > 1017 km g.

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 [28529227Jump To The Next Citation Point28]. For example, for the inspiral of two 106 M ⊙ black holes with aligned spins at a distance of 3 Gpc observed by LISA, a bound of 2 × 1016 km 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 [69142].

Bounds obtainable from gravitational radiation effects should be compared with the solid bound λg > 2.8 × 1012 km [250] derived from solar system dynamics, which limit the presence of a Yukawa modification of Newtonian gravity of the form

GM--- V(r) = r exp (− r∕λg ), (102 )
and with the model-dependent bound 19 λg > 6 × 10 km from consideration of galactic and cluster dynamics [265].
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