"The Confrontation between General Relativity and Experiment"
Clifford M. Will 

7 Gravitational-Wave Tests of Gravitational Theory

7.1 Gravitational-wave observatories

Soon after the publication of this update, a new method of testing relativistic gravity will be realized, when a worldwide network of upgraded laser interferometric gravitational-wave observatories in the U.S. (LIGO Hanford and LIGO Livingston) and Europe (VIRGO and GEO600) begins regular detection and analysis of gravitational-wave signals from astrophysical sources. Within a few years, they may be joined by an underground cryogenic interferometer (KAGRA) in Japan, and around 2022, by a LIGO-type interferometer in India. These broad-band antennas will have the capability of detecting and measuring the gravitational waveforms from astronomical sources in a frequency band between about 10 Hz (the seismic noise cutoff) and 500 Hz (the photon counting noise cutoff), with a maximum sensitivity to strain at around 100 Hz of h ∼ Δl โˆ•l ∼ 10 −22 (rms), for the kilometer-scale LIGO/VIRGO projects. The most promising source for detection and study of the gravitational wave signal is the “inspiralling compact binary” – a binary system of neutron stars or black holes (or one of each) in the final minutes of a death spiral leading to a violent merger. Such is the fate, for example, of the Hulse–Taylor binary pulsar B1913+16 in about 300 Myr, or the double pulsar J0737–3039 in about 85 Myr. Given the expected sensitivity of the advanced LIGO-Virgo detectors, which could see such sources out to many hundreds of megaparsecs, it has been estimated that from 40 to several hundred annual inspiral events could be detectable. Other sources, such as supernova core collapse events, instabilities in rapidly rotating newborn neutron stars, signals from non-axisymmetric pulsars, and a stochastic background of waves, may be detectable (see [352] for a review).

In addition, plans are being developed for orbiting laser interferometer space antennae, such as DECIGO in Japan and eLISA in Europe. The eLISA system would consist of three spacecraft orbiting the sun in a triangular formation separated from each other by a million kilometers, and would be sensitive primarily in the very low-frequency band between 10−4 and 10 −1 Hz, with peak strain sensitivity of order h ∼ 10−23.

A third approach that focuses on the ultra low-frequency band (nanohertz) is that of Pulsar Timing Arrays (PTA), whereby a network of highly stable millisecond pulsars is monitored in a coherent way using radio telescopes, in hopes of detecting the fluctuations in arrival times induced by passing gravitational waves.

For recent reviews of the status of all these approaches to gravitational-wave detection, see the Proceedings of the 8th Edoardo Amaldi Conference on Gravitational Waves [272].

In addition to opening a new astronomical window, the detailed observation of gravitational waves by such observatories may provide the means to test general relativistic predictions for the polarization and speed of the waves, for gravitational radiation damping and for strong-field gravity. These topics have been thoroughly covered in two recent Living Reviews by Gair et al. [170] for space-based detectors, and by Yunes and Siemens [452] for ground-based detectors. Here we present a brief overview.

7.2 Gravitational-wave amplitude and polarization

7.2.1 General relativity

A generic gravitational wave detector can be modelled as a body of mass M at a distance L from a fiducial laboratory point, connected to the point by a spring of resonant frequency ω0 and quality factor Q. From the equation of geodesic deviation, the infinitesimal displacement ξ of the mass along the line of separation from its equilibrium position satisfies the equation of motion

( ) ¨ξ + 2ω0ξห™+ ω2 ξ = L- F+ (๐œƒ,ฯ•, ψ)¨h+ (t) + F× (๐œƒ,ฯ•,ψ )¨h ×(t) , (124 ) Q 0 2
where F+ (๐œƒ,ฯ•, ψ) and F×(๐œƒ,ฯ•, ψ) are “beam-pattern” factors that depend on the direction of the source (๐œƒ,ฯ•) and on a polarization angle ψ, and h+(t) and h ×(t) are gravitational waveforms corresponding to the two polarizations of the gravitational wave (for pedagogical reviews, see [386, 324*]). In a source coordinate system in which the x –y plane is the plane of the sky and the z-direction points toward the detector, these two modes are given by
1( xx yy ) xy h+(t) = -- hTT(t) − hTT (t) , h× (t) = hTT (t), (125 ) 2
where ij hTT represent transverse-traceless (TT) projections of the calculated waveform of Eq. (84*), given by
[( )( ) ( ) ( )] hij = hkl δik − Nˆi ˆN k δjl − NˆjNˆl − 1- δij − NˆiNˆj δkl − NˆkNˆl , (126 ) TT 2
where ˆ j N is a unit vector pointing toward the detector. The beam pattern factors depend on the orientation and nature of the detector. For a wave approaching along the laboratory z-direction, and for a mass whose location on the x– y plane makes an angle ฯ• with the x-axis, the beam pattern factors are given by F+ = cos2ฯ• and F× = sin 2ฯ•. For a laser interferometer with one arm along the laboratory x-axis, the other along the y-axis, with ξ defined as the differential displacement along the two arms, the beam pattern functions are
1- 2 F+ = 2(1 + cos ๐œƒ) cos2ฯ• cos2ψ − cos๐œƒ sin 2ฯ• sin 2ψ, 1 F× = -(1 + cos2๐œƒ) cos2ฯ• sin 2ψ + cos ๐œƒsin2 ฯ•cos 2ψ. (127 ) 2
Here, we assume that ω0 ≈ 0 in Eq. (124*), corresponding to the essentially free motion of the suspended mirrors in the horizontal direction. For a laser interferometer in which the angle between the arms is χ, the overall response is reduced by sinχ; for a space-based interferometer such as eLISA, โˆ˜ χ = 60.

The waveforms h+ (t) and h ×(t) depend on the nature and evolution of the source. For example, for a binary system in a circular orbit, with an inclination i relative to the plane of the sky, and the x-axis oriented along the major axis of the projected orbit, the quadrupole approximation of Eq. (86*) gives

h+ (t) = − 2โ„ณ--(2πโ„ณfb )2โˆ•3(1 + cos2 i)cos2Φb (t), (128 ) R 2โ„ณ-- 2โˆ•3 h× (t) = − R (2πโ„ณfb ) (2cosi)cos 2Φb (t), (129 )
where ∫t ′ ′ Φb(t) = 2π fb(t )dt is the orbital phase.

7.2.2 Alternative theories of gravity

A generic gravitational wave detector whose scale is small compared to the gravitational wavelength measures the local components of a symmetric 3 × 3 tensor which is composed of the “electric” components of the Riemann curvature tensor, R0i0j, via the equation of geodesic deviation, given, for a pair of freely falling particles by ¨xi = − R0i0jxj, where xi denotes the spatial separation. In general there are six independent components, which can be expressed in terms of polarizations (modes with specific transformation properties under rotations and boosts); for a wave propagating in the z-direction, they can be displayed by the matrix

( AS + A+ A × AV1 ) jk ( ) S = A × AS − A+ AV2 . (130 ) AV1 AV2 AL
Three modes (A +, A ×, and A S) are transverse to the direction of propagation, with two representing quadrupolar deformations and one representing a monopolar transverse “breathing” deformation. Three modes are longitudinal, with one (AL) an axially symmetric stretching mode in the propagation direction, and one quadrupolar mode in each of the two orthogonal planes containing the propagation direction (AV1 and AV2). Figure 10* shows the displacements induced on a ring of freely falling test particles by each of these modes. General relativity predicts only the first two transverse quadrupolar modes (a) and (b) independently of the source; these correspond to the waveforms h+ and h × discussed earlier (note the cos 2ฯ• and sin 2ฯ• dependences of the displacements).
View Image
Figure 10: The six polarization modes for gravitational waves permitted in any metric theory of gravity. Shown is the displacement that each mode induces on a ring of test particles. The wave propagates in the +z direction. There is no displacement out of the plane of the picture. In (a), (b), and (c), the wave propagates out of the plane; in (d), (e), and (f), the wave propagates in the plane. In GR, only (a) and (b) are present; in massless scalar–tensor gravity, (c) may also be present.

Massless scalar–tensor gravitational waves can in addition contain the transverse breathing mode (c). This can be obtained from the physical waveform hα β, which is related to &tidle;hαβ and φ to the required order by

hαβ = &tidle;hαβ + Ψ ηαβ, (131 )
where Ψ = φ − 1. In this case, &tidle; A+ (− ) ∝ h+ (−), while AS ∝ Ψ (see Eqs. (99*), (100*), (101*) and (102*) for the leading contributions to these fields). In massive scalar–tensor theories, the longitudinal mode (d) can also be present, but is suppressed relative to (c) by a factor (λโˆ•λC )2, where λ is the wavelength of the radiation, and λC is the Compton wavelength of the massive scalar.

More general metric theories predict additional longitudinal modes, up to the full complement of six (TEGP 10.2 [420*]). For example, Einstein-Æther theory generically predicts all six modes [205].

A suitable array of gravitational antennas could delineate or limit the number of modes present in a given wave. The strategy depends on whether or not the source direction is known. In general there are eight unknowns (six polarizations and two direction cosines), but only six measurables (R0i0j). If the direction can be established by either association of the waves with optical or other observations, or by time-of-flight measurements between separated detectors, then six suitably oriented detectors suffice to determine all six components. If the direction cannot be established, then the system is underdetermined, and no unique solution can be found. However, if one assumes that only transverse waves are present, then there are only three unknowns if the source direction is known, or five unknowns otherwise. Then the corresponding number (three or five) of detectors can determine the polarization. If distinct evidence were found of any mode other than the two transverse quadrupolar modes of GR, the result would be disastrous for GR. On the other hand, the absence of a breathing mode would not necessarily rule out scalar–tensor gravity, because the strength of that mode depends on the nature of the source.

For laser interferometers, the signal controlling the laser phase output can be written in the form

1-( j k j k) jk S (t) = 2 e1e1 − e2e2 S , (132 )
where e1 and e2 are unit vectors directed along the two arms of the interferometer. The final result is
S(t) = FSAS + FLAL + FV1AV1 + FV2AV2 + F+A+ + F×A ×, (133 )
where the angular pattern functions FA (๐œƒ,ฯ•,ψ ) are given by
1- 2 FS = − 2 sin ๐œƒcos 2ฯ•, 1 FL = --sin2๐œƒ cos2ฯ•, 2 FV1 = − sin๐œƒ (cos ๐œƒcos 2ฯ•cos ψ − sin2ฯ• sinψ ), FV2 = − sin๐œƒ (cos ๐œƒcos 2ฯ•sin ψ + sin 2ฯ• cosψ ), 1 F+ = --(1 + cos2 ๐œƒ)cos2 ฯ•cos 2ψ − cos๐œƒ sin 2ฯ• sin 2ψ, 2 F = 1-(1 + cos2 ๐œƒ)cos2 ฯ•sin2 ψ + cos๐œƒ sin 2ฯ• cos2ψ, (134 ) × 2
(see [324] for detailed derivations and definitions). Note that the scalar and longitudinal pattern functions are degenerate and thus no array of laser interferometers can measure these two modes separately.

Some of the details of implementing such polarization observations have been worked out for arrays of resonant cylindrical, disk-shaped, spherical, and truncated icosahedral detectors (TEGP 10.2 [420*], for recent reviews see [256, 403]). Early work to assess whether the ground-based or space-based laser interferometers (or combinations of the two types) could perform interesting polarization measurements was carried out in [404, 70, 267, 171, 411]; for a recent detailed analysis see [301]. Unfortunately for this purpose, the two LIGO observatories (in Washington and Louisiana states, respectively) have been constructed to have their respective arms as parallel as possible, apart from the curvature of the Earth; while this maximizes the joint sensitivity of the two detectors to gravitational waves, it minimizes their ability to detect two modes of polarization. In this regard the addition of Virgo, and the future KAGRA and LIGO-India systems will be crucial to polarization measurements. By combining signals from various interferometers into a kind of “null channel” one can test for the existence of modes beyond the + and × modes in a model independent manner [78]. The capability of space-based interferometers to measure the polarization modes was assessed in detail in [388, 302]. For pulsar timing arrays, see [245, 14, 74].

7.3 Gravitational-wave phase evolution

7.3.1 General relativity

In the binary pulsar, a test of GR was made possible by measuring at least three relativistic effects that depended upon only two unknown masses. The evolution of the orbital phase under the damping effect of gravitational radiation played a crucial role. Another situation in which measurement of orbital phase can lead to tests of GR is that of the inspiralling compact binary system. The key differences are that here gravitational radiation itself is the detected signal, rather than radio pulses, and the phase evolution alone carries all the information. In the binary pulsar, the first derivative of the binary frequency fห™b was measured; here the full nonlinear variation of fb as a function of time is measured.

Broad-band laser interferometers are especially sensitive to the phase evolution of the gravitational waves, which carry the information about the orbital phase evolution. The analysis of gravitational wave data from such sources will involve some form of matched filtering of the noisy detector output against an ensemble of theoretical “template” waveforms which depend on the intrinsic parameters of the inspiralling binary, such as the component masses, spins, and so on, and on its inspiral evolution. How accurate must a template be in order to “match” the waveform from a given source (where by a match we mean maximizing the cross-correlation or the signal-to-noise ratio)? In the total accumulated phase of the wave detected in the sensitive bandwidth, the template must match the signal to a fraction of a cycle. For two inspiralling neutron stars detected by the advanced LIGO/Virgo systems, around 16 000 cycles should be detected during the final few minutes of inspiral; this implies a phasing accuracy of 10− 5 or better. Since v ∼ 1โˆ•10 during the late inspiral, this means that correction terms in the phasing at the level of v5 or higher are needed. More formal analyses confirm this intuition [99, 153, 97, 323].

Because it is a slow-motion system (−3 v ∼ 10), the binary pulsar is sensitive only to the lowest-order effects of gravitational radiation as predicted by the quadrupole formula. Nevertheless, the first correction terms of order v and v2 to the quadrupole formula were calculated as early as 1976 [405] (see TEGP 10.3 [420*]).

But for laser interferometric observations of gravitational waves, the bottom line is that, in order to measure the astrophysical parameters of the source and to test the properties of the gravitational waves, it is necessary to derive the gravitational waveform and the resulting radiation back-reaction on the orbit phasing at least to 3PN order beyond the quadrupole approximation.

For the special case of non-spinning bodies moving on quasi-circular orbits (i.e., circular apart from a slow inspiral), the evolution of the gravitational wave frequency f = 2fb through 2PN order has the form

[ ( ) 96π 2 5โˆ•3 743 11 2โˆ•3 ห™f = -5--f (πโ„ณf ) 1 − 336-+ 4-η (πmf ) + 4π (πmf ) ( ) ] 34103- 13661- 59- 2 4โˆ•3 5โˆ•3 + 18144 + 2016 η + 18 η (πmf ) + ๐’ช[(πmf ) ] , (135 )
where 2 η = m1m2 โˆ•m. The first term is the quadrupole contribution (compare Eq. (88*)), the second term is the 1PN contribution, the third term, with the coefficient 4π, is the “tail” contribution, and the fourth term is the 2PN contribution. Two decades of intensive work by many groups have led to the development of waveforms in GR that are accurate to 3.5PN order, and for some specific effects, such as those related to spin, to 4.5PN order (see [51] for a thorough review).

Similar expressions can be derived for the loss of angular momentum and linear momentum. Expressions for non-circular orbits have also been derived [175, 107]. These losses react back on the orbit to circularize it and cause it to inspiral. The result is that the orbital phase (and consequently the gravitational wave phase) evolves non-linearly with time. It is the sensitivity of the broad-band laser interferometric detectors to phase that makes the higher-order contributions to dfโˆ•dt so observationally relevant.

If the coefficients of each of the powers of f in Eq. (135*) can be measured, then one again obtains more than two constraints on the two unknowns m1 and m2, leading to the possibility to test GR. For example, Blanchet and Sathyaprakash [59, 60] have shown that, by observing a source with a sufficiently strong signal, an interesting test of the 4π coefficient of the “tail” term could be performed (but see [22] for a more sophisticated analysis).

Another possibility involves gravitational waves from a small mass orbiting and inspiralling into a (possibly supermassive) spinning black hole. A general non-circular, non-equatorial orbit will precess around the hole, both in periastron and in orbital plane, leading to a complex gravitational waveform that carries information about the non-spherical, strong-field spacetime around the hole. According to GR, this spacetime must be the Kerr spacetime of a rotating black hole, uniquely specified by its mass and angular momentum, and consequently, observation of the waves could test this fundamental hypothesis of GR [345, 322].

7.3.2 Alternative theories of gravity

In general, alternative theories of gravity will predict rather different phase evolution from that of GR, notably via the addition of dipole gravitational radiation. For example, the dipole gravitational radiation predicted by scalar–tensor theories modifies the gravitational radiation back-reaction, and thereby the phase evolution. Including only the leading 0PN and –1PN (dipole) contributions, one obtains,

fห™= 96π-f2(πα โ„ณf )5โˆ•3κ1-[1 + b(πmf )− 2โˆ•3] , (136 ) 5 12
where โ„ณ = η3โˆ•5m, and b is the coefficient of the dipole term, given to first order in ζ by −5โˆ•3 2 b = (5โˆ•24 )ζ α ๐’ฎ, where κ1 is given by Eq. (104*), −1โˆ•2 ๐’ฎ = α (s1 − s2) and ζ = 1โˆ• (4 + 2ω0 ). Double neutron star systems are not promising because the small range of masses available near 1.4M โŠ™ results in suppression of dipole radiation by symmetry. For black holes, s = 0.5 identically, consequently double black hole systems turn out to be observationally identical in the two theories. Thus mixed systems involving a neutron star and a black hole are preferred. However, a number of analyses of the capabilities of both ground-based and space-based (eLISA) observatories have shown that observing waves from neutron-star–black-hole inspirals is not likely to bound scalar–tensor gravity at a level competitive with the Cassini bound, with future solar-system improvements, or with binary pulsar observations [422, 236, 106, 353, 433*, 41*, 42*, 445]. A possible exception is DECIGO/BBO, a proposed space gravitational-wave observatory with peak sensitivity between the eLISA and LIGO/Virgo bands; observations of inspirals of neutron stars onto hypothetical intermediate mass (4 ∼ 10 M โŠ™) black holes could improve upon the Cassini bound by several orders of magnitude [446].

These considerations suggest that it might be fruitful to attempt to parametrize the phasing formulae in a manner reminiscent of the PPN framework for post-Newtonian gravity. A number of approaches along this line have been developed, including the parametrized post-Einsteinian (PPE) framework [451, 347], a Bayesian parametrized approach [250], and a parametrization based on the post-Newtonian expansions discussed above [288]. The discovery of relationships between the moment of inertia, the gravitational Love number, and the quadrupole moment of neutron stars (“I-Love-Q” relations) in general relativity has opened the possibility of testing theories using gravitational waves in a manner that is relatively free of contamination from the neutron-star equation of state [448, 447].

7.4 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 [420*]).

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,

v2 m2 c4 -2g= 1 − --g2-, (137 ) c E
where mg and E are the graviton rest mass and energy, respectively.

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 or a prompt gamma-ray burst. For a source at a distance D, the resulting value of the difference 1 − vgโˆ•c is

v ( 200 Mpc ) ( Δt ) 1 − -g-= 5 × 10−17 --------- --- , (138 ) c D 1 s
where Δt ≡ Δta − (1 + Z)Δte is the “time difference”, where Δta and Δte 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 − vgโˆ•c.

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

( )1โˆ•2( )1 โˆ•2 12 ---D-----100-Hz- --1-- λg > 3 × 10 km 200 Mpc f fΔt . (139 )

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 [423*]. 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” f Δt in Eq. (139*) can be measured to an accuracy 1โˆ•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 ≈ 1โˆ•10. The result is 13 λg > 10 km. For supermassive binary black holes (4 10 to 7 10 M โŠ™) observed by the proposed laser interferometer space antenna (LISA), D ≈ 3 Gpc, f ≈ 10− 3 Hz, and fΔt ∼ ρ−1 ≈ 1โˆ•1000. The result is λg > 1017 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 [423, 433, 41*, 42, 23, 377, 445]. For example, for the inspiral of two 106M โŠ™ black holes with aligned spins at a distance of 3 Gpc observed by LISA, a bound of 2 × 1016 km could be placed [41]. Other possibilities include using binary pulsar data to bound modifications of gravitational radiation damping by a massive graviton [154], using LISA observations of the phasing of waves from compact white-dwarf binaries, eccentric galactic binaries, and eccentric inspiral binaries [98, 209], using pulsar timing arrays [244], and using DECIGO/BBO to observe neutron-star intermediate-mass black-hole inspirals [446].

Bounds obtainable from gravitational radiation effects should be compared with the solid bound λg > 2.8 × 1012 km [381] derived from solar system dynamics, which limit the presence of a Yukawa modification of Newtonian gravity of the form

V(r) = GM---exp (− rโˆ•λ ), (140 ) r g
and with the model-dependent bound λ > 6 × 1019 km g from consideration of galactic and cluster dynamics [401].

Mirshekari et al. [287] studied bounds that could be placed on more general graviton dispersion relations that could emerge from alternative theories with Lorentz violation, in which the effective propagation speed is given by

v2g- m2gc4 α− 2 c2 = 1 − E2 − ๐”ธE , (141 )
where ๐”ธ and α are parameters that depend on the theory.

  Go to previous page Scroll to top Go to next page