Observations of stellar mass black holes inspiraling into SMBHs, the extreme mass ratio inspirals (EMRIs), have an even greater potential for testing general relativity. The stellar mass black hole spends thousands of precessing (both of periastron and the orbital plane) orbits along highly-eccentric trajectories and slowly inspirals into the larger black hole. The emitted gravitational radiation literally carries the signature of the spacetime geometry around the central object. So fitting the orbit to theoretical templates could reveal small deviations of this geometry from that of Kerr. For example, if we know (from fitting the waveform) the mass and spin of the central black hole, then all its higher multipole moments are determined. If we can measure some of these and they deviate from Kerr, then that would indicate that either the central object is not a black hole or that general relativity needs to be corrected [178, 70].

Current tests of general relativity rely on experiments in the solar system (using the sun’s gravitational field) and observations of binary pulsars. In dimensionless units, the gravitational potential on the surface of the sun is about one part in a million and even in a binary pulsar the potential that each neutron star experiences due to its companion is no more than one part in ten thousand. These are mildly relativistic fields, with the corresponding escape velocity being as large as a thousandth and a hundredth that of light, respectively.

Thus, gravitational fields in the solar system or in a binary pulsar are still weak by comparison to the largest possible values. Indeed, close to the event horizon of a black hole, gravitational fields can get as strong as they can ever get, with the dimensionless potential being of order unity and the escape velocity equal to that of the speed of light. Although general relativity has been found to be consistent with experiments in the solar system and observations of binary pulsars, phenomena close to the event horizons of black holes would be a great challenge to the theory. It would be very exciting to test Einstein’s gravity under such circumstances.

The large SNR that is expected from SMBH binaries makes it possible to test Einstein’s theory under extreme conditions of gravity [47, 48]. To see how one might test the post-Newtonian structure of Einstein’s theory, let us consider the waveform from a binary in the frequency domain. Since an inspiral wave’s frequency changes rather slowly (adiabatic evolution) it is possible to apply a stationary phase approximation to compute the Fourier transform of the waveform given in Equation (124):

with the Fourier amplitude and phase given by Here is the symmetric mass ratio defined before (see Equation 31), is a function of the various angles, as in Equation (124), and and are the fiducial epoch of merger and the phase of the signal at that epoch, respectively. The coefficients in the PN expansion of the Fourier phase are given by These are the PN coefficients in Einstein’s theory; in an alternative theory of gravity they will be different. In Einstein’s theory the coefficients depend only on the two mass parameters, the total mass and symmetric mass ratio . One of the tests we will discuss below concerns the consistency of the various coefficients. Note, in particular, that in Einstein’s gravity the 0.5 PN term is absent, i.e., the coefficient of the term is zero. Even with the very first observations of inspiral events, it should be possible to test if this is really so.Figure 16 shows one such test that is possible with SMBH binaries [47, 48]. The observation of these systems in LISA makes it possible to measure the parameters associated with different physical effects. For example, the rate at which a signal chirps (i.e., the rate at which its frequency changes) depends on the binary’s chirpmass. Given the chirpmass, the length of the signal depends on the system’s symmetric mass ratio (the ratio of reduced mass to total mass). Another example would be the scattering of gravitational waves off the curved spacetime geometry of the binary, producing the tail effect in the emitted signal, which is determined principally by the system’s total mass [87, 86]. Similarly, spin-orbit interaction, spin-spin coupling, etc. depend on other combinations of the masses.

The binary will be seen with a high SNR, which means that we can measure the mass parameters associated with many of these physical effects. If each parameter is known precisely, we can draw a curve corresponding to it in the space of masses. However, our observations are inevitably subject to statistical (and possibly systematic) errors. Therefore, each parameter corresponds to a region in the parameter space (shown in Figure 16 for the statistical errors only). If Einstein’s theory of gravitation is correct, the regions corresponding to the different parameters must all have at least one common region, a region that contains the true parameters of the binary. This is because Einstein’s theory, or an alternative, has to be used to project the observed data onto the space of masses. If the region corresponding to one or more of these parameters does not overlap with the common region of the rest of the parameters, then Einstein’s theory, or its alternative, is in trouble.

In Brans–Dicke theory the system is expected to emit dipole radiation and the PN series would begin an order earlier than in Einstein’s theory. In the notation introduced above we would have an term, which has the form [75, 76]

Here is the the difference in the scalar charges of the two bodies and is the Brans–Dicke parameter. Although this term is formally two orders lower than the lowest-order quadrupole term of Einstein’s gravity (i.e., it is order smaller), numerically its effect will be far smaller than the quadrupole term because of the rather large bound on . Nevertheless, its importance lies in the fact that there is now a new parameter on which the phase depends. Berti, Buonanno and Will conclude that LISA observations of massive black-hole binaries will enable scientists to set limits on .A massive graviton theory would also alter the phase. The dominant effect is at 1 PN order and would change the coefficient to

where is the symmetric mass ratio. This term alters the time of arrival of waves of different frequencies, causing a dispersion, and a corresponding modulation, in the wave’s phase, depending on the Compton wavelength and the distance to the binary. Hence, by tracking the phase of the inspiral waves, one can bound the graviton’s mass. Will [393] finds that one can bound the mass to 1.7 × 10In Section 3 we pointed out that LISA should be able to see many hundreds of signals emitted by compact objects – black holes, neutron stars, even white dwarfs – orbiting around and being captured by massive black holes in the centers of galaxies. But for LISA to reach its full potential, we must model the orbits and their emitted radiation accurately. Since the wave trains may be many hundreds or thousands of cycles long in the LISA band, the challenge of constructing template waveforms that remain accurate to within about one radian over the whole evolution is significant.

The range of mass ratios is also wide. LISA’s central black holes might have masses between and
. Inspiraling neutron stars and white dwarfs might have masses between and .
Inspiraling stellar-population black holes might be in the range of , while intermediate-mass
black holes formed by the first generation of stars (Population III stars) might have masses
around or even . So the mass ratios might be anything in the range 10^{–7} to
1.

The techniques that must be used to compute these signals depend on the mass ratio. Ratios near one
are treated by post-Newtonian methods until the objects are so close that only numerical relativity can
follow their subsequent evolution. For ratios below 10^{–4} (a dividing line that is rather very uncertain and
that depends on the bandwidth being used to observe the system, i.e., on how long the approximation must
be valid for), systems are treated by fully-relativistic perturbation theory, expanding in the mass
ratio. Intermediate mass ratios have not been studied in much detail yet; they can probably
be treated by post-Newtonian methods up to a certain point, but it is not yet clear whether
their final stages can be computed accurately by either numerical relativity or perturbation
theory.

Post-Newtonian methods have been extensively discussed above. The basics of perturbation theory underlying this problem are treated in two Living Reviews [294, 322]. Once one has sufficiently good waveform templates, there remain the challenge of finding weak signals in LISA’s noise. This depends on a number of factors, including the rates of sources. A recent study by a number of specialists [170] has concluded that the event rate is high enough and the detection methods robust enough for us to be very optimistic that LISA will detect hundreds of these sources. In fact, the opposite problem might materialize: LISA might find it has a confusion problem for the detection of these sources, as for the galactic binaries. Recent estimates of the magnitude of the problem [68] suggest that LISA’s noise may at worst be raised effectively by a factor of two, but in return one gets a large number of sources of this kind.

It seems inevitable that general relativity’s description of nature will one day yield to a quantum-based description, involving uncertainties in geometry and probabilities in the outcome of gravitational observations. This is one of the most active areas of research in fundamental physics today, and there are many speculations about how quantum effects might show up in gravitational wave observations.

The simplest idea might be to try to find evidence for “gravitons” directly in gravitational waves, by analogy with the way that astronomers count individual photons from astronomical sources. But this seems doomed to failure. The waves that we can observe have very low frequency, so the energy of each graviton is extremely small. And the total energy flux of the waves is, as we have seen, enormous. So the number of gravitons in a detectable gravitational wave is far more than the number of photons in the light from a distant quasar.

Quantum gravity might involve new gravity-like fields, whose effects might be seen indirectly in the inspiral signals of black holes or neutron stars, as we have noted above. String theory might lead to the production of cosmic strings, which might be observed through their gravitational wave emission [143]. If our universe is just a 4-dimensional subspace of a large-scale 10 or 11-dimensional space, then dynamics in the larger space might produce gravitational effects in our space, and in particular gravitational waves [306].

It might be possible to observe the quantum indeterminacy of geometry directly using gravitational wave detectors, if Hogan’s principle of holographic indeterminacy is valid [195]. Hogan speculates that quantum geometry might be manifested by an uncertainty in the position of a beam splitter, and that this could be the explanation for an unexpectedly large amount of noise at low frequencies in the GEO600 detector. In this connection it is interesting to construct, from fundamental constants alone, a quantity with the dimensions of amplitude spectral noise density . This has units of , so one can define the “Planck noise power” . Then the amplitude noise is . This is comparable to or larger than the instrumental noise in current interferometric gravitational wave detectors, as shown in Figure 5. This in itself does not mean that Planckian noise will show up in gravitational wave detectors, but Hogan argues that the particular design of GEO600 might indeed make it subject to this noise more strongly than other large interferometers.

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