The basic equation we need is for the effect of a plane linear gravitational wave on a beam of light.
Suppose the angle between the direction of the beam and the direction of the plane wave is .
We imagine a very simple experiment in which the light beam originates at a clock, whose
proper time is called
, and is received by a clock, whose proper time is
. The beam and
gravitational-wave travel directions determine a plane, and we denote the polarization component
of the gravitational wave that acts in this plane by
, as measured at the location of
the originating clock. The proper distance between the clocks, in the absence of the wave, is
. If the originating clock puts timestamps onto the light beam, then the receiving clock
can measure the rate of arrival of the timestamps. If there is no gravitational wave, and if the
clocks are ideal, then the rate will be constant, which can be normalized to unity. The effect
of the gravitational wave is to change the arrival rate as a function of the emission rate by
In order to use such an arrangement to detect gravitational waves, one needs two very stable clocks. The
best clocks today are stable to a few parts in 1016 [42], which implies that the minimum amplitude of
gravitational waves that could be detected by such a two-clock experiment is
. However, this
equation is also fundamental to the detection of gravitational waves by pulsar timing, in which the
originating ‘clock’ is a pulsar. By correlating many pulsar signals, one can beat down the single-pulsar noise.
This is described below in Section 4.4.2.
An arrangement that uses only one clock is one that sends a beam out to a receiver, which then reflects
or retransmits (transponds) the beam back to the sender. The sender has the clock, which measures
variations in the round-trip time. This method has been used with interplanetary spacecraft, which has the
advantage that the only clock is on the ground, which can be made more stable than one carried in a
spacecraft (see Section 4.4.1). For the same arrangement as above, the return time varies at the
rate
But the sensitivity of such a one-path system as a gravitational wave detector is still limited by the stability of the clock. For that reason, interferometers have become the most sensitive beam detectors: effectively one arm of the interferometer becomes the ‘clock’, or at least the time standard, that variations in the other arm are compared to. Of course, if both arms are affected by a gravitational wave in the same way, then the interferometer will not see the wave. But this happens only in very special geometries. For most wave arrival directions and polarizations, the arms are affected differently, and a simple interferometer measures the difference between the round-trip travel time variations in the two arms. For the triangular space array LISA, the measured signal is somewhat more complex (see Section 4.4.3 below), but it still preserves the principle that the time reference for one arm is a combination of the others.
Ground-based interferometers are the most sensitive detectors operating today, and are likely to make the
first direct detections [199]. The largest detectors operating today are the LIGO detectors [304], two of
which have arm lengths of 4 km. This is much smaller than the wavelength of the gravitational wave, so the
interaction of one arm with a gravitational wave can be well approximated by the small-
approximation
to Equation (43
), namely
With these definitions, the wave amplitude is the one that has
and
as the axes
of its ellipse. The full wave amplitude is described, as in Equation (6
), by the wave tensor
. The interferometer responds to the difference between these times,
. By analogy with the wave tensor, we define the detector tensor by [147]
It is conventional to re-express this measurable in terms of the stretching of the arms of the interferometer. Within our approximation that the arms are shorter than a wavelength, this makes sense: it is possible to define a local inertial coordinate system that covers the entire interferometer, and within this coordinate patch (where light travels at speed 1) time differences measure proper length differences. The differential return time is twice the differential length change of the arms:
For a bar detector of length lying along the director
, the detector tensor is
When dealing with observations by more than one detector, it is not convenient to tie the alignment of
the basis vectors in the sky plane with those in the detector frame, as we have done in the
left-hand panel of Figure 3, since the detectors will have different orientations. Instead it will
usually be more convenient to choose polarization tensors in the sky plane according to some
universal reference, e.g., using a convenient astronomical reference frame. The right-hand panel of
Figure 3
shows the general situation, where the basis vectors
and
are rotated by an angle
from the basis used in the left-hand panel. The polarization tensors on this new basis,
Then one can write Equation (51) as
These are the antenna-pattern response functions of the interferometer to the two polarizations
of the wave as defined in the sky plane [362]. If one wants the antenna pattern referred to
the detector’s own axes, then one just sets
. If the arms of the interferometer are not
perpendicular to each other, then one defines the detector-plane coordinates
and
in
such a way that the bisector of the angle between the arms lies along the bisector of the angle
between the coordinate axes [337]. Note that the maximum value of either
or
is
1.
The corresponding antenna-pattern functions of a bar detector whose longitudinal axis is aligned along
the direction, are
Any one detector cannot directly measure both independent polarizations of a gravitational wave at the same time, but responds rather to a linear combination of the two that depends on the geometry of the detector and source direction. If the wave lasts only a short time, then the responses of three widely-separated detectors, together with two independent differences in arrival times among them, are, in principle, sufficient to fully reconstruct the source location and gravitational wave polarization. A long-lived wave will change its location in the antenna pattern as the detector moves, and it will also be frequency modulated by the motion of the detector; these effects are in principle sufficient to determine the location of the source and the polarization of the wave.
Since the polarization angle of an incoming gravitational wave would generally be expected to be
unrelated to its direction of arrival, depending rather on the internal orientations in the source, it is useful
to characterize the directional sensitivity of a detector by averaging over the polarization angle . If the
wave has a given amplitude
and is linearly polarized, then, if we are interested in a single detector’s
response, it is always possible to align the polarization angle
in the sky plane with that of the wave, so
that the wave has pure
-polarization. Then the rms response function of the detector is
The polarization amplitudes of the radiation from an inspiraling binary, a rotating neutron star, or a ringing black hole, take a simple form as follows:
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where is an overall (possibly time-dependent) amplitude,
is the signal’s phase and
is the
angle made by the characteristic direction in the source (e.g., the orbital or the spin angular
momentum) with the line of sight. In this case, the response takes a particularly simple form:
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Note that , just as
, takes values in the range [0, 1]. In this case the average response has to be
worked out by considering all possible sky locations, polarizations (which drops out of the calculation) and
source orientations. More precisely, the rms response is
The right-hand panel of Figure 4 shows the percentage area of the sky over which the antenna pattern
of an interferometric detector is larger than a certain fraction
of the peak value. The response is better
than the rms value over 40% of the sky, implying that gravitational wave detectors are fairly
omni-directional. In comparison, the sky coverage of most conventional telescopes (radio, infrared, optical,
etc.) is a tiny fraction of the area of the sky.
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