4.2 Principles of the operation of beam detectors

Interferometers use laser light to measure changes in the difference between the lengths of two perpendicular (or nearly perpendicular) arms. Typically, the arm lengths respond differently to a given gravitational wave, so an interferometer is a natural instrument to measure gravitational waves. But other detectors also use electromagnetic radiation, for example, ranging to spacecraft in the solar system and even pulsar timing.

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 πœƒ. We imagine a very simple experiment in which the light beam originates at a clock, whose proper time is called t, and is received by a clock, whose proper time is tf. 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 h (t) +, as measured at the location of the originating clock. The proper distance between the clocks, in the absence of the wave, is L. 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

dtf-= 1 + 1-(1 + cos πœƒ){h [t + (1 − cosπœƒ)L ] − h (t)} . (42 ) dt 2 + +
This is very simple: the beam of light leaves the emitter at the time when the gravitational wave of phase t passes the emitter, and it reaches the receiver at the time when the gravitational wave of phase t + (1 − cosπœƒ)L is passing the receiver. So in the plane wave case, only the amplitudes of the wave at the emitting and receiving events affect the time delay.

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 [42Jump To The Next Citation Point], which implies that the minimum amplitude of gravitational waves that could be detected by such a two-clock experiment is h ∼ 10 −15. 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 treturn varies at the rate

dt 1 --return-= 1 + -{(1 − cos πœƒ)h+(t + 2L) − (1 + cosπœƒ)h+ (t) dt 2 + 2cos πœƒh+[t + L (1 − cosπœƒ)]}. (43 )
This is known as the three-term relation, the third term being the wave strength at the time the beam returns back to the sender.

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.

4.2.1 The response of a ground-based interferometer

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 [304Jump To The Next Citation Point], 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-L approximation to Equation (43View Equation), namely

dtreturn 2 -------= 1 + sin πœƒLhΛ™+ (t). (44 ) dt
(See [71] for first corrections to the short-arm approximation.) To analyze the full detector, where the second arm will normally point out of the plane we have been working in up till now, it is helpful to go over to a tensorial expression, independent of special coordinate orientations. The gravitational wave will act in the plane transverse to the propagation direction; let us call that direction ˆN and let us set up radiation basis vectors R ˆex and R ˆey in the transverse plane, such that R ˆex lies in the plane formed by the wave propagation direction and the arm of our gravitational wave sensor, which lies along the x-axis of the detector plane, whose unit vector is ˆex. (For a picture of this geometry, see the left-hand panel of Figure 3View Image, where for the moment we are ignoring the y-arm of the detector shown there.)

With these definitions, the wave amplitude h+ is the one that has R ˆex and R ˆey as the axes of its ellipse. The full wave amplitude is described, as in Equation (6View Equation), by the wave tensor

h(t) = h+ (t)e+ + h ×(t)e×, (45 )
where e+ and e× are the polarization tensors associated with these basis vectors (compare Equation (4View Equation)):
R R R R R R R R e+ = (ˆex ⊗ ˆex − ˆey ⊗ ˆey ), e × = (ˆex ⊗ ˆey + ˆey ⊗ ˆex ). (46 )
The unique way of expressing Equation (44View Equation) in terms of h is
( dtreturn) ------- = 1 + Lˆex ⋅ Λ™h ⋅ ˆex. (47 ) dt x−arm
This does not depend on any special orientation of the arm relative to the wave direction, and does not depend on the basis we chose in the transverse plane, so we can use it as well for the second arm of the interferometer, no matter what its orientation. Let us assume it lies along the unit vector by ˆey. (We do not, in fact, have to assume that the two arms are perpendicular to each other, but it simplifies the diagram a little.) The return-time derivative along the second arm is then given by
( ) dtreturn Λ™ dt = 1 + Lˆey ⋅h ⋅ ˆey x−arm

. The interferometer responds to the difference between these times,

( ) ( ) ( ) dδtreturn dtreturn dtreturn ---dt--- = --dt--- − ---dt-- = Lˆex ⋅ Λ™h ⋅ ˆex − Lˆey ⋅ Λ™h ⋅ ˆey x− arm y−arm

. By analogy with the wave tensor, we define the detector tensor d by [147]

d = L (eˆ ⊗ ˆe − ˆe ⊗ ˆe ). (48 ) x x y y
(If the arms are not perpendicular this expression would still give the correct tensor if the unit vectors lie along the actual arm directions.) Then we can express the differential return time rate in the simple invariant form
( ) dδtreturn --dt---- = d :hΛ™, (49 )
where the notation d : h ≡ dlmhlm denotes the Euclidean scalar product of the tensors d and h. Equation (49View Equation) can be integrated over time to give the instantaneous path-length (or time-delay, or phase) difference between the arms, as measured by the central observer’s proper time clock:
δtreturn(t) = d : h. (50 )
This is a robust and compact expression for the response of any interferometer to any wave in the long-wavelength (short-arm) limit. Its dependence on the wave direction is called its antenna pattern.

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:

δL (t) = 1d : h. (51 ) 2

For a bar detector of length L lying along the director ˆa, the detector tensor is

d = L ˆa ⊗ ˆa, (52 )
although one must be careful that the change in proper length of a bar is not simply given by Equation (51View Equation), because of the restoring forces in the bar.

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 3View Image, 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 3View Image shows the general situation, where the basis vectors ˆα and ˆβ are rotated by an angle ψ from the basis used in the left-hand panel. The polarization tensors on this new basis,

πœ–+ = (ˆα ⊗ ˆα − βˆ⊗ ˆβ), πœ–× = (αˆ⊗ ˆβ + ˆβ ⊗ ˆα), (53 )
are found by the following transformation from the previous ones:
πœ–+ = e+ cos2ψ + e×sin 2ψ, πœ– = − e sin 2ψ + e cos2 ψ. (54 ) × + ×

Then one can write Equation (51View Equation) as

δL (t) --L---= F+ (πœƒ, Ο•,ψ)h+ (t) + F ×(πœƒ,Ο•,ψ )h×(t), (55 )
where F+ and F× are the antenna pattern functions for the two polarizations defined on the sky-plane basis by
F+ ≡ d : e+, F × ≡ d : e×. (56 )
Using the geometry in the right-hand panel of Figure 3View Image, one can show that
( ) F+ = 1-1 + cos2 πœƒ cos2 Ο•cos 2ψ − cosπœƒ sin 2Ο• sin 2ψ, 2 1( 2 ) F× = 2 1 + cos πœƒ cos2 Ο•sin2 ψ + cosπœƒ sin 2Ο• cos2ψ. (57 )
View Image

Figure 3: The relative orientation of the sky and detector frames (left panel) and the effect of a rotation by the angle ψ in the sky frame (left panel).

These are the antenna-pattern response functions of the interferometer to the two polarizations of the wave as defined in the sky plane [362Jump To The Next Citation Point]. If one wants the antenna pattern referred to the detector’s own axes, then one just sets ψ = 0. If the arms of the interferometer are not perpendicular to each other, then one defines the detector-plane coordinates x and y 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 F+ or F× is 1.

The corresponding antenna-pattern functions of a bar detector whose longitudinal axis is aligned along the z direction, are

F = sin2 πœƒcos 2ψ, F = sin2πœƒ sin 2ψ. (58 ) + ×

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.

View Image

Figure 4: The antenna pattern of an interferometric detector (left panel) with the arms in the x-y plane and oriented along the two axes. The response F for waves coming from a certain direction is proportional to the distance to the point on the antenna pattern in that direction. Also shown is the fractional area in the sky over which the response exceeds a fraction πœ– of the maximum (right panel).

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 ψ. If the wave has a given amplitude h and is linearly polarized, then, if we are interested in a single detector’s response, it is always possible to align the polarization angle ψ 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

( ∫ ) -- 2 1βˆ•2 F = F+ d ψ . (59 )
The function -- F is often simply called the antenna pattern. For a resonant bar, the antenna pattern is
F-= sin2πœƒ, (60 )
and for an interferometer, it is given by
-2 1( 2 )2 2 2 2 F = --1 + cos πœƒ cos 2Ο• + cos πœƒsin 2Ο•. (61 ) 4
The antenna pattern of an interferometric detector is plotted in the left panel of Figure 4View Image, which clearly shows the quadrupolar nature of the detector. Note that the response of an interferometer is the best for waves coming from a direction orthogonal to the plane containing the detector, and it is zero for waves in the plane of an interferometer’s arms (i.e., πœƒ = π βˆ•2) that arrive from a direction bisecting the two arms (i.e., Ο• = πβˆ•4) or from directions differing from this by a multiple of πβˆ•2. What is the response of an antenna to a linearly-polarized source at a random location in the sky? This is given by the rms value of -- F over the sky,
[ 1 ∫ -- ]1βˆ•2 --- F 2sinπœƒd πœƒdΟ• , (62 ) 4 π
which is smaller than the maximum response by a factor of 2βˆ•√15-- (52%) for a bar detector and by ∘ ---- 2βˆ•5 (63%) for an interferometer.

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:

( ) h+ = h0-1 + cos2 ι cosΦ (t), h× = h0 cosιsinΦ (t), 2

where h 0 is an overall (possibly time-dependent) amplitude, Φ(t) is the signal’s phase and ι 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:

h(t) = F+h+ + F×h × = Ah0 cos(Φ (t) − Φ0 ), (63 )
( 2 2)1βˆ•2 A×-- 1- 2 A = A + + A × , tan Φ0 = A , A+ = 2F+ (1 + cos ι ), A× = F× cosι. +

Note that A, just as F, 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

-- 1 ∫ π ∫ π ∫ 2π ( 2 2 ) A = ---2 sinιdι sinπœƒd πœƒ dφ A+ + A× . (64 ) 8π 0 0 0
For an interferometer the above integral gives 2/5. Thus, the rms response is still 40% of the peak response.

The right-hand panel of Figure 4View Image shows the percentage area of the sky over which the antenna pattern of an interferometric detector is larger than a certain fraction πœ– 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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