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2 Response of a Detector to a Gravitational-Wave Signal

There are two main methods to detect gravitational waves which have been implemented in the currently working instruments. One method is to measure changes induced by gravitational waves on the distances between freely moving test masses using coherent trains of electromagnetic waves. The other method is to measure the deformation of large masses at their resonance frequencies induced by gravitational waves. The first idea is realized in laser interferometric detectors and Doppler tracking experiments  [7055] whereas the second idea is implemented in resonant mass detectors  [10Jump To The Next Citation Point] .

Let us consider the response to a plane gravitational wave of a freely falling configuration of masses. It is enough to consider a configuration of three masses shown in Figure  1View Image to obtain the response for all currently working and planned detectors. Two masses model a Doppler tracking experiment where one mass is the Earth and the other one is a distant spacecraft. Three masses model a ground-based laser interferometer where the masses are suspended from seismically isolated supports or a space-borne interferometer where the three masses are shielded in satellites driven by drag-free control systems.

View Image

Figure 1: Schematic configuration of freely falling masses as a detector of gravitational waves. The masses are labelled 1, 2, and 3. The optical and radio paths are denoted by Li , where the index i corresponds to the opposite mass. The unit vectors n^i point between pairs of masses, with the orientation indicated.
Let n0 be the frequency of the coherent beam used in the detector (laser light in the case of an interferometer and radio waves in the case of Doppler tracking). Let us assume for simplicity that the distance between the masses is the same and equal to L . Let ^ni, i = 1,2,3, be the unit vectors along the lines joining the test masses, let O be the point lying in the plane of the three masses and equidistant from the masses, and let pi, i = 1,2,3, be the vectors of length -- l = L/ V~ 3 joining O and the masses. Let y21 be the relative change Dn/n0 of frequency induced by a transverse, traceless, plane gravitational wave on the coherent beam travelling from the mass 2 to the mass 1, and let y31 be a relative change of frequency induced on the beam travelling from the mass 3 to the mass 1. The equations for y21 and y31 are given by (see  [318] and also  [71] for a coordinate free derivation)
( )( ) y21(t) = 1 - ^k .^n3 Y3(t + ^k .p2 - L) - Y3(t + ^k .p1) , (1) ( )( ) y31(t) = 1 + ^k .^n2 Y2(t + ^k .p3 - L) - Y2(t + ^k .p1) , (2)
where
----Pj(t)--- 1- T Yj(t) := 1 - (^k .^nj)2 , Pj(t) := 2 ^nj .H(t) .^nj, j = 1,2,3. (3)
In Equation (3View Equation) H is the three-dimensional matrix of the spatial metric perturbation produced by the wave in the proper reference frame of the detector, ^k is the unit vector in this reference frame directed from the center O to the source of the gravitational wave and T denotes matrix transposition.

In the source frame the three-dimensional matrix H of the spatial metric perturbation produced by the gravitational wave is given by

( ) h+(t) h ×(t) 0 H(t) = h×(t)- h+(t) 0 , (4) 0 0 0
where h+ and h× are the two polarizations of the wave. In general the detector is moving with respect to the source of a gravitational wave and the signal registered by the detector will be modulated by this motion. To obtain the matrix H it is convenient to introduce a coordinate system with respect to which the source is fixed. The origin of this coordinate system is usually chosen to be the solar system barycenter (SSB). Then we obtain H by the transformation
H(t) = O -21(t) O1(t)H(t) O-1 1(t)O2(t). (5)
Here O1 is the transformation matrix from the source frame to SSB and O2 is the transformation matrix from the detector frame to SSB (see  [351941Jump To The Next Citation Point50Jump To The Next Citation Point] for details).

The difference of the phase fluctuations Df(t) measured, say, by a photo detector, is related to the corresponding relative frequency fluctuations Dn by

Dn--= --1--dDf(t)-. (6) n0 2pn0 dt
For a standard Michelson, equal-arm interferometric configuration Dn is given in terms of one-way frequency changes y21 and y31 [see Equations (1View Equation) and (2View Equation)] by the expression  [80]
Dn--= (y (t) + y (t- L)) - (y (t) + y (t- L)) . (7) n0 31 13 21 12

In the long-wavelength approximation Equation (7View Equation) reduces to

( ) Dn--= 2L 1-^nT .H(t) .^ni- 1nT .H(t) .nj . (8) n0 2 i 2 j
Consequently the phase change is given by
Df(t) = 4pn0Lh(t), (9)
where the function
h(t) = 1-^nT .H(t) .^n - 1^nT .H(t) .^n (10) 2 i i 2 j j
is the response of the interferometer to a gravitational-wave signal in the long-wavelength approximation. In this approximation the response of a laser interferometer is usually derived from the equation of geodesic deviation (where the response is defined as the relative change of the length of the two arms, i.e., h(t) := DL(t)/L). There are important cases where the long-wavelength approximation is not valid. These include the space-borne LISA detector for gravitational-wave frequencies larger than a few mHz and satellite Doppler tracking measurements.

In the case of a bar detector the long-wavelength approximation is very accurate and the detector’s response hB(t) = DL(t)/L, where DL is the change of the length L of the bar, is given by

hB(t) = ^nT .H(t) .^n, (11)
where ^n is the unit vector along the symmetry axis of the bar.

In most cases of interest the response of the detector to a gravitational-wave signal can be written as a linear combination of four constant amplitudes (k) a,

4 h(t;a(k),qm) = sum a(k)h(k)(t;qm) = aT .h(t;qm), (12) k=1
where the four functions h(k) depend on a set of parameters qm but are independent of the parameters a(k) . The parameters a(k) are called extrinsic parameters whereas the parameters qm are called intrinsic . In the long-wavelength approximation the functions h(k) are given by
h(1)(t;qm) = u(t;qm)cos f(t;qm), h(2)(t;qm) = v(t;qm)cos f(t;qm), (13) h(3)(t;qm) = u(t;qm)sinf(t;qm), (4) m m m h (t;q ) = v(t;q )sinf(t;q ),
where f(t;qm) is the phase modulation of the signal and u(t;qm), v(t;qm) are slowly varying amplitude modulations.

Equation (12View Equation) is a model of the response of the space-based detector LISA to gravitational waves from a binary system  [50Jump To The Next Citation Point], whereas Equation (13View Equation) is a model of the response of a ground-based detector to a continuous source of gravitational waves like a rotating neutron star  [41Jump To The Next Citation Point] . The gravitational-wave signal from spinning neutron stars may consist of several components of the form (12View Equation). For short observation times over which the amplitude modulation functions are nearly constant, the response can be approximated by

h(t;A0, f0,qm) = A0 g(t;qm)cos (f(t;qm)- f0) , (14)
where A0 and f0 are constant amplitude and initial phase, respectively, and m g(t;q ) is a slowly varying function of time. Equation (14View Equation) is a good model for a response of a detector to the gravitational-wave signal from a coalescing binary system  [7918] . We would like to stress that not all deterministic gravitational-wave signals may be cast into the general form (12View Equation).
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