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2 Response of a Detector to a Gravitational Wave

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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 [8265] whereas the second idea is implemented in resonant mass detectors [13Jump 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 UpdateJump To The Next Update Information 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 three freely falling masses as a detector of gravitational waves. The masses are labelled 1, 2, and 3, their positions with respect to the origin O of the coordinate system are given by vectors xa (a = 1,2,3). The Euclidean separations between the masses are denoted by La, where the index a corresponds to the opposite mass. The unit vectors na point between pairs of masses, with the orientation indicated.

In Figure 1View Image we have introduced the following notation: O denotes the origin of the TT coordinate system related to the passing gravitational wave, xa (a = 1,2,3) are 3-vectors joining O and the masses, na and La (a = 1, 2,3) are, respectively, 3-vectors of unit Euclidean length along the lines joining the masses and the coordinate Euclidean distances between the masses, where a is the label of the opposite mass. Let us also denote by k the unit 3-vector directed from the origin O to the source of the gravitational wave. We first assume that the spatial coordinates of the masses do not change in time.

Let ν0 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 y21 be the relative change Δ ν∕ν0 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 the relative change of frequency induced on the beam travelling from the mass 3 to the mass 1. The frequency shifts y21 and y31 are given by [371083]

( ) ( ) y21(t) = 1 − k ⋅ n3 Ψ3 (t + k ⋅ x2 − L3 ) − Ψ3 (t + k ⋅ x1 ) , (1 )
( ) ( ) y31(t) = 1 + k ⋅ n2 Ψ2 (t + k ⋅ x3 − L2 ) − Ψ2 (t + k ⋅ x1 ) , (2 )
where (here T denotes matrix transposition)
T Ψa (t) := -n(-a ⋅-H-(t)-⋅ na)-, a = 1,2,3. (3 ) 2 1 − (k ⋅ na )2
In Equation (3View Equation) H is the three-dimensional matrix of the spatial metric perturbation produced by the wave in the TT coordinate system. If one chooses spatial TT coordinates such that the wave is travelling in the +z direction, then the matrix H 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.

Real gravitational-wave detectors do not stay at rest with respect to the TT coordinate system related to the passing gravitational wave, because they also move in the gravitational field of the solar system bodies, as in the case of the LISA spacecraft, or are fixed to the surface of Earth, as in the case of Earth-based laser interferometers or resonant bar detectors. Let us choose the origin O of the TT coordinate system to coincide with the solar system barycenter (SSB). The motion of the detector with respect to the SSB will modulate the gravitational-wave signal registered by the detector. One can show that as far as the velocities of the masses (modelling the detector’s parts) with respect to the SSB are nonrelativistic, which is the case for all existing or planned detectors, the Equations (1View Equation) and (2View Equation) can still be used, provided the 3-vectors x a and n a (a = 1,2, 3) will be interpreted as made of the time-dependent components describing the motion of the masses with respect to the SSB.

It is often convenient to introduce the proper reference frame of the detector with coordinates (xˆα). Because the motion of this frame with respect to the SSB is nonrelativistic, we can assume that the transformation between the SSB-related coordinates α (x ) and the detector’s coordinates αˆ (x ) has the form

ˆt = t, xˆi = xˆi^(t) + O ˆij(t)xj, (5 ) O
where the functions ˆi xO^(t) describe the motion of the origin O^ of the proper reference frame with respect to the SSB, and the functions Oˆij(t) account for the different orientations of the spatial axes of the two reference frames. One can compute some of the quantities entering Equations (1View Equation) and (2View Equation) in the detector’s coordinates rather than in the TT coordinates. For instance, the matrix H^ of the wave-induced spatial metric perturbation in the detector’s coordinates is related to the matrix H of the spatial metric perturbation produced by the wave in the TT coordinate system through equation
^H (t) = (O (t)−1)T ⋅ H (t) ⋅ O (t)−1, (6 )
where the matrix O has elements ˆ O ij. If the transformation matrix O is orthogonal, then O −1 = OT, and Equation (6View Equation) simplifies to
H^(t) = O(t) ⋅ H (t) ⋅ O (t)T. (7 )
See [234250Jump To The Next Citation Point60Jump To The Next Citation Point] for more details.

For a standard Michelson, equal-arm interferometric configuration Δ ν is given in terms of one-way frequency changes y 21 and y 31 (see Equations (1View Equation) and (2View Equation) with L = L = L 2 3, where we assume that the mass 1 corresponds to the corner station of the interferometer) by the expression [93]

Δν ----= (y31(t) + y13(t − L )) − (y21(t) + y12(t − L )). (8 ) ν0
In the long-wavelength approximation Equation (8View Equation) reduces to
( ) Δ-ν- T dH--(t) T dH-(t) ν0 = L n3 ⋅ dt ⋅ n3 − n 2 ⋅ dt ⋅ n2 . (9 )
The difference of the phase fluctuations Δ φ(t) measured, say, by a photo detector, is related to the corresponding relative frequency fluctuations Δ ν by
Δ ν 1 dΔ φ(t) ----= ------------. (10 ) ν0 2π ν0 dt
By virtue of Equation (9View Equation) the phase change can be written as
Δ φ(t) = 4πν0L h(t), (11 )
where the function h,
1-( T T ) h (t) := 2 n3 ⋅ H (t) ⋅ n3 − n 2 ⋅ H (t) ⋅ n2 , (12 )
is the response of the interferometer to a gravitational wave in the long-wavelength approximation. In this approximation the response of a laser interferometer is usually derived from the equation of geodesic deviation (then the response is defined as the difference between the relative wave-induced changes of the proper lengths of the two arms, i.e., h(t) := ΔL (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 is defined as hB(t) := ΔL (t)∕L, where ΔL is the wave-induced change of the proper length L of the bar. The response hB is given by

hB (t) = nT ⋅ H (t) ⋅ n, (13 )
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 can be written as a linear combination of four constant amplitudes a(k),

(k) μ ∑4 (k) (k) μ T μ h(t;a ,ξ ) = a h (t;ξ ) = a ⋅ h(t;ξ ), (14 ) k=1
where the four functions h(k) depend on a set of parameters ξμ but are independent of the parameters (k) a. The parameters (k) a are called extrinsic parameters whereas the parameters μ ξ are called intrinsic. In the long-wavelength approximation the functions (k) h are given by
h(1)(t;ξμ) = u(t;ξμ)cos φ(t;ξμ), h(2)(t;ξμ) = v(t;ξμ)cos φ(t;ξμ), (3) μ μ μ (15 ) h (t;ξ ) = u(t;ξ )sinφ (t;ξ ), h(4)(t;ξμ) = v(t;ξμ)sinφ (t;ξμ),
where φ(t;ξμ) is the phase modulation of the signal and u(t;ξμ), v(t;ξμ) are slowly varying amplitude modulations.

Equation (14View Equation) is a model of the response of the space-based detector LISA to gravitational waves from a binary system [60Jump To The Next Citation Point], whereas Equation (15View Equation) is a model of the response of a ground-based detector to a continuous source of gravitational waves like a rotating neutron star [50Jump To The Next Citation Point]. The gravitational-wave signal from spinning neutron stars may consist of several components of the form (14View Equation). For short observation times over which the amplitude modulation functions are nearly constant, the response can be approximated by

μ μ μ h (t;A0, φ0,ξ ) = A0 g(t;ξ )cos (φ(t;ξ ) − φ0 ), (16 )
where A0 and φ0 are constant amplitude and initial phase, respectively, and g(t;ξμ) is a slowly varying function of time. Equation (16View Equation) is a good model for a response of a detector to the gravitational wave from a coalescing binary system [9222]. We would like to stress that not all deterministic gravitational-wave signals may be cast into the general form (14View Equation).

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