2.3 Solar-system-based detectors

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 the 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 employed in Section 2.1 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 particles (modeling the detector’s parts) with respect to the SSB are non-relativistic, which is the case for all existing or planned detectors, Eqs. (12) 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 particles with respect to the SSB.

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

ˆt = t, ˆxi(t,xk ) = xˆi (t) + Oi (t)xj, (18 ) ^O j
where the functions ˆxi^(t) O describe the motion of the origin O^ of the proper reference frame with respect to the SSB, and the functions Oi(t) j account for the different orientations of the spatial axes of the two reference frames. One can compute some of the quantities entering Eqs. (12) 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, (19 )
where the matrix O has elements Oij. If the transformation matrix O is orthogonal, then O −1 = OT, and Eq. (19View Equation) simplifies to
^H(t) = O(t) ⋅ H(t) ⋅ O (t)T. (20 )
See [31, 54, 68Jump To The Next Citation Point, 78Jump To The Next Citation Point] for more details.

For a ground-based laser-interferometric detector, the long-wavelength approximation can be employed (however, see [27, 115, 114] for a discussion of importance of high-frequency corrections, which modify the interferometer response function computed within the long-wavelength approximation). In the case of an interferometer in a standard Michelson and equal-arm configuration (such configurations can be represented by Figure 1View Image with the particle 1 corresponding to the corner station of the interferometer and with L2 = L3 = L), the observed relative frequency shift Δν (t)∕ν0 is equal to the difference of the round-trip frequency shifts in the two detector’s arms [136]:

Δ ν(t) --ν---= y131(t) − y121(t). (21 ) 0
Let (xd,yd,zd ) be the components (with respect to the TT coordinate system) of the 3-vector rd connecting the origin of the TT coordinate system with the corner station of the interferometer. Then x1 = (xd,yd,zd)T, kT ⋅ x1 = − zd, and, making use of Eq. (17View Equation), the relative frequency shift (21View Equation) can be written as
Δ ν(t) L ( ( z ) ( z ) ) ------ = -- nT2 ⋅ ˙H t − -d- ⋅ n2 − nT3 ⋅ ˙H t −-d ⋅ n3 . (22 ) ν0 c c c
The difference Δ ϕ(t) of the phase fluctuations measured, say, by a photo detector, is related to the corresponding relative frequency fluctuations Δ ν(t) by
Δ ν(t) 1 dΔ ϕ (t) ------ = ------------. (23 ) ν0 2π ν0 dt
One can integrate Eq. (22View Equation) to write the phase change Δϕ (t) as
Δ ϕ(t) = 4πν0Lh (t), (24 )
where the dimensionless function h,
( ( ) ( ) ) h(t) := 1 nT⋅ H t − zd- ⋅ n2 − nT ⋅ H t − zd ⋅ n3 , (25 ) 2 2 c 3 c
is the response function of the interferometric detector to a plane gravitational wave in the long-wavelength approximation. To get Eqs. (24View Equation) – (25View Equation) directly from Eqs. (22View Equation) – (23View Equation) one should assume that the quantities n2, n3, and zd [entering Eq. (22View Equation)] do not depend on time t. But the formulae (24View Equation) – (25View Equation) can also be used in the case when those quantities are time dependent, provided the velocities ˙xa of the detector’s parts with respect to the SSB are non-relativistic. The error we make in such cases is on the order of 𝒪 (hv ), where v is a typical value of the velocities ˙ xa. Thus, the response function of the Earth-based interferometric detector equals
( ) 1 T ( zd(t)) T ( zd(t)) h(t) = -- n2(t) ⋅ H t − ----- ⋅ n2(t) − n3(t) ⋅ H t −----- ⋅ n3(t) , (26 ) 2 c c
where all quantities here are computed in the SSB-related TT coordinate system. The same response function can be written in terms of a detector’s proper-reference-frame quantities as follows
1 ( ( z (t)) ( z (t)) ) h(t) = -- ˆnT2 ⋅ ˆH t −-d--- ⋅ ˆn2 − ˆnT3 ⋅ ˆH t −-d--- ⋅ ˆn3 , (27 ) 2 c c
where the matrices ˆH and H are related to each other by means of formula (20View Equation). In Eq. (27View Equation) the proper-reference-frame components ˆn 2 and ˆn 3 of the unit vectors directed along the interferometer arms can be treated as constant (i.e., time independent) quantities.

From Eqs. (26View Equation) and (3View Equation) it follows that the response function h is a linear combination of the two wave polarizations h+ and h ×, so it can be written as

( zd(t)) ( zd(t)-) h(t) = F+ (t)h+ t − c + F×(t)h× t − c . (28 )
The functions F+ and F× are the interferometric beam-pattern functions. They depend on the location of the detector on Earth, the position of the gravitational-wave source in the sky, and the polarization angle of the wave (this angle describes the orientation, with respect to the detector, of the axes relative to which the plus and cross polarizations of the wave are defined, see, e.g., Figure 9.2 in [135Jump To The Next Citation Point]). Derivation of the explicit formulae for the interferometric beam patterns F + and F × can be found, e.g., in Appendix C of [66Jump To The Next Citation Point].

In the long-wavelength approximation, the response function of the interferometric detector can be derived directly from the equation of geodesic deviation [126]. Then the response is defined as the relative difference between the wave-induced changes of the proper lengths of the two arms, i.e., h (t) := (ΔLˆ (t) − Δ ˆL (t))∕ˆL 2 3 0, where Lˆ + Δ ˆL (t) 0 2 and ˆL + Δ ˆL (t) 0 3 are the instantaneous values of the proper lengths of the interferometer’s arms and ˆ L0 is the unperturbed proper length of these arms.

In the case of an Earth-based resonant-bar detector the long-wavelength approximation is very accurate and the dimensionless detector’s response function can be defined as h(t) := Δ ˆL (t)∕ˆL0, where ΔLˆ(t) is the wave-induced change of the proper length ˆ L0 of the bar. The response function h can be computed in terms of the detector’s proper-reference-frame quantities from the formula (see, e.g., Section 9.5.2 in [135Jump To The Next Citation Point])

T ( zd(t)) h (t) = ˆn ⋅ ˆH t − ----- ⋅ ˆn, (29 ) c
where the column matrix ˆn consists of the components (computed in the proper reference frame of the detector) of the unit vector n directed along the symmetry axis of the bar. The response function (29View Equation) can be written as a linear combination of the wave polarizations h+ and h ×, i.e., the formula (28View Equation) is also valid for the resonant-bar response function but with some bar-pattern functions F+ and F× different from the interferometric beam-pattern functions. Derivation of the explicit form of the bar patterns can be found, e.g., in Appendix C of [66Jump To The Next Citation Point].
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