2.4 General form of the response function

In many cases of interest the response function of the detector to a plane gravitational wave can be written as a linear combination of n waveforms hk(t;ξ) [which all depend on the same set of parameters ξ = (ξ1,...,ξm)] with constant amplitudes ak (k = 1,...,n),
n ∑ h (t;๐œƒ ) = akhk (t;ξ ), (30 ) k=1
where the vector ๐œƒ collects all the signal’s parameters. It is convenient to introduce column matrices
( ) ( ) a1 h1 (t;ξ ) a := |( .. |) , h (t;ξ ) := |( .. |) . (31 ) . . an hn (t;ξ )
Then one can briefly write ๐œƒ = (a,ξ) and the response (30View Equation) can be written as
h(t;๐œƒ ) = aT ⋅ h(t;ξ). (32 )
The functions hk (k = 1,...,n) are independent of the parameters a. The parameters a are called extrinsic (or amplitude) parameters whereas the parameters ξ are called intrinsic.

Eq. (32View Equation) with n = 4 is a model of the response of the space-based detector LISA to gravitational waves from a binary system [78Jump To The Next Citation Point]. Also for n = 4 the same equation models the response of a ground-based detector to a continuous source of gravitational waves like a rotating neutron star [68Jump To The Next Citation Point]. For ground-based detectors the long-wavelength approximation can be applied and within this approximation the functions hk (k = 1,...,4) are given by

h1(t;ξ) = u(t;ξ)cos ฯ•(t;ξ), h2(t;ξ) = v(t;ξ)cosฯ• (t;ξ ), (33 ) h3(t;ξ) = u(t;ξ)sinฯ• (t;ξ ), h4(t;ξ) = v(t;ξ)sinฯ• (t;ξ),
where ฯ•(t;ξ) is the phase modulation of the signal and u(t;ξ) and v(t;ξ ) are slowly varying amplitude modulations. The gravitational-wave signal from spinning neutron stars may consist of several components of the form (32View Equation). For short observation times over which the amplitude modulation functions are nearly constant, the response of the ground-based detector can further be approximated by
h(t;A0,ฯ•0, ξ) = A0g(t;ξ) cos(ฯ•(t;ξ) − ฯ•0), (34 )
where A 0 and ฯ• 0 are constant amplitude and initial phase, respectively, and g(t;ξ) is a slowly varying function of time. Eq. (34View Equation) is a good model for the response of a detector to the gravitational wave from a coalescing compact binary system [135, 30]. We would like to stress that not all deterministic gravitational-wave signals may be cast into the general form (32View Equation).


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