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6 Optimal LISA Sensitivity

All the above interferometric combinations have been shown to individually have rather different sensitivities [7Jump To The Next Citation Point], as a consequence of their different responses to gravitational radiation and system noises. Since LISA has the capability of simultaneously observing a gravitational wave signal with many different interferometric combinations (all having different antenna patterns and noises), we should no longer regard LISA as a single detector system but rather as an array of gravitational wave detectors working in coincidence. This suggests that the presently adopted LISA sensitivity could be improved by optimally combining elements of the TDI space.

Before proceeding with this idea, however, let us consider again the so-called “second generation” TDI Sagnac observables: (a1,a2, a3). The expressions of the gravitational wave signal and the secondary noise sources entering into a 1 will in general be different from those entering into a, the corresponding Sagnac observable derived under the assumption of a stationary LISA array [1Jump To The Next Citation Point7Jump To The Next Citation Point]. However, the other remaining, secondary noises in LISA are so much smaller, and the rotation and systematic velocities in LISA are so intrinsically small, that index permutation may still be done for them [34Jump To The Next Citation Point]. It is therefore easy to derive the following relationship between the signal and secondary noises in a1, and those entering into the stationary TDI combination a [2534],

a1(t) -~ a(t) - a(t - L1 - L2 - L3), (64)
where Li, i = 1,2,3, are the unequal-arm lengths of the stationary LISA array. Equation (64View Equation) implies that any data analysis procedure and algorithm that will be implemented for the second-generation TDI combinations can actually be derived by considering the corresponding “first generation” TDI combinations. For this reason, from now on we will focus our attention on the gravitational wave responses of the first-generation TDI observables (a, b,g,z).

As a consequence of these considerations, we can still regard (a, b,g,z) as the generators of the TDI space, and write the most general expression for an element of the TDI space, j(f), as a linear combination of the Fourier transforms of the four generators (a,b, g,z),

j(f ) =_ a1(f,c) a(f) + a2(f,c) b(f) + a3(f,c)g(f ) + a4(f,c)z(f ), (65)
where the {ai(f, c)}4i=1 are arbitrary complex functions of the Fourier frequency f, and of a vector c containing parameters characterizing the gravitational wave signal (source location in the sky, waveform parameters, etc.) and the noises affecting the four responses (noise levels, their correlations, etc.). For a given choice of the four functions 4 {ai}i=1, j gives an element of the functional space of interferometric combinations generated by (a,b, g,z). Our goal is therefore to identify, for a given gravitational wave signal, the four functions {ai}4i=1 that maximize the signal-to-noise ratio SNR2j of the combination j,
| | integral fu ||a a + a b + a g + a z || 2 SNR2j = <-|-1--s----2-s----3-s----4-s|-> df. (66) fl ||a1 an + a2bn + a3 gn + a4zn||2
In Equation (66View Equation) the subscripts s and n refer to the signal and the noise parts of (a,b,g, z), respectively, the angle brackets represent noise ensemble averages, and the interval of integration (fl,fu) corresponds to the frequency band accessible by LISA.

Before proceeding with the maximization of the SNR2j we may notice from Equation (43View Equation) that the Fourier transform of the totally symmetric Sagnac combination, z, multiplied by the transfer function 2pif(L1+L2+L3) 1 - e can be written as a linear combination of the Fourier transforms of the remaining three generators (a,b, g). Since the signal-to-noise ratio of j and (1 - e2pif(L1+L2+L3))j are equal, we may conclude that the optimization of the signal-to-noise ratio of j can be performed only on the three observables a,b, g. This implies the following redefined expression for SNR2 j:

| | integral fu ||a a + a b + a g || 2 SNR2 = <|-1--s----2-s----3--s|>-df. (67) j fl ||a a + a b + a g || 2 1 n 2 n 3 n
The SNR2 j can be regarded as a functional over the space of the three complex functions {a}3 i i=1, and the particular set of complex functions that extremize it can of course be derived by solving the associated set of Euler-Lagrange equations.

In order to make the derivation of the optimal SNR easier, let us first denote by x(s) and x(n) the two vectors of the signals (a ,b ,g) s s s and the noises (a ,b ,g ) n n n, respectively. Let us also define a to be the vector of the three functions 3 {ai}i=1, and denote with C the Hermitian, non-singular, correlation matrix of the vector random process xn,

< > (C) =_ x(n)x(n)* . (68) rt r t
If we finally define (A)ij to be the components of the Hermitian matrix x(si)x(js)*, we can rewrite SNR2j in the following form,
2 integral fuaiAija*j SNR j = a-C--a*-df, (69) fl r rt t
where we have adopted the usual convention of summation over repeated indices. Since the noise correlation matrix C is non-singular, and the integrand is positive definite or null, the stationary values of the signal-to-noise ratio will be attained at the stationary values of the integrand, which are given by solving the following set of equations (and their complex conjugated expressions):
[ *] -@-- aiAijaj- = 0, k = 1,2,3. (70) @ak arCrta*t
After taking the partial derivatives, Equation (70View Equation) can be rewritten in the following form,
[apApqa* ] (C -1)ir (A)rj (a*)j = -------q*- (a*)i, i = 1,2,3, (71) alClma m
which tells us that the stationary values of the signal-to-noise ratio of j are equal to the eigenvalues of the the matrix C -1 .A. The result in Equation (70View Equation) is well known in the theory of quadratic forms, and it is called Rayleigh’s principle [1823].

In order now to identify the eigenvalues of the matrix C- 1 .A, we first notice that the 3 × 3 matrix A has rank 1. This implies that the matrix C -1 .A has also rank 1, as it is easy to verify. Therefore two of its three eigenvalues are equal to zero, while the remaining non-zero eigenvalue represents the solution we are looking for.

The analytic expression of the third eigenvalue can be obtained by using the property that the trace of the 3 × 3 matrix C -1 .A is equal to the sum of its three eigenvalues, and in our case to the eigenvalue we are looking for. From these considerations we derive the following expression for the optimized signal-to-noise ratio 2 SNR jopt:

integral 2 fu(s)* -1 (s) SNR jopt = fl xi (C )ij xj df. (72)
We can summarize the results derived in this section, which are given by Equations (67View Equation, 72View Equation), in the following way:
  1. Among all possible interferometric combinations LISA will be able to synthesize with its four generators a, b, g, z, the particular combination giving maximum signal-to-noise ratio can be obtained by using only three of them, namely (a, b,g).
  2. The expression of the optimal signal-to-noise ratio given by Equation (72View Equation) implies that LISA should be regarded as a network of three interferometer detectors of gravitational radiation (of responses (a,b,g)) working in coincidence [1221Jump To The Next Citation Point].


 6.1 General application
 6.2 Optimization of SNR for binaries with known direction but with unknown orientation of the orbital plane

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