The expression of the optimal signaltonoise ratio assumes a rather simple form if we diagonalize this correlation matrix by properly “choosing a new basis”. There exists an orthogonal transformation of the generators which will transform the optimal signaltonoise ratio into the sum of the signaltonoise ratios of the “transformed” three interferometric combinations. The expressions of the three eigenvalues (which are real) of the noise correlation matrix can easily be found by using the algebraic manipulator Mathematica, and they are equal to
Note that two of the three real eigenvalues, (, ), are equal. This implies that the eigenvector associated to is orthogonal to the twodimensional space generated by the eigenvalue , while any chosen pair of eigenvectors corresponding to will not necessarily be orthogonal. This inconvenience can be avoided by choosing an arbitrary set of vectors in this twodimensional space, and by orthonormalizing them. After some simple algebra, we have derived the following three orthonormalized eigenvectors: Equation (75) implies the following three linear combinations of the generators : where , , and are italicized to indicate that these are “orthogonal modes”. Although the expressions for the modes and depend on our particular choice for the two eigenvectors (), it is clear from our earlier considerations that the value of the optimal signaltonoise ratio is unaffected by such a choice. From Equation (76) it is also easy to verify that the noise correlation matrix of these three combinations is diagonal, and that its nonzero elements are indeed equal to the eigenvalues given in Equation (74).In order to calculate the sensitivity corresponding to the expression of the optimal signaltonoise ratio, we have proceeded similarly to what was done in [1, 7], and described in more detail in [32]. We assume an equalarm LISA (), and take the onesided spectra of proof mass and aggregate opticalpathnoises (on a single link), expressed as fractional frequency fluctuation spectra, to be and , respectively (see [7, 3]). We also assume that aggregate optical path noise has the same transfer function as shot noise.
The optimum SNR is the square root of the sum of the squares of the SNRs of the three “orthogonal modes” . To compare with previous sensitivity curves of a single LISA Michelson interferometer, we construct the SNRs as a function of Fourier frequency for sinusoidal waves from sources uniformly distributed on the celestial sphere. To produce the SNR of each of the modes we need the gravitational wave response and the noise response as a function of Fourier frequency. We build up the gravitational wave responses of the three modes from the gravitational wave responses of . For Fourier frequencies in the to LISA band, we produce the Fourier transforms of the gravitational wave response of from the formulas in [1, 32]. The averaging over source directions (uniformly distributed on the celestial sphere) and polarization states (uniformly distributed on the Poincaré sphere) is performed via a Monte Carlo method. From the Fourier transforms of the responses at each frequency, we construct the Fourier transforms of . We then square and average to compute the meansquared responses of at that frequency from realizations of (source position, polarization state) pairs.
We adopt the following terminology: We refer to a single element of the module as a data combination, while a function of the elements of the module, such as taking the maximum over several data combinations in the module or squaring and adding data combinations belonging to the module, is called as an observable. The important point to note is that the laser frequency noise is also suppressed for the observable although it may not be an element of the module.
The noise spectra of are determined from the raw spectra of proofmass and opticalpath noises, and the transfer functions of these noises to . Using the transfer functions given in [7], the resulting spectra are equal to
Let the amplitude of the sinusoidal gravitational wave be . The SNR for, e.g. , , at each frequency is equal to times the ratio of the rootmeansquared gravitational wave response at that frequency divided by , where is the bandwidth conventionally taken to be equal to 1 cycle per year. Finally, if we take the reciprocal of and multiply it by 5 to get the conventional sensitivity criterion, we obtain the sensitivity curve for this combination which can then be compared against the corresponding sensitivity curve for the equalarm Michelson interferometer.In Figure 6 we show the sensitivity curve for the LISA equalarm Michelson response () as a function of the Fourier frequency, and the sensitivity curve from the optimum weighting of the data described above: . The SNRs were computed for a bandwidth of 1 cycle/year. Note that at frequencies where the LISA Michelson combination has best sensitivity, the improvement in signaltonoise ratio provided by the optimal observable is slightly larger than .

In order to better understand the contribution from the three different combinations to the optimal combination of the three generators, in Figure 8 we plot the signaltonoise ratios of as well as the optimal signaltonoise ratio. For an assumed , the SNRs of the three modes are plotted versus frequency. For the equalarm case computed here, the SNRs of and are equal across the band. In the long wavelength region of the band, modes and have SNRs much greater than mode , where its contribution to the total SNR is negligible. At higher frequencies, however, the combination has SNR greater than or comparable to the other modes and can dominate the SNR improvement at selected frequencies. Some of these results have also been obtained in [21].
http://www.livingreviews.org/lrr20054 
© Max Planck Society and the author(s)
Problems/comments to 