The expression of the optimal signal-to-noise 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 signal-to-noise ratio into the sum of the signal-to-noise 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
In order to calculate the sensitivity corresponding to the expression of the optimal signal-to-noise ratio, we have proceeded similarly to what was done in [1, 7], and described in more detail in . We assume an equal-arm LISA (), and take the one-sided spectra of proof mass and aggregate optical-path-noises (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 mean-squared 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 proof-mass and optical-path noises, and the transfer functions of these noises to . Using the transfer functions given in , the resulting spectra are equal to
In Figure 6 we show the sensitivity curve for the LISA equal-arm 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 signal-to-noise 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 signal-to-noise ratios of as well as the optimal signal-to-noise ratio. For an assumed , the SNRs of the three modes are plotted versus frequency. For the equal-arm 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 .
© Max Planck Society and the author(s)