Before proceeding with this idea, however, let us consider again the so-called “second-generation” TDI Sagnac observables: . The expressions of the gravitational-wave signal and the secondary noise sources entering into will in general be different from those entering into , the corresponding Sagnac observable derived under the assumption of a stationary LISA array [2*, 15*]. 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 [58*]. It is therefore easy to derive the following relationship between the signal and secondary noises in , and those entering into the stationary TDI combination [45, 58],81*) 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 .
As a consequence of these considerations, we can still regard as the generators of the TDI space, and write the most general expression for an element of the TDI space, , as a linear combination of the Fourier transforms of the four generators ,83*) the subscripts s and n refer to the signal and the noise parts of , respectively, the angle brackets represent noise ensemble averages, and the interval of integration corresponds to the frequency band accessible by LISA.
Before proceeding with the maximization of the we may notice from Eq. (44*) that the Fourier transform of the totally symmetric Sagnac combination, , multiplied by the transfer function can be written as a linear combination of the Fourier transforms of the remaining three generators . Since the signal-to-noise ratio of and are equal, we may conclude that the optimization of the signal-to-noise ratio of can be performed only on the three observables . This implies the following redefined expression for :
In order to make the derivation of the optimal SNR easier, let us first denote by and the two vectors of the signals and the noises , respectively. Let us also define to be the vector of the three functions , and denote with the Hermitian, non-singular, correlation matrix of the vector random process ,87*) can be rewritten in the following form, 87*) is well known in the theory of quadratic forms, and it is called Rayleigh’s principle [36, 42].
In order now to identify the eigenvalues of the matrix , we first notice that the matrix has rank 1. This implies that the matrix 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 matrix 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 :84*) and (89*), in the following way:
- Among all possible interferometric combinations LISA will be able to synthesize with its four generators , , , , the particular combination giving maximum signal-to-noise ratio can be obtained by using only three of them, namely .
- The expression of the optimal signal-to-noise ratio given by Eq. (89*) implies that LISA should be regarded as a network of three interferometer detectors of gravitational radiation (of responses ) working in coincidence [20, 40*].
As an application of Eq. (89*), here we calculate the sensitivity that LISA can reach when observing sinusoidal signals uniformly distributed on the celestial sphere and of random polarization. In order to calculate the optimal signal-to-noise ratio we will also need to use a specific expression for the noise correlation matrix . As a simplification, we will assume the LISA arm lengths to be equal to their nominal value , the optical-path noises to be equal and uncorrelated to each other, and finally the noises due to the proof-mass noises to be also equal, uncorrelated to each other and to the optical-path noises. Under these assumptions the correlation matrix becomes real, its three diagonal elements are equal, and all the off-diagonal terms are equal to each other, as it is easy to verify by direct calculation [15*]. The noise correlation matrix is therefore uniquely identified by two real functions and in the following way:
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 to92*) implies the following three linear combinations of the generators : 93*) it is also easy to verify that the noise correlation matrix of these three combinations is diagonal, and that its non-zero elements are indeed equal to the eigenvalues given in Eq. (91*).
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 [2*, 15*], and described in more detail in [56*]. 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 [15*, 5]). 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 7000 Fourier frequencies in the to LISA band, we produce the Fourier transforms of the gravitational-wave response of from the formulas in [2*, 56]. 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 [15*], the resulting spectra are equal to
In Figure 7* 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 Figure 8* we plot the ratio between the optimal SNR and the SNR of a single Michelson interferometer. In the long-wavelength limit, the SNR improvement is . For Fourier frequencies greater than or about equal to , the SNR improvement is larger and varies with the frequency, showing an average value of about . In particular, for bands of frequencies centered on integer multiples of , contributes strongly and the aggregate SNR in these bands can be greater than 2.
In order to better understand the contribution from the three different combinations to the optimal combination of the three generators, in Figure 9* 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 [40*].
6.2 Optimization of SNR for binaries with known direction but with unknown orientation of the orbital plane
Binaries are important sources for LISA and therefore the analysis of such sources is of major importance. One such class is of massive or super-massive binaries whose individual masses could range from to and which could be up to a few Gpc away. Another class of interest are known binaries within our own galaxy whose individual masses are of the order of a solar mass but are just at a distance of a few kpc or less. Here the focus will be on this latter class of binaries. It is assumed that the direction of the source is known, which is so for known binaries in our galaxy. However, even for such binaries, the inclination angle of the plane of the orbit of the binary is either poorly estimated or unknown. The optimization problem is now posed differently: The SNR is optimized after averaging over the polarizations of the binary signals, so the results obtained are optimal on the average, that is, the source is tracked with an observable which is optimal on the average [40*]. For computing the average, a uniform distribution for the direction of the orbital angular momentum of the binary is assumed.
When the binary masses are of the order of a solar mass and the signal typically has a frequency of a few mHz, the GW frequency of the binary may be taken to be constant over the period of observation, which is typically taken to be of the order of an year. A complete calculation of the signal matrix and the optimization procedure of SNR is given in [39*]. Here we briefly mention the main points and the final results.
A source fixed in the Solar System Barycentric reference frame in the direction is considered. But as the LISA constellation moves along its heliocentric orbit, the apparent direction of the source in the LISA reference frame changes with time. The LISA reference frame has been defined in [39*] as follows: The origin lies at the center of the LISA triangle and the plane of LISA coincides with the plane with spacecraft 2 lying on the axis. Figure (10*) displays this apparent motion for a source lying in the ecliptic plane, that is with and . The source in the LISA reference frame describes a figure of 8. Optimizing the SNR amounts to tracking the source with an optimal observable as the source apparently moves in the LISA reference frame.
Since an average has been taken over the orientation of the orbital plane of the binary or equivalently over the polarizations, the signal matrix is now of rank 2 instead of rank 1 as compared with the application in the previous Section 6.1. The mutually orthogonal data combinations , , are convenient in carrying out the computations because in this case as well, they simultaneously diagonalize the signal and the noise covariance matrix. The optimization problem now reduces to an eigenvalue problem with the eigenvalues being the squares of the SNRs. There are two eigen-vectors which are labeled as belonging to two non-zero eigenvalues. The two SNRs are labelled as and , corresponding to the two orthogonal (thus statistically independent) eigenvectors . As was done in the previous Section 6.1 F the two SNRs can be squared and added to yield a network SNR, which is defined through the equation
The eigenvectors and the SNRs are functions of the apparent source direction parameters in the LISA reference frame, which in turn are functions of time. The eigenvectors optimally track the source as it moves in the LISA reference frame. Assuming an observation period of an year, the SNRs are integrated over this period of time. The sensitivities are computed according to the procedure described in the previous Section 6.1. The results of these findings are displayed in Figure 11*.
It shows the sensitivity curves of the following observables:
- The Michelson combination (faint solid curve).
- The observable obtained by taking the maximum sensitivity among , , and for each direction, where and are the Michelson observables corresponding to the remaining two pairs of arms of LISA . This maximum is denoted by (dash-dotted curve) and is operationally given by switching the combinations , , so that the best sensitivity is achieved.
- The eigen-combination which has the best sensitivity among all data combinations (dashed curve).
- The network observable (solid curve).
It is observed that the sensitivity over the band-width of LISA increases as one goes from Observable 1 to 4. Also it is seen that the does not do much better than . This is because for the source direction chosen , is reasonably well oriented and switching to and combinations does not improve the sensitivity significantly. However, the network and observables show significant improvement in sensitivity over both and . This is the typical behavior and the sensitivity curves (except ) do not show much variations for other source directions and the plots are similar. Also it may be fair to compare the optimal sensitivities with rather than . This comparison of sensitivities is shown in Figure 12*, where the network and the eigen-combinations are compared with .