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6.1 General application

As an application of Equation (72View Equation), 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 C. As a simplification, we will assume the LISA arm lengths to be equal to their nominal value L = 16.67 s, 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 [7Jump To The Next Citation Point]. The noise correlation matrix C is therefore uniquely identified by two real functions Sa and Sab in the following way:
( ) Sa Sab Sab C = Sab Sa Sab . (73) Sab Sab Sa

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 (a,b,g) 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 {mi}3i=1 (which are real) of the noise correlation matrix C can easily be found by using the algebraic manipulator Mathematica, and they are equal to

m = m = S - S , m = S + 2S . (74) 1 2 a ab 3 a ab
Note that two of the three real eigenvalues, (m1, m2), are equal. This implies that the eigenvector associated to m3 is orthogonal to the two-dimensional space generated by the eigenvalue m1, while any chosen pair of eigenvectors corresponding to m 1 will not necessarily be orthogonal. This inconvenience can be avoided by choosing an arbitrary set of vectors in this two-dimensional space, and by ortho-normalizing them. After some simple algebra, we have derived the following three ortho-normalized eigenvectors:
v = V~ 1-(- 1,0,1) v = V~ 1-(1, -2,1) v = V~ 1-(1,1,1). (75) 1 2 2 6 3 3
Equation (75View Equation) implies the following three linear combinations of the generators (a, b,g):
( ) ( ) A = V~ 1-(g - a) E = V~ 1- a - 2b + g T = V~ 1-a + b + g , (76) 2 6 3
where A, E, and T are italicized to indicate that these are “orthogonal modes”. Although the expressions for the modes A and E depend on our particular choice for the two eigenvectors (v1,v2), it is clear from our earlier considerations that the value of the optimal signal-to-noise ratio is unaffected by such a choice. From Equation (76View Equation) 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 Equation (74View Equation).

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 [1Jump To The Next Citation Point7Jump To The Next Citation Point], and described in more detail in [32Jump To The Next Citation Point]. We assume an equal-arm LISA (L = 16.67 s), 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 Spryoof mass= 2.5× 10-48[f /1 Hz]-2 Hz -1 and Soyptical path = 1.8 × 10-37[f/1 Hz]2 Hz -1, respectively (see [7Jump To The Next Citation Point3]). 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” (A, E, T). 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 (A, E, T ) 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 (A,E, T ) from the gravitational wave responses of (a, b,g). For 7000 Fourier frequencies in the ~ 10- 4 Hz to ~ 1 Hz LISA band, we produce the Fourier transforms of the gravitational wave response of (a,b,g) from the formulas in [1Jump To The Next Citation Point32]. 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 (a, b,g) responses at each frequency, we construct the Fourier transforms of (A, E,T ). We then square and average to compute the mean-squared responses of (A, E, T) at that frequency from 4 10 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 (A, E,T ) are determined from the raw spectra of proof-mass and optical-path noises, and the transfer functions of these noises to (A,E, T ). Using the transfer functions given in [7], the resulting spectra are equal to

S (f) = S (f) = 16sin2(pf L)[3 + 2cos(2pf L) + cos(4pf L)]Sproof mass(f) A E y + 8sin2(pf L)[2 + cos(2pf L)]Soyptical path(f ), (77) 2[ 2 proof mass optical path ] ST (f) = 2[1 + 2 cos(2pf L)] 4sin (pf L)S y + Sy (f) . (78)
Let the amplitude of the sinusoidal gravitational wave be h. The SNR for, e.g. A, SNRA, at each frequency f is equal to h times the ratio of the root-mean-squared gravitational wave response at that frequency divided by V~ -------- SA(f )B, where B is the bandwidth conventionally taken to be equal to 1 cycle per year. Finally, if we take the reciprocal of SNR /h A and multiply it by 5 to get the conventional SNR = 5 sensitivity criterion, we obtain the sensitivity curve for this combination which can then be compared against the corresponding sensitivity curve for the equal-arm Michelson interferometer.

In Figure 6View Image we show the sensitivity curve for the LISA equal-arm Michelson response (SNR = 5) as a function of the Fourier frequency, and the sensitivity curve from the optimum weighting of the data described above: V ~ ----2--------2-------2-- 5h/ SNR A + SNR E + SNR T. 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 V~ 2-.

View Image

Figure 6: The LISA Michelson sensitivity curve (SNR = 5) and the sensitivity curve for the optimal combination of the data, both as a function of Fourier frequency. The integration time is equal to one year, and LISA is assumed to have a nominal armlength L = 16.67 s.
View Image

Figure 7: The optimal SNR divided by the SNR of a single Michelson interferometer, as a function of the Fourier frequency f. The sensitivity gain in the low-frequency band is equal to V~ -- 2, while it can get larger than 2 at selected frequencies in the high-frequency region of the accessible band. The integration time has been assumed to be one year, and the proof mass and optical path noise spectra are the nominal ones. See the main body of the paper for a quantitative discussion of this point.
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Figure 8: The SNRs of the three combinations (A,E, T ) and their sum as a function of the Fourier frequency f. The SNRs of A and E are equal over the entire frequency band. The SNR of T is significantly smaller than the other two in the low part of the frequency band, while is comparable to (and at times larger than) the SNR of the other two in the high-frequency region. See text for a complete discussion.
In Figure 7View Image 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 V~ -- 2. For Fourier frequencies greater than or about equal to 1/L, the SNR improvement is larger and varies with the frequency, showing an average value of about V~ -- 3. In particular, for bands of frequencies centered on integer multiples of 1/L, SNRT 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 8View Image we plot the signal-to-noise ratios of (A, E, T) as well as the optimal signal-to-noise ratio. For an assumed h = 10- 23, the SNRs of the three modes are plotted versus frequency. For the equal-arm case computed here, the SNRs of A and E are equal across the band. In the long wavelength region of the band, modes A and E have SNRs much greater than mode T, where its contribution to the total SNR is negligible. At higher frequencies, however, the T 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 [21Jump To The Next Citation Point].


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