6 Optimal LISA Sensitivity
All the above interferometric combinations have been shown to individually have rather different
sensitivities , 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: . 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 [1, 7].
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 . It is therefore easy to derive the following relationship between the signal and
secondary noises in , and those entering into the stationary TDI combination [25, 34],
where , , are the unequal-arm lengths of the stationary LISA array. Equation (64) 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 ,
where the are arbitrary complex functions of the Fourier frequency , and of a vector
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 , gives an element of the functional space of interferometric
combinations generated by . Our goal is therefore to identify, for a given gravitational wave
signal, the four functions that maximize the signal-to-noise ratio of the combination ,
In Equation (66) 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 Equation (43) 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 :
The can be regarded as a functional over the space of the three complex functions , 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 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 ,
If we finally define to be the components of the Hermitian matrix , we can rewrite
in the following form,
where we have adopted the usual convention of summation over repeated indices. Since the noise correlation
matrix 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):
After taking the partial derivatives, Equation (70) can be rewritten in the following form,
which tells us that the stationary values of the signal-to-noise ratio of are equal to the eigenvalues of the
the matrix . The result in Equation (70) is well known in the theory of quadratic forms, and it is
called Rayleigh’s principle [18, 23].
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 :
We can summarize the results derived in this section, which are given by Equations (67, 72), in the
- 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 Equation (72) implies that LISA
should be regarded as a network of three interferometer detectors of gravitational radiation (of
responses ) working in coincidence [12, 21].