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 [32
]. 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 [7],
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 [21
].
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