Pre-Phase A Report - Lisa - Nasa

4.4 Data analysis 113
density Sn(f) , it can be shown [116] that
where the symmetric inner product is defined as
〈s, h〉 =2
P(µ, s) ∼ exp 〈 s, h(µ) 〉 , (4.77)
0
∞
s(f) h ∗ (f)+ h(f)s ∗ (f)
Sh(f)
df . (4.78)
From that definition it follows that, for a waveform h(µ), the signal-to-noise ratio is
approximately given by
S
〈h(µ),h(µ)〉
[h(µ)] =
N rms (〈h(µ),n〉) = 〈h(µ),h(µ)〉 . (4.79)
The error in measurement is taken to be the width of the probability density function
P(µ, s) for the measured value µ, i.e. the variance-covariance matrix
Σij = (µi − µi)(µj − µj) P(µ, s) d n µ. (4.80)
For high signal-to-noise, Σij is well approximated by (Γ −1 )ij, whereΓij is the so-called
Fisher matrix, given by
where ∂i ≡ ∂/∂µi.
Γij =2
0
∞
∂i h(f) ∂j h ∗ (f)+∂j h(f) ∂i h ∗ (f)
Sh(f)
df , (4.81)
4.4.3 Polarization resolution and amplitude extraction
One can clearly estimate the amplitude of the waveform directly from the signal-to-noise
of the detection; they are directly proportional. Given the output of both interferometers I
and II, LISA should be able to extract both the amplitude and polarisation of the incoming
wave, to an accuracy of order the inverse of the signal-to-noise ratio (though again the
accuracy that is achievable also depends on correlations with the other parameters that
one is trying to extract). Even if only a single interferometer output is available, LISA
can still extract the amplitude and polarisation of the wave due to the rotation of the
detector during its orbit. But clearly the yearly rotation of the detector is less helpful
for determining the polarisation of shorter-lived sources such as merging MBH binaries,
where most of the signal-to-noise will typically be accumulated in the final week before
merger.
The same Fisher matrix calculation that tells us the angular resolution of the detector
will simultaneously tell us how accurately the polarisation and amplitude of the source
can be determined.
Corrected version 2.08 3-3-1999 9:33