Pre-Phase A Report - Lisa - Nasa

4.4 Data analysis 105
• LISA observes primarily long-lived sources, while ground-based detectors are expected
to observe mainly bursts that are so short that frequency modulation is
unimportant. LISA is able to find directions and polarisations primarily from the
phase- and amplitude-modulation produced by its motion during an observation.
Ground-based detectors will, of course, look for radiation from rotating neutron
stars, and for this case the detection and signal reconstruction problem are similar
to that for LISA, but LISA’s lower data rate and lower frequency makes the analysis
considerably easier.
• If LISA sees a gravitational wave background, it cannot identify it by crosscorrelation
with another independent detector. We will show in Section 4.4.5 below
how LISA can discriminate one background from another and from instrumental
noise.
In what follows we will consider in turn the methods used for data analysis and the
expected manner and accuracy of extraction of the different kinds of information present
in the signal.
4.4.1 Data reduction and filtering
Noise. The fundamental principle guiding the analysis of LISA data is that of matched
filtering. Assuming that the LISA detector noise n(t) is stationary (an assumption that
is only a first approximation, but which will have to be tested), the noise power can be
characterised by its spectral density, defined as
Sh(f) =
∞
−∞
〈 n(t)n(t + τ) 〉 e −2πifτ , (4.56)
where the autocorrelation of the noise 〈 n(t)n(t + τ) 〉 depends only on the offset time τ
because the noise is stationary. The subscript “h” onSh refers to the gravitational wave
amplitude, and it means that the detector output is assumed normalised and calibrated
so that it reads directly the apparent gravitational wave amplitude.
So far we have not assumed anything about its statistics, the probability density function
(PDF) of the noise. It is conventional to assume it is Gaussian, since it is usually composed
of several influences, and the central limit theorem suggests that it will tend to a Gaussian
distribution. However, it can happen that at some frequenciesthe noise is dominated by
a single influence, and then it can be markedly non-Gaussian. This has been seen in
ground-based interferometers. An important design goal of LISA will be to ensure that
the noise is mainly Gaussian, and during the analysis the characterisation of the noise
statistics will be an important early step.
Maximum likelihood and the matched filter. The most common way of assessing
whether a signal of some expected form is present in a data stream is to use the maximum
likelihood criterion, which is that one uses as the detection statistic the ratio of the probability
that the given data would be observed if the signal were present to the probability
that it would be observed if the signal were absent. This ratio has a PDF that depends
on the PDF of the noise.
Corrected version 2.08 3-3-1999 9:33