LISALISA - iucaa

1.2 Low-frequency sources of gravitational radiation
of spurious signals is sufficiently low. For an observation of length T, the frequency resolution
is just 1/T, and so a longer observation needs to fight against the noise in a smaller bandwidth.
Since noise power is proportional to bandwidth, the rms noise amplitude is proportional to the
square root of the bandwidth, and the result is that the noise at any frequency falls as 1/ √ T. In a
1-year observation, the frequency resolution is 3×10 −8 Hz, and there are (1Hz)/(3×10 −8 Hz) =
3×10 7 resolvable frequencies in the LISA band.
For expected signals due to binaries in our galaxy, the intrinsic wave amplitude h is essentially
constant during a 1-year observation. Such sources are placed in the diagram to show this h on
the vertical scale. But because of LISA’s motion, LISA almost never responds to this maximum
amplitude; rather, the full signal-to-noise ratio SNR over a year is lower by a factor which depends
on the exact position of the source relative to LISA’s orbit. Since the rms antenna sensitivity
for LISA averaged over a year is nearly isotopic [12], we can approximate this effect by using the
reduction factor averaged over the entire sky, which is 1/ √ 5 [6]. This means that, if a source
lies above the 1-σ noise level by a certain factor s, the expected SNR will be typically s/ √ 5.
To be specific, the threshold sensitivity curve in Figure 1.3 is drawn to correspond to a SNR of
5 in a 1-year observation. (Accordingly, it is drawn at a factor of 5 √ 5 ≈ 11 above the 1-year,
1-σ noise level.) This SNR of 5 is a confidence level: for a 1-year observation, the probability
that Gaussian noise will fluctuate to mimic a source at 5 standard deviations in the LISA search
for sources over the whole sky is less than 10 −5 , so one can be confident that any source above
this threshold curve can be reliably detected. To estimate the expected SNR for any long-lived
source in the diagram, one multiplies the factor by which it exceeds the threshold curve by the
threshold level of 5. The threshold curve is drawn on the assumption that the dominant noise is
the 1-σ instrumental noise level. If any of the random gravitational-wave backgrounds described
above are larger, then the threshold must likewise go up, remaining a factor of 5 √ 5 above the
rms gravitational-wave noise.
It is important when looking at Figure 1.3 to realise that even sources near the threshold curve
will be strongly detected: the X-ray binary 4U1820-30 is only a factor of 2 above the curve, but
that implies an expected SNR in amplitude of 10, or in power of 100. Any observation by LISA
above the threshold curve will not only be a detection: there will be enough signal to extract
other information as well, and that will be important in our discussion below.
Note also that sources can be detected below the threshold curve if we have other information
about them. For example, if a binary system is already known well enough to determine its
orbital period and position, then the confidence level can be lowered to something like 3σ,
where the probability would still be less than 10 −4 that (on Gaussian statistics) the noise was
responsible for the observation.
The phase-modulation of a signal produced by LISA’s orbital motion will require that, in the
data analysis, a compensating correction be applied to the data in order to give a signal its
expected SNR as indicated in the diagram. This correction will depend on the assumed location
of the source on the sky. At 0.1 Hz, there may be as many as 10 5 distinguishable locations,
and so there are 10 5 different chances for noise to mimic a source at any level. This factor has
been taken into account in adopting the threshold level of 5 standard deviations in the diagram:
the chances that Gaussian noise will produce a false alarm anywhere in these different locations
at this level is still less than 10 −4 . The data analysis will of course test whether the noise is
Gaussian, and may then set the threshold differently if necessary.
In Chapter 4 we describe in some detail how LISA’s sensitivity is calculated, but here it is
appropriate to note where its main features come from. The best sensitivity is between 3 and
10 mHz. In this range the sensitivity is limited by photon shot noise plus a combination of other
noise sources that are assumed for simplicity to be roughly white, including variations in the
Corrected version 1.04 19 13-9-2000 11:47