The assumption of stationarity is not strictly valid in the case of real gravitational-wave detectors; however, if their performance doesn’t vary greatly over time scales much larger than typical observation time scales, stationarity could be used as a working rule. While this may be good enough in the case of binary inspiral and coalescence searches, it is a matter of concern for the observation of continuous and stochastic gravitational waves. In this review, for simplicity, we shall assume that the detector noise is stationary. In this case the one-sided noise PSD, defined only at positive frequencies, is the Fourier transform of the noise auto-correlation function:
where a factor of 1/2 is included by convention. By using the Fourier transform of It is obvious that has dimensions of time but it is conventional to use the dimensions of Hz–1,
since it is a quantity defined in the frequency domain. The square root of
is the noise amplitude,
, and has dimensions of Hz–1/2. Both noise PSD and noise amplitude measure the noise in a
linear frequency bin. It is often useful to define the power per logarithmic bin
, where
is called the effective gravitational-wave noise, and it is a dimensionless quantity. In
gravitational-wave–interferometer literature one also comes across gravitational-wave displacement noise or
gravitational-wave strain noise defined as
, and the corresponding noise spectrum
, where
is the arm length of the interferometer. The displacement noise gives
the smallest strain
in the arms of an interferometer that can be measured at a given
frequency.
As mentioned earlier, the performance of a gravitational wave detector is characterized by the one-sided noise PSD. The noise PSD plays an important role in signal analysis. In this review we will only discuss the PSDs of interferometric gravitational-wave detectors.
The sensitivity of ground based detectors is limited at frequencies less than a Hertz by the time-varying
local gravitational field caused by a variety of different noise sources, e.g., low frequency seismic vibrations,
density variation in the atmosphere due to winds, etc. Thus, for data analysis purposes, the noise PSD is
assumed to be essentially infinite below a certain lower cutoff . Above this cutoff, i.e., for
,
Table 1 lists the noise PSD
for various interferometric detectors and some of these are plotted in
Figure 5
.
For LISA, Table 1 gives the internal instrumental noise only, taken from [162]. It is based on the noise budget obtained in the LISA Pre-Phase A Study [72]. However, in the frequency range 10–4 – 10–2 Hz, LISA will be affected by source confusion from astrophysical backgrounds produced by several populations of galactic binary systems, such as closed white-dwarf binaries, binaries consisting of Cataclysmic Variables, etc. At frequencies below about 1 mHz, there are too many binaries for LISA to resolve in, say, a 10-year mission, so that they form a Gaussian noise. Above this frequency range, there will still be many resolvable binaries which can, in principle, be removed from the data.
Detector | ![]() |
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GEO | 40 | 150 | 1.0 × 10–46 | ![]() |
ILIGO | 40 | 150 | 9.0 × 10–46 | ![]() |
TAMA | 75 | 400 | 7.5 × 10–46 | ![]() |
VIRGO | 20 | 500 | 3.2 × 10–46 | ![]() |
ALIGO | 20 | 215 | 1.0 × 10–49 | ![]() |
ET | 10 | 200 | 1.5 × 10–52 | ![]() |
LISA | 10–5 | 10–3 | 9.2 × 10–44 | ![]() |
Nelemans et al. [272] estimate that the effective noise power contributed by binaries in the galaxy is
normalized to the same The fraction of uncontaminated frequency bins as a function of frequency remains to be specified.
Let
be the number of binaries in the galaxy per unit frequency. Since the size of the frequency bin
for an observation that lasts a time
is
, the expected number of binaries per frequency bin
is
Barack and Cutler multiply this by a “fudge factor” to allow for the fact that any binary may
contaminate several bins, so that
is the expected number of contaminated bins per binary. If this
is small, then it will equal the fractional contamination at frequency
. In that case, the fraction of
uncontaminated bins is just
. However, if the expected contamination per bin approaches or
exceeds one, then we have to allow for the fact that the binaries are really randomly distributed in
frequency, so that the expected fraction not contaminated comes from the Poisson distribution,
Because LISA will observe binaries for several years, the accuracy with which it will know the frequency, say, of a binary, will be much better than the frequency resolution of LISA during the observation of a transient source, such as many of the IMBH events considered by Barack and Cutler. Therefore, there is a good chance that, in the global LISA data analysis, the effective noise can be reduced below the one-year noise levels that are normally used in projecting the sensitivity of LISA and the science it can do.
In radio astronomy one talks about the sensitivity of a telescope in terms of the limiting detectable energy
flux from an astronomical source. We can do the same here too. Given the gravitational wave amplitude
we can use Equation (17
) to compute the flux of gravitational waves. One can translate the noise power
spectrum
, given in units of Hz–1 at frequency
, to Jy (Jansky), with the conversion factor
. In Figure 6
, the left panel shows the noise power spectrum in astronomical units of Jy
and the right panel depicts the noise spectrum in units of Hz–1 together with lines of constant
flux.
What is striking in Figure 6 is the magnitude of flux. While modern radio interferometers are sensitive
to flux levels of milli and micro-Jy the gravitational wave interferometers need their sources
to be 24 – 27 orders of magnitude brighter. Turning this argument around, the gravitational
wave sources we expect to observe are not really weak, but rather extremely bright sources.
The difficulty in detecting them is due to the fact that gravitation is the weakest of all known
interactions.
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