### 4.5 Characterizing the sensitivity of a gravitational wave antenna

The performance of a gravitational wave detector is characterized by the power spectral density (henceforth denoted PSD) of its noise background. One can construct the noise PSD as follows; a gravitational wave detector outputs a dimensionless data train, say , which in the case of an interferometer is the relative strain in the two arms, scaled to represent the value of that would produce that strain if the wave is optimally oriented with respect to the detector. In the absence of any gravitational wave signal, the detector output is just an instance of noise , that is, . The noise auto-correlation function is defined as
where an overline indicates the average over an ensemble of noise realizations. In general, depends both on and . However, if the detector output is a stationary noise process, i.e., its performance is, statistically speaking, independent of time, then depends only on .

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 , that is , in Equation (65) and substituting the resulting expression in Equation (66), it is easy to shown that for a stationary noise process background
where denotes the complex conjugate of . The above equation justifies the name PSD given to .

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.

#### 4.5.1 Noise power spectral density in interferometers

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.

Table 1: Noise power spectral densities of various interferometers in operation and under construction: GEO600, Initial LIGO (ILIGO), TAMA, VIRGO, Advanced LIGO (ALIGO), Einstein Telescope (ET) and LISA (instrumental noise only). For each detector the noise PSD is given in terms of a dimensionless frequency and rises steeply above a lower cutoff . The parameters in the ET design sensitivity curve are , , , , , , , , , , , , , . (See also Figure 5.)
 Detector /Hz /Hz /Hz–1 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 as we use for LISA in Table 1. This power is a mean frequency average based on projections of the population LISA will find, but, of course, above about 1 mHz, LISA will resolve many binaries and identify most of the members of this population. Barack and Cutler [69] have provided a prescription for including this effect when adding in the confusion noise. They make the conservative assumption that individual binaries contaminate the instrumental noise (see Table 1) in such a way that, effectively, one or a few frequency resolution bins need to be cut out and ignored when detecting other signals, including, of course, other binary signals. This would have approximately the same effect as if the overall instrumental noise at that frequency were raised by an amount obtained simply by dividing the noise by the fraction of bins free of contamination. Of course, when this fraction reaches zero (below 1 mHz), this approximation is not valid, and instead one should just add the full binary confusion noise in Equation (68) to the instrumental noise. A smooth way of merging these two regimes is to set
where is from Table 1 and is from Equation (68). This prescription uses the contaminated instrumental noise, when it is below the total noise power from the binaries, but then uses the total binary confusion power when the prescription for allowing for contamination breaks down.

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,

Inserting this into Equation (69) gives a reasonable approximation to the effective instrumental noise if binaries cannot be removed in a clean way from the data stream when looking for other signals.

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.

#### 4.5.2 Sensitivity of interferometers in units of energy flux

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.