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 x(t), which in the case of an interferometer is the relative strain in the two arms, scaled to represent the value of h 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 n(t), that is, x(t) = n(t). The noise auto-correlation function κ is defined as
---------- κ ≡ n(t1)n(t2), (65 )
where an overline indicates the average over an ensemble of noise realizations. In general, κ depends both on t1 and t2. However, if the detector output is a stationary noise process, i.e., its performance is, statistically speaking, independent of time, then κ depends only on τ ≡ |t1 − t2|.

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:

1 ∫ ∞ 2πifτ Sh(f) ≡ -- κ(τ)e d τ, f ≥ 0, 2 − ∞ ≡ 0, f < 0, (66 )
where a factor of 1/2 is included by convention. By using the Fourier transform of n(t), that is ∫∞ 2πift &tidle;n (f ) ≡ −∞ n(t)e dt, in Equation (65View Equation) and substituting the resulting expression in Equation (66View Equation), it is easy to shown that for a stationary noise process background
----------- 1 &tidle;n(f)&tidle;n ∗(f ′) = --Sh(f)δ(f − f ′), (67 ) 2
where &tidle;n∗(f) denotes the complex conjugate of &tidle;n (f ). The above equation justifies the name PSD given to Sh(f ).

It is obvious that Sh (f ) 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 Sh(f) is the noise amplitude, ∘ ------ Sh(f), 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 h2n(f) ≡ f Sh(f), where hn (f ) 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 h ℓ(f ) ≡ ℓhn(f), and the corresponding noise spectrum S ℓ(f ) ≡ ℓ2Sh (f), where ℓ is the arm length of the interferometer. The displacement noise gives the smallest strain δℓ∕ℓ in the arms of an interferometer that can be measured at a given frequency.

View Image

Figure 5: The right panel plots the noise amplitude spectrum, ∘ ------- f Sh(f), in three generations of ground-based interferometers. For the sake of clarity, we have only plotted initial and advanced LIGO and a possible third generation detector sensitivities. VIRGO has similar sensitivity to LIGO at the initial and advanced stages, and may surpass it at lower frequencies. Also shown are the expected amplitude spectrum of various narrow and broad-band astrophysical sources. The left panel is the same as the right except for the LISA detector. The SMBH sources are assumed to lie at a redshift of z = 1, but LISA can detect these sources with a good SNR practically anywhere in the universe. The curve labelled “Galactic WDBs” is the confusion background from the unresolvable Galactic population of white dwarf binaries.

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 fs. Above this cutoff, i.e., for f ≥ fs, Table 1 lists the noise PSD Sh(f) for various interferometric detectors and some of these are plotted in Figure 5View Image.

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 Sh(f) 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 x = f∕f0 and rises steeply above a lower cutoff fs. The parameters in the ET design sensitivity curve are α = − 4.1, β = − 0.69, a0 = 186, b0 = 233, b1 = 31, b2 = − 65, b3 = 52, b4 = − 42, b5 = 10, b6 = 12, c1 = 14, c2 = − 37, c3 = 19, c4 = 27. (See also Figure 5View Image.)
Detector fs/Hz f0/Hz S0/Hz–1 Sh (x)∕S0
GEO 40 150 1.0 × 10–46 − 30 −1 20(1−x2+0.5x4) (3.4x ) + 34x + (1+0.5x2)
         
ILIGO 40 150 9.0 × 10–46 (4.49x )−56 + 0.16x −4.52 + 0.52 + 0.32x2
         
TAMA 75 400 7.5 × 10–46 x−5 + 13x −1 + 9(1 + x2 )
         
VIRGO 20 500 3.2 × 10–46 (7.8x)−5 + 2x− 1 + 0.63 + x2
         
ALIGO 20 215 1.0 × 10–49 x −4.14 − 5x−2 + 111(1−x2+0.25x4) 1+0.5x
         
ET 10 200 1.5 × 10–52 α β b0(1+b1x+b2x2+b3x3+b4x4+b5x5+b6x6) x + a0x + 1+c1x+c2x2+c3x3+c4x4
         
LISA 10–5 10–3 9.2 × 10–44 (x∕10)− 4 + 173 + x2

Nelemans et al. [272] estimate that the effective noise power contributed by binaries in the galaxy is

gal ( f )7∕3 Sh = 2.1 × 10 −38 -- Hz− 1, fs = 10− 3Hz, (68 ) fs
normalized to the same fs 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 Sinstr h (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 η 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 (68View Equation) to the instrumental noise. A smooth way of merging these two regimes is to set
( ) full 1-instr instr gal Sh = min ηSh , Sh + Sh , (69 )
where Sinstr h is from Table 1 and Sgal h is from Equation (68View Equation). 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 η of uncontaminated frequency bins as a function of frequency remains to be specified. Let dN ∕df 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 Tobs is 1∕Tobs, the expected number of binaries per frequency bin is

1 dN (f ) ΔN (f) = -----------. Tobs df

Barack and Cutler multiply this by a “fudge factor” κ > 1 to allow for the fact that any binary may contaminate several bins, so that κΔN (f) is the expected number of contaminated bins per binary. If this is small, then it will equal the fractional contamination at frequency f. In that case, the fraction of uncontaminated bins is just 1 − κΔN (f). 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,

η = exp(− κΔN ). (70 )
Inserting this into Equation (69View Equation) 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 h we can use Equation (17View Equation) to compute the flux of gravitational waves. One can translate the noise power spectrum Sh(f ), given in units of Hz–1 at frequency f, to Jy (Jansky), with the conversion factor 3 2 4c f ∕ (πG ). In Figure 6View Image, 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.

View Image

Figure 6: The sensitivity of interferometers in terms of the limiting energy flux they can detect, Jy/Hz, (left panel) and in terms of the gravitational wave amplitude with lines of constant flux levels (right panel).

What is striking in Figure 6View Image 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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