Extracted from: Internal report LNF-96/001
Detection of impulsive, monochromatic and stochastic gravitational waves by means of resonant antennas
INFN, Sezione di Roma "La Sapienza"
G. V. Pallottino
Dipartimento di Fisica "La Sapienza"
INFN, Sezione di Roma "La Sapienza"
Dipartimento di Fisica "Tor Vergata"
Laboratori Nazionali di Frascati dell' INFN
In this note we express in a simple and unitary form, although sometimemes with approximations only aimed to help clarity, the sensitivity of resonant antennas to various types of gravitational waves. As a matter of fact, in the last years, some of those detectors begun to operate with very satisfactory performance and high duty cycle over relatively long periods of time, and more are close to operate
As a model for these detectors, we shall consider the simplest resonant antenna, a cylinder of high Q material, strongly coupled to a non resonant transducer followed by a very low noise electronic amplifier.
In practice, the detectors now operating use resonant transducers (and therefore there are two modes coupled to the gravitational field) to obtain high coupling and high Q, followed by a dc SQUID superconducting amplifier or by a microwave parametric amplifier.
The equation for the end bar displacement x is
where f is the applied force, m the oscillator reduced mass (for a cylinder m = M/2) and b1=wo/2Q.
We consider here only the noise which can be easily modeled. The noise of the detector is the sum of two terms : the thermal (Brownian) noise of the basic detector and the electronic noise contributed by the readout system. By referring the overall noise to the displacement of the bar ends, we obtain the power spectrum: (2)
where Te is the equivalent temperature which includes the effect of the back-action from the electronic amplifier and G is the spectral ratio between electronic and brownian noise 
Tn is the amplifier noise temperature and b the transducer coupling to the bar (b ≈ 10-2-10-3). The power spectrums are expressed in two-sided form.
When a gravitational wave with amplitude h and optimum polarization impinges perpendicularly to the bar axis, the bar displacement corresponds  to the action of a force
The bar end spectral displacement due to a continuous spectrum of g.w. is similar to that due to the action of the Brownian force. Therefore, if only the Brownian noise were present, we would have an infinite bandwidth, in terms of signal to noise ratio (SNR).
2. The power spectrums
For a g.w. excitation with power spectrum Sh(w), the spectrum of the corresponding bar end displacement is
We can then write the SNR
The g.w. spectrum that can be detected with SNR=1 (that is the detector noise spectrum referred to the input) is obtained by introducing this condition in the above:
where v is the sound velocity in the bar material (v=5400 m/s in aluminum).
We remark the the best spectral sensitivity, obtained at the resonance frequency of the detector, only depends, according to eq. (7), on the temperature T, on the mass M and on the quality factor Q of the detector, provided T=Te, that is the coupling between bar and read-out system is sufficiently small. Note that those conditions are rather different from that required for optimum pulse sensitivity (see later).
The bandwidth can be calculated with the formula
It is expected that the bandwidth would become of the order of 10 Hz by improving the amplifier noise temperature Tn from 20 mK to 0.5 mK
We come now into applying the Wiener optimum filter for detecting small signals in the noise. It can be shown  that the SNR for a gravitational wave signal h(t) whose Fourier transform we indicate with H(w) is given by
with Sh(n) given by (7).
3. Thorne definitions
Thorne defines, for broad band detectors, the following characteristic frequency of the g.w.
and a characteristic strength :
hc = √[Sh(nc) nc] (11)
as can be seen by putting in (9) and (10) SNR=1.
4.1 Resonant detectors
We solve (9) with SNR=1 by noticing that Sh has a minimum around the resonance (see fig. 1) and that for a short burst H(w)=constant=Ho. From (9) we obtain
Ho2=2 Sh(w0)/ (p Dn) (13)
with ∆n given by (8). A factor of p has been introduced because we need the equivalent frequency band-width for a bilateral Ho.
Introducing (7) we get
if the effective temperature .
With the values given in fig.1 we get Teff=0.8 mK.
With the antenna Explorer we have already reached values of the order of 5 mK (with Te = 2.5 K) in close agreement with our expectation.
Formula (13) is still valid with ∆n ≈ no= nc as can be seen from (9)
We have roughly for SNR=1
hc = H no = √ (Sh no) (16)
like (11), which is the widely used formula. We have to remark that the definition (11) is not consistent with the case of a resonant detector.