Extracted from: Internal report LNF-96/001

(1996)

Detection of impulsive, monochromatic and stochastic gravitational waves by means of resonant antennas

P. Astone

INFN, Sezione di
Roma "La Sapienza"

G. V. Pallottino

Dipartimento di
Fisica "La Sapienza"

INFN, Sezione di
Roma "La Sapienza"

G. Pizzella

Dipartimento di
Fisica "Tor Vergata"

Laboratori
Nazionali di Frascati dell' INFN

1. __Introduction__

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

(1)

where f is the applied force, m the oscillator reduced mass (for a
cylinder m = M/2) and b_{1}=w_{o}/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 T_{e} 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 [7]

(3)

T_{n} 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 [8] to the action of a force

(4)

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 S_{h}(w), the spectrum
of the corresponding bar end displacement is

(5)

We can then write the SNR

(6)

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:

(7)

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=T_{e}, 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

(8)

It is expected that the bandwidth would become of the
order of 10 Hz by improving the amplifier noise temperature T_{n }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 [9] that the SNR for a gravitational wave signal h(t) whose Fourier transform we indicate with H(w) is given by

(9)

with S_{h}(n) given by (7).

3. __Thorne definitions __

Thorne defines, for broad band detectors, the following characteristic frequency of the g.w.

(10)

and a characteristic strength :

h_{c}
= √[S_{h}(n_{c}) n_{c}] (11)

Thus

(12)

as can be seen by putting in (9) and (10) SNR=1.

4. __Bursts__

4.1__ Resonant detectors__

We solve (9) with SNR=1 by noticing that S_{h}
has a minimum around the resonance (see fig. 1) and that for a short burst H(w)=constant=H_{o}.
From (9) we obtain

Ho*2*=2 S*h*(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 H_{o}.

Introducing (7) we get

(14)

where

(15)

if the effective temperature [12].

With the values given in fig.1 we get T_{eff}=0.8
mK.

With the antenna Explorer we have already reached values of the order of 5 mK (with T_{e} = 2.5 K) in close agreement with our expectation.

4.2 __Interferometers__

Formula (13) is still valid with ∆n ≈ n_{o}= n_{c }as can be seen from (9)

_{}

We have roughly for SNR=1

h_{c}
= H n_{o} = √ (S_{h} n_{o}) (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.