## 3. Sensitivity for stochastic wavesBy equating to unity the ratio of the noise spectrum (due to the
thermal noise of the detector and electronic noise contributed by the
readout system), and the spectrum of the bar end displacement due to a
g.w. excitation with power spectrum ,
we obtain (Astone et al. 1993) the where is the equivalent
temperature of the detector that includes the heating effect
(back-action) due to the electronic amplifier, Being , at a resonance we have (neglecting the transducer constant and the gain of the
amplifiers), where is the velocity
of sound in the bar. We notice that, for a bar of given material and
resonance frequency , the best
spectral sensitivity, obtained on resonance, only depends, according
to Eq. (4), on the temperature The bandwidth, with , estimated at half-height of the power spectrum, is given by (Pallottino et al. 1984) The previous formulas are valid for a bar equipped with a non-resonant transducer. In the case of a detector with a resonant transducer, we have to take into account the stochastic force acting on the transducer oscillator. If the transducer is well tuned to the bar, the effect of this additional force is equivalent to double the force spectrum. This means that the final spectral sensitivity is reduced by a factor of 2: and , given by Eq. (4) are twice than before. For any arbitrary tuning of the transducer formulas (3) and (4) can be also used, but the equivalent force spectra for the two modes are different This means that at one mode we can obtain a better spectral sensitivity at the expense of a reduced sensitivity at the other mode (Astone et al. 1997a). The g.w. spectrum for the where is the spectrum of the noise forces driving the mechanical oscillator, , and is the angular resonance frequency of the mode considered. represents the transfer function of the antenna as seen through the lock-in amplifier, and represents the filtering action of the lock-in (neglecting the gain of the amplifiers). is the amplitude decay time of the mode and is the integration time of the lock-in amplifier and it is set equal to the sampling time . The corresponding spectra for a flat g.w. spectrum , will be In order to estimate at each mode we must divide the power spectrum obtained from the two discrete sequences and by the square modulus of ( is the inverse filter that cancels the dynamic of the antenna and of the lock-in integrator). In Fig. 1 we report the square modulus of the FFT (periodogram)
relative to the data of the mode at
for a time period of one hour starting on 17 UT, 7 January 1992, and
in Fig. 2 the corresponding estimation of the square root of the g.w.
spectrum detectable with SNR=1 (amplitude strain
), obtained from the square root of
the product between each data of the periodogram and the corresponding
value of the inverse transfer function
computed with
and
. We note that the minimum value of
is on resonance and corresponds to
the value of , that it is only a
factor two greater than the
The bandwidth is an important parameter to measure the g.w. stochastic background, by crosscorrelating the output of two identical antennas close to each other, within a distance much smaller than the g.w. wavelength (Astone et al. 1996). In Fig. 3 we show the amplitude (strain) at computed by the hourly periodogram of the mode+ for a period of 84 hours starting on 19 UT, 5 January 1992. We note that we have a duty cycle of of useful data.
In Fig. 4 we report the value of at with , , , and relative to one hour of data starting on 22 UT, 8 April 1992.
The sensitivities and bandwidth for the gravitational wave resonant
detectors of the Roma group are shown in Table 1.
© European Southern Observatory (ESO) 1999 Online publication: March 1, 1999 |