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Astron. Astrophys. 343, 19-22 (1999)

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3. Sensitivity for stochastic waves

By 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 [FORMULA], we obtain (Astone et al. 1993) the g.w. spectrum detectable with [FORMULA] , that is the detector noise spectrum referred to the input


where [FORMULA] is the equivalent temperature of the detector that includes the heating effect (back-action) due to the electronic amplifier, k is the Boltzmann constant, [FORMULA] is the spectral ratio between electronic and brownian noise (Pizzella 1975), Q is the overall quality factor, L is the length of the bar, and [FORMULA] is the resonance frequency of the bar.

Being [FORMULA], at a resonance [FORMULA] we have


(neglecting the transducer constant and the gain of the amplifiers), where [FORMULA] is the velocity of sound in the bar. We notice that, for a bar of given material and resonance frequency [FORMULA], the best spectral sensitivity, obtained on resonance, only depends, according to Eq. (4), on the temperature T, the mass M and the quality factor Q of the detector, provided [FORMULA], that is the coupling between bar and read-out system is sufficiently small.

The bandwidth, with [FORMULA], 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: [FORMULA] and [FORMULA], 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 ALTAIR detector is estimated by analyzing the outputs x(t) and y(t) of the lock-in amplifiers, operating at the two resonant modes. For each mode, the two outputs of the instruments provide two independent noise processes with spectra (Astone et al. 1994b)


where [FORMULA] is the spectrum of the noise forces driving the mechanical oscillator, [FORMULA], and [FORMULA] is the angular resonance frequency of the mode considered.

[FORMULA] represents the transfer function of the antenna as seen through the lock-in amplifier, and [FORMULA] represents the filtering action of the lock-in (neglecting the gain of the amplifiers). [FORMULA] is the amplitude decay time of the mode and [FORMULA] is the integration time of the lock-in amplifier and it is set equal to the sampling time [FORMULA].

The corresponding spectra for a flat g.w. spectrum [FORMULA], will be


In order to estimate [FORMULA] at each mode we must divide the power spectrum obtained from the two discrete sequences [FORMULA] and [FORMULA] by the square modulus of [FORMULA] ([FORMULA] 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 [FORMULA] 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 [FORMULA]), obtained from the square root of the product between each data of the periodogram and the corresponding value of the inverse transfer function [FORMULA] computed with [FORMULA] and [FORMULA]. We note that the minimum value of [FORMULA] is on resonance and corresponds to the value of [FORMULA], that it is only a factor two greater than the Explorer one, at [FORMULA]. On the other hand, the bandwidth of ALTAIR is a factor two greater than that of the Explorer .

[FIGURE] Fig. 1. The square modulus of the FFT (periodogram), for one hour of data of ALTAIR, at the mode+ ([FORMULA]).

[FIGURE] Fig. 2. Sensitivity to stochastic g.w. background with SNR=1 for ALTAIR at [FORMULA]. [FORMULA], [FORMULA], spectrum averaged over one hour. The normalization in terms of h units is obtained computing from the integral value of the periodogram the corrisponding value of [FORMULA] in kelvin, and imposing that the value of [FORMULA] is given by Eq. (6) using the value of [FORMULA].

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) [FORMULA] at [FORMULA] 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 [FORMULA] of useful data.

[FIGURE] Fig. 3. Sensitivity to stochastic g.w. background with SNR=1 for ALTAIR at 1784.85 Hz, computed by hourly periodograms starting on 19 UT, 5 January 1992, [FORMULA], [FORMULA].

In Fig. 4 we report the value of [FORMULA] at [FORMULA] with [FORMULA], [FORMULA], [FORMULA], and [FORMULA] relative to one hour of data starting on 22 UT, 8 April 1992.

[FIGURE] Fig. 4. Sensitivity to stochastic g.w. background with SNR=1 for ALTAIR at [FORMULA]. [FORMULA], [FORMULA], spectrum averaged over one hour.

The sensitivities and bandwidth for the gravitational wave resonant detectors of the Roma group are shown in Table 1. ALTAIR operates at about [FORMULA] and all the others operate at about [FORMULA]. The other operating antennas: Allegro (Mauceli et al. 1996), Niobe (Blair et al. 1995) and Auriga (Vitale et al.1997) have parameters very similar to those of Explorer and Nautilus.


Table 1. Sensitivity and bandwidth of cryogenic gravitational wave detectors of the Roma group.
* experimental strain sensitivity is dimensionless h

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999