UPPER LIMIT FOR THE GRAVITATIONAL WAVE
STOCHASTIC BACKGROUND WITH THE EXPLORER AND NAUTILUS RESONANT DETECTORS

P.Astone^{(1)}, M.Bassan^{(2,1)},
P.Bonifazi^{(1,3)}, P.Carelli^{(1,4)}, E.Coccia^{(2,1)},
C.Cosmelli^{(1,5)}, V. Fafone^{(2,1)}, S.Frasca^{(1,5)},
S.Marini^{(1)}, G.Mazzitelli^{(1)}, P.Modestino^{(1)},
I.Modena^{(2,1)}, A.Moleti^{(1,2)}, G.V.Pallottino^{(1,5)},
M.A.Papa^{(1,2)}, G.Pizzella^{(2,1)}, P.Rapagnani^{(1,5)},
F.Ricci^{(1,5)}, F.Ronga^{(1)}, R.Terenzi^{(1,3)},
M.Visco^{(1,3)}, L.Votano^{(1)}

(1) Istituto Nazionale di Fisica
Nucleare, Sezione di Roma1, Italy

(2) University of Rome "Tor
Vergata" and INFN Sezione di Roma2, Italy

(3) Consiglio
Nazionale delle Ricerche IFSI, Frascati, Italy

(4) University of L'Aquila and Sezione
INFN, Italy

(5) University of Rome "La
Sapienza" and Sezione INFN di Roma1, Italy

Among the possible gravitational waves
(gw) signals, a cosmic stochastic background is one of the most interesting, as
it might give information on the very early stages of the Universe and its
formation. Several sources of stochastic background have been considered in the
past years [1]. We recall the effect of the superposition of many continuous
waves generated by the pulsars, the overlapping bursts due to gravitational
collapses and to coalescence of binary systems.

Nucleosynthesis considerations put an
upper limit on the ratio W of the gw energy density to the critical density needed for a closed
universe. The upper limit is W ≤ 10^{-5}.
As well known the critical density is given by

_{}

where H is the Hubble constant.

Recently a source based on the string
theory has been more deeply investigated [2,3]. The interesting feature of this
theory, from the actual observer point of view, is that it might predict relic
gw whose density W increases in a certain range with the frequency f to
the third power (remaining below the nucleosynthesis limit). In fact the
previous models tend to predict gw in the frequency range below 1 Hz, lower
than the operating frequency of the present detectors already in operation
(resonant bars) or entering in operation in the next four to five years
(long-arm interferometers [4]). Only the newly proposed space experiment LISA
could explore, with good sensitivity, a frequency range below 1 Hz, but such an
experiment, if approved, will fly after the year 2016.

The predictions of the new string
cosmology models depend on a number of parameters, such as the maximum
frequency and the precise dependency of W on the frequency
when it gets near to the maximum value. A measurement, even an upper limit,
would help very much in delineating the exact model. At this stage is therefore
very important to turn to the experiment.

The Rome group has operated the
cryogenic resonant antenna EXPLORER [5] since 1990 and the ultracryogenic
resonant antenna NAUTILUS [6] since 1995. Therefore it is worthwhile to study
the recorded data to look for useful experimental information on relic gw.

The study of this problem has shown that
the presently available resonant detectors are suited for this type of
measurement. As a matter of fact it turns out, as shown below, that the sensitivity of the resonant antennas
to a stochastic background (for the present detectors operating with dcSQUID
electronic amplifiers) depends essentially on the quantity

(2)

where T, M and Q are the detector thermodynamic
temperature, mass and quality factor. The bandwidth of the apparatus enters to
a minor extent, as it will be shown later. Thus what is the drawback of a
resonant detector, namely the small bandwidth, does not jeopardize the
measurement of the stochastic background.

The equation for the
end bar displacement x is

(3)

where F is the applied force, m the oscillator reduced
mass (for a cylinder m = M/2), w_{o}=2pf_{o} is the angular
resonance frequency and Q is the merit factor.

We consider here
only the noise which can be easily modeled. This 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 [6] the noise power spectrum : (4)

with

(5)

where T_{e} is the equivalent temperature that
includes the effect of the back-action from the electronic amplifier and G (usually G<<1) is the spectral ratio between electronic
and brownian noise [7]

(6)

T_{n} is the amplifier noise temperature and b the coupling parameter of the transducer 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 [7] to the action of a force

(7)

For a gw excitation with power spectrum
S_{h}(f), the spectrum of the corresponding bar end displacement is

(8)

We notice that the power spectrum of the
bar displacement for a constant spectrum of gw 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).

By taking the ratio of the noise
spectrum (4) and the signal spectrum (8) we obtain the signal to noise ratio
(SNR)

(9)

By equating to unity the above ratio we
obtain the gw spectrum detectable with SNR=1 ,that is the detector noise
spectrum referred to the input

(10)

At the resonance f_{o} we have
(being G<<1)

(11)

We remark that the equivalent
temperature T_{e }reduces to T if
the backaction from the electromechanical transducer can be neglected, as in
the case of a dcSQUID.

The above quantity S_{h}(f)
must be related to the

_{}

predicted by the theory. It turns out that [3] we have

_{}

The target sensitivity of ultracryogenic
antennas like NAUTILUS [8] or AURIGA [9] with f=920 Hz, M=2300 kg, T=0.1 K and
Q=5 10^{6} is S_{h}(f)=(8.6 10^{-23}/√Hz)^{2}. This is not sufficient to reach the limit imposed by the
nucleosynthesis bound of W≤10^{-5}, but it gives an upper limit.

We give now the results of measurements
made with the antennas EXPLORER in the years 1991 (Fig.1) and 1994 (Fig.2) and
NAUTILUS operating at T=1.3 K in the
year 1995 (Fig.3). NAUTILUS is capable to operate below 0.1 K, but in 1995 we
operated it at 1.3 K because we had some excess noise. We recall that both
EXPLORER and NAUTILUS employ a resonant electromechanical transducer, thus
showing two resonances which may have different sensitivity according to the
noise on each one and to the tuning of the transducer to the bar.

Fig.1

Sensitivity to stochastic gw background with SNR=1 for EXPLORER.

T=2.9 K, M=2300 kg, Q=10^{6}, average spectrum over
31.4 days (1991).

Fig.2

Sensitivity to stochastic gw background with SNR=1 for EXPLORER.

T=2.4 K, M=2300 kg, Q=5 10^{6}, average spectrum
over 36 hours (1994).

With one detector only, the sensitivity
does not depend on the length of the measuring time. Increasing the time of
measurement would just reduce the error in the spectral determination, leaving
practically unchanged the level of the spectrum. We notice that in 1994 we
obtained at both the resonances (907 Hz and 923.32 Hz) a measurement 6 10^{-22}/√Hz. The upper limit from these measurements
turns out to be still very high, about W=300.

Fig.3

Sensitivity to stochastic gw background with SNR=1 for NAUTILUS.

T=1.3 K, M=2300 kg, Q=2.6 10^{6}, average spectrum
over 2.3 hours (1995).

At the frequency of 923.8 Hz we obtain
from NAUTILUS 7 10^{-22}/√Hz.

Better sensitivity can be obtained by
cross correlating the output of two antennas, because the local noises are
uncorrelated and the sensitivity improves with a longer measuring time. It can
be shown [10] that, in such a case, if the two identical antennas with
respective spectral outputs S_{1h}
and S_{2h} are close to each
other, within a distance much smaller than the gw wavelength [11], the
sensitivity is

_{}

where t_{m}
is the measuring time and ∆f is the antenna bandwidth.

We see in the above the effect of the
bandwidth which enters as the 1/4 power for the usually given sensitivity
expressed as 1/√Hz. With the present resonant detectors at T=0.1 K having
∆f=1 Hz and for a measuring time of one year one can reach dS_{h}(f)=(1.1 10^{-24}/√Hz)^{2 }corresponding to W=1.3 10^{-3}. at 920 Hz.

In order to reach the limit of W=10^{-5}, two resonant
detectors each one cooled to 10 mK and with a ten times larger mass would be
required. If such two resonant detectors operate at their quantum limit then
the bandwidth may become as large as ∆f=50 Hz thus allowing to reach W=5 10^{-6 }at f=920 Hz.

Acknowledgments

We thank R.Brustein, M.Gasperini and
G.Veneziano for stimulating discussions and suggestions. Thanks to M.Cerdonio
and E.Picasso for providing a stimulating forum for discussing the gw
stochastic background detection.

References

1. K.S.Thorne, in 300 years of Gravitation,
S.W.Hawking and W.Israel, Eds.
(Cambridge Univ. Press, Cambridge, 1987)

2. R.Brustein, M.Gasperini, M.Giovannini,
G.Veneziano

"Relic gravitational waves from string cosmology"

Physics Letters B 361(1995) 45-51

3. R.Brustein, M.Gasperini, G.Veneziano

"Spectrum of Cosmic Gravitational Waves Background
from String Cosmology"

CERN TH/96-37, February 1996

4. E. E.
Flanagan

Phys. Review D48 (1993) 2389

5. P.Astone, M.Bassan, P.Bonifazi,
P.Carelli, M.G.Castellano, G.Cavallari,
E.Coccia, C.Cosmelli,
V.Fafone, S.Frasca, F.Ricci, M.Visco,

"Long-term
operation of the Rome Explorer cryogenic gravitational
wave detector".

Physical Review D, 47, 362, January (1993).

6 G.V.Pallottino,
G.Pizzella,

"Sensitivity of a Weber type resonant
antenna to monochromatic gravitational
waves".

Il Nuovo Cimento, 7C, 155 (1984).

7 G.Pizzella,

"Gravitational-Radiation
experiments".

Il
Nuovo Cimento 5, 369 (1975).

8 P.Astone, M.Bassan, P.Bonifazi,
M.G.Castellano, P.Carelli, E.Coccia,
C.Cosmelli, S.Frasca, V.Fafone, E.Majorana, A.Marini, G.Mazzitelli,I.Modena, G.Modestino, G.V.Pallottino, M.A.Papa, G.Pizzella, P.Puppo, P.Rapagnani, F.Ricci,
F.Ronga, R.Terenzi, M.Visco,
L.Votano

"The Nautilus experiment"

First Edoardo Amaldi Conference on Gravitational wave
Experiments

Frascati, 14-17 June 1994

9. M.Cerdonio, L.Franceschini, G.Fontana,
R.Mezzena, S.Paoli, G.A.Prodi, S.Vitale,
J.P.Zendri, M.Biasotto, M.Lollo, F.Bronzini, R.Macchietto,
G.Maron, A.Ortolan, M.Strollo, G.Vedovato, M.Bonaldi,
P.Falferi, E.Cavallini, P.L.Fortini, E.Montanari, L.Taffarello, A.Colombo, D.Pascoli, B.Tiveron

"Status of the Auriga gravitational
wave antenna"

First Edoardo Amaldi Conference on Gravitational wave Experiments

Frascati, 14-17 June 1994

10. J.S.Bendat and A.G.Piersol

"Measurement
and analysis of random data"

John
Wiley & Sons,, New York, 1966

11. P.F.Michelson

"On detecting stochastic background gravitational
radiation with terrestrial
detectors"

Mon. Not. R. astr. Soc.(1987) 227, 933-941