Search for Coincident Excitation of the Widely Spaced Resonant Gravitational Wave Detectors EXPLORER , NAUTILUS and NIOBE
P.Astone^{a}, M.Bassan^{b}, D.G.Blair^{c},
P.Bonifazi^{d,a}, P.Carelli^{e}, E.Coccia^{b}, C.Cosmelli^{f}, V.Fafone^{g},
L.Febo ^{f}, S.Frasca^{f}, I.S.Heng^{c}, E.N.Ivanov^{c},
A.Marini^{g}, E.Mauceli^{g}, S.M.Merkowitz^{g},
Y.Minenkov^{b}, I.Modena^{b}, G.Modestino^{g}, A.Moleti^{b},
G.V.Pallottino^{f}, M.A.Papa^{b}, G.Pizzella^{b,g},
F.Ronga^{g}, R.Terenzi^{b,d},
M.E.Tobar^{c}, P.J.Turner^{c}, F.J.van Kann^{c}, M.Visco^{b,d}, L.Votano^{g}.
^{a
}Istituto
Nazionale di Fisica Nucleare (INFN), Rome, Italy
^{b
} Physics Dept, University “Tor Vergata” and
INFN, Rome, Italy
^{c
} University of Western Australia, Perth,
Australia
^{d
} Istituto di Fisica dello Spazio
Interplanetario CNR, Rome, Italy
^{e
} University of L’Aquila and INFN, L’Aquila,
Italy
^{f
} Physics Dept, University “La Sapienza” and
INFN, Rome, Italy
^{g
} Laboratorio Nazionale INFN, Frascati, Italy
PACS:
04.80.y
Abstract
We report the search for coincidences among
three resonant mass detectors: EXPLORER^{ }at CERN and NAUTILUS in
Frascati of the Rome group and NIOBE in Perth of the UWA group. The three
detectors have a sensitivity for short bursts of GW in the h ≈ 10^{18} range, about one thousand times
better in energy than Weber's original detectors. The analysis is based on the comparison of candidate event lists
recorded by the detectors in the period December 1994 through October 1996. The events have been obtained by applying a
pulse detection filter to the raw data and using a predetermined threshold. Due
to the different periods of data taking it was not possible to search for
triple coincidences. We searched for coincidences between EXPLORER and NAUTILUS
during the years 1995 and 1996 for a total time coverage of 1372 hours and
between EXPLORER and NIOBE in 1995 for a coverage of 1362 hours. The results
have been: a weak indication of a coincidence excess with respect to the
accidental ones between EXPLORER and NAUTILUS and no coincidence excess between
EXPLORER and NIOBE.
Corresponding
author G.Pizzella
Institute for
Nuclear Physics, Frascati
Via E.Fermi,
40  00044 Frascati (Rome), Italy
email:
pizzella@lnf.infn.it
Telephone: 0694032796, fax: 0694032427
1.
Introduction
Almost
30 years ago Weber reported [1] coincident excitation of a pair of resonant
mass gravitational wave detectors. Numerous subsequent attempts failed to
confirm his observations. The painstaking development of much more sensitive
detectors has continued ever since.
After
the initial results with room temperature resonant detectors the new generation
of cryogenic gravitational wave (GW) antennas entered long term data taking
operation in 1990 (EXPLORER [2]), in 1991 (ALLEGRO [3]), in 1993 (NIOBE [4]),
in 1994 (NAUTILUS [5]) and in 1997 (AURIGA [6]. Several problems were
encountered. The most important one was the problem of nonstationary noise
causing varying sensitivity performance. The existence of nonstationary noise
means that criteria must be developed for extracting candidate events and
vetoing data when performance is sufficiently degraded.
Moreover,
the periods of data taking were not continuous because of short interruptions
for maintenance operations as well as long stops for instrumental improvements.
Resonant
GW antennas can detect signals of various types in the kHz frequency range: 1)
short bursts of radiation, such as that expected from the asymmetrical collapse
of a stellar core or the plunge of a neutron star into a black hole, 2)
periodic GW such as those emitted by nonaxisymmetric pulsars and 3) stochastic
GW due to various possible cosmological sources.
In this paper we report a search for
coincident excitations due to short GW bursts using three resonant
detectors that are located at large
distances: EXPLORER at CERN (Geneva, Switzerland), NAUTILUS in Frascati (near
Rome, Italy) and NIOBE in Perth (Australia), in the years: December 1994, 1995
and 1996. The data was chosen so that they could all be treated with the same
algorithm optimized for the search for GW bursts.
There
was no "a priori" reason for requiring the data from each detector to
be filtered by the same filtering algorithm. However it is essential to avoid
"a posteriori" choices of the algorithm to maintain the statistical
validity of the analysis. The most reasonable way to achieve this is to insist
that all data is filtered by the algorithm providing the best sensitivity, if
possible the same algorithm. We chose the adaptive Wiener algorithm which has
been extensively studied [7, 8] and provides good performance even in the
presence of non stationary noise.
In
table 1 we give the periods when the
data from each detector was available. It is important to note that while NIOBE
was operational in 1996 this data was not used in the present search because we
wanted to use data treated by the same Wiener algorithm and the 1996 NIOBE data
was not yet available with this chosen algorithm. For the same reason the
Allegro data, treated with a different algorithm, was not considered in this
analysis. Analysis of this data will be the topic of a future paper.
Preliminary
results of coincidence between EXPLORER and NIOBE were discussed at
conferences. In this case less powerful, and different, filter algorithms were
used for each detector giving results inconsistent with those reported here.
Again we prefer here to present data filtered using the same algorithm, which also provides better
sensitivity.
detector
/ year 
1994 
1995 
1996 
EXPLORER 
120
Dec 
24
Feb 31 Dec 
19
Feb 5 Nov 
NAUTILUS 
115
Dec 

10
Jan 5 Nov 
NIOBE 

29
June  6 Oct 

Table 1. Periods of available data from the three detectors
From table 1 we notice that there were no periods when data from the three detectors was available simultaneously, so it was possible to only search for coincidences between EXPLORER and NAUTILUS and between EXPLORER and NIOBE.
Coincidence
analysis provides a strong attenuation of the effects of the intrinsic detector
noise as well as external local disturbances. Due to the uncertain nature of
possible gravitational waves signals, in comparing the data from the
three antennas we have privileged on the statistical aspects of the data
analysis. The main problem in any coincidence experiment, as in any statistical
analysis, are the analyst's choices which may reduce or even cancel the final
meaning of any result. As detailed in the following sections we have been
careful to establish the criteria of the search before beginning the
statistical analysis.
In
our case the choices are: a) which detectors will be compared, b)the time
periods for the data used in the analysis, c)
the energy threshold used for defining the events, d) the choice of the
time windows for the coincidences. The first two choices have been already
discussed. The c) and d) choices will be considered in the following sections.
2.
Experimental details
The
three detectors, EXPLORER, NAUTILUS and NIOBE, each consists of a massive bar
resonating at its first longitudinal mode of vibration (916 Hz for EXPLORER and
NAUTILUS, 700 Hz for NIOBE). For the first two detectors the bars are cylinders
of Al 5056 with mass of 2270 kg.
EXPLORER is cooled to T = 2.6 K and NAUTILUS to T = 0.1 K. NAUTILUS is the
first resonant antenna making use of ultra lowtemperature techniques in order
to reduce the thermal noise. The NIOBE antenna is a cylinder of niobium cooled
to T=4.2 K. The characteristics of the three detectors are given in table 2.
Antenna 
NIOBE 
EXPLORER 
NAUTILUS 
Mass (kg) bar transducer 
1510 0.45 
2270 0.4 
2260 0.32 
material 
Nb 
Al 
Al 
Length (m) 
2.75 
2.97 
2.97 
mode frequencies f and f+ (Hz) 
694.6 713.0 
904.7 921.3 
908.3 923.8 
Loaded Quality Factor (10^{6}) 
30 
1.01 
2.3 
Typical Noise Temperature (mK) 
5 
15 
12 
Typical Pulse sensitivity h 
1 10^{18} 
1 10^{18} 
9 10^{19} 
Antenna position and
orientation 

Longitude 
115.8˚E 
6.25˚E 
12.67 ˚E 
Latitude 
32˚S 
46.25˚N 
41.8 ˚N 
Azimuth 
0˚ 
39.3˚ 
39.3˚ 
Table 2.
Antenna parameters. From these parameters, the relation between the noise
temperature T_{eff }and the pulse sensitivity of the
antenna, for 1 ms bursts, is calculated
to be h = 1.5 x 10^{17}
√T_{eff }for NIOBE and
h= 8.1 x 10^{}^{18}√T_{eff
} for EXPLORER and
NAUTILUS.
For
converting the mechanical vibrations into electrical signals we use resonant
transducers resonating at the antenna frequency. For EXPLORER and NAUTILUS a capacitive
transducer is used followed by a superconducting transformer and a dc SQUID
amplifier. For NIOBE a parametric transducer is used consisting of a 10
GHz microwave reentrant cavity followed by a cryogenic GaAs FET amplifier.
The bar with the resonant transducer form a
coupled oscillator system, which has two resonant modes, whose frequencies we
indicate with f and f+.
All
antennas use similar data acquisition systems. The signal from the transducer
is fed into a pair of lockin amplifiers tuned to the resonant frequency of
each antenna mode. The output of the lockin amplifiers are sampled at 10Hz
(NIOBE) and 3.44Hz (EXPLORER and NAUTILUS). The raw data for each detector was
filtered according to predetermined procedures. For all detectors an offline
adaptive WienerKolmogorov filter was used, using parameters obtained from the
spectral response of the antenna. Events with energies larger than seven times
the mean noise energy of the previous 600 seconds of data are automatically
extracted as candidate event list.
For EXPLORER, the timing information is
checked against a set of 1 Hz pulses from a combined source based on the HBG
time standard broadcasting station and a rubidium clock^{ }and gives
the Universal Time with accuracy of 0.1 seconds. The NAUTILUS timing is checked
against the data of time standard broadcasting stations. The UWA timing has
been verified by a Global Positioning System (GPS). However for this initial
experiment the beginning of each run for UWA is determined manually with maximum
error of ±0.5 s. In addition the beginning time is truncated to the second.
Therefore all together there is a time error of the order of ±1 s (after adding
0.5 s for the truncation) that should be considered with respect to the
Universal Time.
The
detectors have been calibrated as follows. The NIOBE calibration relies on the
intrinsic selfcalibration of a parametric transducer referred to the tuning
coefficient of the microwave cavity transducer element. This is measured by
determining the change in resonant frequency when the cavity gap spacing is
altered^{ }[7, 8] . EXPLORER is calibrated by using a small
precalibrated piezoelectric transducer to apply wave packets of known
frequency, amplitude and duration as well as by using a near gravitational field
induced by a rotor^{ }[11]. NAUTILUS is calibrated also by using a
small precalibrated piezoelectric transducer.
3.
The algorithms for short burst detection
The
candidate events for coincidence analysis are obtained in two steps: a) by
filtering the data recorded by the detectors to obtain estimates of the energy
innovation, i.e. the change in the energy status of the bar; and b) by applying
a suitable energy threshold to the energy innovation. We use optimum filters
designed to improve the signal to noise (SNR) ratio for short bursts of
gravitational radiation, where short means a duration smaller than the sampling
time and the time constants of the apparatus.
The estimate of the energy innovations,
for each normal mode of the detector, is obtained as follows. We have four
channels: x, y, x+, y+, because each mode (,+) is represented by the two
quadrature components (x,y) after the lockin. These data are processed
offline using the adaptive Wiener filter^{
}[10,11], which accounts for the non stationary nature of the
noise of the apparatus (the adaptive Wiener filter generates a new filter
function every two hours, based on the measured output noise spectrum).
The
corresponding filtered sequences (x_{w}, y_{w}, x_{w+}, y_{w+})
after proper normalization based on the calibration of the apparatus, are used
to construct the energy innovations _{} and _{} of each normal mode:
_{} (1)
The
two independent estimates, at each time instant, are finally combined by taking
their minimum, which represents the energy innovation r^{2} of
the detector. This quantity is usually expressed in kelvin. The effective
temperature T_{eff} for each
normal mode is defined as the mean value of r^{2}:
it represents the energy sensitivity of the detector for pulse detection in the
sense that it gives the smallest energy innovation that can be measured with
SNR=1.
The
variable r^{2},
in the absence of signals and of non gaussian disturbances, follows the
Boltzmann distribution:
_{} (2)
where
N_{o} is the total
number of samples.
Candidate
events are obtained by scanning the time sequence of the energy innovations. An
event starts when one sample exceeds the local value of the noise by a
convenient (see later) factor here chosen to be 7.0 and lasts until one of the
samples that follows fall below this threshold with a subsequent dead time of
10 s. Since the noise, in general, is not stationary, the noise T_{eff
} is calculated and
adjourned using a moving average over a time of ten minutes. Each event is
therefore characterized by its duration and by the time and the energy E_{max} of
its maximum. The reason of the choice of the moving average and of the value
7.0 chosen for the threshold is to set a convenient value to the number of
candidate events. Considering a correlation time of about one second for the
data (see below), we have N=86400 exp(7)=80 candidate events per day, to be
used for coincidence analysis. With this moving threshold the event rate does
not increase during the noisier periods. Clearly, however, it is important that
T_{eff} remains below
reasonable limits for the data to be useful, as discussed below.
4.
The data selection
The
gravitational wave antennas are very sensitive instruments, more sensitive than
any other instrument which can be used to check their performances. Thus, while
seismometers, electromagnetic detectors, power line fluctuation monitors and
cosmic ray detectors are sometimes useful for eliminating unwanted noise, many
times the noise source is unknown^{ }[12].
For this reason the coincidence technique is of the uttermost importance.
The
sensitivity of this technique depends on the signal to noise ratio (SNR) of the
possible GW signals. One approach is to select periods when the SNR is within
an acceptable range and to eliminate those periods when the noise is very
large. This procedure is very delicate, and it must be applied "a
priori" in order no to invalidate the statistical analysis by considering
a number of different alternatives.
We
have therefore decided to simply a priori eliminate from the analysis the hours
where the detector operation is clearly unacceptable. For EXPLORER and
NAUTILUS, those hours with average
noise T_{eff} above the
value 100 mK were eliminated. For NIOBE the noise temperature is smaller. Thus
we scaled the above value by a factor (8.1/15)^{2}= 0.29 under the assumption that the common excitations are
due to GWs (0.29 corresponds to the ratio of the cross sections, which depend
on the mass of the bar and on the velocity of sound in the material). For the period under consideration, from December 1994 (when
NAUTILUS started to record data) through October 1996 (when this data analysis
started) we have 1372 hours when both EXPLORER and NAUTILUS were operating with
hourly noise temperature below 100 mK. For the period when EXPLORER and NIOBE
overlapped (day 180 to day 280 of year 1995) we have 1362 hours when EXPLORER
had hourly noise temperature below 100 mK and NIOBE below 29 mK. The objection
can be made that at such high level of noise the operation of the detectors was
rather poor, so we considered also more conservative data selection: hours with
noise temperature below 50 mK and 25 mK for EXPLORER and NAUTILUS, hours with
noise temperature below, respectively, 14.5 mK and 7.1 mK for NIOBE. All these three selections must be presented
together in the final result.
As
far as eliminating data recorded in correlation with seismic or electromagnetic
disturbances as detected by the auxiliary channels, this was done for the
antenna EXPLORER, reducing the data by about
10%. For NAUTILUS this was not done, as the auxiliary channels were not
operative yet. For NIOBE the level of correlation with the auxiliary channels
is sufficiently low that veto are not normally needed.
We
show in figs.1, 2 and 3 the distribution of the EXPLORER, NAUTILUS and NIOBE
events during the selected periods of time.
Fig.1 Distribution of the event energy for EXPLORER during the hours with noise temperature smaller than 100 mK
Fig.2 Distribution of the event energy for NAUTILUS during the hours with noise temperature smaller than 100 mK
Fig.3 Distribution of the event energy for NIOBE during the hours with noise temperature smaller than 29 mK
5.
Choice of the coincidence window
We
considered it important to establish the time window for the coincidences
before starting the analysis, as to avoid introducing any bias by "a
posteriori" selection.
By
taking a very small window, one may lose a GW signal because of various
reasons: a) the basic time uncertainty introduced by the filter whose integration time determines the time
uncertainty of the events; b) the effect of the noise, that we know considerably
affects the observations even in conditions of relatively high SNR [13]; c)
possible timing errors. If the window is too long, on the other hand, the
number of accidentals becomes too large.
To
establish the coincidence window for EXPLORER and NAUTILUS, before performing
the coincidence analysis of the experimental data, we simulated the raw data of
EXPLORER and NAUTILUS using the actual parameters (resonance frequencies,
quality factors, temperatures, electronic noise) of the detectors, obtaining a
simulated noise temperature of 14 mK very close to the experimental values. To
these data, representing the noise, we added 223 signals (detector responses to
short bursts), occurring exactly at the same times for the two detectors. This
was repeated with different amplitude for the events. The resulting sequences
were then processed with our standard procedures, i.e. filtered to obtain the
energy innovations and treated with the moving threshold to obtain the events.
The last step was to search for coincidences between the two simulated event
lists by using time windows of various duration.
w (second) 
1
K SNR=70 
0.5
K SNR=35 
0.2
K SNR=14 
0.1
K SNR=7 
± 0.1 
0.65 
0.57 
0.30 
0.076 
± 0.3 
0.97 
0.94 
0.59 
0.19 
± 0.6 
1.00 
0.99 
0.71 
0.26 
± 1.0 
1.00 
1.00 
0.74 
0.27 
± 3.0 
1.00 
1.00 
0.78 
0.31 
Table 3. Simulation for determining an appropriate coincidence window. Efficiency of detection as a function of burst energy and coincidence window
The
results of this simulation are given in Table 3, where we show the fractional
number of coincidences actually found
as a function of both the amplitude of the events (given as burst energy
in unit of kelvin) and the coincidence window ±w. For these simulations the
expected number of accidental coincidences due to the noise processes is
smaller than one.
The
efficiency of the procedure for determining the best value of the window w, when applied to the real data, depends,
of course, on the absolute numbers of coincidences and accidentals.
The
number of the accidentals for the real case can be estimated with the aid of
the following formula
_{} (3)
where
t_{m} is the time
of measurement and N_{expl}
and N_{naut} are the
numbers of events respectively of EXPLORER and NAUTILUS during this time.
A
possible coincidence excess has to be compared with the standard deviation
√<n>_{th},
to determine the excess in terms of number of standard deviations. We do not
know the coincidence excess, but we can divide the efficiency of detection (as
obtained from table 3) by √<n>_{th},
in this way defining a quantity n_{s}
which characterizes the conditions for the highest statistical significance of
detection. We obtain the result shown in fig.4.
We
notice that for very small signals, very near the threshold, that is 7x14 =100 mK, the maximum n_{s}
occurs for w ≈ 0.5 s and that for large signals it is convenient ( as
intuitively expected) to take w as small as possible. Considering that: i) the sampling time for this data is 0.2908
s, ii) from fig.4, the best window for signals with energy up
to 200 mK (twice the threshold) is w=0.3 s, we consider reasonable to take as
coincidence window for the EXPLORER and NAUTILUS coincidence search the
sampling time of the detector, that is w=± 0.2908 s.
As far as the window for the search for coincidences between EXPLORER and NIOBE we are forced to take w=±1 s, because of the uncertainty in the time measurement.
Fig.4. Number
n_{s} = efficiency
of signal detection, obtained from the efficiency given in table 3, normalized
to the standard deviation, as function of the coincidence window for different
signal amplitude
6.
Coincidences
6.1
EXPLORER and NAUTILUS
We
have searched for coincidences for EXPLORER and NAUTILUS within a window
of ±0.2908 seconds for the three data
selections performance data selections discussed above. In each case, we have
determined the experimental number <n>_{exp} of
accidentals by counting the coincidences obtained after time shifting the
events of one detector with respect to the other: this was done 10000 times, using time shifts from 10000 s
to +10000 s in steps of 2 s. The step is larger than the correlation time for
the measured data so that the 10000 determinations of the accidental coincidences
are independent.
The
result of our coincidence analysis is shown in
Table 4 .
T_{eff} max [mK] 
# hours in common 
< T_{eff}> [mK] Expl 
< T_{eff}> [mK] Naut 
n_{c} 
<n>_{exp} 
<n>_{th} 
p % 
p_{exp} % 
25 
162 
12 
7 
4 
1.81 
1.80 
10 
10.25 
50 
701 
16 
16 
19 
10.8 
11.1 
1.5 
1.44 
100 
1372 
23 
28 
31 
24.3 
25.2 
10.7 
10.83 
Table
4. Result of the coincidence search between EXPLORER and NAUTILUS
In
the third column we have reported the average value of T_{eff}
for the corresponding hour selection for EXPLORER and in the fourth column for
NAUTILUS. In the fifth column we have the number of coincidences at zero delay,
in the sixth the number <n>exp of
accidentals computed from the 10,000 time shifts, and in the seventh the
corresponding expected number of accidentals <n>_{th} as
computed with eq.(3), where t_{m} is
the number of common hours and N_{expl}
and N_{naut} are the
numbers of events respectively of EXPLORER and NAUTILUS during such hours. In the last two columns, finally, we report the calculated Poisson probability
p to have n_{c} coincidences or more while expecting
<n>_{exp} on average,
and the experimental probability p_{exp}
obtained by counting how many of the
10,000 time shifts gave a number of accidentals equal or greater than the
number n_{c} at zero delay
coincidences.
We
notice that <n>_{th} is
in very good agreement with <n>_{exp},
and the calculated poissonian probability p (theoretical probability) also
agrees well with the experimental probability p_{exp}.
The
19 coincidences for the data selection referring to T_{eff
}<50 mK are listed below in Table 5 with the indication of
the time difference Dt
between the events of the two detector (
Dt=time of
EXPLORERtime of NAUTILUS) and the energy of the event. The time is counted
from 1 January 1994.
day 
hour 
min 
sec 
∆t 
Expl [K] 
Naut [K] 
868 
1 
27 
48.35 
0.26 
0.25 
1.13 
913 
23 
44 
46.40 
0.28 
0.83 
2.08 
951 
0 
35 
30.49 
0.28 
2.88 
0.12 
965 
21 
28 
5.370 
0.14 
0.44 
17.72 
974 
13 
26 
27.93 
0.15 
0.19 
0.75 
974 
21 
35 
47.53 
0.12 
0.07 
0.22 
989 
2 
38 
49.81 
0.12 
1.90 
8.23 
997 
12 
40 
24.26 
0.24 
1.69 
0.41 
999 
21 
56 
3.930 
0.16 
1.44 
0.66 
1001 
1 
3 
48.95 
0.16 
0.88 
0.23 
1002 
7 
32 
22.20 
0.25 
1.38 
0.11 
1002 
22 
42 
31.35 
0.03 
0.99 
9.95 
1009 
21 
15 
52.87 
0.06 
1.53 
2.03 
1011 
18 
28 
56.83 
0.28 
0.81 
7.86 
1013 
5 
34 
55.88 
0.15 
0.08 
0.49 
1014 
2 
7 
59.76 
0.15 
0.11 
0.29 
1014 
3 
29 
51.05 
0.13 
0.07 
4.14 
1018 
3 
16 
12.98 
0.27 
0.98 
0.46 
1028 
20 
48 
58.12 
0.03 
3.05 
0.29 
Table 5. List of coincidences for the hours with T_{eff}
< 50 mK
It
is noted in the Table 5 that some coincident event energies recorded by
EXPLORER and NAUTILUS differ by more
than an order of magnitude. We have discussed earlier that the effect of the
noise acts to spread signal energy, so that a relative large range of energies
can be expected. This, however, only partially explains the above finding, as
the basic reason is the effect of the previously mentioned nonstationary
noise. We sometimes have, in fact, periods with duration of a few minutes with
relatively high noise, also during hours with T_{eff} < 50 mK.
These small periods of relative higher noise
were not eliminated from the analysis, in order to avoid any deviation from the
strict rules established in section 4.
However,
the objection can be raised that coincidences with a very large ratio of the
two energies have small chance to have physical meaning and then the question
arises of what happens if the search of coincidences eliminates all pairs whose
two energies have a ratio outside certain limits. It is clear that this
condition cannot be applied only to the coincidences listed in table 5, because
such a condition also affects the number of accidental coincidences. We have
then decided, following a suggestion by the referee and violating exceptionally
the rule to not make any additional selection of the experimental data after
the periods of the data to be used in the analysis have been established at the
beginning, to search for coincidences also applying a veto on the energy ratio.
The
result of this new special search is shown in fig.5.
In
this figure the real and accidental coincidences versus the energy ratio veto
for the three cases of the data selection are shown. We note that in all cases
a coincidence excess remains for which the probability to occur by chance is
between 1.5 and 10 %.
It is important to check that the probability estimation is correct. We checked that, when shifting in time 10,000 times by steps of 2 s, the accidental coincidences are independent of time shift and properly distributed with stationary poissonian law. This is shown in the following figures for the data selected with hourly averaged T_{eff }< 50 mK.
Fig.5 Real and
accidental coincidences versus the veto for the ratio of the energies of the
Explorer and Nautilus coincident events. Only coincidences with energy ratio
smaller than the value indicated on the abscissa are taken. The pairs a), b)
and c) refer to the selected periods with hourly noise T_{eff} smaller,
respectively, of 25 mK, 50 mK and 100 mK. For each pair of curves the upper one
indicates the real coincidences and the lower one the accidentals.
In
fig.6 we give the numbers of coincidences for each one of the 10,000 time
delays, including the number n_{c}=19
obtained at zero delay. The time distribution appears to be rather uniform.
In
fig.7 we give the distribution of the accidental coincidences for the 10,000
delays. The continuos line is calculated with the Poisson law with average
value 10.8 deduced from table 4.
Fig.6. Delay distribution of the EXPLORER and NAUTILUS coincidences for the 702 hours with noise temperature smaller than 50 mK. The large dot at zero delay indicates the number n_{c}=19 of coincidences
Fig.7. The continuous line is the Poisson probability distribution (multiplied by the 10,000 trials) with average value of 10.8 multiplied by 10,000. The dots show the experimental frequency distribution for the data shown in fig.5.
Finally we have calculated the autocorrelation of the 10,000 accidentals versus the number of 2s steps shown in fig.6, in order to check that the 10,000 accidental coincidences are independent one from each other. The result of this calculation gives values close to zero for all delays greater than or equal to 2 s.
6.2 EXPLORER and NIOBE
The
search for coincidences between EXPLORER and NIOBE in the overlapping period
day 180 to 280 of year 1995 according to the above procedures has given
the result shown in Table 6. In this
cases 1000 time delays were used. The common period of best performance was
much larger than in the case of NAUTILUS and EXPLORER. Unfortunately the larger
time window substantially increases the number of accidental coincidences by a
factor 1/0.29, so that a small excess similar to that observed by EXPLORER and
NAUTILUS would not be discernible.
T_{eff} max [mK] 
# hours in common 
< T_{eff}> [mK] Expl 
< T_{eff}> [mK] Niobe 
n_{c} 
<n>_{exp} 
<n>_{th} 
p % 
p_{exp} % 
25 
855 
11 
4 
70 
85 
86 
96 
95.7 
50 
1202 
13 
5 
101 
119 
119 
96 
95.9 
100 
1362 
15 
7 
121 
134 
133 
89 
87.4 
Table 6. Result of the coincidence search between EXPLORER
and NIOBE
This
emphasizes the importance of improving the time resolution and also of using
other methods to reduce the accidentals, as we plan to do in further analyses.
7.
Conclusions
We
have searched for coincidences among the three distant detectors, EXPLORER,
NAUTILUS and NIOBE, analyzing all the available data and taking care to
establish the rules for the statistical analysis before actually performing it.
The result for EXPLORER and NAUTILUS is shown in Table 4. An indication for a
weak coincidence excess is found. No confirmation from the EXPLORER and NIOBE
coincidence search has been found, but in this last case, the larger coincidence window increases the
number of accidentals by a factor 1/0.29.
In
the analysis presented here we have taken extreme care to minimize the number
of choices which could influence the result. We realize that some of these a
priori choices, in particular as regards the data selection, could be objected
on the ground of data quality arguments. For example, instead of considering
the hourly noise temperature, we could have selected the data on the basis of
the “local” noise behavior, thereby better accounting for the nonstationary
noise. We also understand that it is
possible to apply additional filters to reduce the background, such as direction
filters, filters based on the shape (or the duration) of the events, or the
energy ratio filter which vetoes coincidence events for which the event
energies are very different. This last filter must be implemented with care,
however, since its implementation depends strongly on the observed pulse energy
distribution in each antenna. In addition, when the detector frequencies are
different, the implementation of such a filter requires assumptions about the
expected spectral characteristics of the signal.
In
view of these difficulties we have deferred a more detailed analysis, basically
aimed at refining the present findings on more physical bases, until
statistically appropriate protocols are established.
We thank the CERN and the LNF cryogenic facilities for the
efficient supply of the cryogenic liquids for the detector EXPLORER. We also
thank G. Federici, G. Martinelli, E. Serrani and
R. Simonetti for their continuous technical support.
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