A search for arrival direction clustering in the HiRes-I monocular data above 1019.5 eV

Abstract

In the past few years, small scale anisotropy has become a primary focus in the search for source of ultra-high energy cosmic rays (UHECRs). The Akeno Giant Air Shower Array (AGASA) has reported the presence of clusters of event arrival directions in their highest energy data set. The High Resolution Fly’s Eye (HiRes) has accumulated an exposure in one of its monocular eyes at energies above 1019.5 eV comparable to that of AGASA. However, monocular events observed with an air fluorescence detector are characterized by highly asymmetric angular resolution. A method is developed for measuring autocorrelation with asymmetric angular resolution. It is concluded that HiRes-I observations are consistent with no autocorrelation and that the sensitivity to clustering of the HiRes-I detector is comparable to that of the reported AGASA data set. Furthermore, we state with a 90% confidence level that not more than 13% of the observed HiRes-I events above 1019.5 eV could be sharing common arrival directions. However, because a measure of autocorrelation makes no assumption of the underlying astrophysical mechanism that results in clustering phenomena, we cannot claim that the HiRes monocular analysis and the AGASA analysis are inconsistent beyond a specified confidence level.

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arXiv:astro-ph/0404366v2 27 May 2004
A Search for Arrival Direction Clustering in
the HiRes-I Monocular Data above 1019.5eV
R.U. Abbasi,aT. Abu-Zayyad,aJ.F. Amann,bG. Archbold,a
R. Atkins,aJ.A. Bellido,cK. Belov,aJ.W. Belz,dS. BenZvi,e
D.R. Bergman,fG.W. Burt,aZ. Cao,aR.W. Clay,c
B. Connolly,eB.R. Dawson,cW. Deng,aY. Fedorova,a
J. Findlay,aC.B. Finley,eW.F. Hanlon,aC.M. Hoffman,b
M.H. Holzscheiter,bG.A. Hughes,fP. H¨ untemeyer,a
C.C.H. Jui,aK. Kim,aM.A. Kirn,dE.C. Loh,a
M.M. Maestas,aN. Manago,gL.J. Marek,bK. Martens,a
J.A.J. Matthews,hJ.N. Matthews,aA. O’Neill,eC.A. Painter,b
L. Perera,fK. Reil,aR. Riehle,aM. Roberts,hM. Sasaki,g
S.R. Schnetzer,fK.M. Simpson,cG. Sinnis,bJ.D. Smith,a
R. Snow,aP. Sokolsky,aC. Song,eR.W. Springer,a
B.T. Stokes,a,∗J.R. Thomas,aS.B. Thomas,aG.B. Thomson,f
D. Tupa,bS. Westerhoff,eL.R. Wiencke,aand A. Zechf
The High Resolution Fly’s Eye Collaboration
aUniversity of Utah, Department of Physics and High Energy Astrophysics
Institute, Salt Lake City, UT 84112, USA
bLos Alamos National Laboratory, Los Alamos, NM 87545, USA
cUniversity of Adelaide, Department of Physics, Adelaide, SA 5005, Australia
dUniversity of Montana, Department of Physics and Astronomy, Missoula,
MT 59812, USA.
eColumbia University, Department of Physics and Nevis Laboratories, New York,
NY 10027, USA
fRutgers — The State University of New Jersey, Department of Physics and
Astronomy, Piscataway, NJ 08854, USA
gUniversity of Tokyo, Institute for Cosmic Ray Research, Kashiwa City,
Chiba 277-8582, Japan
hUniversity of New Mexico, Department of Physics and Astronomy, Albuquerque,
NM 87131, USA
Preprint submitted to Elsevier Science2 February 2008

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Abstract
In the past few years, small scale anisotropy has become a primary focus in the
search for source of Ultra-High Energy Cosmic Rays (UHECRs). The Akeno Giant
Air Shower Array (AGASA) has reported the presence of clusters of event arrival di-
rections in their highest energy data set. The High Resolution Fly’s Eye (HiRes) has
accumulated an exposure in one of its monocular eyes at energies above 1019.5eV
comparable to that of AGASA. However, monocular events observed with an air
fluorescence detector are characterized by highly asymmetric angular resolution. A
method is developed for measuring autocorrelation with asymmetric angular resolu-
tion. It is concluded that HiRes-I observations are consistent with no autocorrelation
and that the sensitivity to clustering of the HiRes-I detector is comparable to that
of the reported AGASA data set. Furthermore, we state with a 90% confidence level
that no more than 13% of the observed HiRes-I events above 1019.5eV could be
sharing common arrival directions. However, because a measure of autocorrelation
makes no assumption of the underlying astrophysical mechanism that results in
clustering phenomena, we cannot claim that the HiRes monocular analysis and the
AGASA analysis are inconsistent beyond a specified confidence level.
Key words: cosmic rays, anisotropy, clustering, autocorrelation, HiRes, AGASA
PACS: 98.70.Sa, 95.55.Vj, 96.40.Pq, 13.85.Tp
1 Introduction
Over the past decade, the search for sources of Ultra-High Energy Cosmic
Rays (UHECRs) has begun to focus upon small scale anisotropy in event ar-
rival directions. This refers to statistically significant excesses occurring at the
scale of ≤ 2.5◦. The interest in this sort of anisotropy has largely been fueled
by the observations of the Akeno Giant Air Shower Array (AGASA). In 1999
[1] and again in 2001 [2], the AGASA collaboration reported observing what
eventually became seven clusters (six “doublets” and one “triplet”) with esti-
mated energies above ∼ 3.8×1019eV. Several attempts that have been made
to ascertain the significance of these clusters returned chance probabilities of
4 × 10−6[3] to 0.08 [4].
By contrast, the monocular (and stereo) analyses that have been presented by
the High Resolution Fly’s Eye (HiRes) demonstrate that the level of autocor-
relation observed in our sample is completely consistent with that expected
from background coincidences [5,6,7]. Any analysis of HiRes monocular data
∗Corresponding author. E-mail address: stokes@cosmic.utah.edu (B.T. Stokes)
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needs to take into account that the angular resolution in monocular mode is
highly asymmetric.
It is very difficult to compare the results of the HiRes monocular and AGASA
analyses. They are very different in the way that they measure autocorrelation.
Differences in the published energy spectra of the two experiments suggest
an energy scale difference of 30% [8,9]. Additionally, the two experiments
observe UHECRs in very different ways. The HiRes experiment has an energy-
dependent aperture and an exposure with a seasonal variability [8]. These
differences make it very difficult get an intuitive grasp of what HiRes should
see if the AGASA claim of autocorrelation is justified. In order to develop this
sort of intuition, we apply the same analysis to both AGASA and HiRes data.
2The HiRes-I Monocular Data
The data set that we consider consists of events that were included in the
HiRes-I monocular spectrum measurement [8,10]. This set contains 52 events
observed between May 1997 and February 2003 with measured energies greater
than 1019.5eV. The data set represents a cumulative exposure of ∼ 3000 km2·sr·yr
at 5 × 1019eV. This data was subject to a number of quality cuts that are
detailed in the above-mentioned papers [8,10]. We previously verified that this
data set was consistent with Monte Carlo predictions in many ways including
impact parameter (Rp) distributions [8] and zenith angle distributions [11].
For this study, we presumed an average atmospheric clarity [12].
In order to calculate the autocorrelation function for this subset of data,
we must first parameterize the HiRes-I monocular angular resolution. For a
monocular air fluorescence detector, angular resolution consists of two com-
ponents, the plane of reconstruction, that is the plane in which the shower is
observed, and the angle ψ within the plane of reconstruction (see figure 1).
We can determine the plane of reconstruction very accurately. However, the
value of ψ is more difficult to determine accurately because it is dependent on
the precise results of the profile-constrained fit [8,10].
The HiRes-I angular resolution is therefore described by an elliptical, two-
dimensional Gaussian distribution with the two Gaussian parameters, σψand
σplane, being defined by the two angular resolutions. For the range of estimated
energies considered in this paper, σψ= [4.9,6.1]◦and σplane[0.4,1.5]◦. In fig-
ure 2, the arrival directions of the HiRes-I events are plotted in equatorial
coordinates along with their 1σ error ellipses.
In order to understand the systematic uncertainty in the angular resolution
estimates, we consider a comparison of estimated arrival directions that suc-
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Fig. 1. The geometry of reconstruction for a monocular air fluorescence detector
Fig. 2. The arrival directions of the HiRes-I monocular with reconstructed energies
above 1019.5eV events and their 1σ angular resolution
cessfully reconstructed in both HiRes-I monocular mode and HiRes stereo
mode. Because of the dearth of events with estimated energies above 1019.5eV
that reconstructed satisfactorily in both stereo and mono mode, we consider
all mono/stereo candidate events with estimated energies above 1018.5eV. In
stereo mode, the shower detector planes of the two detectors are intersected,
thus the geometry is much more precisely known and the total angular reso-
lution is of order 0.6◦, a number that is largely correlated to σplaneand thus is
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0
20
40
05
Arrival Direction Error (Degrees)
101520253035404550
Number of Events
(a)
0
1
2
3
4
05
Arrival Direction Error (Degrees)
10 152025 3035404550
Data/MC Ratio
(b)
Fig. 3. Arrival direction error comparison between real data (mono vs. stereo) and
simulated data for events with estimated energies above 1018.5eV. The solid line
histogram corresponds to the arrival direction error distribution of the monocular
reconstructed Monte Carlo simulated data. The crosses correspond to the arrival
directions error distribution observed for actual data by comparing the arrival di-
rections estimated by the monocular and stereo reconstructions. The solid line in
the ratio component corresponds to the fit y = ax+ b where a = 0.000 ± 0.011 and
b = 0.98 ± 0.11.
negligible when added in quadrature to the larger term, σψ. This allows us to
perform a comparison of the angular resolution estimated through simulations
to the observed angular resolution values of actual data. In figure 3, we show
the distribution of angular errors for real and simulated data. The uncertainty
in the slope of the ratio (figure 3b) leads to an 7.5% uncertainty in the angular
resolution.
3The Published AGASA Data
The AGASA data with energies above 40 EeV has been published up to the
year 2000 [2] and all but one of these events used for this calculation has a
measured energy greater than 4 × 1019eV. The AGASA estimated angular
errors [1] are shown in figure 4. The AGASA angular errors (figure 4) are fit
to a two-component Gaussian distribution:
n = N◦(EEeV)
?
0.33∆θe−(∆θ)2/2σ2
1+ 0.67∆θe−(∆θ)2/2σ2
2
?
(1)
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o
o
o
o
o
18.0
θ
∆
8
6
4
2
0
Log(Energy[eV])
Opening Angle
90%
68%
18.519.520.5

19.020.0
Fig. 4. The AGASA angular resolution as a function of estimated energy [1]
Fig. 5. The arrival directions of the published AGASA events with their 68% angular
resolution
where σ1= 6.52◦− 2.16◦log10EEeV, σ2= 3.25◦− 1.22◦log10EEeV, and N◦(E)
is a numerically determined normalization constant. Figure 5 shows the arrival
directions of the published AGASA events plotted in equatorial coordinates
with their 68% angular resolution.
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(a)
0
1
2
3
4
5
6
7
8
9
10
-1-0.500.51
cosθ
Normalized Bin Density
(b)
0
1
2
3
4
5
6
7
8
9
10
0.9850.990.9951
cosθ
Normalized Bin Density
Fig. 6. An example of the autocorrelation function for a simulated data set that
contains ∼ 10 clusters in a total of 60 events—(a) the full autocorrelation func-
tion for θ = [0◦,180◦]; (b) the critical region of the the autocorrelation function:
θ = [0◦,10◦].
4The Autocorrelation Function
We measure the degree of autocorrelation in both samples by means of an
autocorrelation function. It is calculated as follows:
(1) For each event, an arrival direction is sampled on a probabilistic basis
from the error space defined by the angular resolution of the event.
(2) The opening angle is measured between the arrival directions of a pair of
events.
(3) The cosine of the opening angle is then histogrammed.
(4) The preceding steps are repeated until all possible pairs of the events are
considered.
(5) The preceding steps are repeated until the error space, in the arrival
direction of each event, is thoroughly sampled.
(6) The histogram is normalized and the resulting curve is the autocorrelation
function.
Figure 6a shows an example of the autocorrelation function for a highly clus-
tered set of simulated data. The sharper the peak at cosθmin is, the more
highly autocorrelated the data set is. There are many ways that one could
quantify the degree of autocorrelation that a set possesses. The most obvi-
ous way is to look at the value of the bin which contains cosθmin. However,
this method has some pitfalls. First of all, the value of the last bin is depen-
dent upon the chosen bin width. Also, the value of the last bin is not stable
unless the angular resolution is sampled at a level that is computationally
unfeasible. Finally, the value of the last bin over a large number of similarly
autocorrelated sets does not produce a Gaussian distribution (see figure 7a),
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(a)
0
20
40
60
80
100
120
140
160
0
Normalized Bin Density, cosθmin
1020
Number of Simulated Sets
(b)
0
0.992
20
40
60
80
100
120
0.9940.996
<cosθ>[0°,10°]
Number of Simulated Sets
Fig. 7. Distributions of normalized bin densities of cosθminand <cosθ>[0◦,10◦]val-
ues for a large number of simulated sets with the same level of clustering as in fig-
ure 6—(a) Distribution of observed normalized bin densities of cosθmin, note that it
is not Gaussian (χ2/dof = 5.44); (b) : <cosθ>[0◦,10◦]distribution (χ2/dof = 1.09).
thus complicating the interpretation of the results of an analysis employing
cosθminas an observable.
A more well-behaved measure of the autocorrelation of a specific set of data is
the value of <cosθ> for θ ≤ 10◦. This value is also a measure of the sharpness
of the autocorrelation peak at cosθ = 1. However, this method of quantifica-
tion does not depend on bin width and it does produce Gaussian distributions
when it is applied to large numbers of sets with similar degrees of autocorrela-
tion as is demonstrated in figure 7b. An additional advantage to this method
is that by considering the continuous autocorrelation function over a specified
interval, both the peak at the smallest values of θ and the corresponding sta-
tistical deficit in the autocorrelation function at slightly higher values of θ are
taken into account. Thus we simultaneously measure both the positive and
negative aspects of the autocorrelation signal. The interval of [0◦,10◦] was
chosen because in simulations it was found to optimize the autocorrelation
signal for clusters resulting from point sources spread isotropically across the
sky.
Using the description of the HiRes-I monocular angular resolution above, we
then calculate the autocorrelation function via the method described above.
In figure 8, we show the result of this calculation. For this sample, we obtain
<cosθ>[0◦,10◦]= 0.99234.
We also calculate the autocorrelation function for the published AGASA
events. We show the result in figure 9. For this sample, we obtain <cosθ>[0◦,10◦]=
0.99352.
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(a)
0
1
2
3
4
5
6
7
8
9
10
-1 -0.500.51
cosθ
Normalized Bin Density
(b)
0
1
2
3
4
5
6
7
8
9
10
0.985 0.990.9951
cosθ
Normalized Bin Density
Fig. 8. The autocorrelation for the HiRes-I events above 1019.5eV—(a) the full au-
tocorrelation function for θ = [0◦,180◦]; (b) the critical region of the autocorrelation
function: <cosθ>[0◦,10◦]= 0.99234.
(a)
0
1
2
3
4
5
6
7
8
9
10
-1-0.500.51
cosθ
Normalized Bin Density
(b)
0
1
2
3
4
5
6
7
8
9
10
0.9850.990.9951
cosθ
Normalized Bin Density
Fig. 9. The autocorrelation for the published AGASA events—(a) the full autocor-
relation function for θ = [0◦,180◦]; (b) the critical region of the autocorrelation
function: <cosθ>[0◦,10◦]= 0.99352.
5Quantifying the Relative Sensitivity of HiRes-I and AGASA to
Autocorrelation
In order to quantify the relative sensitivity of the AGASA and HiRes-I data
sets, we must first understand the exposures of both detectors. For HiRes-I,
we assemble a library of approximately 8×104simulated events with energies
above 1019.5eV. We then pair each event with times during which the detector
was operating. A mirror-by-mirror correction is applied where simulated events
are rejected if the mirror(s) that would have observed the event in question was
not operating at the time that event would have occurred. Once 107pairings
of simulated events and times are assembled, a surface plot is created of the
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Fig. 10. Hires-1 estimated relative exposure, ρH(δ,α), for events above 1019.5eV in
equatorial coordinates (right ascension right to left). The lightest region corresponds
to a normalized event density of 2.5. The observable sky extends from δ = −30◦to
δ = 90◦.
event density on a bin by bin basis. The value of each bin is then normalized
so that the mean value of all the bins in the observable sky δ = [−30◦,90◦] is 1.
The resulting surface plot is shown in a Hammer-Aitoff projection in figure 10.
We have previously shown that this method produced zenith angle, azimuthal
angle, and sidereal time distributions that were consistent with that observed
in the actual data [11]. The highest exposure areas have a normalized relative
exposure: ρH(δ,α) =∼ 2.5.
For the AGASA detector, we refer to the distribution of event declinations
presented in Uchiori et al. [13]. By following the lead of Evans et al. [14], we
fit a normalized polynomial to this distribution:
N(δ)=0.323616 + 0.0361515δ − 5.04019 × 10−4δ2+
5.539141 × 10−7δ3; (2)
where N(δ) holds for δ = [−8◦,87.5◦]the maximum value of N(δ) is 1. We also
know that:
A◦
87.5◦
?
−8◦
N(δ)dδ =
87.5◦
?
−8◦
ρA(δ)cosδ dδ,
(3)
where A◦is a numerically determined normalization constant. We then derive:
ρA(δ) = A◦N(δ)secδ;A◦= 1.0251.
(4)
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Fig. 11. AGASA estimated relative exposure, ρA(δ), for events above 1019.5eV in
equatorial coordinates (right ascension right to left). The lightest region corresponds
to a normalized event density of ∼ 1.6. The observable sky extends from δ = −8◦
to δ = 87.5◦.
(a)
0
50
100
150
200
250
300
350
0.9920.994
<cosθ>[0°,10°]
Number of Simulated Sets
(b)
0
0.99
100
200
300
400
500
600
0.992 0.994
<cosθ>[0°,10°]
Number of Simulated Sets
Fig. 12. Distribution of <cosθ>[0◦,10◦]values for simulated isotropic data sets—(a)
HiRes-I; (b) AGASA. In each figure, the vertical line represents the the value of
<cosθ>[0◦,10◦]for the observed data.
The value of each bin is once again normalized so that the mean value of
all the bins in the observable sky δ = [−8◦,87.5◦] is 1. The resulting surface
plot is shown in a Hammer-Aitoff projection of a equatorial coordinates in
figure 11. The highest exposure areas have ρA(α) =∼ 1.6. In figure 12, we
show the distribution of <cosθ>[0◦,10◦]values for isotropic data sets with each
of the two different exposure models (HiRes-I and AGASA). The AGASA data
set manifests ∼ 10−3chance probability above background. For the AGASA
data, we also calculated the autocorrelation function without consideration
to angular resolution and employed the more conventional θmin observable.
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After varying the bin width for θmin and accounting for the trials factor,
we independently concluded that the chance probability is ∼ 10−3for the
optimal bin width, θmin= [0◦,2.5◦]. We thus conclude that factoring angular
resolution into our analysis and employing <cosθ>[0◦,10◦]as an observable in
no way diminishes the sensitivity to autocorrelation in the reported AGASA
data.
There are a few important differences between the exposure of the HiRes-
I and AGASA detectors. First of all, the exposure of the HiRes-I detector
is more asymmetric than the exposure of the AGASA detector. This is not
only due to seasonal variations in the HiRes detector, but also due to its
ability to constantly observe the region around δ = 90◦due to a higher zenith
angle acceptance. This higher zenith angle acceptance also allows the HiRes
detector to observe a greater region of the southern hemisphere. In general,
while AGASA reports observations for 56.9% of the total sky, the HiRes-I
detector reports observations for 75% of the total sky.
To simulate clustering we use the following prescription:
(1) An event is chosen based upon the distribution in α and δ that is dictated
by ρ. In the case of HiRes-I, this is simply done by selecting a simulated
event from our library and then assigning it a time that is a known good-
weather ontime for the mirror(s) that observed that event. In the case
of the AGASA detector, this is done by selecting a random value for δ
that conforms to the distribution in equation (4) and then assigning it
a random value in α between 0h and 24h and sampling a value for the
energy from the energies of the reported events.
(2) This event does not represent the source location itself, but is assumed
to have arrived from the source location with some error. We construct
a ”true” source location by sampling the error space of this event.
(3) For each additional event assigned to that source, a simulated event is
selected with a “true” arrival direction that is the same as that of the
initial event.
To study the relative sensitivity of AGASA and HiRes-I, we measure the value
of <cosθ>[0◦,10◦]for multiple simulated sets with a variable number of doublets
inserted. We then construct an interpolation of the mean value and standard
deviation of <cosθ>[0◦,10◦]from a given number of observed doublets for each
experiment. This will allow us to state the number of doublets required for
each experiment in order for the 90% confidence limit of <cosθ>[0◦,10◦]to be
above the background value of 0.99250. Figure 13 shows the result of these
simulations. In general, for the HiRes-I data set, the 90% confidence lower
limit corresponds to the mean expected background signal with the inclu-
sion of 6.25 doublets. For AGASA,the 90% confidence lower limit corresponds
to the mean expected background signal with the inclusion of 5.5 doublets.
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(a)
0.991
0.992
0.993
0.994
0.995
0510
1σ
2σ
Number of Doublets
<cosθ>[0°,10°]
(b)
0.991
0.992
0.993
0.994
0.995
0510
90% Conf. Lower Limit
95% Conf. Lower Limit
Number of Doublets
<cosθ>[0°,10°]
(c)
0.991
0.992
0.993
0.994
0.995
05 10
1σ
2σ
Number of Doublets
<cosθ>[0°,10°]
(d)
0.991
0.992
0.993
0.994
0.995
05 10
90% Conf. Lower Limit
95% Conf. Lower Limit
Number of Doublets
<cosθ>[0°,10°]
Fig. 13. Relative sensitivity of HiRes-I and AGASA to doublets—(a) Simulations
with the HiRes-I detector and 52 events; (b) 90% confidence above background: 6.25
doublets, 95% confidence above background: 8.25 doublets. (c) simulations with the
AGASA detector and 59 events; (d) 90% confidence above background: 5.5 doublets,
95% confidence above background: 7.0 doublets. In each figure, the horizontal line
indicates the expected value of <cosθ>[0◦,10◦]for an isotropic background
This demonstrates that while AGASA has a slightly better ability to perceive
autocorrelation, the sensitivity of the two experiments is comparable.
We now apply the actual Hires-I <cosθ>[0◦,10◦]to the sensitivity curve shown
in figure 13. In figure 14 we can see the result of these simulations. The ob-
served HiRes-I signal corresponds to the 90% confidence upper limit with the
inclusion of only 3.5 doublets beyond random background coincidence.
If we repeat this analysis with first, a 7.5% reduction in the estimated angular
resolution values and second, a 7.5% increase in the estimated angular resolu-
tion values, we obtain a range for the 90% confidence upper limit of [2.75,4.0]
doublets and a range for the 95% confidence upper limit of [4.5,5.5] doublets.
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(a)
0.991
0.992
0.993
0.994
0.995
05 10
1σ
2σ
Number of Doublets
<cosθ>[0°,10°]
(b)
0.991
0.992
0.993
0.994
0.995
05 10
90% Conf. Lower Limit
95% Conf. Lower Limit
Number of Doublets
<cosθ>[0°,10°]
Fig. 14. Sensitivity of the HiRes-I monocular observations to doublets—(a) Simula-
tions with the HiRes-I detector and 52 events; (b) 90% confidence above observed
signal: 3.5 doublets, 95% confidence above observed signal: 5 doublets. In each plot,
the horizontal line represents the value of <cosθ>[0◦,10◦]for the observed HiRes-I
data
A final area of concern is the systematic uncertainty in the determination of
atmospheric clarity. Because hourly atmospheric observations are not avail-
able for the entire HiRes-I monocular data set, we have relied upon the use of
an average atmospheric profile for the reconstruction of our data [12]. While
different atmospheric conditions have negligible impact on the determination
of the arrival direction for events with measured energies this high, differing
conditions can have an impact on energy estimation and thus the number of
events that are included in our data set. Over the 1σ error space for our esti-
mation of atmospheric conditions, the total number of events in our data set
fluctuates on the interval [41,65]. The value of the observable, <cosθ>[0◦,10◦],
has a fluctuation on the interval [0.99226,0.99249] owing to addition and sub-
traction of events from the data set. Note that in neither case does the value
of <cosθ>[0◦,10◦]exceed the mean value (0.99250) expected for a background
set.
6 Conclusion
We conclude that the HiRes-I monocular detector sees no evidence of cluster-
ing in its highest energy events. Furthermore, the HiRes-I monocular data has
an intrinsic sensitivity to global autocorrelation such that we can claim at the
90% confidence level that there can be no more than 3.5 doublets above that
which would be expected by background coincidence in the HiRes-I monocu-
lar data set above 1019.5eV. From this result, we can then derive, with a 90%
confidence level, that no more than 13% of the observed HiRes-I events could
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be sharing common arrival directions. This data set is comparable to the sen-
sitivity of the reported AGASA data set if one assumes that there is indeed a
30% energy scale difference between the two experiments. It should be empha-
sized that this conclusion pertains only to point sources of the sort claimed by
the AGASA collaboration. Furthermore, because a measure of autocorrelation
makes no assumption of the underlying astrophysical mechanism that results
in clustering phenomena, we cannot claim that the HiRes monocular analysis
and the AGASA analysis are inconsistent beyond a specified confidence level.
7 Acknowledgments
This work is supported by US NSF grants PHY 9322298, PHY 9321949, PHY
9974537, PHY 0071069, PHY 0098826, PHY 0140688, PHY 0245428, PHY
0307098 by the DOE grant FG03-92ER40732, and by the Australian Research
Council. We gratefully acknowledge the contributions from the technical staffs
of our home institutions. We gratefully acknowledge the contributions from the
University of Utah Center for High Performance Computing. The cooperation
of Colonels E. Fisher and G. Harter, the US Army and the Dugway Proving
Ground staff is appreciated.
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