**1.3. Distribution of arrival directions**

The distribution of arrival directions is perhaps the most helpful observable in yielding clues about the CR origin. On the one hand, if cosmic rays cluster within a small angular region (see e.g. [46]) or show directional alignment with powerful compact objects (see e.g. [47]), one might be able to associate them with isolated sources in the sky. On the other hand, if the distribution of arrival directions exhibits a large-scale anisotropy, this could indicate whether or not certain classes of sources are associated with large-scale structures (such as the Galactic plane or the Galactic halo).

Cosmic ray air shower detectors which experience stable operation over a
period of a year or more can have a uniform exposure in right ascension,
. A traditional technique to
search for large-scale anisotropies is then to fit the right ascension
distribution of events to a sine
wave with period 2 / *m*
(*m*^{th} harmonic) to determine the
components (*x*, *y*) of the Rayleigh vector
[48]

(3) |

The *m*^{th} harmonic amplitude of *N* measurements of
_{i} is
given by the Rayleigh vector length
= (*x*^{2} +
*y*^{2})^{1/2}. The expected length of such vector
for values randomly sampled from a uniform phase distribution is
_{0} = 2 /
*N*.
The chance probability
of obtaining an amplitude with length larger than that measured is
*p*(
) =
*e*^{-k0}, where *k*_{0} =
^{2} /
_{0}^{2}.
To give a specific example, a vector of length
*k*_{0} 6.6
would be required to
claim an observation whose probability of arising from random fluctuation
was 0.0013 (a "3"
result) [6].

AGASA has revealed a correlation of the arrival direction of the cosmic
rays to the Galactic Plane (GP) at the
4 level
[49].
The energy bin width which gives the maximum
*k*_{0}-value corresponds to the region 10^{17.9}
eV - 10^{18.3} eV where *k*_{0} = 11.1,
yielding a chance probability of
*p*(
_{E ~
EeV}^{AGASA})
1.5 ×
10^{-5}. The GP excess, which is roughly 4% of
the diffuse flux, is mostly
concentrated in the direction of the Cygnus region, with a second
spot towards the Galactic Center (GC)
[50]. Evidence at
the 3.2 level for GP
enhancement in a similar energy range has also been reported by the HiRes
Collaboration
[51].
The existence of a point-like excess in
the direction of the GC has been confirmed via independent analysis
[52]
of data collected with SUGAR. This is a remarkable level of
agreement among experiment using a variety of techniques.

At lower energies (~ PeV), the Rayleigh analysis shows no evidence of
anisotropy
[53].
Hence, the excess from the GP is very suggestive of neutrons as candidate
primaries,
because the directional signal requires relatively-stable neutral
primaries, and time-dilated neutrons can reach
the Earth from typical Galactic distances when the neutron energy exceeds
10^{18} eV. Arguably, if the Galactic messengers
are neutrons, then those with energies below 10^{18} eV will
decay in flight, providing a flux of cosmic antineutrinos
above 1 TeV that should be observable at kilometer-scale neutrino
telescopes
[54].
A measurement of the
-flux will supply a
strong confirmation of the GP neutron hypothesis.

For the ultra high energy
( 10^{19.6}
eV) regime, all experiments to date have reported *k*_{0}
<< 6.6,
*m* < 5
[55,
56,
57,
58].
^{(1)}
This does not imply an isotropic
distribution, but it merely means that available data are too sparse to
claim a statistically significant measurement of anisotropy.
In other words, there may exist anisotropies at a level too low to
discern given existing statistics
[59].

The right harmonic analyses are completely blind to intensity
variations which depend only on declination,
. Combining
anisotropy searches in
over a range of declinations could
dilute the results, since significant but out of phase Rayleigh
vectors from different declination bands can cancel each other
out. Moreover, the analysis methods that consider distributions
in one celestial coordinate, while integrating away the second,
have proved to be potentially misleading
[60]. An
unambiguous interpretation of anisotropy data requires two
ingredients: *exposure to the full celestial sphere and
analysis in terms of both celestial coordinates.* In this
direction, a recent study
[61]
of the angular power spectrum of the distribution of arrival directions
of CRs with energy > 10^{19.6} eV, as seen by the AGASA and SUGAR
experiments, shows no departures from either homogeneity or
isotropy on angular scale greater than 10°. Finally, the
recently analyzed HiRes data is also statistically consistent with
an isotropic distribution
[62].

All in all, the simplest interpretation of the existing data is that,
beyond the ankle,
a new population of extragalactic CRs emerges to dominate the more steeply
falling Galactic population. Moreover, there are two extreme explanations
for the near observed isotropy beyond 10^{19.6} eV: one is to
argue a cosmological
origin for these events, and the other is that we have nearby sources
(say, within the Local Supercluster) with a tangled magnetic field in the
Galaxy, and beyond, which bends the particle orbits camouflaging the exact
location of the sources.

Although there seems to be a remarkable agreement among experiment on
predictions about isotropy on large scale
structure, this is certainly not the case when
considering the two-point correlation function on a small angular
scale. The analyses carried out by AGASA Collaboration seem to indicate
that the pairing of events on the celestial sky could be occurring at
higher than chance coincidence
[46,
63].
Specifically, when showers with separation angle less
than the angular resolution
_{min} =
2.5° are paired up, AGASA finds five doublets and one triplet
among the 58 events reported with mean energy above 10^{19.6} eV.
The probability of observing these clusters by chance coincidence under an
isotropic distribution was quoted as smaller than 1%. A third independent
analysis
[64],
using the Goldberg-Weiler formalism
[65],
confirmed the result reported by AGASA Collaboration
and further showed that the chance probability is extremely sensitive to
the angular binning. The "world" data set has also been studied
[66].
Six doublets and two triplets out of 92 events with energies
> 10^{19.6} eV were found, with the chance
probability being less than 1% in the restricted region within
± 10° of the super-Galactic plane. The angular
two-point correlation function of a
combined data sample of AGASA (*E* > 4.8 × 10^{19}
eV) and Yakutsk (*E* > 2.4 × 10^{19} eV) was analyzed
[67].
For a uniform distribution of sources, the probability of chance
clustering is reported to be as small as 4 × 10^{-6}. Far from
confirming what seemed a fascinating discovery, the recent analysis reported
by the HiRes Collaboration showed that the data is consistent with no
small-scale anisotropy among the highest energy events
[68].

The discovery of such clusters would be a tremendous breakthrough for the
field, but the case for them is not yet proven. To calculate a meaningful
statistical significance in such an analysis, it is important to define
the search procedure *a priori* in order to ensure it is not
inadvertently devised especially to suite the particular data set after
having studied it. In the analyses carried out by AGASA Collaboration
[46,
63],
for instance, the angular bin size was not defined ahead of time.
Very recently, with the aim to avoid accidental bias on the
number of trials performed in selecting the angular bin, the original
claim of AGASA Collaboration
[46]
was re-examined considering only the events observed after the
claim
[69].
This study showed that the
evidence for clustering in the AGASA data set is weaker than was
previously claimed, and consistent with the null hypothesis of isotropically
distributed arrival directions.

Summing up, the clustering on small angular scale at the upper end of the spectrum remains an open question, and the increase in statistics and improved resolution attainable with PAO is awaited to solve the issue.

^{1} For the Fly's Eye data-sample
the Rayleigh vector was computed using weighted showers, because it has
had a nonuniform exposure in sideral time. A shower's weight depends on
the hour of its sideral arrival time, and the 24 different weights are
such that every time bin has the same weighted number of showers.
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