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\begin{document}
\vspace*{4cm}
\title{A COMMENT ON CATALOG SEARCHES}
\author{P.~TINYAKOV}
\address{Service Physique Theorique CP225,
Universit\'e Libre de Bruxelles, \\
Boulevard du Triomphe 1050 Bruxelles}
\maketitle\abstracts{We illustrate in a concrete example that a mere
positional correlation of highest-energy cosmic rays with active
galactic nuclei (AGN), although suggests, does not necessarily imply
that the latter are sources of the cosmic rays. Different
interpretations of this correlation are possible, and signatures
other than positional correlations are needed to discriminate
between them. We point out that some of these signatures seem to
disfavor the AGN interpretation with already existing data. }
In this talk I would like to clarify two points related to the
correlations between the ultra-high energy comic rays (UHECR) and
nearby active galactic nuclei (AGN) from the catalog
\cite{2006A&A...455..773V}, which were recently found by the Pierre
Auger Observatory (PAO) \cite{Cronin:2007zz,Abraham:2007si}.
\paragraph{1} Contrary to naive expectation, a correlation of cosmic
rays with AGN (or any other objects) {\it does not} automatically
imply that the latter are cosmic ray sources. This is not related to
the significance of the correlation, but follows from the very nature
of the statistical test performed to establish the correlation. In
positional correlation analysis one compares the distribution of the
data events over the sky with the {\em isotropic} distribution. If the
two distributions are found to be incompatible, this means simply that
the data are {\em not isotropic}. The actual sources should be
identified by different methods.
To illustrate the relevance of this point consider a concrete
example. The same set of cosmic ray events which correlate with AGN in
the PAO analysis may be cross-correlated, by the same method, with
just one object for which we take Cen A, an active galaxy in the
direction of the Centaurus supercluster. Cen A is a radio-galaxy which
is exceptionally close to us: the distance to Cen A is about
3.5~Mpc. It possesses jets and radio lobes, the usual attributes of a
potential acceleration site.
There is an excess of events in the data in the direction of Cen
A. The significance of the excess at a given angular scale $\delta$
can be characterized by the probability $P(\delta)$ that equal or
larger excess occurs by chance as a result of a fluctuation in the
uniform distribution. The smaller is the probability to obtain a given
excess by chance, the more significant it is. This probability may be
determined by the Monte-Carlo simulation. The result of the
simulation is shown in Fig.~\ref{fig:pearl}. One can see that the
excess is most significant at about 20$^\circ$. Out of 27 events in
total, 9 events fall within 20$^\circ$ from Cen A while only $1.5$ are
expected for the uniform distribution. Note that the events
contributing to this correlation with Cen A are the same events that
contribute to the correlation with AGN if the latter are assumed to
be sources.
\begin{figure}
\begin{center}
\psfig{figure=cenA-corr.eps,height=2.5in}
\end{center}
\caption{The probability $P$ that the observed excess of
events within angular distance $\delta$ around Cen~A has occured by
chance. The values of $P$ are indicative only since their calculation
accounts neither for the statistical penalty associated with the
choice of angular scale nor for the bias in the sample.
\label{fig:pearl}
}
\end{figure}
Such a situation is explained in the following way. The distribution
of the nearby AGN is rather inhomogeneous. Moreover, Cen A is
projected onto one of the largest nearby structures, the Centaurus
supercluster, as can be seen on Fig.2.
\begin{figure}
\begin{center}
\psfig{figure=grained2.eps,height=2.5in}
\end{center}
\caption{Hammer projection of the celestial sphere in supergalactic
coordinates. Crosses show positions of nearby AGN. The color
saturation of a given cross indicates the expected cosmic-ray flux
with the account of the PAO exposure and the $1/r^2$ suppression, $r$
being the distance to the source. Open circles represent 27
highest-energy cosmic rays detected by PAO. Shading shows the expected
cosmic-ray flux from sources that follow the local matter distribution
smoothed at the angular scale of $3.1^\circ$ and convoluted with the
PAO exposure (darker regions correspond to higher cosmic-ray
flux). Blue lines cut out the region with Galactic latitude $|b|<
15^\circ$ where the latter flux cannot be determined because of
incompleteness of the source catalog. The positions of the Centaurus
(Cen) and Virgo (Vir) superclusters are indicated.
\label{fig:skymap}
}
\end{figure}
For this reason, the same data show correlations with both Cen A and
AGN. Importantly, if either AGN or Cen A are indeed sources of
highest-energy cosmic rays, {\it both} correlation signals will {\em
increase} with the accumulation of statistics. So, a mere increase of
significance will not allow to discriminate between the two
possibilities.
\paragraph{2} It follows from the above that alternative
signatures are needed to distinguish between the two cases. We
present here one of such signatures.
The idea is that the cosmic ray flux predicted by the AGN hypothesis
can be computed and compared to the observed one. In this way the AGN
hypothesis itself will be subject to a test, not the hypothesis of the
isotropic distribution.
The computation can be performed in a straightforward way taking into
account the distance to AGN and the attenuation of protons of
different energies (see Refs.\cite{Gorbunov:2007ja,Gorbunov:2008ef}
for details). The results are presented in Fig.~\ref{fig:skymap} in
the form of red crosses which show the positions of the nearby
AGN. The intensity of a cross represents its expected contribution to
the flux. This figure should be understood in a statistical sense: the
fluxes of individual sources cannot, of course, be predicted without
the detailed modeling of corresponding AGN (for which modeling there
is probably not enough information anyway). However, in large groups
of galaxies like galaxy clusters individual differences in luminosity
will average away and only the common factors determined by the
distance will remain. The relative contributions to the total flux
from such groups can thus be reliably predicted.
One can observe the overdensity of the events in the direction of the
Centaurus supercluster. The second region where a high flux is
expected, the Virgo cluster, is completely devoid of events. This is a
strange feature that does not look compatible with the AGN
hypothesis.
The latter statement can be quantified by comparing the expected and
observed distributions of events in the angular distance from the
center of the Virgo cluster, as well as their distributions in
Galactic and supergalactic longitudes and latitudes. The comparison
may be performed by the Kolmogorov-Smirnov test. The results of
different tests show different degree of incompatibility between the
predicted and observed distributions with the probability that it has
occured as a result of a fluctuation varying from 10\% to
$10^{-4}$. Taking into account the strongest discrepancy and the
number of tests performed, we estimate the significance of the tension
between the AGN hypothesis and the data to be of order 99\%.
One of the drawbacks of the analysis just described is the
incompleteness of the AGN catalog. To check how much our results
depend on this incompleteness we have replaced the catalog of AGN by a
complete catalog of galaxies containing objects up to 270~Mpc
\cite{Kalashev:2007ph}. The above tests performed with the AGN catalog
replaced by the complete galaxy catalog show similar results. We think
therefore that incompleteness of the catalog is not an issue.
Another drawback, which unfortunately cannot be avoided at present, is
the {\em a posteriori} nature of the tests performed. To avoid this
problem, the tests which we have described will have to be repeated
with the new independent data. This is why now, before the new data
arrive, it is particularly important to formulate other hypotheses and
procedures to test them which may then be performed in a more reliable
{\em a priori} way with independent data sets.
\section*{Acknowledgments}
This work is supported in part by Belgian Science Policy under IUAP
VI/11, by IISN and by the FNRS contract 1.5.335.08.
%\bibliographystyle{h-physrev4}
\section*{References}
\bibliography{uhe}
\end{document}
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