CORRELATIONS OF THE ARRIVAL DIRECTIONS OF ULTRA-HIGH ENERGY COSMIC RAYS WITH EXTRAGALACTIC OBJECTS AS OBSERVED BY THE TELESCOPE ARRAY EXPERIMENT

We use the catalogs of extragalactic objects that are derived from measurements as listed in Table 1. In several catalogs, the objects near the Galactic plane are excluded to avoid incompleteness from the experimental limitation by the authors of each catalog. We also exclude the observed SD events in the corresponding regions.

The target objects and the cut criteria in each of the catalogs are summarized below. These criteria (e.g., significance level) were chosen by the authors of each catalog. The 3CRR catalog contains radio galaxies detected at 178 MHz with fluxes greater than 10 Jy (Laing et al. 1983). Objects in the direction of the Galactic disk (|b| < 10°) were not included. The 2MRS (Huchra et al. 2012) catalog is derived from the 2MASS observation with a detection range between 1–2 μm and K_s ⩽ 11.75 mag. This catalog also loses completeness near the Galactic plane, so the authors of the catalog excluded regions with |b| < 5° for 30° ⩽ l ⩽ 330° and |b| < 8° otherwise. The SB-58M catalog consists of objects that were detected with a significance greater than 4.8σ in the energy range of 14–195 keV in the first 58 months of observation by the Swift BAT. We select the extragalactic objects in this catalog for this work. The catalog of SB-AGN contains AGNs with at least 5σ significance in the energy range of 15–55 keV in the first 60 months of observation by the Swift BAT. The 2LAC (Ackermann et al. 2011) data set consists of AGNs detected with at least 4σ significance in the energy range of 100 MeV–100 GeV in the first 24 months of observation by the Fermi-LAT. The Galactic disk region with |b| < 10° is cut away. We also examine the VCV catalog, which is a compilation of several AGN surveys. The number of objects and the SD events after the cuts are applied in each case are given in Table 2.

For a given set of parameters (E_min, ψ, z_max), there are N events with energies E ⩾ E_min. We can count the number of events, k, out of N that are correlated with objects in a catalog with redshifts z ⩽ z_max and within the angular separation ψ. We can calculate the probability, P, that k or more correlated events are found from an isotropic UHECR flux under the same conditions. We carry out a parameter scan in (E_min, ψ, z_max) space to find the set of parameters that maximizes the correlation between the TA events and the catalog objects, i.e., minimizes P. To determine the probability, P, we first obtain the probability, p, that a random event is correlated with at least one object by chance for a given (ψ, z_max). To do so, we generate 10⁴ randomly distributed events in the same experimental region of each of the catalogs as described in Table 2 and cross-correlate them with the objects.

Then P can be obtained as a cumulative binomial probability:

$$\begin{eqnarray} &&P= \sum ^{N}_{j=k}C^{N}_{j}p^{j}(1-p)^{N-j}. \end{eqnarray} \tag{ 1 }$$

The scan over parameters is performed as follows. The value of E_min is set by the energy of the Nth highest energy event. We scan over all values of N such that E_min is greater than 40 EeV. Note that this energy is less than the energy (50 EeV) at which the TA energy spectrum begins to fall off steeply (Abu-Zayyad et al. 2013). We set the upper boundary of the parameter z_max as 0.03, which corresponds to the distances smaller than 120 Mpc. This is comparable to the GZK horizon: at E > 40 EeV about 40% of the UHECR flux is collected from within this distance (Koers & Tinyakov 2009). Although choosing a larger maximum distance would increase this fraction, note that the UHECR deflections also increase with distance, as well as the penalty for scanning. The selected step size of z_max is 0.001, which is the typical accuracy in the redshift measurements. The separation angle, ψ, is varied from 1° to 15°. The maximum search window of ψ = 15° is selected as appropriate for lower energy events (∼40 EeV) arriving from the distance of 100 Mpc. The selected step size in ψ is chosen as 01 for ψ < 8° and 1° for 8° ⩽ ψ ⩽ 15°. The parameter ranges and step sizes are summarized in Table 3.

The minimum P obtained from this procedure does not represent the correlation probability directly, because the parameter scanning enhances the correlation probability artificially (Tinyakov & Tkachev 2004). Therefore, a penalty for parameter scanning (PPS) should be evaluated and the true probability of correlation must include this penalty. This will be described in Section 4.

The results of the parameter scan are listed in Table 4. The smallest value of $P_{\rm min}^{\rm obs}$ among all the catalogs is 1.3 × 10⁻⁵ found in the SB-AGN catalog with the best parameters (E_min, ψ, z_max)_best = (62.20 EeV, 10°, 0.020). A sky map of the TA events and the objects under the condition (E_min, ψ, z_max)_best that gives the smallest $P_{\rm min}^{\rm obs}$ is shown in Figure 2. All the observed UHECRs with E ⩾ E_min correlate with at least one object with z ⩽ z_max in the SB-AGN catalog. Figures 3 and 4 show the probability as a function of each parameter (E_min, ψ, z_max)_best, while fixing the values of the other two at the optimum value for this data comparison.

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Now let us consider the PPS. We evaluate the probability, P_PPS, of finding a correlation by chance with $P_{\rm min}^{\rm sim}$ smaller than that obtained from the data as follows (for a more detailed description of the penalty calculation see, e.g., Tinyakov & Tkachev 2004). We generate 10⁴ random sets of N "cosmic ray events," where N is the same as the number of the observed events with energies greater than 40 EeV. For each of the mock event sets, the parameter scanning is carried out using exactly the same method as for the observed data set, and $P_{\rm min}^{\rm sim}$ was calculated. Note that the parameters (E_min, ψ, z_max)_best that yield $P_{\rm min}^{\rm sim}$ are different for each of the 10⁴ trials. The distribution of $P_{\rm min}^{\rm sim}$ in the case of the SB-AGN catalog is shown in Figure 5 together with $P_{\rm min}^{\rm obs}$. One can see that rather small values P_min ⩽ 1.3 × 10⁻⁵ can happen even though the simulated UHECR distribution is isotropic.

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If we repeat the same experiment and the parameter scanning many times, the value of our result $P_{\rm min}^{\rm obs} = 1.3 \times 10^{-5}$ could be just a chance occurrence. The probability including the PPS is evaluated as P_PPS = 0.01 for the SB-AGN catalog, and the values for all the catalogs are listed in Table 4. The smallest value of P_PPS among the catalogs is 0.01 from the SB-AGN catalog. This does not yet include the penalty for searching in several catalogs.

If we have several catalogs, regardless of whether they are independent or partially overlapping, there is a possibility of finding a catalog that gives the same or smaller $P_{\rm min}^{\rm obs}$ value by chance, even though there are no correlations between the events and the objects. The straightforward way to calculate the penalty factor associated with the partially overlapping catalogs, as is the case in our analysis, is to include all the catalogs in the Monte Carlo simulation. Therefore, we have repeated the simulation with 10⁴ mock sets as described above but with the scanning performed in all six catalogs. Calculating the fraction of mock sets that show equal or better correlation than the data, we find the final probability with a PPS and a PCS, P_{PPS + PCS} = 0.09. Therefore, we conclude that no significant correlation between UHECRs and the astronomical objects is found in the current TA data set.

First, we consider the effect of finite resolution in the scanning parameters. The uncertainty in determination of the arrival directions and energy only makes correlations worse because of direction smearing and the contamination of energy events lower than E_min. Therefore, the obtained $P_{\rm min}^{\rm obs}$ already includes these resolution effects. This also concerns the uncertainty in the redshifts of the catalog objects.

Consider the effect of the systematic uncertainty in energy determination. As mentioned above, this uncertainty is 22% (Abu-Zayyad et al. 2011). Note, however, that the present analysis with the parameter scanning is independent of the absolute energy scale: the energies of the events are no more than keys for event sorting, and a systematic energy shift does not affect the scanning in E_min, hence the number of events involved in the correlation with the objects and the probability $P_{\rm min}^{\rm obs}$.

The last issue to discuss is the incompleteness of the catalogs, which remains even after we cut out the regions around the Galactic plane. The objects in the VCV catalog are inhomogeneous because it is a mere compilation of objects detected under different conditions. The completeness of the other catalogs, in particular the 2LAC, could be affected by our cuts, particularly by the selection of objects with the known redshift. While the incompleteness may make the interpretation of correlations ambiguous if they are present, it does not affect the calculation of $P_{\rm min}^{\rm obs}$. In fact, the effect of the incompleteness cancels out in $P_{\rm min}^{\rm obs}$ since the same set of objects is used to cross-correlate with the data and each mock event set. Therefore, the incompleteness of the catalogs cannot produce spurious correlations (although it may, in principle, be responsible for their absence).

So far we have treated the objects in each catalog equally, regardless of their class. Now let us examine whether there is a specific type of object that has stronger correlations with UHECRs than others. We will consider the case of the SB-AGN catalog that shows strongest correlations with UHECRs.

First, we count the number of objects of each class in the TA FOV with redshifts smaller than 0.02. Some of the objects are labeled "unclassified" in the catalog. For these we used the information from other surveys (Noguchi et al. 2010; Parisi 2011; Veron-Cetty & Veron 2010; Tueller et al. 2010). The fractions of Seyfert 2, 1, 1.5, 1.9, and LINER galaxies in the SB-AGN catalog satisfying the above conditions are 0.441, 0.235, 0.132, 0.044, and 0.044, respectively (the total fraction of other class AGNs: 0.044; the fraction of the unclassified AGN: 0.059). The total number of AGNs that are correlated with UHECRs is 22 with the parameters in Table 4 (note that the number of UHECR events and that of AGNs are not the same because some of the events fall within the given angular distance from several sources). Among these 22 AGNs, the fractions of Seyfert 2, 1, 1.5, 1.9, LINER, and unclassified galaxies are 0.455, 0.182, 0.227, 0.045, 0.045, and 0.045, respectively. We see that the largest difference is for the Seyfert 1.5 galaxies. The probability, P, of finding 5 or more correlated Seyfert 1.5 galaxies out of 22 can be evaluated by the cumulative binomial probability with an expectation of 0.13 and is P = 0.16. Therefore, no significant correlation with a specific type of AGN in the SB-AGN catalog is found.