We apply a recently proposed cross-correlation power spectrum technique to study relationship between the ultra-high energy cosmic ray (UHERC) flux from the Pierre Auger Observatory and galaxies from the 2MASS Redshift Survey. Using a simple linear bias model relative to the galaxy auto power spectrum, we are able to constrain the value of bias to be less than 1% for UHECR with energies 4 EeV - 8 EeV, less than 2.3% for UHECR with energies above 8 EeV and less than 21% for UHECR with energies above 52 EeV (all 95% confidence limit). We study energy dependence of the bias, but the small sample size does not allow us to reach any statistically significant conclusions. For the cosmic ray events above 52 EeV we discover a curious excess cross-correlation at $sim 1^circ$ degree scales. Given similar cross-correlation is not visible at larger angular scales, statistical fluctuation seems like the most plausible explanation.
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