Toward a robust inference method for the likelihood of low-luminosity gamma-ray bursts to be progenitors of ultrahigh-energy cosmic rays correlating with starburst galaxies

Very recently, the Pierre Auger Collaboration reported a $4.5\sigma$ correlation between the arrival directions of the highest energy cosmic rays and nearby starburst galaxies. The cosmic rays producing the anisotropy signal have been proposed to originate in low-luminosity gamma-ray bursts (llGRBs). On the basis of the well-justified assumption that at redshift $z<0.3$ the host metallicity is a good indicator of the llGRB production rate, we show that the association of llGRBs and the starbursts correlating with Auger data is excluded at the 95% confidence level.